Hard-gapped Holographic Superconductors

In this work we discuss the zero temperature limit of a"p-wave"holographic superconductor. The bulk description consists of a non-Abelian SU(2) gauge fields minimally coupled to gravity. We numerically construct the zero temperature solution which is the gravity dual of the superconducting ground state of the"p-wave"holographic superconductors. The solution is a smooth soliton with zero horizon size and shows an emergent conformal symmetry in the IR. We found the expected superconducting behavior. Using the near horizon analysis we show that the system has a"hard gap"for the relevant gauge field fluctuations. At zero temperature the real part of the conductivity is zero for an excitation frequency less than the gap frequency. This is in contrast with what has been observed in similar scalar- gravity-gauge systems (holographic superconductors). We also discuss the low but finite temperature behavior of our solution.


Introduction
In recent years string theory or, more specifically, gauge-gravity duality has seen interesting applications in the field of condensed matter physics. One of the earliest such applications is the discovery of a superconductor-like phase transition in AdS with a Reissner-Nordström black hole and a charged scalar field minimally coupled to a U (1) gauge field [1,2].
Earlier works on this system at low (but non zero) temperatures indicated the presence of a gap in the spectrum, with an exponentially decaying population of normal carriers following the Boltzmann distribution. In various recent works [3,4,5,6,7,8,9], authors have addressed the problem in the zero temperature limit. More precisely, they consider a charged scalar field and gauge fields in an AdS4 background at zero temperature. The solution turns out to be a solitonic solution with a zero sized horizon. The authors find that the effective potential for small gauge field fluctuations vanishes near the black hole horizon. This implies that the normal component of the A.C. conductivity never vanishes, even at zero temperature, which in turn indicates that the superconductor is gapless. This is a new and interesting result different from what would have been naively expected.
In this paper we consider non-Abelian gauge fields in the background of an extremal AdS4 black hole. The setup is quite similar to so-called "p-wave" holographic superconductors [10] and known to have a superconducting transition. 5 . We work at the zero temperature and found a fully gravity back-reacted solitonic solution to this system similar to [3]. Then we consider the frequency dependent conductivity by considering certain vector fluctuations on this background. This is an anisotropic system which shows different conductivity at different directions. We find the expected superconducting behavior for relevant currents. Interestingly we find that the corresponding effective potential does not vanish at the horizon (it goes to zero near the boundary). Following the argument in [3], we thus arrive at the conclusion that the holographic non-Abelian superconductor does have a finite gap for the relevant gauge field fluctuations.
The plan of the paper is as follows, in section 2 we discuss the general setup and Einstein's equation for a non-Abelian gauge field coupled to gravity. In section 3 we discuss the instability of the RN black hole solution. In section 4 we construct the zero temperature solitonic solution which is the superconducting ground state of the theory. We also discuss issues related to conductivity and show the existence of a hard gap. In section 5 we propose a possible form of the solution at non-zero temperature using the method of matched asymptotic expansion.

Setup
The Einstein-YM action for a non-Abelian gauge field with a negative cosmological constant is given by [11], where Fµν is the field strength of an SU (2) gauge field.
To tally with our plan of considering anisotropy in the spatial direction, we choose the following ansatz for our metric and the gauge fields 6 , It is straightforward to find the non-zero components of the field strength are The energy momentum tensor is and the non-zero components of the energy momentum tensor are, It is to be noted that here Txx = Tyy due our anisotropic ansatz 7 . The Maxwell's equations of A(r), B(r) are 6 Due to a repulsive term coming from the non-Abelian interactions, it is expected that a isotropic ansatz will have a quartic instability and would possibly have more free energy than the anisotropic ones [10,12]. We left the detailed discussion of these issues for a future study. 7 HSH thanks Matthew Roberts and Gary Horowitz for discussions on this point.
The potential term for A(r) is different from the similar term in EYMH case due to the existence of a factor 1 r 2 . The diagonal Einstein equations give, The above equations are invariant under the following scaling symmetries: The second scaling symmetry may be used to set χ = 0 at infinity and the third scaling symmetry may be used to set c = 1 at infinity, so that the asymptotic metric is that of AdS4.
The fields have the following asymptotic behavior: where µ is the chemical potential and ρ is the charge density in the boundary theory. In what follows we will only consider the solutions for the field B which vanishes near the boundary, i.e. B0 = 0.

Reissner-Nordström black hole solution and Instability
To see when one expects hairy black holes at low temperature, one can study linearized perturbations of the extremal Reissner-Nordström AdS (RN-AdS) black hole [3]. The general RN-AdS solution is given by, The temperature of the black hole (12) is For AdS-RN, T = (12 − ρ 2 )/16π. It is known that fluctuations of Bx in this background develops a tachyonic mode [10] at low temperaure and for sufficiently large q. The resulting solution is a superconducting black hole with vector hairs. Following [3], we argue that such an instability persists at the extremal limit. At the extremal limit (T = 0), we have ρ = 2 √ 3. The near-horizon limit of the extremal solution is AdS2 × R 2 with a metric, Plugging this into the wave equation of B(r), and changing variablesr = r − 1, we recover a wave equation for AdS2 with a new effective mass, The instability to form SU (2) vector hair at low temperature then is just the instability of scalar fields below the Breitenlohner-Freedman (BF) bound for AdS2: m 2 BF = −1/4. Thus the condition for instability is It is reasonable to expect that such an instability takes the system to its superconducting ground state.  Figure 1: α vs q plot shows a almost linear relation with a slight dip for α for a value of q < 2.5.

