Generalized Superconductors and Holographic Optics

We study generalized holographic s-wave superconductors in four dimensional R-charged black hole and Lifshitz black hole backgrounds, in the probe limit. We first establish the superconducting nature of the boundary theories, and then study their optical properties. Numerical analysis indicates that a negative Depine-Lakhtakia index may appear at low frequencies in the theory dual to the R-charged black hole, for certain temperature ranges, for specific values of the charge parameter. The corresponding cut-off values for these are numerically established in several cases. Such effects are seen to be absent in the Lifshitz background where this index is always positive.


Introduction
The use of the AdS/CFT correspondence [1], [2], [3] to understand strongly coupled dynamics of material systems has attracted a lot of attention of late. Several exciting directions of research have appeared, and there are indications that useful insight into condensed matter systems might be possible using the gauge gravity duality [4], [5]. In this context, one of the novel results in the last few years is the possibility of providing a holographic description of the phenomenon of superconductivity [6]. While the analysis of [7] provided evidence of a spontaneous breaking of gauge invariance in an Abelian Higgs model in AdS backgrounds, the work of [6] gave a holographic description of the quantum dynamics of the condensation of a charged operator on the boundary field theory, via a classical instability of the dual AdS black hole background. Indeed, the discovery of the fact that AdS black holes support scalar hair, and its implications in understanding strongly coupled condensed matter systems, have been the focus of intense research over the past few years. In particular, shortly after the initial results on holographic superconductors appeared, the work of [8] generalized the original holographic description of [6] to cases where the global gauge symmetry is broken via a Stuckelberg mechanism. These authors constructed gauge invariant Lagrangians, of the Stuckelberg form, and demonstrated the possibility of a superconducting phase transition. Interestingly, their work points to the existence of a first order phase transition, and a metastable region in the superconducting phase.
Another interesting development in the application of holography has been in the context of optics. This is motivated in part to understand an exotic property of matter, namely negative refraction [9], [10], via the AdS/CFT correspondence. Indeed, materials with such properties (called meta-materials) have been artificially engineered, and find a variety of application in physics and engineering. The main property of these materials is that their dielectric permittivity, ε, and the permeability, µ, can both be negative, and in that case, the negative sign in the defining equation for the refractive index needs to be chosen. This corresponds to a situation where the phase velocity of light waves in the medium is in a direction opposite to its energy flux. 1 In [14], a holographic description of meta-materials was given, and it was shown that strongly coupled field theories generically admit a negative refractive index in the hydrodynamic limit (see also [15]). This was generalized for gauge theories dual to R-charged black hole backgrounds, in [16]. In the works of [17], [18], optical properties of holographic superconductors, were studied, and it was shown that in the probe limit, they did not exhibit phenomena of negative refraction, when the bulk theory is a five dimensional AdS black hole. 2 However, it was shown in [18] that inclusion of back reaction did allow for a negative refractive index at low frequencies.
It is important and interesting to extend this investigation to other backgrounds in different dimensions, and study situations that involve some bulk control parameters. In this paper, we consider optical properties of s-wave holographic superconductors in the probe limit, where the bulk theory is four dimensional. 3 In particular, here we consider a generalized holographic superconductor [8], with a background R-charged black hole [21]. It is assumed that the background is fixed, and that there is no back reaction. In this approximation, we first establish the superconducting property of the boundary theory, i.e the existence of scalar hair below a certain temperature. Next, we study the optical properties of this theory, and calculate the Depine-Lakhtakia (DL) [22] index by evaluating the retarded correlator at the boundary. We find that the DL index can be negative for sufficiently small frequencies, in contrast with the results reported in [17], [18] for five-dimensional examples. We also establish the dependence of the refractive index on the temperature and the black hole charge parameter. We find preliminary evidence that the index of refraction may be negative only in a certain window of temperatures. This paper is organized as follows. In the next section, we first describe the model under consideration, to fix our notations and conventions. We then proceed to show that the model has superconducting behavior at the boundary. In section 3, the optical properties of the boundary theory are established. We end with some discussions and our conclusions in section 4.
2 Superconducting meta-materials have attracted a lot of attention of late, see, e.g [19]. 3 In the normal phase, optical properties of four dimensional RN-AdS black holes have been reported in [20].

