Momentum dependent gap in holographic superconductors revisited

We reconsider the angular dependence in gap structure of holographic superconductors, which has not been treated carefully so far. For the vector field model, we show that the normalizable ground state is in the p-wave state because s-wave state is not normalizable. On the other hand, in the scalar order model, the ground state is in the $s$-wave. The angle dependent gap function is explicitly constructed in these models. We also suggest the modified ansatz of the vector order which enables to discuss the order $p_{x}\pm ip_{y}$ gap, which has not been possible so far. We have also analytically investigated the critical temperature and the behavior of the gap near there. Interestingly, for the fixed conformal dimension of the Cooper pair operator, the critical temperature in vector model is higher than that of the scalar model.


Introduction
The gauge/gravity duality [1]- [4] has been applied to strongly correlated systems as an efficient tool to handle the strong coupling. In the context of this duality, a theory of superconductivity was set up [5] using the spontaneous symmetry breaking of the U (1) symmetry of an Abelian Higgs model coupled to AdS the gravity [6], [7], after which huge number of investigations on s-wave holographic superconductors [12]- [22] has been reported in past decade. Although the original model [5] allowed to estimate the gap and the critical temperature of conductivity of isotropic system, typical high T c superconductors show the momentum dependent gap structure ∆ k [8,9,11] which has been considered as one of the most important finger prints of high T c superconductors. To address p-wave superconductors, Gubser [23] first introduced the non-Abelian gauge field for holographic superconductor model, which is followed by many investigations on p-wave [24]- [35] or d-wave [36]- [42] holographic superconductors using abelian vector and tensor fields. However, to our surprise, none of the investigations addressed the momentum k dependence of the superconducting gap, because in all the previous works, non-vanishing components of vector A ν or tensors B µν were assumed to be isotropic. Although the angle dependence of the gap was introduced in a notably exceptional paper [36], the angle dependence in that work was introduced by considering the 'fermion spectrum' explicitly rather than through the gap equation, that is, the equation of motion of the complex scalar function.
In this paper, we study the momentum dependence of the gap function in holographic set-up. We consider angle dependent fields in two holographic models, namely, scalar field model and vector field model. We first consider the vector field model in order to see whether the role of the spin-1 field model is essential. We have explicitly shown that the normalized ground state of the vector model is provided only by the p-wave state and the s-wave state is not normalizable. Similarly in the scalar field model, we have shown that the ground state of s-wave superconductors is from s-wave, while the p-wave and d-wave states give excited states. We will also show that the traditional formalism of holographic p-wave superconductor is rather twisted in the gap structure so that one can not introduce the order parameter of p x ± ip y type. We will show how to fix this problem.
Our investigation is done by constructing the angle dependent gap function in momentum space. For the vector field model, we will be able to construct the gap functions analytically by imposing the vortex free condition. For the scalar field model case, we can find general l-wave type superconductors as excited states. In general, the gap equation in holography is non-linearly coupled with that of the photon field in the AdS, which is not solvable by usual separation of variable, which is useful in linear equations. To overcome this difficulty, we consider the system near the critical temperature (T c ) and expand the field and equation of motion by the small parameter 1 − T /T c and consider the system of equations order by order.
We have also investigated T c itself in the probe limit of the gravity field for each models. We have used matrix-eigenvalue algorithm with the Pincherle's Theorem to calculate the T c values for scalar field model. For vector field model, we have used Sturm-Liouville eigenvalue method. Interestingly, we have observed that the critical temperature for vector field model is higher than the critical temperature of scalar field model for any fixed value of the dimension of the Cooper pair operator. We considered the possibility that p-wave condensation can be discussed in scalar field model context provided s-wave condensation is forbidden under a special constraint due to e.g. the lattice symmetry. We compared the gap functions for p-wave states coming from scalar model and that from the vector model. This paper is organized as follows. We have started our discussion on different holographic set up for s, p, d-wave superconductors of the BCS theory in section 2. In section 3, we studied field equations of various holographic models and described their equation of motion in the unified fashion. In section 4, we have studied the vector field model with angular dependence and show that the normalizable solution is available only from the p-wave solution. We also show how to introduce the p x ± ip y type of gap of p-wave superconductivity in holographic set up. In section 5, we compare the critical temperature for ground state of scalar field model and vector field model. We then compare the excited p-wave state from scalar field model with p-wave state in the vector field model.
We summarize our findings in section 6.

