Hindawi Publishing CorporationJournal of GravityVolume 2013 Article ID 782512 10 pageshttpdxdoiorg1011552013782512

Research ArticleMore on Superconductors via GaugeGravity Duality withNonlinear Maxwell Field

Davood Momeni1 Muhammad Raza2 and Ratbay Myrzakulov1

1 Eurasian International Center for Theoretical Physics Eurasian National University Astana 010008 Kazakhstan2Department of Mathematics COMSATS Institute of Information Technology (CIIT) Sahiwal Campus Pakistan

Correspondence should be addressed to Davood Momeni dmomeniyahoocom

Received 22 February 2013 Revised 16 March 2013 Accepted 16 March 2013

Academic Editor Sergei Odintsov

Copyright copy 2013 Davood Momeni et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We have developed the recent investigations on the second-order phase transition in the holographic superconductor using theprobe limit for a nonlinear Maxwell field strength coupled to a massless scalar field By analytical methods based on the variationalSturm-Liouville minimization technique we study the effects of the spacetime dimension and the nonlinearity parameter onthe critical temperature and the scalar condensation of the dual operators on the boundary Further as a motivated result weanalytically deduce theDCconductivity in the low and zero temperatures regime Especially in the zero temperature limit and in twodimensional toy model we thoroughly compute the conductivity analytically Our work clarifies more features of the holographicsuperconductors both in different space dimensions and on the effect of the nonlinearity in Maxwellrsquos strength field

1 Introduction

In the recent years using the holographic picture of theworld the AdSCFT (anti de Sitterconformal field the-ory) correspondence [1ndash3] has been applied to study somestrongly correlated systems in condensed matter physicsespecially for strongly coupled systems with the scale-invariance Particularly people studied the low temperaturequantum critical systems near critical point (see eg [45] and references therein) The critical phenomena whichhappen here is a second-order phase transition from nor-mal phase to the superconducting phase in which belowa specific temperature 119879

119888 the DC conductivity becomes

infinite Such second-order phase transitions happen in thehigh-temperature superconductors and can be described verywell by the AdSCFT dictionary [6 7] From the classicaland phenomenological point of view superconductivityin the high-temperature type II superconductors modeledby a phenomenological Landau-Ginzburg Lagrangian ThisLagrangian contains a general complex value scalar field Ψplays the role of a condensate in a superconductive phaseBasically to have a scalar condensation in the boundaryquantum field theory using CFT on the boundary of the bulk

Hartnoll et al [8] introduced a 119880(1) abelian gauge field 119860120583

and a typical conformal coupled charged complex scalar fieldin the bulk black hole background The conformal mass isabove the Breitenlohner-Freedman (BF) bound [9] To solvethe negative mass problem finally Gubser [10] showed thatthe vector potential modifies themass term of scalar field andwe have a possibility to have hairy black holes in some partsof the parameter space

The full description of the superconductivity in the probelimit or away this limit needs to provide the numericalsolutions of a couple of nonlinear differential equations Bysimplicity they can be solved using the shooting approachby expanding in series the functions and matching these byvarying the free parameters of the series in a typical pointbetween the horizon and the spatial infinity Parallel to thenumerical studies recently some analytical approaches havebeen proposed to find the universal properties of second-order phase transitions in holographic superconductors [11ndash18] In particular the authors in [18] used the variationalfunctional method on the real valued functions for theSturm-Liouville eigenvalue problem to compute analyticallythe basic properties of the holographic superconductors ina three-dimensional boundary using the solutions of the

2 Journal of Gravity

bulk gauge fields in the four dimensional bulk theory In[18 19] the authors have shown that also it is possible tocalculate critical exponent and critical temperature by thisapproach The eigenvalue of this variational problem is afunction of the critical chemical potential 120583

119888 and conse-

quently it is related to the 119879119888 Different modes of super

criticality s-wave p-wave and d-wave have been studied [20]Also one can apply this method to superconductors withexternal magnetic fields which gives the analytical results ingood agreement with numerical results produced previously[21 22]

Furthermore a number of aspects of external and bulkmagnetic fields in holographic superconductors have beeninvestigated [23ndash26] The phase transition can be interpretedin terms of the string interactions [27ndash31] The effects of thenonlinear electrodynamics in the holographic superconduc-tors have been investigated recently [32ndash36]

There are many interests in the modified gravity the-ories For example on Gauss-Bonnet and Weyl correctedsuperconductors on which we are working with a higherderivative corrected bulk black hole like Weyl corrections[37] numerically Furthermore we have studied the Weylcorrections to the superconductors analytically [38 39]Moreover we showed that there exists a family of p-waveholographic with Weyl corrections [40]

In the present paper we would like to study the ((119889 minus

2) + 1)-dimensional holographic superconductor in theprobe limit for a power law Maxwell field strength (119865120583]119865

120583])120575

coupled to a scalar field We focus just on the s-wave casesWe must clarify the motivation of the s-wave approximationin holographic models of superconductors In the relativisticmodels of the gravity it is highly known that s-wave approx-imation is not a good approximation for example in thecosmological models and black holes [41] The meaning ofthe s-wave here does not back to the reduction of the actionfrom four to two dimensional like the dilatonic action fromthe four dimensional spinor (Majorana) action We meanby s-wave in the context of holographic superconductors ascalar order parameter whose expectation value breaks theU(1) but not rotational symmetry Moreover we can have theYang-Mills fields with 119878119880(2) symmetry which additionallythey can generate another symmetry breaking of an axialvector type The last case resembles the p-wave models Wemention here that the three dimensional non-linear model ofthe superconductors whichwe used in this paper is a realisticmodel and it will be more interesting that we can find a directrelation between this nonlinear model and the results of ahigher dimensional model by a principle like the detailedbalance

Another additional point is to restrict ourselves just to thecase of a single horizon The problem of the multihorizoncases needs more investigation for example the case ofthe Nariai black holes The theory here will be so differentThis later appeared in the lower dimensional models Forexample the case of the quantum corrected BTZ like blackhole is a good example [42] In the holographic set upfor superconductors one must identify a temperature inhis gravitational bulk model to the CFT temperature on

the boundary If the black hole has only one horizon inthis case we can use the Hawking-Bekenstein (horizon)or Kodama- Hayward temperature [43] as a reasonablecandidate But if our asymptotically AdS bulk has morethan one horizon for example in the case of the chargedBTZ like black holes then we take the temperature of thereal physical horizon (the temperature which is obtained bycalculation the surface gravity of the biggest null hypersurfaceorthogonal surface) as the candidate for temperature of theCFT In fact the effects of the quantum corrections andchargedMaxwell field on the background of the bulk are veryinteresting problems and can be investigated in more detailsAlso it is possible to relate the instability of such chargeddilaton configurations in the AdS spacetime [44] to thesymmetry breaking mechanism of the superconductors Theidea has motivation enough as a new work In this paper weinvestigate analytically the effect of the spacetime dimension119889 and the power 120575 on the critical temperature 119879

119888 Although

our problem is the especial massless case of the modelwhich has been investigated recently [45] we study thesecorrections to the superconductors analytically Additionallywe want to compute the DC conductivity for this kind ofthe superconductor using the perturbation method In thisapproach we apply an external linear electromagnetic fieldThis field is periodic in time By calculating the responsein the first order linear approximation we compute theconductivity for the low temperature case especially for zerotemperature configuration

Our plan in this paper is as the following In Section 2 weclarify our motivation for considering the nonlinearMaxwellaction instead of the linear theory In Section 3 we introduceour model for holographic superconductors In Section 4 weapply the variational method to obtain the critical temper-ature of the system In Section 5 we calculate the criticalexponent for the condensation operator In Sections 6 and 7we compute the conductivity for low and zero temperaturecases We summarize and conclude in the final section

2 Motivation for Nonlinear Maxwell Effects inHolographic Superconductors

The linear approximations in themathematical physics as weknow have limitations both in predictions of the model andespecially on matching with the full description of the modelusing the numerical resultsThe linearMaxwell theory fails insome domains and it is needed to consider the general formof the Lagrangian instead of the linear one The Lagrangianof the Maxwell model is

L119872= minus

1

41198652

(1)

where 1198652 = 119865120583]119865120583] A natural extension of (1) is obtained by

replacing a general function of119865 in the formΦ(119865) Recently ithas been shown that such nonlinear general forms have a richfamily of black holes in 119891(119877) gravity [46] There are differentreasons for investigating these forms The oldest one may bethe Born-Infeld (BI) alternative for linear Maxwellrsquos theory

Journal of Gravity 3

In string theory language this BI action can be replaced bythe tachyonic action The BI Lagrangian reads [47]

LBI = 1205782

(1 minus radic1 +1198652

21205782) (2)

Here 120578 is the stringrsquos tension parameter This form reduces to(1) in the limit of 120578 rarr infin Indeed we can expand in series(2) in the following form

LBI =infin

sum

119899=1

1198881198991205782(1minus119899)

1198652119899

(3)

The leading order term 119899 = 1 is just in the form of (1) Evenif we do not work with BI theory this nonlinearity meets usfrom a geometrical point of view Suppose that we want towrite a conformal invariance (CI) Lagrangian constructedfrom the 119880(1) gauge fields 119860120583 Such CI is the invariance ofthe whole theory under geometrical transformation 119892

120583] rarr

1198902120590

119892120583] in a 119889 ge 4 Riemannian manifold without torsion or

nonmetricity fields However previously a version of suchLagrangian has been found in Weitznbock spacetime withtorsion and nonmetricity fields [48] It is easy to show that theLagrangianL prop 119865

1198892 is invariant under CI transformationsWhen 119889 = 4 the proper Lagrangian is L prop 119865

2 butin 119889 = 5 the suitable form is L prop 119865

52 The last formbelongs to the nonlinear noninteger Maxwell family Thusfrom geometrical view the nonlinearity is welcome in ourLagrangian dynamical theory Further even if we do notknow any on BI or CI when we are working with vacuumeffects there is a simple generalization ofMaxwell Lagrangianin a logarithmic form

Llog = minus1205782 log(1 + 119865

2

41205782) (4)

As BI case in the limit of |119865| ≪ 120578 by expanding this equationthe leading order term 119899 = 1 is the linear theory (1) In briefaccording to the above discussion it seems that considerationof the nonlinear effects of theMaxwell field as119865120575 is importantand the significant differences between the usual linear theory1198652 and nonlinear theory 119865

120575 can be shown Holographicsuperconductors provide a rich background for testing suchtype of new physics In this paper we will describe a (119889 minus

1) dimensional holographic superconductor (HSC) via 119889-dimensional gravity dual described by a 119889-dimensional AdSblack hole on the static patch We set the Stuckelberg field tozero and work with massless scalar fields

3 Field Equations

We write the following action [32ndash36] for a power Maxwellfield 119865

120583] which it is coupled minimally to a massless scalerfield 120595 in a 119889-dimensional asymptotic AdS

119889spacetime [45]

119878 = int119889119889

119909radicminus119892 [119877+(119889 minus 1) (119889 minus 2)minus120585(119865120583]119865120583])120575

minus1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

)

(5)

Here we set the radius of the AdS 119871 = 1 119863120583= 120597120583minus 119894119890119860

120583

119865120583] equiv 2120597

[120583119860]] = 120597

120583119860] minus 120597]119860120583 119860120583 is 119880(1) vector potential

The exponent (power) 120575 can be noninteger as we explained itin the previous section Further when we work withMaxwelllinear electrodynamics 120575 = 1 119889 = 4 then 120585 = 14 Howeverin our case with 120575 = 1 in general 120585 isin R remains as a freeparameter in our model We set the electric charge 119890 = 1In normal phase and in the absence of the scalar field weput 120595 = 0 This action has been used before in literaturefor full description of the nonlinear Maxwell field in thegravitational action In the probe limit when thematter actionand the gravitational part decouple the system of the fieldequations has an exact solution which will be discussed hereThe solution of the generalized Maxwell-Einstein equationsis the simple 119889-dimensional static Reissner-Nordstrom-Anti-de Sitter (RNAdS) black hole with the following metricform

