aa r X i v : . [ h e p - t h ] A ug Preprint typeset in JHEP style - HYPER VERSION
Refractive index in holographicsuperconductors
Xin Gao
Key Laboratory of Frontiers in Theoretical Physics, Institute of TheoreticalPhysics, Chinese Academy of Sciences, 100190 Beijing, China [email protected]
Hongbao Zhang
Crete Center for Theoretical Physics, Department of Physics,University of Crete, 71003 Heraklion, Greece [email protected]
Abstract:
With the probe limit, we investigate the behavior of the electric permit-tivity and effective magnetic permeability and related optical properties in the s-waveholographic superconductors. In particular, our result shows that unlike the strong cou-pled systems which admit a gravity dual of charged black holes in the bulk, the electricpermittivity and effective magnetic permeability are unable to conspire to bring aboutthe negative Depine-Lakhtakia index at low frequencies, which implies that the nega-tive phase velocity does not appear in the holographic superconductors under such asituation. ontents
1. Introduction 12. Holographic model of superconductors 2
3. Refractive index in holographic superconductors 5
4. Conclusions 12
1. Introduction
Since the advent of AdS/CFT correspondence, not only do the main efforts includethe examination of its validity case by case, but also involve its applications in variousfields, in which condensed matter physics has recently received a special attention. Themotives are twofold. On the one hand, there exists many strong coupled systems incondensed matter physics, which are intractable by the traditional approaches. WhileAdS/CFT correspondence, as kind of strong/weak duality, can provide a powerful toolto attack them. On the other hand, unlike other disciplines such as particle physics andcosmology, condensed matter physics allows one to engineer matter such that variousvacuum states and phases can be created in the laboratory, which may in turn providethe first experimental evidence for AdS/CFT correspondence. For a review of thissubject, please refer to [1, 2, 3, 4].In particular, most recently such a bulk/boundary duality has been applied toexplore the optical properties of strong coupled media which admit charged black holesas a dual gravitational description and it is shown that this type of media generallyhave a negative refractive index at low frequencies[5]. Along this line, this paper isa first attempt to investigate how the refractive index and related optical propertiesbehave in the holographic superconductors and to see whether the negative refractiveindex shows up at low frequencies. – 1 –he gravity dual of superconductors was firstly established to model the s-wavesuperconductors[6, 7]. Later both the p-wave and d-wave holographic superconductorswere also realized[8, 9, 10, 11, 12, 13]. Please refer to [14] and references therein for areview of various properties related to such holographic superconductors.In the next section, we shall recall the holographic model of s-wave superconductors,where to make our life easier we would like to work in the probe limit and focus on thespecial case in which the complex scalar field is massless. After the setup, in Section 3we will present our numerical results for the behavior of the relevant optical quantitiesin such holographic superconductors. Conclusions and discussions will be addressed inthe end.
