Noncommutative Extension of AdS-CFT and Holographic Superconductors

In this Letter, we consider a Non-Commutative (NC) extension of AdS-CFT correspondence and its effects on holographic superconductors. NC corrections are incorporated via the NC generalization of Schwarzschild black hole metric in AdS with the probe limit. We study NC effects on the relations connecting the charge density and the critical temperature of the Holographic Superconductors. Furthermore, condensation operator of the superconductor has been analyzed. Our results suggest that generically, NC effects increase the critical temperature of the holographic superconductor.


Introduction
In recent years AdS/CFT correspondence, proposed by Maldacena [1], has captured the attention of both High Energy and Condensed Matter theorists since it can address issues in strongly interacting systems in the latter, (that are otherwise intractable in conventional Condensed Matter framework), by exploiting results obtained in weakly coupled systems in the former. In particular, there exists explicit mapping between relevant operators and parameters of a field theory in the bulk AdS space-time to those of a Conformal Field Theory living in the (one dimension lowered) boundary.
Armed with the AdS/CFT dictionary, in its simplest form, it was possible to have glimpses of the unique features of high T c superconductors (referred to as Holographic Superconductors) from the study of scalar electrodynamics in a Schwarzschild-AdS background, as shown by Gubser [2].
More specifically, the model describing phase transitions in high T c superconductors consists of a charged scalar field minimally coupled to an Abelian gauge field in an AdS Black Hole background [3]. The Black Hole admits scalar hair at a temperature T below a certain critical temperature T c by the mechanism of breaking of a local U (1) symmetry near the event horizon of the Black Hole [2].
The emergence of a hairy AdS Black Hole implies the formation of a charged scalar condensate in the dual CFTs by the AdS/CFT correspondence. The minimal model has been generalized in different directions in the AdS sector to study its effects on the CFT. An interesting extension was discussed in [4] that considered non-linear Born-Infield model instead of the usual Maxwell lagrangian. The analysis showed that the critical temperature of the Holographic Superconductor decreased with increase of the non-linear coupling. In the present article we study a completely different type of generalization of the gravity-field theory sector -Non-Commutative (NC) extension of the spacetime. Our results indicate that NC effects are small but can lower the critical temperature of the Holographic Superconductor. ratio of NC parameter θ to Black Hole mass is within a certain ratio.
Our results smoothly reduce to the conventional results for θ = 0.
Noncommutativity in spacetime was introduced long ago by Snyder [5] in the hope of removing short distance singularities in quantum field theory but it was not successful. Later NC field theory was resurrected by Seiberg and Witten [6] who demonstrated that in the weak energy limit open strings attached to D-branes induced noncommutativity in the D-branes. In [6] rules were provided for extending QFTs to NC QFTs where normal products between local fields were replaced by *products so that NC QFTs can be studied perturbatively for small NC parameter θ. Furthermore, NC gauge theories had to be treated in a special way by incorporating the Seiberg-Witten map [6]. (For a review see [7].) Later Nicolini, Smailagic and Spalucci [8] were able to construct an NC extension of Schwarzschild-AdS metric by directly solving the Einstein equation with a smeared matter source that takes into account the NC effect. The Black Hole singularity was successfully removed in this scenario. NC effects on salient Black Hole properties, such as Hawking radiation, have been studied using this θ -corrected metric [9].
In the present paper we aim to study the bulk NC effect on Holographic Superconductors.
Interestingly, the NC generalization in the bulk requires both Seiberg-Witten mapped extension [6,7] for the gauge sector and Nicolini-Spalucci extension [8] for the Black Hole sector. As far as we are aware of, this form of generalized NC extension has not been studid before. We will show there are some subtleties involved in the lowest order approximation in the NC parameter θ. It is well known that asymptotic behavior of the scalar and gauge fields in the bulk dictate properties of the boundary CFT. It is perhaps significant that, (at least to the lowest non-trivial order of θ), NC effect does not change the asymptotic behavior of bulk fields qualitatively, that is, the functional forms remain unchanged but the numerical parameters undergo NC corrections.
This allows us to use the same AdS/CFT dictionary in order to compute the θ-corrected relation between the critical temperature and charge density of the Holographic Superconductor and the condensate-temperature relation.
The paper is organized as follows: In Section 2, we introduce the AdS Black Hole metric and define the action for an Abelian gauge field (coupled with a scalar) in this noncommutative background spacetime. In Section 3, we study the asymptotic behavior of the gauge and scalar field.
Then we proceed to study the relation between critical temperature and charge density in Section 4. In following Section 5, we calculate the critical exponents and condensation values. Finally we discuss and summarize our findings and conclude in Section 6.