119892120583] = diag(minus119891 (119903) 1

119891 (119903) 1199032

Σ119889minus2

) (6)

where Σ119889minus2

is the metric on a (119889 minus 2) dimensional sphere andthe metric function is

119891 (119903) = 1199032

(1 minus (119903+

119903)

119889minus1

) (7)

Here 119903+is the black hole horizon which in general is the

largest root of the algebraic equation119891(119903+) = 0 If we set 120575 = 1

in (5) we recover the usual holographic superconductorsIn the probe limit by ignoring the backreaction effects ofthe matter fields in the background metric 119892

120583] and withspherically symmetric staticmetric and by choosing a suitablegauge fixing for the gauge field 119860

120583we can take the functions

(120595 120601) isin R and one variable functions We assume that thefunctions 120595 and 120601 have finite numbers of poles on the realaxis It means the analytical solutions can be written in theforms of the algebraic series expressions To obtain the fieldequations we assume that 119860

120583= 120601(119903)120575

120583119905 so the nonzero

components of the 119865120583] read as

119865119903119905= minus119865119905119903= 1206011015840

(8)

Hence we have

119865120583]119865120583] = minus2120601

10158402

(9)

Also the Ricci 119877 of a 119889-dimensional spacetime with sphericalsymmetry reads

119877 = minus(11989110158401015840

+2 (119889 minus 2) 119891

1015840

119903minus(119889 minus 2) (119889 minus 3) 119891

1199032) (10)

Further we compute

1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

= 11989112059510158402

minus12060110158402

119891 (11)

4 Journal of Gravity

Finally by plugging the above expressions into the action (5)making a partial integration we get the following effectiveLagrangian

L = 119903119889minus2

[minus(11989110158401015840

+2 (119889minus2) 119891

1015840

119903minus(119889minus2) (119889minus3) 119891

1199032)

+ (119889minus1) (119889minus2) minus 120585(minus212060110158402

)120575

minus(11989112059510158402

minus12060110158402

119891)]

(12)

As a first step it is necessary to eliminate the 11989110158401015840 term byintegration part by part After it to write the field equationwe use the Euler-Lagrange equation as the following

119889

119889119903(120597L

120597119902119903

) =120597L

120597119902 119902 = 120601 120595 (13)

The field equations derived from (12) read

12059510158401015840

+ (1198911015840

119891+119889 minus 2

119903)1205951015840

+1206012

1198912120595 = 0 (14)

12060110158401015840

+119889 minus 2

2120575 minus 1

1206011015840

119903minus 119862120575

1205952

120585119891120601(1206011015840

)2(1minus120575)

= 0 (15)

here 119862120575= ((minus2)

2minus120575

120575(2120575 minus 1))minus1 To avoid the pure complex

numbers in our field equations we assume that120575 = 2minus(12119873)The case with 120575 = 1119863 = 4 is the usual four-dimensionalHSCwhich describes the three dimensional superconductors

These field equations are the special massless case of themodel which has been investigated recently [45 49] To avoidthe divergence near the singularity 119891(119903

+) = 0 we write the

boundary conditions for (14) (15) by

120601 (119903+) = 1205951015840

(119903+) = 0 (16)

The asymptotic solutions for system (14) (15) on the AdSboundary 119903 rarr infin are

120595 = 119863minus+

119863+

119903119889minus1 (17)

120601 = 119860 minus119861

119903120578 (18)

in (18) 119863plusmn= ⟨Oplusmn⟩ 120578 = (119889 minus 2120575 minus 1)(2120575 minus 1) where ⟨O

plusmn⟩

denotes the vacuum expectation value of the dual operatorOplusmnon the boundary and the chemical potential and charge

density of the dual theory are 119860 = 120583 119861 = 120588(2120575minus1)

minus1

respectively

We must clarify the reason for the modification of theform of the electric potential 120601 modified In the asymptoticregime we know that the metric function 119891 behaves like119891 sim 119903

2 Also the scalar field has the following asymptoticform 120595 sim 0 so (15) gives us

12060110158401015840

infinsim minus

119889 minus 2

2120575 minus 1

1206011015840

infin

119903 (19)

The solution (19) reads

120601infin(119903) = 119888

1+1198880(2120575 minus 1)

2120575 minus 119889 + 1

1

119903(119889minus2120575minus1)(2120575minus1) (20)

This solution coincides completely with the solution pre-sented in (18)Wementionhere that the above function120601

infin(119903)

in the limit of the linear electrodynamic theory 120575 = 1 hasthe true asymptotic form of 120601

infin(119903) sim 119903

3minus119889 119889 = 2 For 119889 = 2

the expression of 120601infin(119903) is in the form of a diverging log term

120601infin(119903) sim log(119903) and the application of the AdSCFT fails At

least we do not know the unique and true dictionary of theAdSCFT in this lower dimensional bulk theory

The only thing left is to identify the parameters with thephysical quantities in the dual theory that is the chemicalpotential 120583 and the charge density 120588 We did it before so theasymptotic behaviors have the same forms

The asymptotic solutions for 120601 120595 are the same as theprevious expressions which have been presented in [45] Fornormalization purposes we set119863

minus= 0

4 Variational Method

To solve the solutions of the field equations given by (14)and (15) the well-known technique is numerical algorithmsHowever from these numerical solutions it is not so easyand straightforward to read ⟨O⟩ Another method is usingthe matching method It is a potentially powerful methodEven so the resultsmust be interpreted very carefully near theboundaries The first step for solving system (14) (15) usingvariational approach [18] is rewriting the equations in a newdimensionless coordinate 119911 = 119903

+119903 in the following forms

12059510158401015840

+ (1198911015840

119891minus119889 minus 4

119911)1205951015840

+ (119903+

1199112)

2 1206012

1198912120595 = 0 (21)

12060110158401015840

minus120578 minus 1

1199111206011015840

minus 119862120575

1199032120575

+

1205851199114120575

1205952

119891120601(1206011015840

)2(1minus120575)

= 0 (22)

Now the prime is for differentiation with respect to thedimensionless radial coordinate 119911 Near the critical point119879 =

119879119888 the following solution is valid

120601 asymp 120601119861(1 minus 119911

120578

) (23)

Here 120601119861= 120583119888= 120588(2120575minus1)

minus1

(119903+)120578+1 is the value of the 120601 at the

horizon 119903 = 119903+ 120575 = (119889minus1)2 and the critical point corresponds

by the critical chemical potential120583 = 120583119888 Further we canwrite

120595 asymp ⟨119874+⟩ (

119911

119903+

)

119889minus1

119865 (119911) (24)

near the AdS boundary 119911 rarr 0 with 119865(0) = 1 1198651015840(0) = 0 Thetrial variational function satisfies the following second-orderSturm-Liouville self sdjoint differential equation

[120583 (119911) 1198651015840

(119911)]1015840

minus 119876 (119911) 119865 (119911) + (120601119861

119903+

)

2

119875 (119911) 119865 (119911) = 0

(25)

Journal of Gravity 5

where

120583 (119911) = 1199112119889minus2

(119911119889minus1

minus 1)

119875 (119911) = 120583 (119911) (1 minus 119911120578

1 minus 119911119889minus1)

2

119876 (119911) = minus 120583 (119911) [(119889 minus 1) (119889 minus 2)

1199112minus119889 minus 1

119911

times (2 + (119889 minus 3) 119911

119889minus1

119911 (1 minus 119911119889minus1)+119889 minus 4

119911)]

(26)

our strategy is to obtain the minimum value of 120601119861119903+from

the minimization of the following functional

Θ (120575 119889) equiv (120601119861

119903+

)

2

Min=

int1

0

(120583 (119911) 1198651015840

(119911)2

+ 119876 (119911) 1198652

(119911)) 119889119911

int1

0

119875 (119911) 1198652 (119911) 119889119911

(27)

Theminimum of the critical temperature 119879119888is obtained from

the 119879 = (119889 minus 1)119903+4120587 It reads

119879119888= 120574120588120578+1

(28)

and 120574 = ((119889 minus 1)4120587)((120601119861119903+)Min)minus(120578+1) Here we take trial

function 119865(119911) = 1 minus 1205721199112 in (27) It is useful to set 120588 = 1

The values of the critical temperature 119879119888for 119889 = 4 5 by

minimizing the functional (27) with trial function 119865(119911) andusing (28) (in case 120588 = 1) are given by the following

For 119889 = 4 120575 = 1 119879119888= 00844 for 120575 = 34 119879

119888=

01692For 119889 = 5 120575 = 1 119879

119888= 001676 for 120575 = 34 119879

119888=

02503 for 120575 = 54 119879119888= 00954

These values are in good agreements with the numericalvalues [50] and also coincide to the analytical values given in[45]

5 Calculating the Critical Exponent

We begin from (22) by writing it near the critical point (CP)and in limit 119879 rarr 119879

119888 The first step is rewriting the solution

in a perturbative scheme with respect to the perturbationparameter 120598 = ⟨119874

+⟩2 as it was described by Kanno [51] Near

the CP the solution of the field 120601 is written as

120601 (119911) = 1206010+ 120598120594 (119911) (29)

where 1206010= 120581119879119888(1 minus 119911

120578

) Using (24) and (29) in first orderwith respect to the 119874(120598) we obtain the following ordinarydifferential equation

12059410158401015840

(119911) minus120578 minus 1

1199111205941015840

(119911) = 119864 (119911) (30)

where

119864 (119911) =1198621205751199032

+119911minus120575minus2

1198652

(119911)

120585119891 (119911)1206012

0(1206010)10158402(1minus120575)

119865 (119911) |119911rarr0

asymp 1

(31)

Since 120601(119911) = (119860(119879119888)120578+1

119879120578

)(1 minus 119911120578

) 119860 = (119889 minus 1)4120587

writing the solution for 120601 in 119911 = 0 we have

120601 (0) = 1206010(0) + 120598120594 (0) (32)

The general solution for (30) is given by

120594 (119911) = radic119911 (11988811198691(119909) + 119888

21198841(119909))

+ radic1199111205871205782minus2120575

119879119888

4minus2120575

1205814minus2120575

119891 (119909)

(33)

with

119891 (119909) = 1205721198691(119909) int

1198841(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

+ 1205731198841(119909) int

1198691(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

(34)

Here 119909 = 2radic(1 minus 120578)119911 119869119899(119909) 119884119899(119909) are Bessel functions of

first and second kindsFinding the value of 120594(0) from (33) and solving it for

⟨119874+⟩ = radic120598 we obtain (we take 119879

119888= 1)

⟨119874+⟩ prop 119879

119889minus120575minus1205782

[1 minus 119879120578

]12

(35)

When the power 120575 decreases the value of the ⟨119874+⟩

increases Thus we conclude that the effect of the power 120575 in119889 = 4model is in the direction of the increase of ⟨119874

+⟩

However in the five dimensions the analysis is a littlebit different As we observe in 119889 = 5 when the power 120575increases the value of the ⟨119874

+⟩ increases Thus we can say

that the effect of the power 120575 in 119889 = 5model is in the directionof the increase of ⟨119874

+⟩

6 Calculating the Low Temperature DCConductivity

In this section we compute the low-temperature DC conduc-tivityWe concentrate on the general space time dimension 119889and we will try to calculate the conductivity 120590 as a functionof the rescaled frequency as follows

=120596

⟨119874+⟩1Δ

Δ = 119889 minus 1 (36)

In this limit the asymptotic form of the scalar field is

120595 (119911) =119887119889minus1

radic2119911119889minus1

119865 (119911) 119887 equiv ⟨119874+⟩ (37)

We follow the method in [18] Assuming that thereexists an external magnetic field 119860(119903 119905) = 119860(119903)119890

minus119894120596119905Note that here the applied Maxwell field is linear It isnot related to the nonlinear Maxwellrsquos field in the bulkaction In fact the nonlinearity of the Maxwell field nowis stored in the background metric As we know the fieldequations have some terms which involve the exponent 120575This parameter denotes the nonlinearity which is hiddenin the structure of the background metric and through it