Zero temperature solution
Being a single state without any degeneracy, a superconducting ground state does not have any entropy associated with it. This fact concur with the observations in [3], [4,5] that the zero temperature limit of the superconducting black holes in EYMH theory have zero horizon size. Motivated by these facts we assume that same is true for a non-Abelian hairy black hole also and the ground state is a geometry with zero horizon size. We choose the following ansatz, near r → 0 + , (similar to that of EYMH case [3]) All the terms with subscript 1 are sub-leading and go to zero, faster than the leading part where it is applicable, as r → 0. Putting Eq. (17) in Eq. (8), we get find out the various terms in Eq. (17). For example from the equation of motion for A, where we have used the observation A0 → 0 at the horizon. Following similar procedures, we get where by rescaling (10) we set the coefficients A0 = 1 and χ0 = 0, c0 = 1. After we solve the equations we again use the rescalings of g, c, χ to make the Asymptotic geometry the same as that of AdS4. For a given q, we choose α in such a fashion that B vanishes near the boundary [3]. The numerical evaluation shows us an almost linear relation between q and α (Fig 1),with a slight dip for a lower value of q. Fitting a linear relation between q and α with a range of 3 < q < 8, we get α ≈ 0.088 + 0.66q. Our numerics is less reliable in the region q < 0.95. It is not clear what exactly happens in this region. As q → √ 3 2 ≈ 0.866, the geometry of our solution seems to approaches extremal RN geometry (Fig 1). However the questions of phase transitions remain unclear. It is expected that there is a second order phase transition at q = √ 3 2 from RN black hole to our solution. However one may not rule out the possibility of a first or any other type of phase transition(s). At least in the range of our numerics it seems that our solutions have lower free energy than the extremal RN black hole in a grand canonical ensemble. As expected the solitonic solution is more massive than the RN black hole, but they carry more charge for the same boundary chemical potential and consequently have a lower free energy.
General properties of our solution is similar to EYMH case [3]. B, χ approaches a constant near r = 0, g = r 2 and the gauge field A and its derivative (electric field) vanishes. The metric approaches AdS4 with the same value of the cosmological constant at infinity. The extremal horizon is just the Poincare horizon of near horizon geometry AdS4. In terms of the dual field theory, this means that the full conformal symmetry is restored in the infrared. Due to the asymmetry of our solution in x and y Our solution may be viewed as static, charged solitons, where the electrostatic repulsion is balancing the gravitational and non-Abelian interactions. As can be checked easily from the asymptotic behaviors the energy of the soliton is finite. 8 One interesting feature is the existence of an "essential singularity"(i.e. an e − # r behavior) of A(r) at r = 0. Similar singularities appear for other quantities in Eq. (20). Hence all the sufficiently high derivative of the metric and gauge fields vanishes at r = 0. This is in contrast with the EYMH case [3], where sufficiently large curvature invariants diverge at r = 0 for a generic α and behavior of the fields near r = 0 is power law. In terms of the "tortoise" co-ordinates (27), the essential singularity is just the reflection of the fact that A is massive near the horizon and decays exponentially.