Noncommutative AdS Black Hole in charged background
We construct the action of NC charged scalar electrodynamics [6] in NC AdS Black Hole background [8]. We consider the Lorentz covariant form Here θ µν , an antisymmetric Lorentz tensor, is represented in terms of a block diagonal form Thus to maintain the covariance of (1) a foliation of spacetime into noncommutative planes has been done, as defined by (2). But Lorentz invariance and unitarity implies that noncommutativity does not privilege any of such planes, which enforces a unique noncommutative parameter θ 1 = θ 2 = ... θ n/2 = θ.
In this approach [8]  Thus the noncommutative Black Hole metric is considered in the form [8] where For small θ the exponential damping reduces the metric to its original form. With this metric (3), the gauge field action is constructed as where, to O(θ), star product is defined as The NC extension of gauge fields needs extra care and Seiberg-Witten map [6] has to be employed.
The deformed gauge potential to O(θ) iŝ by virtue of which the electromagnetic tensor becomeŝ Notice that the NC Abelian electromagnetic tensor has a non-Abelian flavor. Using (6) and (7), the fourth term in the action (5) becomes As is customary in this context, we consider gauge field A µ to have only temporal component [2,3], i.e. A µ = (φ(r), 0, 0, 0) and ψ = ψ(r). Our action (5) becomes Lagrange's equation of motion for scalar field ψ, yields The Lagrange's equation for the gauge potential φ is given by Equations (12) and (2) are the governing equations of our model.

Asymptotic behavior of ψ and φ
Let us find the asymptotic behavior of (12) and (2). We restrict ourselves to the lowest non-trivial effects of θ. Note that the NC extension in Black Hole metric does not allow a naive θ-expansion and whereas for the NC-gauge sector it is permissible. Hence we expand the differential equation for electric field φ i.e. (2) to the order O(θ) and O 2M √ πθ e − r 2 4θ [9]. The second, third, fourth and fifth term of (12) can be written to the first order of O 2M √ πθ where f (r) = K − 2M r + r 2 L 2 . This approximation is justified as we discussed in the previous section about how the noncommutative parameter θ is related to the physical energy scale beyond which one cannot probe.
Similarly the third, eighth, ninth and tenth term of (2) can be written as With the above approximation, the equation for ψ and φ can be written as and Since f (r) = K − 2M r + r 2 L 2 we have 1 f (r) ≈ L 2 r 2 (considering terms up to 1 r 2 only). Replacing all f (r) by r 2 L 2 , the above equation for ψ can be written (keeping terms only up to 1 r 2 ) as It is known that ψ = C r + D r 2 is a solution of Therefore we assume that the solution of (25) to be of the form Substituting (27) into (25) we see that the differential equation for ψ 1 becomes where we considered terms up to O( 1 r 2 ). The solution of (28) is given by Therefore the asymptotic behavior of ψ can be expressed as As mentioned earlier, the θ-contribution has the same structure as the original one. Substituting f (r) = r 2 L 2 and (30) in (3) and considering terms up to 1 r 2 we have The solution of is known to be So, let us consider the solution of (31) in the form Substituting this form in (31) we have the differential equation for φ 1 as Neglecting the last terms, as we have considered only first order term of θ, we arrive at Therefore we finally have the solution Once again the θ-contribution has the same structure as the original one. This asymptotic behavior can be written in the form From (38), it can be clearly observed the the effect of noncommutativity enters through the very definition of the charge density ρ and the chemical potential µ.