Momentum dependence of the Gap in BCS theory
To understand the origin of the momentum dependence gap structure in p and d-wave superconductors, we start with a basic discussion on the BCS gap structure. For arbitrary pairing interaction, the BCS gap equation can be written as [46] where ξ k is energy spectrum. The potential in k-space can be expressed as where l is angular quantum number and V (r), P l , J l are the interaction potential, Legendre polynomials and Bessel function respectively. The different value of l describes the different orbital symmetry which determines the different type superconductors, namely, s-wave(l = 0), p-wave(l = 1), d-wave(l = 2) superconductor. Recently f -wave(l = 3) and g-wave(l = 4) superconductors also has been reported in [47,48] and [49] respectively.
For non-zero value of l, they lead to the momentum dependent BCS gap function. For different orbital symmetry, the angle dependent gap structures are known in literature. The cuprate exhibits d x 2 −y 2 -wave superconductivity. The superconducting order parameter of d x 2 −y 2 -wave superconductivity is [50] ∆ k = ∆(T )(cos k x − cos k y ) ∆(T ) cos(2θ) . (2.3) This gap structure in the holographic set-up will play a very important role to understand the properties of real-world materials since there are many data available: those of angle-resolved photoelectron spectroscopy (ARPES), Raman Spectroscopy, scanning tunnel spectroscopy and neutron magnetic scattering [50].

Unified Field equations of holographic models
We start with a general discussion from and summarizing field equations for various spin fields in a unified form following [44]- [45]. We use planar symmetric AdS 4 -Schwarzschild blackhole as the background: where r h is the horizon radius. For s-wave superconductor, we use Abelian-Higgs model with a complex scalar field [5]: Using the ansatz ψ = ψ(r) and A = A t (r), the equation motion for s-wave holographic superconductors takes form For p-wave superconductors, the holographic p-wave model was first introduced in [23] using a SU (2) Yang-Mills field in AdS 4 -Schwarzschild background with Lagrangian is the field strength for SU (2) gauge field. The equation of motion yields To break the rotation symmetry in the system, the field ansatz is considered as A = A t (r)σ 3 dt + ψ (g) x (r)σ 1 dx in which the condensed phase breaks U (1) symmetry and SO(2) rotational symmetry in xy-plane. Using this ansatz, the equation motion for non-abelian gauge theory reads Later, an alternative p-wave holographic superconductors model was introduced by a charged vector field. The matter Lagrangian density for this model is which gives the following field equations By taking complex vector field ψ µ dx µ = ψ x (r)dx x along x-direction, we break the rotational symmetry of the vector field. Using this vector field ansatz and gauge field ansatz A µ dx µ = A t (r)dt, the field equations takes If we map ψ x (r) = ψ (g) x (r) √ 2 , we will recover the field equations for non-abelian model. This two models are equivalent for m 2 = 0. To construct holographic d-wave model, the vector field model was generalized tensor field model with minimal effective matter Lagrangian density where B µν is a charged tensor field. To realized d-wave condensate, they considered the tensor field ansatz B xx = −B yy = ψ xx (r) which breaks rotational symmetry and flips sign under a π 2 -rotation on the xy-plane. Using this tensor field ansatz and the gauge field ansatz A = A t (r), the field equations read which is called 'effective potential' 2 . From this above unified field equation, we recover the field equations for s, p, d-wave holographic superconductor for the spin values s = 0, 1, 2 1 The Lagrangian density in this modified model is and R µνρλ is the Riemann tensor of the background spacetime. With the tensor fields ansatz B xy = ψ xy (r) and gauge field ansatz A = A t (r), the matter field equation reads which is differ from the eq.(3.17) because of the last term. If we take ψ xy (r) = r 2 √ 2 ψ(r), we will get exactly same field equations for s-wave holographic model. 2 This terminology is quoted because it is not exactly the same as the actual effective potential term derived from a dynamical equation. For our discussion, we consider it as an effective potential for this coupled equation.
respectively. If we calculate the perturbation of Maxwell's field A µ = δ x µ A x exp(−iωt) for conductivity, we will get the same equation structure for different values of spin. The radial (only) dependent field structure in the unified field equations leads to spherically symmetric ground state of superconductors. Since all holographic superconductor models so far, are governed by the above unified field equations which depends only on r, we can say that the momentum dependent order parameter in holographic set-up is missing in the literature.
To gain a better understanding of high T c superconductors through holographic setup, we need to modify the ansatz of the fields which will help us to distinguish s, p, d-wave superconductors in a generic sense. The generic sense means that the distinguishable properties for s, p, d-wave superconductors depends on the values of angular momentum quantum number instead of spin number s in holographic set-up. The angular momentum number determines the actual orbital symmetry which is responsible for the superconductivity.
Although the choice of the ansatz of the field breaks the rotational symmetry in holographic superconductors models, those models do not have any angular momentum number l which is essential to understand the p-wave or d-wave superconductivity. In [36], the spatial angle θ is introduced by the transformation of spin two field and the angle dependence gap is generated using the interaction term between the spin two field with fermions. From the literature of holographic superconductors, it seems that the spin number s of the field in holographic set-up is related to the angular momentum number l of the boundary theory. In order to understand the connection between them, we start with vector field (s = 1) model with angular dependent fields.