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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AstrophysicsJournal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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Biophysics

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ThermodynamicsJournal of

2 Journal of Gravity

bulk gauge fields in the four dimensional bulk theory In[18 19] the authors have shown that also it is possible tocalculate critical exponent and critical temperature by thisapproach The eigenvalue of this variational problem is afunction of the critical chemical potential 120583

119888 and conse-

quently it is related to the 119879119888 Different modes of super

criticality s-wave p-wave and d-wave have been studied [20]Also one can apply this method to superconductors withexternal magnetic fields which gives the analytical results ingood agreement with numerical results produced previously[21 22]

Furthermore a number of aspects of external and bulkmagnetic fields in holographic superconductors have beeninvestigated [23ndash26] The phase transition can be interpretedin terms of the string interactions [27ndash31] The effects of thenonlinear electrodynamics in the holographic superconduc-tors have been investigated recently [32ndash36]

There are many interests in the modified gravity the-ories For example on Gauss-Bonnet and Weyl correctedsuperconductors on which we are working with a higherderivative corrected bulk black hole like Weyl corrections[37] numerically Furthermore we have studied the Weylcorrections to the superconductors analytically [38 39]Moreover we showed that there exists a family of p-waveholographic with Weyl corrections [40]

In the present paper we would like to study the ((119889 minus

2) + 1)-dimensional holographic superconductor in theprobe limit for a power law Maxwell field strength (119865120583]119865

120583])120575

coupled to a scalar field We focus just on the s-wave casesWe must clarify the motivation of the s-wave approximationin holographic models of superconductors In the relativisticmodels of the gravity it is highly known that s-wave approx-imation is not a good approximation for example in thecosmological models and black holes [41] The meaning ofthe s-wave here does not back to the reduction of the actionfrom four to two dimensional like the dilatonic action fromthe four dimensional spinor (Majorana) action We meanby s-wave in the context of holographic superconductors ascalar order parameter whose expectation value breaks theU(1) but not rotational symmetry Moreover we can have theYang-Mills fields with 119878119880(2) symmetry which additionallythey can generate another symmetry breaking of an axialvector type The last case resembles the p-wave models Wemention here that the three dimensional non-linear model ofthe superconductors whichwe used in this paper is a realisticmodel and it will be more interesting that we can find a directrelation between this nonlinear model and the results of ahigher dimensional model by a principle like the detailedbalance

Another additional point is to restrict ourselves just to thecase of a single horizon The problem of the multihorizoncases needs more investigation for example the case ofthe Nariai black holes The theory here will be so differentThis later appeared in the lower dimensional models Forexample the case of the quantum corrected BTZ like blackhole is a good example [42] In the holographic set upfor superconductors one must identify a temperature inhis gravitational bulk model to the CFT temperature on

the boundary If the black hole has only one horizon inthis case we can use the Hawking-Bekenstein (horizon)or Kodama- Hayward temperature [43] as a reasonablecandidate But if our asymptotically AdS bulk has morethan one horizon for example in the case of the chargedBTZ like black holes then we take the temperature of thereal physical horizon (the temperature which is obtained bycalculation the surface gravity of the biggest null hypersurfaceorthogonal surface) as the candidate for temperature of theCFT In fact the effects of the quantum corrections andchargedMaxwell field on the background of the bulk are veryinteresting problems and can be investigated in more detailsAlso it is possible to relate the instability of such chargeddilaton configurations in the AdS spacetime [44] to thesymmetry breaking mechanism of the superconductors Theidea has motivation enough as a new work In this paper weinvestigate analytically the effect of the spacetime dimension119889 and the power 120575 on the critical temperature 119879

119888 Although

our problem is the especial massless case of the modelwhich has been investigated recently [45] we study thesecorrections to the superconductors analytically Additionallywe want to compute the DC conductivity for this kind ofthe superconductor using the perturbation method In thisapproach we apply an external linear electromagnetic fieldThis field is periodic in time By calculating the responsein the first order linear approximation we compute theconductivity for the low temperature case especially for zerotemperature configuration

Our plan in this paper is as the following In Section 2 weclarify our motivation for considering the nonlinearMaxwellaction instead of the linear theory In Section 3 we introduceour model for holographic superconductors In Section 4 weapply the variational method to obtain the critical temper-ature of the system In Section 5 we calculate the criticalexponent for the condensation operator In Sections 6 and 7we compute the conductivity for low and zero temperaturecases We summarize and conclude in the final section

2 Motivation for Nonlinear Maxwell Effects inHolographic Superconductors

The linear approximations in themathematical physics as weknow have limitations both in predictions of the model andespecially on matching with the full description of the modelusing the numerical resultsThe linearMaxwell theory fails insome domains and it is needed to consider the general formof the Lagrangian instead of the linear one The Lagrangianof the Maxwell model is

L119872= minus

1

41198652

(1)

where 1198652 = 119865120583]119865120583] A natural extension of (1) is obtained by

replacing a general function of119865 in the formΦ(119865) Recently ithas been shown that such nonlinear general forms have a richfamily of black holes in 119891(119877) gravity [46] There are differentreasons for investigating these forms The oldest one may bethe Born-Infeld (BI) alternative for linear Maxwellrsquos theory

Journal of Gravity 3

In string theory language this BI action can be replaced bythe tachyonic action The BI Lagrangian reads [47]

LBI = 1205782

(1 minus radic1 +1198652

21205782) (2)

Here 120578 is the stringrsquos tension parameter This form reduces to(1) in the limit of 120578 rarr infin Indeed we can expand in series(2) in the following form

LBI =infin

sum

119899=1

1198881198991205782(1minus119899)

1198652119899

(3)

The leading order term 119899 = 1 is just in the form of (1) Evenif we do not work with BI theory this nonlinearity meets usfrom a geometrical point of view Suppose that we want towrite a conformal invariance (CI) Lagrangian constructedfrom the 119880(1) gauge fields 119860120583 Such CI is the invariance ofthe whole theory under geometrical transformation 119892

120583] rarr

1198902120590

119892120583] in a 119889 ge 4 Riemannian manifold without torsion or

nonmetricity fields However previously a version of suchLagrangian has been found in Weitznbock spacetime withtorsion and nonmetricity fields [48] It is easy to show that theLagrangianL prop 119865

1198892 is invariant under CI transformationsWhen 119889 = 4 the proper Lagrangian is L prop 119865

2 butin 119889 = 5 the suitable form is L prop 119865

52 The last formbelongs to the nonlinear noninteger Maxwell family Thusfrom geometrical view the nonlinearity is welcome in ourLagrangian dynamical theory Further even if we do notknow any on BI or CI when we are working with vacuumeffects there is a simple generalization ofMaxwell Lagrangianin a logarithmic form

Llog = minus1205782 log(1 + 119865

2

41205782) (4)

As BI case in the limit of |119865| ≪ 120578 by expanding this equationthe leading order term 119899 = 1 is the linear theory (1) In briefaccording to the above discussion it seems that considerationof the nonlinear effects of theMaxwell field as119865120575 is importantand the significant differences between the usual linear theory1198652 and nonlinear theory 119865

120575 can be shown Holographicsuperconductors provide a rich background for testing suchtype of new physics In this paper we will describe a (119889 minus

1) dimensional holographic superconductor (HSC) via 119889-dimensional gravity dual described by a 119889-dimensional AdSblack hole on the static patch We set the Stuckelberg field tozero and work with massless scalar fields

3 Field Equations

We write the following action [32ndash36] for a power Maxwellfield 119865

120583] which it is coupled minimally to a massless scalerfield 120595 in a 119889-dimensional asymptotic AdS

119889spacetime [45]

119878 = int119889119889

119909radicminus119892 [119877+(119889 minus 1) (119889 minus 2)minus120585(119865120583]119865120583])120575

minus1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

)

(5)

Here we set the radius of the AdS 119871 = 1 119863120583= 120597120583minus 119894119890119860

120583

119865120583] equiv 2120597

[120583119860]] = 120597

120583119860] minus 120597]119860120583 119860120583 is 119880(1) vector potential

The exponent (power) 120575 can be noninteger as we explained itin the previous section Further when we work withMaxwelllinear electrodynamics 120575 = 1 119889 = 4 then 120585 = 14 Howeverin our case with 120575 = 1 in general 120585 isin R remains as a freeparameter in our model We set the electric charge 119890 = 1In normal phase and in the absence of the scalar field weput 120595 = 0 This action has been used before in literaturefor full description of the nonlinear Maxwell field in thegravitational action In the probe limit when thematter actionand the gravitational part decouple the system of the fieldequations has an exact solution which will be discussed hereThe solution of the generalized Maxwell-Einstein equationsis the simple 119889-dimensional static Reissner-Nordstrom-Anti-de Sitter (RNAdS) black hole with the following metricform

119892120583] = diag(minus119891 (119903) 1

119891 (119903) 1199032

Σ119889minus2

) (6)

where Σ119889minus2

is the metric on a (119889 minus 2) dimensional sphere andthe metric function is

119891 (119903) = 1199032

(1 minus (119903+

119903)

119889minus1

) (7)

Here 119903+is the black hole horizon which in general is the

largest root of the algebraic equation119891(119903+) = 0 If we set 120575 = 1

in (5) we recover the usual holographic superconductorsIn the probe limit by ignoring the backreaction effects ofthe matter fields in the background metric 119892

120583] and withspherically symmetric staticmetric and by choosing a suitablegauge fixing for the gauge field 119860

120583we can take the functions

(120595 120601) isin R and one variable functions We assume that thefunctions 120595 and 120601 have finite numbers of poles on the realaxis It means the analytical solutions can be written in theforms of the algebraic series expressions To obtain the fieldequations we assume that 119860

120583= 120601(119903)120575

120583119905 so the nonzero

components of the 119865120583] read as

119865119903119905= minus119865119905119903= 1206011015840

(8)

Hence we have

119865120583]119865120583] = minus2120601

10158402

(9)

Also the Ricci 119877 of a 119889-dimensional spacetime with sphericalsymmetry reads

119877 = minus(11989110158401015840

+2 (119889 minus 2) 119891

1015840

119903minus(119889 minus 2) (119889 minus 3) 119891

1199032) (10)

Further we compute

1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

= 11989112059510158402

minus12060110158402

119891 (11)

4 Journal of Gravity

Finally by plugging the above expressions into the action (5)making a partial integration we get the following effectiveLagrangian

L = 119903119889minus2

[minus(11989110158401015840

+2 (119889minus2) 119891

1015840

119903minus(119889minus2) (119889minus3) 119891

1199032)

+ (119889minus1) (119889minus2) minus 120585(minus212060110158402

)120575

minus(11989112059510158402

minus12060110158402

119891)]

(12)

As a first step it is necessary to eliminate the 11989110158401015840 term byintegration part by part After it to write the field equationwe use the Euler-Lagrange equation as the following

119889

119889119903(120597L

120597119902119903

) =120597L

120597119902 119902 = 120601 120595 (13)

The field equations derived from (12) read

12059510158401015840

+ (1198911015840

119891+119889 minus 2

119903)1205951015840

+1206012

1198912120595 = 0 (14)

12060110158401015840

+119889 minus 2

2120575 minus 1

1206011015840

119903minus 119862120575

1205952

120585119891120601(1206011015840

)2(1minus120575)

= 0 (15)

here 119862120575= ((minus2)

2minus120575

120575(2120575 minus 1))minus1 To avoid the pure complex

numbers in our field equations we assume that120575 = 2minus(12119873)The case with 120575 = 1119863 = 4 is the usual four-dimensionalHSCwhich describes the three dimensional superconductors

These field equations are the special massless case of themodel which has been investigated recently [45 49] To avoidthe divergence near the singularity 119891(119903

+) = 0 we write the

boundary conditions for (14) (15) by

120601 (119903+) = 1205951015840

(119903+) = 0 (16)

The asymptotic solutions for system (14) (15) on the AdSboundary 119903 rarr infin are

120595 = 119863minus+

119863+

119903119889minus1 (17)