2. Holographic model of superconductors
The bulk dual contents of a holographic superconductor with a s-wave order parameterin 4+1 dimensions include the gravitational field, the U (1) gauge field, and the complexscalar field of mass m and charge q . The corresponding bulk action is given by[11] S bulk = Z d x √− g [ 12 κ ( R + 12 L ) − q ( 14 F ab F ab + | D a Φ | + m | Φ | )] , (2.1)where F ≡ dA , and D ≡ ∇ − iA where ∇ is the covariant derivative compatible withthe metric. In what follows, we will work in the probe limit, i.e., κqL → ds = L u [ − f ( u ) dt + dx + dy + dz + du f ( u ) ] , (2.2)where f ( u ) = 1 − ( uu h ) with u h > u →
0, sucha solution asymptotically becomes anti-de Sitter. Next around this background theequations of motion for the matter field are given by0 = D a D a Φ − m Φ = 1 √− g ( ∂ a − iA a )[ √− gg ab ( ∂ b − iA b )Φ] − m Φ , ∇ a F ab − i ( ¯Φ D b Φ − Φ ¯ D b ¯Φ) = 1 √− g ∂ a ( √− gF ab ) − i [ ¯Φ( ∂ b − iA b )Φ − Φ( ∂ b + iA b ) ¯Φ] . (2.3)By the holographic dictionary, the boundary value of the metric g µν acts as the sourcefor the energy momentum stress tensor T µν on the boundary, the bulk gauge field A µ – 2 –valuated at the boundary serves as the source for a conserved current J µ associatedwith a global U (1) symmetry, and the near boundary data of the scalar field Φ sources ascalar operator O with the conformal scaling dimension ∆ satisfying ∆(∆ −
4) = m L .In what follows, we shall be focusing on the particular case, i.e., m = 0, which yieldsan operator of dimension four by normalizability. Whence the asymptotic solution ofΦ and A µ can be expanded asΦ = q √ L (Φ (0) + Φ (1) u + · · · ) ,A µ = A (0) µ + q L [ A (1) µ u − g (0) ρσ ∂ ρ ( ∂ σ A (0) µ − ∂ µ A (0) σ ) u ln uǫ ] + · · · (2.4)with ǫ ≪ g µν = lim u → u L g µν . Then it follows from the holographicdictionary that the expectation value of the corresponding boundary quantum fieldtheory operators h O i and h J µ i can be obtained by variations of the renormalized actionwith respect to the sources, i.e., h O i = lim u → p − g δSδ ¯Φ (0) = Φ (1) , h J µ i = lim u → p − g δSδA µ = g (0) µν [ A (1) ν + cg (0) ρσ ∂ ρ ( ∂ σ A (0) ν − ∂ ν A (0) σ )] , (2.5)where the constant c is renormalization scheme dependent[15], to be fixed in the laterdiscussions. With the following ansatz, i.e.,Φ = Φ( z ) , A t = φ ( z ) , A x = 0 , A y = 0 , A z = 0 , (2.6)the equations of motion can be reduced to0 = Φ ′′ + ( f ′ f − u )Φ ′ + φ f Φ , φ ′′ − u φ ′ − L Φ f u φ, (2.7)where the prime denotes the differentiation with respect to u and Φ has been takento be real by taking into account the fact that the u component of Maxwell equationsimplies that the phase of Φ is independent of u . Note that the regularization conditionrequires φ = 0 on the horizon. Then multiplying the first equation in (2.7) by f , one– 3 – ΡΡ c < O > Ρ c Figure 1:
The condensate as a function of charge density, where the temperature is fixed tobe 1 as the reference scale. can find Φ ′ = 0 on the horizon. Thus for the above two second differential equations,we have only a two parameter family of solutions, which can be labeled by Φ( u h ) and φ ′ ( u h ). For convenience, we will set q = 1, L = 1, u h = 1 in the later calculations. Theresult for any other u h can be easily restored by the following scaling rules, i.e.,Φ (0) ( u h ) = Φ (0) (1) , Φ (1) ( u h ) = 1 u h Φ (1) (1) ,A (0) µ ( u h ) = 1 u h A (0) µ (1) ,A (1) µ ( u h ) = 1 u h A (1) µ (1) ,T ( u h ) = 1 u h π . (2.8)Firstly, it is easy to find the trivial solution asΦ = 0 , φ = C (1 − u ) , (2.9)which corresponds to the normal phase on the boundary with the chemical potential C .On the other hand, according to the numerical calculations, by fixing the charge densityor chemical potential the non-trivial solution emerges below a critical temperature– 4 – .0 1.5 2.0 2.50.00.20.40.60.81.0 SS c < O > S c Figure 2:
The condensate as a function of chemical potential, where the temperature is fixedto be 1 as the reference scale. T c , and the condensate increases when the temperate is lowered[16]. Its boundarycorrespondence is the superconducting phase.By the above scaling rules, such a non-trivial solution can also be thought of asthe superconducting phase emerging above a critical charge density ρ c or a chemicalpotential Σ c if the temperature is fixed. We would like to plot the correspondingcondensate as a function of charge density and chemical potential in Fig.1 and Fig.2respectively.