The Holographic Setup
In this section we will review the basic set up on the gravity side which gives a superconducting system in the boundary theory. The black hole background which we will be interested in, is the four dimensional R-charged black hole [21] with metric where we define Here, M and κ i are the mass and charge parameters of the black hole. r h is radius of the outer horizon, and k can take two values, 0 and 1, which correspond to noncompact and compact horizons, respectively. In this paper, we will consider examples with planar horizons, i.e k = 0, although all our results can be straightforwardly extended to the case k = 1. In order to obtain a scalar condensate in the boundary dual, the authors in [6], [7] introduced a Abelian-Higgs model in AdS black hole backgrounds with the matter Lagrangian for the bulk theory given by where F = dA, D µ = ∂ µ − iA µ andΨ is the complex scalar field with mass m. As we have mentioned, we will work in the probe limit in which the scalar and Maxwell fields do not back-react on the metric of eq. (1). Re-writing the charged scalar field Ψ asΨ = Ψe iα , the matter Lagrangian can be re-written as, where both Ψ and the phase α are real. The local U(1) gauge symmetry in this theory is given by A µ → A µ + ∂ µ λ together with α → α + λ. As shown in [8], this model can be generalized as, where G is a function of Ψ. Now, by varying the action, it is straightforward to write down the scalar and the Maxwell's equation, and these are given by where we can use the gauge symmetry to fix the phase α = 0. Since we are interested in a superconductor like solution, we will consider the following ansatz Then, the equations of motion reduce to In order to solve these coupled differential equations, suitable conditions at the horizon and at the boundary must be imposed. We will impose regularity conditions for Ψ and Φ at the horizon where these fields behave as Important for us will be the asymptotic expressions for Φ and Ψ near the boundary, where λ ± = 3± √ 9+4m 2 2 , and µ, ρ are the chemical potential and the charge density of the boundary theory, respectively. At the boundary both Ψ − and Ψ + are normalisable, and can act as source (or vacuum expectation value) of the corresponding operators. In this paper we consider Ψ − as a source and hence put Ψ − = 0 as the boundary condition.
The system in eq.(9) are difficult to solve analytically, and we resort to a numerical solution. 4 To simplify the numerical calculations, it is more convenient to use z = r h /r, as is conventional in standard literature. In the z coordinate, the horizon and the boundary are located at z = 1 and z = 0, respectively. For numerical calculations, we have considered m 2 = −2 and the particular form G(Ψ) = Ψ 2 + ξΨ 8 . Although m 2 is negative but it is above the Breitenlohner-Freedman bound in four dimensions, m 2 BF = −9/4 [23]. For this value of m 2 , we get λ ∓ = 1, 2. The condensate of the scalar operator < O 2 > in the boundary theory dual to the scalar field is given by We need to show that our boundary theory is superconducting. In figure (1), we have shown the variation of condensate < O 2 > with respect to the chemical potential. For simplicity, here we have considered only the single R-charged case, κ 1 = 1, with all the other charges set to zero. When the chemical potential exceeds its critical value µ c , the condensate develops a nonzero vacuum expectation value which indicates the appearance of a superconducting phase. Below µ c , the system is in the normal (or insulating) phase. We find that in our case, the critical potential is µ c ∼ 3.502. This behavior indicates that our boundary theory is indeed superconducting. One should note that in the special case ξ = 0, κ 1 = 0, our generalized superconductor reduces to the conventional s-wave superconductor studied in [6]. Also, from fig.(1), we see that for sufficiently high values of ξ, the system exhibits a first order phase transition, similar to what was observed in [24].
Having established that our theory is superconducting, we will now study its optical properties. There is an obvious caveat here, namely the absence of a dynamical photon on the boundary. We will proceed by assuming that the boundary theory is weakly coupled to a dynamical electromagnetic field, and it is the optical properties of the latter that is being computed [14].

Optical Properties of Generalized Superconductors
In this section we will discuss optical properties of generalized superconductors that we have discussed in the previous section. Here, we will follow the convention used in [17], [18]. We first setup the system of differential equations that are used to compute the retarded correlators at the boundary. We then analyze these equations numerically, and establish the optical properties of the boundary system.

General Setup of The Problem
The relevant quantity that is used to establish negative refractive index in a medium is called the Depine-Lakhtakia index [22] η DL , and is given by with negativity of the DL index indicating that the phase velocity in the medium is opposite to the direction of energy flow, i.e the system has negative refractive index. Computation of the DL index involves a number of steps. We begin by writing down the boundary action. Subsequently, the transverse current-current correlators from the boundary action are obtained via the prescription of [26]. Then, we proceed by casting the momentum dependent correlators in the following form:  The permittivity and the effective permeability [25], can be expressed in terms of G 0 T and G 2 T as ǫ (ω) = 1 + 4π Here C em , K and ω are the EM coupling constant (set to unity for numerical computations), the frequency, and the spatial momentum, respectively. To calculate retarded correlators [26] for the transverse currents in superconducting phase, it is enough for us to consider a perturbation of the Maxwell field, say A x , on the superconducting black hole background, In the probe limit, the equation of motion for A x decouples from other field equations,  Eq.(18) must be solved with appropriate boundary conditions. We solve this equation with ingoing wave boundary condition at the horizon i.e As in the previous section, we have confined ourself only to the single R-Charged case, for simplicity. Also note that asymptotically, A x behaves as From the AdS/CFT dictionary, we can identify A (0) x and A (1) x as the dual source and the expectation value of boundary current respectively. Also, the retarded correlators can be computed from the relation [27] In order to obtain the refractive index and other optical quantities in the boundary theory, we first calculate G 0 T and G 2 T . This can be done by expanding A x in powers of K in same way that G T is expanded in eq. (14),  18), and separating the coefficients in powers of K, we obtain the differential equations for A x0 and A x2 as (24) the asymptotic forms of A x0 and A x2 can be found from eq.(20) and therefore, we obtain We substitute eq.(26) into eq.(15), (16) and obtain ε(ω), µ(ω) and n DL in terms of A x2 and A (1) x2 . We solve these equations numerically and as before, use the z-coordinate to simplify the numerics.