120601 = 119860 minus119861

119903120578 (18)

in (18) 119863plusmn= ⟨Oplusmn⟩ 120578 = (119889 minus 2120575 minus 1)(2120575 minus 1) where ⟨O

plusmn⟩

denotes the vacuum expectation value of the dual operatorOplusmnon the boundary and the chemical potential and charge

density of the dual theory are 119860 = 120583 119861 = 120588(2120575minus1)

minus1

respectively

We must clarify the reason for the modification of theform of the electric potential 120601 modified In the asymptoticregime we know that the metric function 119891 behaves like119891 sim 119903

2 Also the scalar field has the following asymptoticform 120595 sim 0 so (15) gives us

12060110158401015840

infinsim minus

119889 minus 2

2120575 minus 1

1206011015840

infin

119903 (19)

The solution (19) reads

120601infin(119903) = 119888

1+1198880(2120575 minus 1)

2120575 minus 119889 + 1

1

119903(119889minus2120575minus1)(2120575minus1) (20)

This solution coincides completely with the solution pre-sented in (18)Wementionhere that the above function120601

infin(119903)

in the limit of the linear electrodynamic theory 120575 = 1 hasthe true asymptotic form of 120601

infin(119903) sim 119903

3minus119889 119889 = 2 For 119889 = 2

the expression of 120601infin(119903) is in the form of a diverging log term

120601infin(119903) sim log(119903) and the application of the AdSCFT fails At

least we do not know the unique and true dictionary of theAdSCFT in this lower dimensional bulk theory

The only thing left is to identify the parameters with thephysical quantities in the dual theory that is the chemicalpotential 120583 and the charge density 120588 We did it before so theasymptotic behaviors have the same forms

The asymptotic solutions for 120601 120595 are the same as theprevious expressions which have been presented in [45] Fornormalization purposes we set119863

minus= 0

4 Variational Method

To solve the solutions of the field equations given by (14)and (15) the well-known technique is numerical algorithmsHowever from these numerical solutions it is not so easyand straightforward to read ⟨O⟩ Another method is usingthe matching method It is a potentially powerful methodEven so the resultsmust be interpreted very carefully near theboundaries The first step for solving system (14) (15) usingvariational approach [18] is rewriting the equations in a newdimensionless coordinate 119911 = 119903

+119903 in the following forms

12059510158401015840

+ (1198911015840

119891minus119889 minus 4

119911)1205951015840

+ (119903+

1199112)

2 1206012

1198912120595 = 0 (21)

12060110158401015840

minus120578 minus 1

1199111206011015840

minus 119862120575

1199032120575

+

1205851199114120575

1205952

119891120601(1206011015840

)2(1minus120575)

= 0 (22)

Now the prime is for differentiation with respect to thedimensionless radial coordinate 119911 Near the critical point119879 =

119879119888 the following solution is valid

120601 asymp 120601119861(1 minus 119911

120578

) (23)

Here 120601119861= 120583119888= 120588(2120575minus1)

minus1

(119903+)120578+1 is the value of the 120601 at the

horizon 119903 = 119903+ 120575 = (119889minus1)2 and the critical point corresponds

by the critical chemical potential120583 = 120583119888 Further we canwrite

120595 asymp ⟨119874+⟩ (

119911

119903+

)

119889minus1

119865 (119911) (24)

near the AdS boundary 119911 rarr 0 with 119865(0) = 1 1198651015840(0) = 0 Thetrial variational function satisfies the following second-orderSturm-Liouville self sdjoint differential equation

[120583 (119911) 1198651015840

(119911)]1015840

minus 119876 (119911) 119865 (119911) + (120601119861

119903+

)

2

119875 (119911) 119865 (119911) = 0

(25)

Journal of Gravity 5

where

120583 (119911) = 1199112119889minus2

(119911119889minus1

minus 1)

119875 (119911) = 120583 (119911) (1 minus 119911120578

1 minus 119911119889minus1)

2

119876 (119911) = minus 120583 (119911) [(119889 minus 1) (119889 minus 2)

1199112minus119889 minus 1

119911

times (2 + (119889 minus 3) 119911

119889minus1

119911 (1 minus 119911119889minus1)+119889 minus 4

119911)]

(26)

our strategy is to obtain the minimum value of 120601119861119903+from

the minimization of the following functional

Θ (120575 119889) equiv (120601119861

119903+

)

2

Min=

int1

0

(120583 (119911) 1198651015840

(119911)2

+ 119876 (119911) 1198652

(119911)) 119889119911

int1

0

119875 (119911) 1198652 (119911) 119889119911

(27)

Theminimum of the critical temperature 119879119888is obtained from

the 119879 = (119889 minus 1)119903+4120587 It reads

119879119888= 120574120588120578+1

(28)

and 120574 = ((119889 minus 1)4120587)((120601119861119903+)Min)minus(120578+1) Here we take trial

function 119865(119911) = 1 minus 1205721199112 in (27) It is useful to set 120588 = 1

The values of the critical temperature 119879119888for 119889 = 4 5 by

minimizing the functional (27) with trial function 119865(119911) andusing (28) (in case 120588 = 1) are given by the following

For 119889 = 4 120575 = 1 119879119888= 00844 for 120575 = 34 119879

119888=

01692For 119889 = 5 120575 = 1 119879

119888= 001676 for 120575 = 34 119879

119888=

02503 for 120575 = 54 119879119888= 00954

These values are in good agreements with the numericalvalues [50] and also coincide to the analytical values given in[45]

5 Calculating the Critical Exponent

We begin from (22) by writing it near the critical point (CP)and in limit 119879 rarr 119879

119888 The first step is rewriting the solution

in a perturbative scheme with respect to the perturbationparameter 120598 = ⟨119874

+⟩2 as it was described by Kanno [51] Near

the CP the solution of the field 120601 is written as

120601 (119911) = 1206010+ 120598120594 (119911) (29)

where 1206010= 120581119879119888(1 minus 119911

120578

) Using (24) and (29) in first orderwith respect to the 119874(120598) we obtain the following ordinarydifferential equation

12059410158401015840

(119911) minus120578 minus 1

1199111205941015840

(119911) = 119864 (119911) (30)

where

119864 (119911) =1198621205751199032

+119911minus120575minus2

1198652

(119911)

120585119891 (119911)1206012

0(1206010)10158402(1minus120575)

119865 (119911) |119911rarr0

asymp 1

(31)

Since 120601(119911) = (119860(119879119888)120578+1

119879120578

)(1 minus 119911120578

) 119860 = (119889 minus 1)4120587

writing the solution for 120601 in 119911 = 0 we have

120601 (0) = 1206010(0) + 120598120594 (0) (32)

The general solution for (30) is given by

120594 (119911) = radic119911 (11988811198691(119909) + 119888

21198841(119909))

+ radic1199111205871205782minus2120575

119879119888

4minus2120575

1205814minus2120575

119891 (119909)

(33)

with

119891 (119909) = 1205721198691(119909) int

1198841(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

+ 1205731198841(119909) int

1198691(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

(34)

Here 119909 = 2radic(1 minus 120578)119911 119869119899(119909) 119884119899(119909) are Bessel functions of

first and second kindsFinding the value of 120594(0) from (33) and solving it for

⟨119874+⟩ = radic120598 we obtain (we take 119879

119888= 1)

⟨119874+⟩ prop 119879

119889minus120575minus1205782

[1 minus 119879120578

]12

(35)

When the power 120575 decreases the value of the ⟨119874+⟩

increases Thus we conclude that the effect of the power 120575 in119889 = 4model is in the direction of the increase of ⟨119874

+⟩

However in the five dimensions the analysis is a littlebit different As we observe in 119889 = 5 when the power 120575increases the value of the ⟨119874

+⟩ increases Thus we can say

that the effect of the power 120575 in 119889 = 5model is in the directionof the increase of ⟨119874

+⟩

6 Calculating the Low Temperature DCConductivity

In this section we compute the low-temperature DC conduc-tivityWe concentrate on the general space time dimension 119889and we will try to calculate the conductivity 120590 as a functionof the rescaled frequency as follows

=120596

⟨119874+⟩1Δ

Δ = 119889 minus 1 (36)

In this limit the asymptotic form of the scalar field is

120595 (119911) =119887119889minus1

radic2119911119889minus1

119865 (119911) 119887 equiv ⟨119874+⟩ (37)

We follow the method in [18] Assuming that thereexists an external magnetic field 119860(119903 119905) = 119860(119903)119890

minus119894120596119905Note that here the applied Maxwell field is linear It isnot related to the nonlinear Maxwellrsquos field in the bulkaction In fact the nonlinearity of the Maxwell field nowis stored in the background metric As we know the fieldequations have some terms which involve the exponent 120575This parameter denotes the nonlinearity which is hiddenin the structure of the background metric and through it

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 3

In string theory language this BI action can be replaced bythe tachyonic action The BI Lagrangian reads [47]

LBI = 1205782

(1 minus radic1 +1198652

21205782) (2)

Here 120578 is the stringrsquos tension parameter This form reduces to(1) in the limit of 120578 rarr infin Indeed we can expand in series(2) in the following form

LBI =infin

sum

119899=1

1198881198991205782(1minus119899)

1198652119899

(3)

The leading order term 119899 = 1 is just in the form of (1) Evenif we do not work with BI theory this nonlinearity meets usfrom a geometrical point of view Suppose that we want towrite a conformal invariance (CI) Lagrangian constructedfrom the 119880(1) gauge fields 119860120583 Such CI is the invariance ofthe whole theory under geometrical transformation 119892

120583] rarr

1198902120590

119892120583] in a 119889 ge 4 Riemannian manifold without torsion or

nonmetricity fields However previously a version of suchLagrangian has been found in Weitznbock spacetime withtorsion and nonmetricity fields [48] It is easy to show that theLagrangianL prop 119865

1198892 is invariant under CI transformationsWhen 119889 = 4 the proper Lagrangian is L prop 119865

2 butin 119889 = 5 the suitable form is L prop 119865

52 The last formbelongs to the nonlinear noninteger Maxwell family Thusfrom geometrical view the nonlinearity is welcome in ourLagrangian dynamical theory Further even if we do notknow any on BI or CI when we are working with vacuumeffects there is a simple generalization ofMaxwell Lagrangianin a logarithmic form

Llog = minus1205782 log(1 + 119865

2

41205782) (4)

As BI case in the limit of |119865| ≪ 120578 by expanding this equationthe leading order term 119899 = 1 is the linear theory (1) In briefaccording to the above discussion it seems that considerationof the nonlinear effects of theMaxwell field as119865120575 is importantand the significant differences between the usual linear theory1198652 and nonlinear theory 119865

120575 can be shown Holographicsuperconductors provide a rich background for testing suchtype of new physics In this paper we will describe a (119889 minus

1) dimensional holographic superconductor (HSC) via 119889-dimensional gravity dual described by a 119889-dimensional AdSblack hole on the static patch We set the Stuckelberg field tozero and work with massless scalar fields

3 Field Equations

We write the following action [32ndash36] for a power Maxwellfield 119865

120583] which it is coupled minimally to a massless scalerfield 120595 in a 119889-dimensional asymptotic AdS

119889spacetime [45]

119878 = int119889119889

119909radicminus119892 [119877+(119889 minus 1) (119889 minus 2)minus120585(119865120583]119865120583])120575

minus1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

)

(5)

Here we set the radius of the AdS 119871 = 1 119863120583= 120597120583minus 119894119890119860

120583

119865120583] equiv 2120597

[120583119860]] = 120597

120583119860] minus 120597]119860120583 119860120583 is 119880(1) vector potential

The exponent (power) 120575 can be noninteger as we explained itin the previous section Further when we work withMaxwelllinear electrodynamics 120575 = 1 119889 = 4 then 120585 = 14 Howeverin our case with 120575 = 1 in general 120585 isin R remains as a freeparameter in our model We set the electric charge 119890 = 1In normal phase and in the absence of the scalar field weput 120595 = 0 This action has been used before in literaturefor full description of the nonlinear Maxwell field in thegravitational action In the probe limit when thematter actionand the gravitational part decouple the system of the fieldequations has an exact solution which will be discussed hereThe solution of the generalized Maxwell-Einstein equationsis the simple 119889-dimensional static Reissner-Nordstrom-Anti-de Sitter (RNAdS) black hole with the following metricform