3. Refractive index in holographic superconductors
Before proceeding, we would like to introduce some quantities we want to calculate andtheir relations without going into details. In the linear response theory, the inducedelectromagnetic current J is related to the external vector potential A as J i ( ω, k ) = G ij ( ω, k ) A j ( ω, k ), where G is the retarded Green function. On the other hand, inthe Laudau-Lifshitz approach to electrodynamics of continuous media[17, 18, 19], forthe isotropic media, the transverse part of the dielectric tensor is determined by the– 5 –ransverse retarded Green function as follows [5, 20] ǫ T ( ω, k ) = 1 + 4 πω G T ( ω, k ) , (3.1)from which the corresponding electric permittivity and effective magnetic permeabilitycan be expressed as ǫ ( ω ) = 1 + 4 πω G T ( ω ) ,µ ( ω ) = 11 − πG T ( ω ) . (3.2)Here G T ( ω ) and G T ( ω ) are the expansion coefficients of the transverse retarded Greenfunction in k , i.e., G T ( ω, k ) = G T ( ω ) + k G T ( ω ) + · · · (3.3)with G T ( ω, k ) = ( δ ij − k i k j k ) G ij ( ω, k ). With this, the refractive index can be given by n ( ω ) = ǫ ( ω ) µ ( ω ). But to identify whether our holographic superconductors displaythe opposite phase velocity to the power flow, we shall appeal to the Depine-Lakhtakiaindex, which is defined as n DL ( ω ) = | ǫ ( ω ) | Re [ µ ( ω )] + | µ ( ω ) | Re [ ǫ ( ω )] . (3.4)As shown in [21], the phase velocity is opposite to the power flow if and only if n DL < A asfollows δA x = a x ( u ) e − iωt + iky . (3.5)Note that such a fluctuation decouples and has no back reaction, satisfying the equationof motion 0 = a ′′ x + ( f ′ f − u ) a ′ x + ( ω f − k f − f u ) a x . (3.6)Whence a x goes like (1 − u ) ± i ω near the horizon, corresponding to the outgoing and ingo-ing boundary conditions respectively. We here choose the ingoing boundary condition,which will lead to the retarded Green function in the dual field theory[22]. Speakingspecifically, with such an ingoing boundary condition, we assume the correspondingasymptotic expansion a x goes like a x = a (0) x + 12 [ a (1) x u − ( ω − k ) a (0) x u ln uǫ ] + · · · (3.7) Note that the expression in [5] is different from ours due to the different conventions used for theretarded Green function. – 6 –here we have used Eq.(3.5) and the second equation in (3.7). By Eq.(2.5), the inducedcurrent is given by J x = a (1) x + c ( ω − k ) a (0) x . (3.8)Whence the retarded transverse Green function can be obtained as G T ( ω, k ) = a (1) a (0) + c ( ω − k ) . (3.9)An analytic solution of Eq.(3.6) does not appear to be available. So in the subse-quent section, we shall resort to Mathematica for numerical calculation, where in orderto obtain G T ( ω ) and G T ( ω ), it is advisable to adopt an alternative approach by firstlyexpanding a x in series of k , i.e., a x = a x + k a x + · · · , (3.10)from which Eq.(3.6) becomes0 = a ′′ x + ( f ′ f − u ) a ′ x + ( ω f − f u ) a x , a ′′ x + ( f ′ f − u ) a ′ x + ( ω f − f u ) a x − f a x . (3.11)Near the boundary, the asymptotic expansion for a x and a x can be read out of Eq.(3.7)as a x = a (0) x + 12 ( a (1) x u − ω a (0) x u ln uǫ ) + · · · a x = a (0) x + 12 [ a (1) x u − ( ω a (0) x − a (0) x ) u ln uǫ ] + · · · . (3.12)Whence we have G T ( ω ) = a (1) x a (0) x + cω ,G T ( ω ) = a (1) x a (0) x [ a (1) x a (1) x − a (0) x a (0) x ] − c, (3.13)where c can be fixed by requiring that G T approaches zero at large frequencies as thesystem has no time to respond to the rapid variation of external fields.– 7 – m @ Ε T D TT C = Ω T C kT C Figure 3:
The imaginary part of the ǫ T for real ω and k at temperature TT c = 0 . Im @ Ε T D TT C = Ω T C kT C Figure 4:
The imaginary part of the ǫ T for real ω and k at temperature TT c = 0 . Here we plot all the results by fixing the charge density to be 1 but varying the tem-perature with respect to the critical temperature and consequently the condensate.As explained in [5], firstly we have to ensure that the system is in thermodynamical– 8 – .5 1.0 1.5 2.0 2.5 3.0 3.5 Ω T C - - - - @ Ε D Figure 5:
The real part of the electric permittivity as a function of frequency with varioustemperatures. Ω T C @ Ε D Figure 6:
The imaginary part of the electric permittivity as a function of frequency withvarious temperatures. equilibrium, i.e., the imaginary part of G T , or equivalently the imaginary part of ǫ T should be greater than zero for real ω and k . This is actually guaranteed by the holo-graphic duality, as the bulk background is static with the black brane. For illustration,we would like to plot the imaginary part of ǫ T in Fig.3 and Fig.4 to demonstrate thatthis is the case.Then we plot the real and imaginary parts of the electric permittivity in Fig.5– 9 – .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Ω T C @ Μ D Figure 7:
The real part of the effective magnetic permeability as a function of frequencywith various temperatures. and Fig.6 respectively. By the scaling rules, to lower the temperature is equivalent toincrease the charge density and chemical potential. Therefore the electric permittivityhere demonstrates the similar behavior at low frequencies as that considered in [5]. Inparticular, the larger are the charge density and chemical potential, the larger is theregime for the negative real part of the electric permittivity. In addition, below thecritical temperature, both of the real and imaginary parts of the electric permittivitydiverge at ω = 0. By the fact that the electric permittivity is related to the conductivityas ǫ ( ω ) = 1 + i πω σ ( ω ), the real part of the electric permittivity diverges as − ω , andaccordingly the imaginary part blows up like δ ( ω ) ω due to the Kramers-Kronig relationsfrom causality[14].Next we plot the real and imaginary parts of the effective magnetic permeability inFig.7 and Fig.8 respectively. The real part of the effective magnetic permeability stilldisplays the similar behavior at low frequencies as that in [5], while the imaginary partshows a totally different behavior, i.e., there is no blow-up at ω = 0.Such a difference eventually leads to the conclusion that at low frequencies thenegative phase velocity does not show up in the holographic superconductors at leastwithin the accuracy of our numerics, which is illustrated by the non-negative Depine-Lakhtakia index in Fig.9. In hindsight, the reason may arise in the fact that the bulkbackground is essentially kind of neutral black hole although the electromagnetic fieldhas an influence on the fluctuation equation (3.6) indirectly through the condensate.At last, we would like to plot the real and imaginary parts of the refractive indexin Fig.10 and Fig.11 respectively. Note that the ratio of the imaginary part of the– 10 – .5 1.0 1.5 2.0 2.5 3.0 3.5 Ω T C - - - - - - - @ Μ D Figure 8:
The imaginary part of the effective magnetic permeability as a function of fre-quency with various temperatures. Ω T C n DL Figure 9:
The Depine-Lakhtakia index as a function of frequency with various temperatures. refractive index to its real part describes the ratio between dissipation and propagation.We see that the there exists a characteristic frequency, below which light can notpropagate in our holographic superconductors while above which light can propagatewithout dissipation. – 11 – .5 1.0 1.5 2.0 2.5 3.0 3.5 Ω T C @ n D Figure 10:
The real part of the refractive index as a function of frequency with varioustemperatures. Ω T C @ n D Figure 11:
The imaginary part of the refractive as a function of frequency with varioustemperatures.