Numerical Analysis and Results
In this subsection we will numerically analyze the optical properties of the boundary superconducting theory. As before, we will consider the single R-charged cases with κ 1 = 1. In figs.(2), (3), (4) and (5), we plot Re(ε), Im(ε), Re(µ), and Im(µ) respectively, as a function of ω/T . Here, we have chosen T = 0.81T c (so that our system is in the superconducting phase) and ξ = 0.2. In figs.(2)-(5) the red, green, blue, brown, orange and pink curves corresponds to κ 1 = 0, 1, 5, 10, 15 and 25 respectively. 5 Our results for the permittivity and the effective permeability are qualitatively similar to those obtained in [17], [18]. As expected, Re(ε) is negative at low frequencies and Im(ε) is always positive and has a pole at zero frequency for all κ 1 .
Our main purpose here is to calculate n DL and this is shown in figure (6). At high frequencies, n DL has the same qualitative features found in [17], [18]. But at low frequencies, a significant difference emerges, with the appearance of negative n DL indicating negative refractive index. To illustrate the point, in fig.(7), we have plotted n DL for low frequencies and one can see the emergence of negative refraction, below a certain value of ω/T . Let us elaborate on this a bit further. For the moment, we focus on the simpler situation, ξ = 0. Here, our numerical analysis indicates that there is no negative refraction for T = 0.85T c , for any value of κ 1 . For T = 0.83T c , we find that negative refraction is allowed for small values of κ 1 , upto κ 1 ∼ 10. When the temperature is further reduced to T = 0.75T c , we find the possibility of negative refraction upto κ 1 ∼ 50. Now, at low values of the temperature, back-reaction effects might be important, and numerical analysis in the probe limit may not be fully trusted. However, we find an indication that there might be a window of temperatures for which negative refraction is allowed. For example, setting ξ = 0, κ 1 = 0, negative refraction sets in as the temperature is lowered from its critical value to ∼ 0.83T c . At T = 0.1T c , however, this is not observed. We emphasize that this is simply an indication, and needs to be investigated further, with inclusion of back-reaction effects.
Further, our results indicate that, at a particular ξ and a fixed temperature, there exists a critical κ 1c above which negative refraction cannot occur, irrespective of how low the frequency is. This can be seen, for example, from fig.(7). A more useful statement is that for a fixed value of ξ and κ 1 , the system might go from a positive refraction "phase" to a negative refraction "phase" by varying the temperature. To justify this, we have plotted n DL in figs. (8) and (9) at a temperature T = 0.86T c . It is observed that negative refraction which occurs for κ 1 = 0, 1, 5, 10 and 15, at T = 0.81T c , ( fig.(7)), disappears at T = 0.86T c . Interestingly, qualitatively similar results have been the topic of discussion in the optics community of late (see, e.g. [28]), although we make no claims beyond a naive similarity of those results with ours.
We have also calculated n DL for other values of ξ. In figs. (10) and (11), we have used ξ = 1 and plotted n DL at T = 0.86T c . Expectedly, the essential features of our analysis are similar with the ξ = 0.2 case. In figure (11), we get negative refraction at low frequencies for κ 1 = 0 and 1. By comparing fig.(9) with fig.11), we see that the possibility for negative refraction for fixed κ 1 , increases with ξ.
In this paper we have only presented the analysis for n DL and other optical quantities for the single R-charged cases, but the same analysis can be generalized to multiple R-charged backgrounds. Preliminary results involving multiple charge

Conclusions
In this paper, we have studied optical properties of generalized holographic superconductors corresponding to four dimensional R-charged black hole backgrounds, in the probe limit. As a special case, this reduces to the four dimensional SAdS background, whose five dimensional form was studied in the same limit in [17]. Our main result is that, at small enough frequencies, below a cut-off value of the charge parameter κ 1c , the superconducting phase can exhibit a negative refractive index. By increasing the temperature, this behavior disappears. Our results indicate that there might be a window of temperatures for which the system exhibits negative refraction. We have analyzed in details the optical properties of the system by varying the control parameters of the theory as well as the temperature. Qualitatively, we get predictions similar to [28], and it will be interesting to analyze this issue further, to see if one can get closer to realistic systems.
Our results should be contrasted with those of [17], [18], where negative refraction was not observed in the probe limit, for five dimensional black hole backgrounds. In particular, it was shown [18] that negative refraction occurs only when the matter fields backreact on the bulk metric. The appearance of negative refraction in our case is possibly due to the different nature of the boundary theory here. Also, as we have seen, the tunable R-charge parameter plays an important role in determining the optical properties of the bulk theory. Our analysis is limited by the fact that we worked in the probe limit. It will be interesting to study the effect of holographic superconductors in R-charged backgrounds, including effects of back reaction, although the numerics might be substantially more complicated. We leave such an analysis for the future.