119892120583] = diag(minus119891 (119903) 1

119891 (119903) 1199032

Σ119889minus2

) (6)

where Σ119889minus2

is the metric on a (119889 minus 2) dimensional sphere andthe metric function is

119891 (119903) = 1199032

(1 minus (119903+

119903)

119889minus1

) (7)

Here 119903+is the black hole horizon which in general is the

largest root of the algebraic equation119891(119903+) = 0 If we set 120575 = 1

in (5) we recover the usual holographic superconductorsIn the probe limit by ignoring the backreaction effects ofthe matter fields in the background metric 119892

120583] and withspherically symmetric staticmetric and by choosing a suitablegauge fixing for the gauge field 119860

120583we can take the functions

(120595 120601) isin R and one variable functions We assume that thefunctions 120595 and 120601 have finite numbers of poles on the realaxis It means the analytical solutions can be written in theforms of the algebraic series expressions To obtain the fieldequations we assume that 119860

120583= 120601(119903)120575

120583119905 so the nonzero

components of the 119865120583] read as

119865119903119905= minus119865119905119903= 1206011015840

(8)

Hence we have

119865120583]119865120583] = minus2120601

10158402

(9)

Also the Ricci 119877 of a 119889-dimensional spacetime with sphericalsymmetry reads

119877 = minus(11989110158401015840

+2 (119889 minus 2) 119891

1015840

119903minus(119889 minus 2) (119889 minus 3) 119891

1199032) (10)

Further we compute

1003816100381610038161003816100381611986312058312059510038161003816100381610038161003816

2

= 11989112059510158402

minus12060110158402

119891 (11)

4 Journal of Gravity

Finally by plugging the above expressions into the action (5)making a partial integration we get the following effectiveLagrangian

L = 119903119889minus2

[minus(11989110158401015840

+2 (119889minus2) 119891

1015840

119903minus(119889minus2) (119889minus3) 119891

1199032)

+ (119889minus1) (119889minus2) minus 120585(minus212060110158402

)120575

minus(11989112059510158402

minus12060110158402

119891)]

(12)

As a first step it is necessary to eliminate the 11989110158401015840 term byintegration part by part After it to write the field equationwe use the Euler-Lagrange equation as the following

119889

119889119903(120597L

120597119902119903

) =120597L

120597119902 119902 = 120601 120595 (13)

The field equations derived from (12) read

12059510158401015840

+ (1198911015840

119891+119889 minus 2

119903)1205951015840

+1206012

1198912120595 = 0 (14)

12060110158401015840

+119889 minus 2

2120575 minus 1

1206011015840

119903minus 119862120575

1205952

120585119891120601(1206011015840

)2(1minus120575)

= 0 (15)

here 119862120575= ((minus2)

2minus120575

120575(2120575 minus 1))minus1 To avoid the pure complex

numbers in our field equations we assume that120575 = 2minus(12119873)The case with 120575 = 1119863 = 4 is the usual four-dimensionalHSCwhich describes the three dimensional superconductors

These field equations are the special massless case of themodel which has been investigated recently [45 49] To avoidthe divergence near the singularity 119891(119903

+) = 0 we write the

boundary conditions for (14) (15) by

120601 (119903+) = 1205951015840

(119903+) = 0 (16)

The asymptotic solutions for system (14) (15) on the AdSboundary 119903 rarr infin are

120595 = 119863minus+

119863+

119903119889minus1 (17)

120601 = 119860 minus119861

119903120578 (18)

in (18) 119863plusmn= ⟨Oplusmn⟩ 120578 = (119889 minus 2120575 minus 1)(2120575 minus 1) where ⟨O

plusmn⟩

denotes the vacuum expectation value of the dual operatorOplusmnon the boundary and the chemical potential and charge

density of the dual theory are 119860 = 120583 119861 = 120588(2120575minus1)

minus1

respectively

We must clarify the reason for the modification of theform of the electric potential 120601 modified In the asymptoticregime we know that the metric function 119891 behaves like119891 sim 119903

2 Also the scalar field has the following asymptoticform 120595 sim 0 so (15) gives us

12060110158401015840

infinsim minus

119889 minus 2

2120575 minus 1

1206011015840

infin

119903 (19)

The solution (19) reads

120601infin(119903) = 119888

1+1198880(2120575 minus 1)

2120575 minus 119889 + 1

1

119903(119889minus2120575minus1)(2120575minus1) (20)

This solution coincides completely with the solution pre-sented in (18)Wementionhere that the above function120601

infin(119903)

in the limit of the linear electrodynamic theory 120575 = 1 hasthe true asymptotic form of 120601

infin(119903) sim 119903

3minus119889 119889 = 2 For 119889 = 2

the expression of 120601infin(119903) is in the form of a diverging log term

120601infin(119903) sim log(119903) and the application of the AdSCFT fails At

least we do not know the unique and true dictionary of theAdSCFT in this lower dimensional bulk theory

The only thing left is to identify the parameters with thephysical quantities in the dual theory that is the chemicalpotential 120583 and the charge density 120588 We did it before so theasymptotic behaviors have the same forms

The asymptotic solutions for 120601 120595 are the same as theprevious expressions which have been presented in [45] Fornormalization purposes we set119863

minus= 0

4 Variational Method

To solve the solutions of the field equations given by (14)and (15) the well-known technique is numerical algorithmsHowever from these numerical solutions it is not so easyand straightforward to read ⟨O⟩ Another method is usingthe matching method It is a potentially powerful methodEven so the resultsmust be interpreted very carefully near theboundaries The first step for solving system (14) (15) usingvariational approach [18] is rewriting the equations in a newdimensionless coordinate 119911 = 119903

+119903 in the following forms

12059510158401015840

+ (1198911015840

119891minus119889 minus 4

119911)1205951015840

+ (119903+

1199112)

2 1206012

1198912120595 = 0 (21)

12060110158401015840

minus120578 minus 1

1199111206011015840

minus 119862120575

1199032120575

+

1205851199114120575

1205952

119891120601(1206011015840

)2(1minus120575)

= 0 (22)

Now the prime is for differentiation with respect to thedimensionless radial coordinate 119911 Near the critical point119879 =

119879119888 the following solution is valid

120601 asymp 120601119861(1 minus 119911

120578

) (23)

Here 120601119861= 120583119888= 120588(2120575minus1)

minus1

(119903+)120578+1 is the value of the 120601 at the

horizon 119903 = 119903+ 120575 = (119889minus1)2 and the critical point corresponds

by the critical chemical potential120583 = 120583119888 Further we canwrite

120595 asymp ⟨119874+⟩ (

119911

119903+

)

119889minus1

119865 (119911) (24)

near the AdS boundary 119911 rarr 0 with 119865(0) = 1 1198651015840(0) = 0 Thetrial variational function satisfies the following second-orderSturm-Liouville self sdjoint differential equation

[120583 (119911) 1198651015840

(119911)]1015840

minus 119876 (119911) 119865 (119911) + (120601119861

119903+

)

2

119875 (119911) 119865 (119911) = 0

(25)

Journal of Gravity 5

where

120583 (119911) = 1199112119889minus2

(119911119889minus1

minus 1)

119875 (119911) = 120583 (119911) (1 minus 119911120578

1 minus 119911119889minus1)

2

119876 (119911) = minus 120583 (119911) [(119889 minus 1) (119889 minus 2)

1199112minus119889 minus 1

119911

times (2 + (119889 minus 3) 119911

119889minus1

119911 (1 minus 119911119889minus1)+119889 minus 4

119911)]

(26)

our strategy is to obtain the minimum value of 120601119861119903+from

the minimization of the following functional

Θ (120575 119889) equiv (120601119861

119903+

)

2

Min=

int1

0

(120583 (119911) 1198651015840

(119911)2

+ 119876 (119911) 1198652

(119911)) 119889119911

int1

0

119875 (119911) 1198652 (119911) 119889119911

(27)

Theminimum of the critical temperature 119879119888is obtained from

the 119879 = (119889 minus 1)119903+4120587 It reads

119879119888= 120574120588120578+1

(28)

and 120574 = ((119889 minus 1)4120587)((120601119861119903+)Min)minus(120578+1) Here we take trial

function 119865(119911) = 1 minus 1205721199112 in (27) It is useful to set 120588 = 1

The values of the critical temperature 119879119888for 119889 = 4 5 by

minimizing the functional (27) with trial function 119865(119911) andusing (28) (in case 120588 = 1) are given by the following

For 119889 = 4 120575 = 1 119879119888= 00844 for 120575 = 34 119879

119888=

01692For 119889 = 5 120575 = 1 119879

119888= 001676 for 120575 = 34 119879

119888=

02503 for 120575 = 54 119879119888= 00954

These values are in good agreements with the numericalvalues [50] and also coincide to the analytical values given in[45]

5 Calculating the Critical Exponent

We begin from (22) by writing it near the critical point (CP)and in limit 119879 rarr 119879

119888 The first step is rewriting the solution

in a perturbative scheme with respect to the perturbationparameter 120598 = ⟨119874

+⟩2 as it was described by Kanno [51] Near

the CP the solution of the field 120601 is written as

120601 (119911) = 1206010+ 120598120594 (119911) (29)

where 1206010= 120581119879119888(1 minus 119911

120578

) Using (24) and (29) in first orderwith respect to the 119874(120598) we obtain the following ordinarydifferential equation

12059410158401015840

(119911) minus120578 minus 1

1199111205941015840

(119911) = 119864 (119911) (30)

where

119864 (119911) =1198621205751199032

+119911minus120575minus2

1198652

(119911)

120585119891 (119911)1206012

0(1206010)10158402(1minus120575)

119865 (119911) |119911rarr0

asymp 1

(31)

Since 120601(119911) = (119860(119879119888)120578+1

119879120578

)(1 minus 119911120578

) 119860 = (119889 minus 1)4120587

writing the solution for 120601 in 119911 = 0 we have

120601 (0) = 1206010(0) + 120598120594 (0) (32)

The general solution for (30) is given by

120594 (119911) = radic119911 (11988811198691(119909) + 119888

21198841(119909))

+ radic1199111205871205782minus2120575

119879119888

4minus2120575

1205814minus2120575

119891 (119909)

(33)

with

119891 (119909) = 1205721198691(119909) int

1198841(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

+ 1205731198841(119909) int

1198691(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

(34)

Here 119909 = 2radic(1 minus 120578)119911 119869119899(119909) 119884119899(119909) are Bessel functions of

first and second kindsFinding the value of 120594(0) from (33) and solving it for

⟨119874+⟩ = radic120598 we obtain (we take 119879

119888= 1)

⟨119874+⟩ prop 119879

119889minus120575minus1205782

[1 minus 119879120578

]12

(35)

When the power 120575 decreases the value of the ⟨119874+⟩

increases Thus we conclude that the effect of the power 120575 in119889 = 4model is in the direction of the increase of ⟨119874

+⟩

However in the five dimensions the analysis is a littlebit different As we observe in 119889 = 5 when the power 120575increases the value of the ⟨119874

+⟩ increases Thus we can say

that the effect of the power 120575 in 119889 = 5model is in the directionof the increase of ⟨119874

+⟩

6 Calculating the Low Temperature DCConductivity

In this section we compute the low-temperature DC conduc-tivityWe concentrate on the general space time dimension 119889and we will try to calculate the conductivity 120590 as a functionof the rescaled frequency as follows

=120596

⟨119874+⟩1Δ

Δ = 119889 minus 1 (36)

In this limit the asymptotic form of the scalar field is

120595 (119911) =119887119889minus1

radic2119911119889minus1

119865 (119911) 119887 equiv ⟨119874+⟩ (37)