4. Conclusions
We have carried out the numerical analysis of the optical properties of the s-wave– 12 –olographic superconductors in the probe limit. In particular, we have calculatedthe electric permittivity, effective magnetic permeability, refractive index, and Depine-Lakhtakia index by the holographic duality, which is generically difficult to be achievedby other approaches. As a result, the electric permittivity and effective magnetic per-meability conspire to make our holographic superconductors robust against a negativephase velocity at low frequencies.We conclude with some generalizations of our work in various directions. Firstly,although the cases for other scalar masses are expected to demonstrate the qualitativelysimilar properties as the massless case, it is interesting to investigate how the specificoptical properties of holographic superconductors depend quantitatively on the mass m . Secondly, it is natural to extend our work to the 2+1 dimensional holographicsuperconductors[23]. More importantly, as the superconductors are believed to havethe potential to support low losses, which is critical for many applications like super-resolution imaging, it is rewarding to search for the negative phase velocity in othersituations. In particular, it is worthwhile to go beyond the probe limit to explorethe full behavior of the theory, where the negative refractive index is expected toappear at low frequencies because the background is essentially kind of charged blackhole. In addition, as suggested by the early experimental implementation of negativerefractive index in the ferromagnet-superconductors[24], it is highly possible to see thenegative refractive show up when the external magnetic field is added to our holographicsuperconductors. Furthermore, it is also intriguing to extend our analysis to both thep-wave and d-wave holographic superconductors, where some new features should comeout due to the nonisotropy of the dual media. Finally, it is also tempted to generalize ourwork to some non-minimal models of holographic superconductors[25, 26, 27, 28, 29, 30]. Acknowledgements
XG would like to thank Yan Liu, Rongxin Miao, Zhangyu Nie, and Yang Zhou foruseful discussions. HZ is much indebted to Antonio Amariti and Davide Forcella forilluminating correspondence during the whole project. He is also grateful to Bom SooKim, Elias Kiritsis, Matthew Lippert, and Rene Meyer for helpful discussions. Inaddition, it is a great pleasure to thank Ioannis Iatrakis and Qiyuan Pan for help withthe numerical calculation. XG was supported in part by the NSFC under grant No.10821504. HZ was partially supported by a European Union grant FP7-REGPOT-2008-1-CreteHEPCosmo-228644. – 13 – eferences [1] S. A. Hartnoll, Class. Quant. Grav. 26, 224002(2009).[2] C. P. Herzog, J. Phys. A42, 343001(2009).[3] J. McGreevy, arXiv:0909.0518[hep-th].[4] S. Sachdev, arXiv:1002.2947[hep-th].[5] A. Amariti, D. Forcella, A. Mariotti, and G. Policastro, arXiv:1006.5714[hep-th].[6] S. S. Gubser, Phys. Rev. D78, 065034(2008).[7] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Phys. Rev. Lett. 101, 031601(2008).[8] S. S. Gubser, Phys. Rev. Lett. 101, 191601(2008).[9] S. S. Gubser and S. S Pufu, JHEP 0811, 033(2008).[10] J. W. Chen et al. , Phys. Rev. D81, 106008(2010).[11] C. P. Herzog, Phys. Rev. D81, 126009(2010).[12] F. Benini, C. P. Herzog, and A. Yarom, arXiv:1006.0731[hep-th].[13] F. Benini et al. , arXiv:1007.1981[hep-th].[14] G. T. Horowitz, arXiv:1002.1722[hep-th].[15] K. Skenderis, Class. Quant. Grav.19, 5849(2002).[16] G. T. Horowitz and M. M. Roberts, Phys. Rev. D78, 126008(2008).[17] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media(PergamonPress, Oxford, 1984).[18] D. B. Melrose and R. C. Mcphedran, Electromagnetic Processes in DispersiveMedia(Cambridge University Press, Cambridge, 1991).[19] V. M. Agranovich and Y. N. Gartstein, Metamaterials 3, 1(2009).[20] M. Dressel and G. Gruner, Electrodynamics of Solids(Cambridge University Press,Cambridge, 2002).[21] R. A. Depine and A. Lakhtakia, Microwave and Optical Technology Letters 41,315(2004).[22] D. T. Son and A. O. Starinets, JHEP 0209, 042(2002). – 14 –
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