Conductivity
In order to calculate the conductivity of this system, we need to turn on a small perturbation in the vector potential. The gauge field and the associated metric perturbations are of the form: where we assume that both the gauge and the metric perturbations are of the same order O(ǫ). We won't be looking in the other component of the gauge field perturbations which decouples from the above. From Maxwell's equation we get, To linear order, Einstein's equations give, 8 However it should be noted that B(r) is not square integrable. This is not a requirement of a valid solution to the Einstein's equations and assuming that our numerics are reliable, such solutions exists. The question is that whether they play a role in the context of AdS/CFT. In a sense whether it belongs to same quantum ensemble as the extremal black hole. As our zero temperature solution is of finite energy this seems to be the case. There are well known solutions such as the BPST instanton [19] which also have a non square integrable gauge field configuration with a bounded action. In our case, just as in case of BPST instanton, the solution may be approached by configurations which are of finite norm with a bounded energy (just replace B → exp(− β r )B(r), β → 0). At finite temperature the gauge field B is likely to be square integrable due to the cutoff provided by the horizon. Our zero temperature solution may also be thought as a zero temperature limiting case of such a solution (see Sec. 5).  Figure 3: Plot of the rescaled potential V (r n ) for q = 1.5, 1.1, 1 (from the right) at T = 0 with rescaled co-ordinate r n . The poincare horizon is at r n = 0 for the zero temperature solutions. V (r n ) is non-zero at the horizon. The dashed curve is for the extremal black hole with a horizon at r = 1.
which combined with the equation of A(r) to give: This can be written as a Schrödinger equation: where and all the derivatives in Eq. (25) are in terms of the new variable new variabler ("tortoise coordinate") given by: Here, c approaches to unity as r → ∞, so that the spacetime is asymptotically AdS4. It follows then from Eq. (11) that the potential V (r) vanishes near the boundary. If we require g ∼ r 2 near the horizon at r = 0 then the first term vanishes, while the second term is finite as B(r = 0) ≡ B0 = 0 and χ is also finite at the horizon. The value of the potential at the horizon decreases as q → √ 3 2 (Fig 3). Note that since c → 1 near the boundary, the quantity ω can be interpreted as the frequency of the incoming wave.
The superconducting nature of the system is argued from the existence of a supercurrent solution. If we choose ω = 0 and integrate A 3 x from the horizon (with a regularity condition at the horizon), we are expected to get a non-trivial Ax. Existence of such a solution implies a δ function for the real part of conductivity at ω = 0 [20, 3] 9 Re(σ(ω)) ∼ δ(ω) + finite.
The nature of the finite part of the conductivity can be inferred from the potential V (r). As argued before the potential vanishes near the boundary and is finite near the horizon. The fact that the potential is nonzero at the horizon makes it possible for this system to exhibit a hard gap. From Eq. (25), the field a has the following asymptotic behaviours near the horizon (r → −∞) and the boundary (r → 0): 9 One should be more careful about the existence of a delta function for the real part of conductivity at ω = 0 and relation between superconductivity. Due to the coupling of gravity modes the potential V (r) is non-trivial even in the normal phase (Fig 3) and consequently a delta function is present in the real part of DC conductivity. Physical interpretation of such a delta function comes from the "translational invariance" of the theory. See [21] for further discussions. whereω = p c 2 0 ω 2 − V0, with c h , a h being the near-horizon values of c and a respectively. Here, we have chosen the incoming boundary condition near the horizon. The conductivity of the system is given by: It follows from Eq. (25) that : which implies that the quantity Λ = a * a ′ − aa * ′ = 2iIm(aa * ′ ) is a constant. Equating the values of Λ near the horizon and the boundary we get: Therefore, the real part of the conductivity will vanish wheneverω is imaginary, i.e. when ω < √ V0/c0, which defines the gap.
As our solution is an anisotropic superconductor (only superconducting in Ay direction), the excitations in A 3 x channel may not be gapped [10]. The gap seems to be a property of the anisotropic non-Abelian system, as the scalar-Abelian gauge field system does not exhibit this gapped behaviour in the extremal limit [3].

Finite temperature analysis
Although the Eq. (8) has been solved in a probe limit, the full set of equations with gravity backreaction (Eq. (9)) is not yet solved at a non-zero temperature. Hence it is an issue whether our zero temperature solution may be realized as a zero temperature limit of the finite temperature hairy black hole (and whether such a low temperature hairy black hole solution at all exists.). One affirmative clue comes from the emergent conformal symmetry in the IR. In the near horizon region, i.e. α ≫ r, our solution approaches a AdS4 geometry with a background gauge field Ax = B0 and At = 0. One may consider a black hole situated deep inside this AdS space, such that the radius of the black hole r0 ≪ α. Here, g(r) near the horizon is given by, g(r) = r 2 (1 − r 0 r ). However this Schwarzschild like solution will get correction due to a small non-zero A(r) near the horizon. The solution of A(r) in the r ≪ α is given by, A(r) is small in the region r ≪ α. In this region we can do a perturbation in A(r) for other quantities. Realizing that our scaling relations Eq. (20) is just a perturbation in A(r), we can match a near horizon solution with the scaling solution (i.e. Eq. (20)) in a region r0 ≪ r ≪ α. Hence we use the method of matched asymptotic expansion to argue that, is a solution to Einstein's equations if r0 ≪ α. The quantities (A, B.c, χ) appear in the right hand side of the above equations are our numerical solution and the quantities with a subscript T are the finite temperature versions. Whether correction to such a solution is finite or not is an open question, which may be partially settled by a convincing finite temperature numerics. However due to the possible "smooth" nature of the equations Eq. (8), i.e. a small change in the near horizon boundary condition leads to a small change near the boundary, it is reasonable to believe that corrections to Eq. (36) are well controlled. What happens to the potential V (r) (see Eq. (26)) in this limit is interesting. At any non-zero r0, the potential V (r) goes to zero at the horizon (r = r0) and the height of the potential is always finite. Hence the transmission through this potential is non-zero. In our language the (i.e. Eq. (33))ω = ω is a real number. Hence the real part of the conductivity is non-zero for any value of ω. However, in the "tortoise co-ordinates" (27) the potential is gradually widened as r0 → 0 (Fig 4). Hence the transmission through the potential gradually decreases as the horizon size is decreased. We expect a exp(− ∆ T ) fall off for the real part of conductivity. A detailed discussion of the various gaps etc is left for a future work.