We follow the method in [18] Assuming that thereexists an external magnetic field 119860(119903 119905) = 119860(119903)119890

minus119894120596119905Note that here the applied Maxwell field is linear It isnot related to the nonlinear Maxwellrsquos field in the bulkaction In fact the nonlinearity of the Maxwell field nowis stored in the background metric As we know the fieldequations have some terms which involve the exponent 120575This parameter denotes the nonlinearity which is hiddenin the structure of the background metric and through it

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ThermodynamicsJournal of

4 Journal of Gravity

Finally by plugging the above expressions into the action (5)making a partial integration we get the following effectiveLagrangian

L = 119903119889minus2

[minus(11989110158401015840

+2 (119889minus2) 119891

1015840

119903minus(119889minus2) (119889minus3) 119891

1199032)

+ (119889minus1) (119889minus2) minus 120585(minus212060110158402

)120575

minus(11989112059510158402

minus12060110158402

119891)]

(12)

As a first step it is necessary to eliminate the 11989110158401015840 term byintegration part by part After it to write the field equationwe use the Euler-Lagrange equation as the following

119889

119889119903(120597L

120597119902119903

) =120597L

120597119902 119902 = 120601 120595 (13)

The field equations derived from (12) read

12059510158401015840

+ (1198911015840

119891+119889 minus 2

119903)1205951015840

+1206012

1198912120595 = 0 (14)

12060110158401015840

+119889 minus 2

2120575 minus 1

1206011015840

119903minus 119862120575

1205952

120585119891120601(1206011015840

)2(1minus120575)

= 0 (15)

here 119862120575= ((minus2)

2minus120575

120575(2120575 minus 1))minus1 To avoid the pure complex

numbers in our field equations we assume that120575 = 2minus(12119873)The case with 120575 = 1119863 = 4 is the usual four-dimensionalHSCwhich describes the three dimensional superconductors

These field equations are the special massless case of themodel which has been investigated recently [45 49] To avoidthe divergence near the singularity 119891(119903

+) = 0 we write the

boundary conditions for (14) (15) by

120601 (119903+) = 1205951015840

(119903+) = 0 (16)

The asymptotic solutions for system (14) (15) on the AdSboundary 119903 rarr infin are

120595 = 119863minus+

119863+

119903119889minus1 (17)

120601 = 119860 minus119861

119903120578 (18)

in (18) 119863plusmn= ⟨Oplusmn⟩ 120578 = (119889 minus 2120575 minus 1)(2120575 minus 1) where ⟨O

plusmn⟩

denotes the vacuum expectation value of the dual operatorOplusmnon the boundary and the chemical potential and charge

density of the dual theory are 119860 = 120583 119861 = 120588(2120575minus1)

minus1

respectively

We must clarify the reason for the modification of theform of the electric potential 120601 modified In the asymptoticregime we know that the metric function 119891 behaves like119891 sim 119903

2 Also the scalar field has the following asymptoticform 120595 sim 0 so (15) gives us

12060110158401015840

infinsim minus

119889 minus 2

2120575 minus 1

1206011015840

infin

119903 (19)

The solution (19) reads

120601infin(119903) = 119888

1+1198880(2120575 minus 1)

2120575 minus 119889 + 1

1

119903(119889minus2120575minus1)(2120575minus1) (20)

This solution coincides completely with the solution pre-sented in (18)Wementionhere that the above function120601

infin(119903)

in the limit of the linear electrodynamic theory 120575 = 1 hasthe true asymptotic form of 120601

infin(119903) sim 119903

3minus119889 119889 = 2 For 119889 = 2

the expression of 120601infin(119903) is in the form of a diverging log term

120601infin(119903) sim log(119903) and the application of the AdSCFT fails At

least we do not know the unique and true dictionary of theAdSCFT in this lower dimensional bulk theory

The only thing left is to identify the parameters with thephysical quantities in the dual theory that is the chemicalpotential 120583 and the charge density 120588 We did it before so theasymptotic behaviors have the same forms

The asymptotic solutions for 120601 120595 are the same as theprevious expressions which have been presented in [45] Fornormalization purposes we set119863

minus= 0

4 Variational Method

To solve the solutions of the field equations given by (14)and (15) the well-known technique is numerical algorithmsHowever from these numerical solutions it is not so easyand straightforward to read ⟨O⟩ Another method is usingthe matching method It is a potentially powerful methodEven so the resultsmust be interpreted very carefully near theboundaries The first step for solving system (14) (15) usingvariational approach [18] is rewriting the equations in a newdimensionless coordinate 119911 = 119903

+119903 in the following forms

12059510158401015840

+ (1198911015840

119891minus119889 minus 4

119911)1205951015840

+ (119903+

1199112)

2 1206012

1198912120595 = 0 (21)

12060110158401015840

minus120578 minus 1

1199111206011015840

minus 119862120575

1199032120575

+

1205851199114120575

1205952

119891120601(1206011015840

)2(1minus120575)

= 0 (22)

Now the prime is for differentiation with respect to thedimensionless radial coordinate 119911 Near the critical point119879 =

119879119888 the following solution is valid

120601 asymp 120601119861(1 minus 119911

120578

) (23)

Here 120601119861= 120583119888= 120588(2120575minus1)

minus1

(119903+)120578+1 is the value of the 120601 at the

horizon 119903 = 119903+ 120575 = (119889minus1)2 and the critical point corresponds

by the critical chemical potential120583 = 120583119888 Further we canwrite

120595 asymp ⟨119874+⟩ (

119911

119903+

)

119889minus1

119865 (119911) (24)

near the AdS boundary 119911 rarr 0 with 119865(0) = 1 1198651015840(0) = 0 Thetrial variational function satisfies the following second-orderSturm-Liouville self sdjoint differential equation

[120583 (119911) 1198651015840

(119911)]1015840

minus 119876 (119911) 119865 (119911) + (120601119861

119903+

)

2

119875 (119911) 119865 (119911) = 0

(25)

Journal of Gravity 5

where

120583 (119911) = 1199112119889minus2

(119911119889minus1

minus 1)

119875 (119911) = 120583 (119911) (1 minus 119911120578

1 minus 119911119889minus1)

2

119876 (119911) = minus 120583 (119911) [(119889 minus 1) (119889 minus 2)

1199112minus119889 minus 1

119911

times (2 + (119889 minus 3) 119911

119889minus1

119911 (1 minus 119911119889minus1)+119889 minus 4

119911)]

(26)

our strategy is to obtain the minimum value of 120601119861119903+from

the minimization of the following functional

Θ (120575 119889) equiv (120601119861

119903+

)

2

Min=

int1

0

(120583 (119911) 1198651015840

(119911)2

+ 119876 (119911) 1198652

(119911)) 119889119911

int1

0

119875 (119911) 1198652 (119911) 119889119911

(27)

Theminimum of the critical temperature 119879119888is obtained from

the 119879 = (119889 minus 1)119903+4120587 It reads

119879119888= 120574120588120578+1

(28)

and 120574 = ((119889 minus 1)4120587)((120601119861119903+)Min)minus(120578+1) Here we take trial

function 119865(119911) = 1 minus 1205721199112 in (27) It is useful to set 120588 = 1

The values of the critical temperature 119879119888for 119889 = 4 5 by

minimizing the functional (27) with trial function 119865(119911) andusing (28) (in case 120588 = 1) are given by the following

For 119889 = 4 120575 = 1 119879119888= 00844 for 120575 = 34 119879

119888=

01692For 119889 = 5 120575 = 1 119879

119888= 001676 for 120575 = 34 119879

119888=

02503 for 120575 = 54 119879119888= 00954

These values are in good agreements with the numericalvalues [50] and also coincide to the analytical values given in[45]

5 Calculating the Critical Exponent

We begin from (22) by writing it near the critical point (CP)and in limit 119879 rarr 119879

119888 The first step is rewriting the solution

in a perturbative scheme with respect to the perturbationparameter 120598 = ⟨119874

+⟩2 as it was described by Kanno [51] Near

the CP the solution of the field 120601 is written as

120601 (119911) = 1206010+ 120598120594 (119911) (29)

where 1206010= 120581119879119888(1 minus 119911

120578

) Using (24) and (29) in first orderwith respect to the 119874(120598) we obtain the following ordinarydifferential equation

12059410158401015840

(119911) minus120578 minus 1

1199111205941015840

(119911) = 119864 (119911) (30)

where

119864 (119911) =1198621205751199032

+119911minus120575minus2

1198652

(119911)

120585119891 (119911)1206012

0(1206010)10158402(1minus120575)

119865 (119911) |119911rarr0

asymp 1

(31)

Since 120601(119911) = (119860(119879119888)120578+1

119879120578

)(1 minus 119911120578

) 119860 = (119889 minus 1)4120587

writing the solution for 120601 in 119911 = 0 we have

120601 (0) = 1206010(0) + 120598120594 (0) (32)

The general solution for (30) is given by

120594 (119911) = radic119911 (11988811198691(119909) + 119888

21198841(119909))

+ radic1199111205871205782minus2120575

119879119888

4minus2120575

1205814minus2120575

119891 (119909)

(33)

with

119891 (119909) = 1205721198691(119909) int

1198841(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

+ 1205731198841(119909) int

1198691(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

(34)

Here 119909 = 2radic(1 minus 120578)119911 119869119899(119909) 119884119899(119909) are Bessel functions of

first and second kindsFinding the value of 120594(0) from (33) and solving it for

⟨119874+⟩ = radic120598 we obtain (we take 119879

119888= 1)

⟨119874+⟩ prop 119879

119889minus120575minus1205782

[1 minus 119879120578

]12

(35)

When the power 120575 decreases the value of the ⟨119874+⟩

increases Thus we conclude that the effect of the power 120575 in119889 = 4model is in the direction of the increase of ⟨119874

+⟩

However in the five dimensions the analysis is a littlebit different As we observe in 119889 = 5 when the power 120575increases the value of the ⟨119874

+⟩ increases Thus we can say

that the effect of the power 120575 in 119889 = 5model is in the directionof the increase of ⟨119874

+⟩

6 Calculating the Low Temperature DCConductivity

In this section we compute the low-temperature DC conduc-tivityWe concentrate on the general space time dimension 119889and we will try to calculate the conductivity 120590 as a functionof the rescaled frequency as follows

=120596

⟨119874+⟩1Δ

Δ = 119889 minus 1 (36)

In this limit the asymptotic form of the scalar field is

120595 (119911) =119887119889minus1

radic2119911119889minus1

119865 (119911) 119887 equiv ⟨119874+⟩ (37)

We follow the method in [18] Assuming that thereexists an external magnetic field 119860(119903 119905) = 119860(119903)119890

minus119894120596119905Note that here the applied Maxwell field is linear It isnot related to the nonlinear Maxwellrsquos field in the bulkaction In fact the nonlinearity of the Maxwell field nowis stored in the background metric As we know the fieldequations have some terms which involve the exponent 120575This parameter denotes the nonlinearity which is hiddenin the structure of the background metric and through it

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 5

where

120583 (119911) = 1199112119889minus2

(119911119889minus1

minus 1)

119875 (119911) = 120583 (119911) (1 minus 119911120578

1 minus 119911119889minus1)

2

119876 (119911) = minus 120583 (119911) [(119889 minus 1) (119889 minus 2)

1199112minus119889 minus 1

119911

times (2 + (119889 minus 3) 119911

119889minus1

119911 (1 minus 119911119889minus1)+119889 minus 4

119911)]

(26)

our strategy is to obtain the minimum value of 120601119861119903+from

the minimization of the following functional

Θ (120575 119889) equiv (120601119861

119903+

)

2

Min=

int1

0

(120583 (119911) 1198651015840

(119911)2

+ 119876 (119911) 1198652

(119911)) 119889119911

int1

0

119875 (119911) 1198652 (119911) 119889119911

(27)

Theminimum of the critical temperature 119879119888is obtained from

the 119879 = (119889 minus 1)119903+4120587 It reads

119879119888= 120574120588120578+1

(28)

and 120574 = ((119889 minus 1)4120587)((120601119861119903+)Min)minus(120578+1) Here we take trial

function 119865(119911) = 1 minus 1205721199112 in (27) It is useful to set 120588 = 1