Relation between critical temperature and charge density
In order to obtain the relation between the critical temperature T c and charge density (ρ), we follow the technique used in [4]. As explained earlier, we expand the differential equation for electric field [9] as At the critical temperature T = T c the scalar field ψ vanishes, ψ = 0. By virtue of this the above equation (4) becomes The solution of (40) is The Horizon r = r + is the solution of f 1 (r) = 0. We are now going to change our solution region from r + ≤ r < ∞ to 1 ≥ z > 0 by the transformation z = r + r . We rewrite the asymptotic behavior Using the transformation z = r +c r the horizon comes to at z = 1 and asymptotic boundary behavior becomes at z = 0. Since (42) represents asymptotic behavior of φ, then at z = 0 equation (42) and (43) must be same. Substituting z = 0 in the above two equation we have a 1 = a. Again differentiating (43) and (42) w.r.to "z" and then equating them at the asymptotic boundary z = 0, Another boundary condition at the horizon z = 1 i.e. φ(z = 1) = 0 gives from (43) to the first order of O(θ) Therefore the solution (43) of scalar potential φ at the critical temperature T = T c to the lowest non-trivial order of θ can be written as where λ = b(1+θ) We are now going to investigate the boundary behavior of the scalar field ψ as T → T c . To the first order of O(θ) and O 2M √ πθ e − r 2 4θ the differential equation for scaler field ψ (12), after replacing z = r + r and using the solution (45) (as T → T c ) becomes where f (z) = K − 2M r + z + r 2 + L 2 z 2 . Near the asymptotic boundary (as z → 0) we define where F (z) satisfies F (0) = 1 and F ′ (0) = 0. Using (47) the above equation (4) becomes Now it is straightforward to cast (4) into a Sturm-Liouville (SL) eigenvalue problem of the generic is an arbitrary function that will be fixed later. The Hawking temperature [11] T is related to r + by the relation T = 3 4π r + and λ is related to charge density ρ by the relation λ = ρ r 2 +c , where ρ = b(1 + θ). Therefore the relation between critical temperature T c and charge density ρ becomes Note that both ρ and λ min contain θ-corrections. Explicitly λ min is found from minimizing the expression for a given choice of F (z). We may take F (z) = 1 − cz 2 as a trial solution [3,10] which satisfy F (0) = 1 and F ′ (0) = 0, where c is the minimizing parameter, in order to find λ from (54). To evaluate λ for different values of θ we may simplify the integration in the denominator of (54) by where λ| θ=0 is the value of λ for θ = 0 [4]. We would like to mention a crucial point here: For the usual AdS scenario without noncommutativity, θ = 0, we From (55), horizon radius r + can be calculated as f (r = r + ) = 0 which gives considering the usual convention L = 1, for the flat case K = 0. One can see here that using (55) and r + the expression for λ becomes From (57) from which the horizon radius can be determined by f 1 (r = r + ) = 0 which gives (for K = 0, L = 1) Clearly the second term in r.h.s. of (59) destroys the simple explicit relation (56). Using the form of f (z) = − 2M z r + + r 2 + z 2 and the relation (59) the expression for eigenvalue λ stands out to be Thus the eigenvalue λ depends on both M and θ through r + . Subsequent results, such as ζ, (the coefficient of √ ρ), in the relation of critical temperature T c and charge density ρ becomes θ as well as M dependent. This is a highly non-trivial feature of the noncommutative AdS model considered here. We speculate that it hints at a generalized form of AdS/CFT duality where the Holographic Superconductor will have other parameters, besides the charge density and chemical potential one generally associates it with. From (60), one can now determine numerical value of ζ in (53), as studied in [10,4]. This is shown in Tables 1, 2, 3 and 4. We will discuss the implications in the last section.

Summary and Discussion
In this paper, we have considered a noncommutative charged AdS Black Hole background and a scalar field coupled to gravity, thus introducing a hairy Black Hole. We have used the AdS/CFT correspondence to study the effects of this noncommutative background on the properties of holographic high T c superconductors. First we studied the asymptotic behavior of the gauge and scalar field and explicitly show the effects of noncommutativity on the physical parameters like charge density and chemical potential. Then we proceeded to analyse the modified relation between critical temperature and charge density. We also calculated the modified expressions for critical exponents and condensation values for this noncommutative scenario.
We have provided some numerical estimates in Table 1. We consider the established lower bound of θ to be θ ≤ (10 T eV ) −2 [12]. In [8] the Black Hole mass M is related to θ by M ≈ √ θ/G where the Newton's constant G has been reinstated. This yields M ≈ 10 33 GeV, which, however, is far below the mass of the astrophysical Black Holes. In Table 1 we have taken values of θ to be lower than the above bound [12] and the corresponding Black Hole masses to be considerably larger than the [8]. We find that larger values of θ tend to lower the critical temperature of the holographic superconductor. Expectedly for very small θ the relation T c = 0.225 √ ρ [10] is recovered.
However, for smaller mass Black Holes, (i.e. less than [8]), and same values of θ, we find that the critical temperature rises above the θ = 0 value. Interestingly, from Table 2, we find that for θ = 0.5 and M = 2.75 (θ)/G (which is close to the mass M = 2.4 (θ)/G considered in [8]) the value of ζ = 0.2443 which is appreciably larger than the θ = 0 result of ζ = 0.225 [10] indicating a larger critical temperature. But once again for larger M , ζ stabilises to 0.2214, which is less than 0.225.
It will be interesting to consider non-zero vector potential in the non-commutative framework to study conductivity and other properties of the holographic superconductors. Furthermore, in the noncommutative extension considered here, other parameters of the bulk theory such as the noncommutative parameter θ and the Black Hole mass are involved. Mapping of these parameters to the boundary theory may lead to generalized forms of holographic superconductors.