The values of the critical temperature 119879119888for 119889 = 4 5 by

minimizing the functional (27) with trial function 119865(119911) andusing (28) (in case 120588 = 1) are given by the following

For 119889 = 4 120575 = 1 119879119888= 00844 for 120575 = 34 119879

119888=

01692For 119889 = 5 120575 = 1 119879

119888= 001676 for 120575 = 34 119879

119888=

02503 for 120575 = 54 119879119888= 00954

These values are in good agreements with the numericalvalues [50] and also coincide to the analytical values given in[45]

5 Calculating the Critical Exponent

We begin from (22) by writing it near the critical point (CP)and in limit 119879 rarr 119879

119888 The first step is rewriting the solution

in a perturbative scheme with respect to the perturbationparameter 120598 = ⟨119874

+⟩2 as it was described by Kanno [51] Near

the CP the solution of the field 120601 is written as

120601 (119911) = 1206010+ 120598120594 (119911) (29)

where 1206010= 120581119879119888(1 minus 119911

120578

) Using (24) and (29) in first orderwith respect to the 119874(120598) we obtain the following ordinarydifferential equation

12059410158401015840

(119911) minus120578 minus 1

1199111205941015840

(119911) = 119864 (119911) (30)

where

119864 (119911) =1198621205751199032

+119911minus120575minus2

1198652

(119911)

120585119891 (119911)1206012

0(1206010)10158402(1minus120575)

119865 (119911) |119911rarr0

asymp 1

(31)

Since 120601(119911) = (119860(119879119888)120578+1

119879120578

)(1 minus 119911120578

) 119860 = (119889 minus 1)4120587

writing the solution for 120601 in 119911 = 0 we have

120601 (0) = 1206010(0) + 120598120594 (0) (32)

The general solution for (30) is given by

120594 (119911) = radic119911 (11988811198691(119909) + 119888

21198841(119909))

+ radic1199111205871205782minus2120575

119879119888

4minus2120575

1205814minus2120575

119891 (119909)

(33)

with

119891 (119909) = 1205721198691(119909) int

1198841(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

+ 1205731198841(119909) int

1198691(119909) (119911

120578

minus 1)2

1199112120578minus12+120575minus2120575120578

119911119889 minus 119911119889119911

(34)

Here 119909 = 2radic(1 minus 120578)119911 119869119899(119909) 119884119899(119909) are Bessel functions of

first and second kindsFinding the value of 120594(0) from (33) and solving it for

⟨119874+⟩ = radic120598 we obtain (we take 119879

119888= 1)

⟨119874+⟩ prop 119879

119889minus120575minus1205782

[1 minus 119879120578

]12

(35)

When the power 120575 decreases the value of the ⟨119874+⟩

increases Thus we conclude that the effect of the power 120575 in119889 = 4model is in the direction of the increase of ⟨119874

+⟩

However in the five dimensions the analysis is a littlebit different As we observe in 119889 = 5 when the power 120575increases the value of the ⟨119874

+⟩ increases Thus we can say

that the effect of the power 120575 in 119889 = 5model is in the directionof the increase of ⟨119874

+⟩

6 Calculating the Low Temperature DCConductivity

In this section we compute the low-temperature DC conduc-tivityWe concentrate on the general space time dimension 119889and we will try to calculate the conductivity 120590 as a functionof the rescaled frequency as follows

=120596

⟨119874+⟩1Δ

Δ = 119889 minus 1 (36)

In this limit the asymptotic form of the scalar field is

120595 (119911) =119887119889minus1

radic2119911119889minus1

119865 (119911) 119887 equiv ⟨119874+⟩ (37)

We follow the method in [18] Assuming that thereexists an external magnetic field 119860(119903 119905) = 119860(119903)119890

minus119894120596119905Note that here the applied Maxwell field is linear It isnot related to the nonlinear Maxwellrsquos field in the bulkaction In fact the nonlinearity of the Maxwell field nowis stored in the background metric As we know the fieldequations have some terms which involve the exponent 120575This parameter denotes the nonlinearity which is hiddenin the structure of the background metric and through it

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Gravity

Moreover it diffuses to the dynamics of the scalar field andthe Abelian gauge field 119880(1) so to compute the conductivitythe applied external magnetic field is linear and satisfies theusual linear Maxwell field 119865120583

120583] = 0 which is nothing but thelinear wave equation

The linear wave equation for this field reduces to

minus1198892

119860

1198891199032+ 119881 (119903) 119860 = 120596

2

119860 (38)

Equation (38) is written in the Schrodingerrsquos formand in terms of the new tortoise coordinate 119903 =

minus(1119903+)Σinfin

119899=0(119911119899(119889minus1)+1

(119899(119889 minus 1) + 1)) The horizon 119903 = 119903+

is located at 119911 = 1 or 119903 rarr minusinfin Here the potential is119881 = 2119891(119903)120595

2

(119903) The ingoing waves in horizon behave as aboundary condition (BC) for solving this wave equation (38)read as

119860 sim 119890minus119894120596119903

sim (1 minus 119911)minus119894120596(119889minus1)119903

+ (39)

The electromagnetic wave equation (38) in the coordinate 119911reads

119860119911119911+ (

2

119911+1198911015840

2119891)119860119911+ (minus

21199032

+120595(119911)2

1199114+1205962

1199032

+

1199114119891)119860 = 0 (40)

By replacing (37) we obtain

119860119911119911+ 1199011(119911) 119860119911+ 1198761(119911) 119860 = 0

1199011(119911) = (

2

119911+1198911015840

2119891)

1198761(119911) = minus

1199032

+1198872(119889minus1)

1199112(119889minus1)

1198652

(119911)

1199114+1205962

1199032

+

1199114119891

(41)

To keep the boundary conditions we put

119860 = (1 minus 119911)minus119894120596(119889minus1)119903

+119890minus119894120596119911(119889minus1)119903

+Θ (119911) (42)

Substituting this ansatz in (41) we obtain

Θ10158401015840

+ 119875 (119911)Θ1015840

+ 119876 (119911)Θ = 0 (43)

119875 (119911) =119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 2119894120596119911

119903+(1 minus 119911) (119889 minus 1)

(44)

119876 (119911) = (119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

times (1199032

+(119889 minus 1)

2

(1 minus 119911)2

)minus1

(45)

By imposing the regularity condition on wave function Θ

at the black hole horizon 119911 = 1 we obtain the followingauxiliary boundary condition

0 = Θ1015840

(1) lim119911rarr1

(1

2119903+(1 minus 119911) (119889 minus 1) 119901

1(119911) + 119894120596119911)

+ Θ (1) lim119911rarr1

(119894119911 (1 minus 119911) 120596119903+(119889 minus 1) 119901

1(119911)

+ 1199032

+(119889 minus 1)

2

(1 minus 119911)2

1198761(119911)

+120596 (119894 (119889 minus 1) 119903+minus 1205961199112

))

(46)

Explicitly by computing the limits we have

119894 (minus119903++ 119894120596 + 119903

+119889) 120596Θ (1) = 0 (47)

One possibility is Θ(1) = 0 Another 120596 = 119894119903+(119889 minus 1)but the

last case from (39) leads to the

119860 sim 119890minus119894120596119903

sim (1 minus 119911) (48)

which has no meaning as the ingoing wave toward thehorizon so we impose Θ(1) = 0 Since we are working inthe low temperature limit we take the limit 119887 rarr infin so werescale the coordinate 119911 by 119911 rarr 119911119887 Also we must put onesuitable trial form for119865(119911) for the case ofΔ = 119889minus1 gt 32 119889 gt

2 we put 119865(119911119887) rarr 119865(0) = 1 We rescale (43) so we obtain

Θ10158401015840

+ 119887119875(119911

119887)Θ1015840

+ 1198872

119876(119911

119887)Θ = 0

1015840

= 119889119889 (119911119887) (49)

Finally we obtain

Θ10158401015840

+119911

1199032+

(1 +2119894119903+

119889 minus 1)Θ1015840

+3119894

119903+(119889 minus 1)

Θ = 0 (50)

The general solution for (50) reads

Θ (119911) = 119911eminus1199112

(12+119894119903+

)1199032

+

times (119888+119872(120583 ]

1

2

(1 + 2119894119903+) 1199112

1199032+

)

+119888minus119880(120583 ]

(1 + 2119894119903+) 1199112

1199032+

))

(51)

120583 =1

2

2119889 minus 2 + 4119894119903+119889 minus 7119894119903

+

(1 + 2119894119903+) (119889 minus 1)

] =3

2 (52)

The DC conductivity in the low temperature limit is definedby

120590 () =119894

Θ1015840

(minus2

)

Θ (minus2

)

(53)

We need the asymptotic limit of (51) Indeed we guess thatlim119911rarrinfin

Θ(119911) asymp Θ(minus2

) where here the is the quasinormalmodes and locates on the real axis so we guessRe[120590()] = 0

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 7

except at the poles of Im[120590()] where asRe[120590()] asymp 120575()First by imposing Θ(1) = 0 we have

Θ (119911) = 119888minus

119911eminus1199112

(12+119894119903+

)1199032

+

119880 (120583 ] (1 + 2 119894 119903+) 1199032+)

times [119872(120583 ](1 + 2119894119903

+) 1199112

1199032+

)119880(120583 ]1 + 2119894119903

+

1199032+

)

minus 119880(120583 ](1 + 2119894119903

+) 1199112

1199032+

)

times119872(120583 ]1 + 2119894119903

+

1199032+

)]

(54)

Here 119872(120583 ] 119909) 119880(120583 ] 119909) denote the Kummer functions[52]

The quasinormal modes are the solutions of the followingequation

119880(120583 ]1 + 2119894119903

+

1199032+

) = 119872(120583 ]1 + 2119894119903

+

1199032+

) (55)

which has no closed analytical solution and can be solved justnumerically

7 Calculating the Zero TemperatureConductivity

In this section we compute analytically the conductivity 120590()in the zero temperature 119879 = 0 limit This case corresponds tothe limiting case 119887 rarr infin We begin by writing (38) in thecoordinate 1199111015840 = 119911119887

1198601199111015840

1199111015840 +

1

21199111015840[

[

2 minus (119889 + 1) (1198871199111015840

)119889minus1

1 minus (1198871199111015840)119889minus1

]

]

times 1198601199111015840 +

1199032

+

(1198871199111015840)4[

1198872

11991110158402

2

1 minus (1198871199111015840)119889minus1

minus(1198872

1199111015840

)2(119889minus1)

119865(1198871199111015840

)2

]119860 = 0

(56)

By taking the limit 119887 rarr infin we have

1198601199111015840

1199111015840 +

119889 + 1

211991110158401198601199111015840 + 1199032

+[minus

2

(1198871199111015840)119889+1

minus 1198874(119889minus2)

11991110158402(119889minus3)

]119860 = 0

(57)

In (57) we set lim119887rarrinfin

119865(1198871199111015840

) = 119865(0) = 1 The closed formof the exact solution for (57) depends on the value of the 119889

Belowwe list the solutions for special dimensions119889 = 2119889 = 3

as follows

119860(1199111015840

) = 1198621199111015840minus14

119873minusradic1+16119903

2

+

2

(2119894119903+

11988732radic1199111015840) 119889 = 2 (58)

119860(1199111015840

) =

119867(0 120573 120575 120574 (11991110158402

+ 1) (11991110158402

minus 1))

radic1199111015840

times (1198881+ 1198882int

1198891199111015840

119911119867 (0 120573 120575 120574 (11991110158402 + 1) (11991110158402 minus 1)))

119889 = 3

(59)

Here

120573 = minus1

4

41198878

1199032

++ 1198874

+ 41199032

+2

1198874

120575 = minus2

1199032

+(1198878

minus 2

)

1198874

120574 = minus1

4

41199032

+2

minus 1198874

+ 41198878

1199032

+

1198874

(60)

Here 119873](119909) is the second kind of the Bessel function and119867(0 120573 120575 120574 119909) denotes the Heun double confluent function[53] by expending (58) (59) in series in the form

119860(1199111015840

) = 1198600+ 1198602(1198871199111015840

)2

minus sdot sdot sdot (61)

The conductivity can be computed via the following simpleformula

120590 () =2

119894

1198602

1198600

+119894

2 (62)

So by computing the asymptotic series the conductivity willbe determined by the analytical expression

It is appropriate here to present the explicit form of theconductivity for one case We choose the case given by (58)In the zero temperature limit we treat the horizon size verytiny so in application we takeradic1 + 161199032

+asymp 1 Also it is better

we define 119909 = 2119903+11988732radic1199111015840 which by the definition of the 1199111015840

reads as 119909 = 2120596119903+119887radic119911 When 119887 rarr infin then 119909 ≪ 1 so we

can expand the Bessel function in terms of the infinitesimalargument 119909 and hence rewrite (58) as the following form

119860 (119911) = 1198624

radic119887

119911119873minus] (119894119909) ] =

1

2 119909 =

2120596119903+

119887radic119911 (63)

Remembering the following identities

119873minus] (119894119909) =

119894minus]minus1

119868minus] (119909) cosh (]119909) minus 119894

]minus1119868]

sinh (]119909) (64)

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Gravity

100

10minus1

10minus2

10minus3

10minus4

10minus5

0 2 4 6 8 10log(120596)

logI[120590(120596)]

Figure 1 Variation of the log(Im[120590()]) for zero temperature limitin 119889 = 2

Also we know that

11986812

(119909) = radic2

120587119909cosh (119909) 119868

minus(12)(119909) = radic

2

120587119909sinh (119909)

(65)

so we have

119873minus(12)

(119894119909) = radic2

119894120587119909[119894 cosh (119909) cosh (1199092) minus sinh (119909)

sinh (1199092)]

(66)

Also in limit 119909 ≪ 1we have

119894 cosh (119909) cosh (1199092) minus sinh (119909)sinh (1199092)

=2119894

119909minus 2 +

7

6119894119909 minus

1

41199092

+59

3601198941199093

+ 119874 (1199094

)

(67)

So for the electromagnetic field we have

119860 (119911) asymp 119862

4

radic119887

radic119894120587119903+[119894radic119911

119903+

minus 2 +7

3119894119903+

radic119911minus2

1199032

+

119911] (68)

Finally we obtain the conductivity using (62) by the followingsimple formula

120590 () =1

2119894+119894

2 (69)

where as we guess Re[120590()] = 0 Figure 1 shows Im[120590()]

as a function of the for zero temperature case and in 119889 =

2 Equation (69) gives the expression for DC conductivity inthe zero temperature limit in AdS

2CFT1model Indeed it is

comparable with the numerical results of the previous papersabout one dimensional holographic superconductors [54]

8 Conclusion

In this paper we investigated the analytical properties ofa holographic superconductor with power Maxwellrsquos fieldWe studied the problem in the probe limit We observedthat it is possible to find the critical temperature 119879

119888and the

condensation ⟨119874+⟩ and the conductivity 120590(120596) via Sturm-

Liouville variational approach We concluded that in 119889 = 4when the power 120575 decreases the value of the ⟨119874

+⟩ increases

Thus we can say that the effect of the power 120575 in 119889 = 4modelis in the direction of the increase of ⟨119874

+⟩ In 119889 = 5 when the

power 120575 increases the value of the ⟨119874+⟩ increases Thus we

can say that the effect of the power 120575 in 119889 = 5 model is inthe direction of the increase of ⟨119874

+⟩ Further we analytically

deduced the low temperature and the zero temperature DCconductivity 120590 as a function of the = 120596⟨119874

+⟩1Δ Our

work helps to give a better understanding of some unfamiliareffects of the holographic superconductors both in differentspace dimensions and nonlinearity in Maxwellrsquos strengthfield

References

[1] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics LettersSection B vol 428 no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti-de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] C P Herzog ldquoLectures on holographic superfluidity and super-conductivityrdquo Journal of Physics A vol 42 no 34 Article ID343001 2009

[5] S A Hartnoll ldquoLectures on holographic methods for con-densed matter physicsrdquo Classical and Quantum Gravity vol 26no 22 Article ID 224002 2009

[6] G Policastro D T Son and A O Starinets ldquoShear viscosity ofstrongly coupled 119873 = 4 supersymmetric yang-mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 4 pages2001

[7] P Kovtun D T Son and A O Starinets ldquoHolography andhydrodynamics diffusion on stretched horizonsrdquo Journal ofHigh Energy Physics vol 7 no 10 pp 1675ndash1701 2003

[8] S A Hartnoll C P Herzog and G T Horowitz ldquoBuilding aholographic superconductorrdquo Physical Review Letters vol 101no 3 Article ID 031601 4 pages 2008

[9] P Breitenlohner and D Z Freedman ldquoStability in gaugedextended supergravityrdquo Annals of Physics vol 144 no 2 pp249ndash281 1982

[10] S S Gubser ldquoBreaking an abelian gauge symmetry near a blackhole horizonrdquo Physical Review D vol 78 Article ID 065034 15pages 2008

[11] X H Ge B Wang S F Wu and G H Yang ldquoAnalytical studyon holographic superconductors in external magnetic fieldrdquoJournal of High Energy Physics vol 1008 p 108 2010

[12] H F Li R G Cai and H Q Zhang ldquoAnalytical studies onholographic superconductors in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 1104 p 028 2011

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 9

[13] R G Cai H F Li and H Q Zhang ldquoAnalytical studieson holographic insulatorsuperconductor phase transitionsrdquoPhysical Review D vol 83 no 12 Article ID 126007 7 pages2011

[14] C M Chen and M F Wu ldquoAn analytic analysis of phase tran-sitions in holographic superconductorsrdquo Progress of TheoreticalPhysics vol 126 pp 387ndash395 2011

[15] XHGe andHQ Leng ldquoAnalytical calculation on criticalmag-netic field in holographic superconductors with backreactionrdquoProgress of Theoretical Physics vol 128 pp 1211ndash1228 2012

[16] M R Setare D Momeni R Myrzakulov and M RazaldquoHolographic superconductors in the AdS black hole with amagnetic chargerdquo Physica Scripta vol 86 Article ID 0450052012

[17] DMomeniM R Setare andNMajd ldquoHolographic supercon-ductors in a model of non-relativistic gravityrdquo Journal of HighEnergy Physics vol 05 p 118 2011

[18] G Siopsis J Therrien and S Musiri ldquoHolographic supercon-ductors near the Breitenlohner-Freedman boundrdquoClassical andQuantum Gravity vol 29 no 8 Article ID 085007

[19] H B Zeng X Gao Y Jiang and H S Zong ldquoAnalyticalcomputation of critical exponents in several holographic super-conductorsrdquo Journal of High Energy Physics 2011

[20] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic p-wave superconductorsrdquo Journal ofHigh Energy Physics vol 08 p 104 2012

[21] E Nakano and W Y Wen ldquoCritical magnetic field in aholographic superconductorrdquo Physical Review D vol 78 no 4Article ID 046004 3 pages 2008

[22] T Albash and C V Johnson ldquoA holographic superconductor inan external magnetic fieldrdquo Journal of High Energy Physics vol2008 p 121 2008

[23] O Domenech M Montull A Pomarol A Salvio and P JSilva ldquoEmergent gauge fields in holographic superconductorsrdquoJournal of High Energy Physics vol 2010 p 033 2010

[24] M Montull O Pujolas A Salvio and P J Silva ldquoFlux peri-odicities and quantum hair on holographic superconductorsrdquoPhysical Review Letters vol 107 no 18 Article ID 181601 2011

[25] M Montull O Pujolas A Salvio and P J Silva ldquoMagneticresponse in the holographic insulatorsuperconductor transi-tionrdquo Journal of High Energy Physics vol 1204 p 135 2012

[26] Salvio A ldquoHolographic superfluids and superconductors indilaton-gravityrdquo Journal of High Energy Physics vol 1209 p 1342012

[27] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2011

[28] N Bobev N Halmagyi K Pilch and N P Warner ldquoSuper-gravity instabilities of non-supersymmetric quantum criticalpointsrdquo Journal of High Energy Physics 46 pages 2010

[29] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 25 pages 2012

[30] T Fischbacher K Pilch and N P Warner ldquoNew supersymmet-ric and stable non-supersymmetric phases in supergravity andholographic field theoryrdquo httparxivorgabs10104910

[31] N Bobev A Kundu K Pilch andN PWarner ldquoMinimal holo-graphic superconductors from maximal supergravityrdquo Journalof High Energy Physics 2011

[32] D Roychowdhury ldquoAdSCFT superconductors with powermaxwell electrodynamics reminiscent of the meissner effectrdquoPhysics Letters B vol 718 no 3 pp 1089ndash1094 2013

[33] D Roychowdhury ldquoEffect of external magnetic field on holo-graphic superconductors in presence of nonlinear correctionsrdquoPhysical Review D vol 86 no 10 Article ID 106009 12 pages2012

[34] R Banerjee S Gangopadhyay D Roychowdhury and ALala ldquoHolographic s-wave condensate with non-linearelectrodynamics a nontrivial boundary value problemrdquohttparxivorgabs12085902

[35] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofgauss-bonnet holographic superconductors in born-Infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 1562012

[36] S Gangopadhyay and D Roychowdhury ldquoAnalytic study ofproperties of holographic superconductors in born-infeld elec-trodynamicsrdquo Journal of High Energy Physics vol 05 p 0022012

[37] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[38] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 p 2889 2011

[39] D Momeni M R Setare and R Myrzakulov ldquoCondensationof the scalar field with stuckelberg and weyl corrections inthe background of a planar AdS-schwarzschild black holerdquoInternational Journal of Modern Physics A vol 27 Article ID1250128 2012

[40] D Momeni N Majd and R Myrzakulov ldquop-wave holographicsuperconductors with Weyl correctionsrdquo Europhysics Lettersvol 97 no 6 Article ID 61001 2012

[41] SNojiri and SDOdintsov ldquoAnomaly-induced effective actionsin even dimensions and reliability of s-wave approximationrdquoPhysics Letters B vol 463 no 1 Article ID 990414 pp 57ndash621999

[42] S Nojiri and S D Odintsov ldquoCan quantum-corrected btz blackhole anti-evaporaterdquo Modern Physics Letters A vol 13 no 33Article ID 980603 p 2695 1998

[43] S A Hayward ldquoUnified first law of black-hole dynamics andrelativistic thermodynamicsrdquo Classical and Quantum Gravityvol 15 no 10 p 3147 1998

[44] S Nojiri and S D Odintsov ldquoQuantum (in)stability of 2Dcharged dilaton black holes and 3D rotating black holesrdquoPhysical Review D vol 59 no 4 Article ID 044003 11 pages1999

[45] J Jing Q Pan and S Chen ldquoHolographic superconductors withPower-Maxwell fieldrdquo Journal of High Energy Physics vol 2011p 045 2011

[46] S H Mazharimousavi andM Halilsoy ldquoBlack hole solutions inf(R) gravity coupled with non-linear Yang-Mills fieldrdquo PhysicalReview D vol 84 Article ID 064032 2011

[47] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society A vol 144 no 852 pp 425ndash4511934

[48] Y N Obukhov ldquoConformal invariance and space-time torsionrdquoPhysics Letters A vol 90 no 1-2 pp 13ndash16 1982

[49] Q Pan J Jing and B Wang ldquoHolographic superconductormodels with the Maxwell field strength correctionsrdquo PhysicalReview D vol 84 2011

[50] G T Horowitz andMM Roberts ldquoHolographic superconduc-tors with various condensatesrdquo Physical Review D vol 78 no12 Article ID 126008 8 pages 2008

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Gravity

[51] S Kanno ldquoA note on gauss-bonnet holographic superconduc-torsrdquo Classical and Quantum Gravity vol 28 no 12 Article ID127001 2011

[52] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1972

[53] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress 1995

[54] J Ren ldquoOne-dimensional holographic superconductor fromAdS3CFT

2correspondencerdquo Journal of High Energy Physics

vol 1011 p 055 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of