Vector field model with angle dependent Gap
Here, we assume that there is strong asymmetry in the direction of the c-axis so that we can just consider 2+1 dimensional direction. To introduce the angle dependent gap structure in holographic superconductors, we consider the polar coordinate of 2+1-dimensional boundary where the system lives. Accordingly we write the AdS 4 -Schwarzschild black hole metric (3.1) in polar coordinate (t, r, u, θ) reads from which the Hawking temperature can be read as The p-wave holographic superconductors model has been described by the matter Lagrangian density (3.10) which consists of gauge field and vector field. The field equation are given by eq.(3.11) and eq.(3.12). We now modify the complex vector field and gauge field ansatz which reads We consider field along one direction since we want to break the rotational symmetry of the vector field. The ansatz changes in polar coordinate Using the above relation, we find the relation between ψ u and ψ θ which is Using the eq.(3.11) and the above field ansatz, the gauge field equation becomes From eq.(3.12), the matter field equation reads Setting ν = u and substituting √ −g = r 2 u, we obtain Similarly we obtain field equation for ψ θ by setting ν = θ,

The angular dependent part of the matter field
We now impose vortex free condition (perpendicular to the plane) which is Using the relation between ψ u and ψ θ , the above condition becomes Using this condition, we want to solve the u, θ dependence part of the matter field. We can write From eq.(4.12) and eq.(4.13), we now try to solve the boundary wave state R u (u, θ) with help of the separation constant l p Therefore the solution reads R u (u, θ) = u lp−1 (cos θ) lp . where we deleted one multiplicative integration constants absorbing them into Ψ. Using the relation (4.6) and ψ θ (r, u, θ) = Ψ(r)R θ (u, θ), we find R θ (u, θ) = u lp (cos θ) lp−1 sin θ. Therefore, we can write The separation constant l p should be integer: this can be seen from the fact that the matter field should be one valued under the rotation of 2π of θ, which is the same as the twice of the π rotation under which From this, we can identify l p as the angular momentum in this set-up. For l p = 0, which is not normalizable because the normalization condition is which is logarithmically divergent. Therefore the vector field model does not give us the normalizable ground state for the s-wave state. For the p-wave of l p = 1, the solution is given by which is normalizable solution. Notice that C should be chosen to be −1 to be consistent with eq.(4.5). Therefore the ground state of the vector field model comes from the l p = 1, while the s-wave solution of the model is not normalizable. which is first main result of this paper. Notice also that ψ x (r, u, θ) = Ψ(r) for l p = 1 but only for this case. That is, the solution ψ x (r, x, y)dx is reduced to ψ x (r)dx for the ground state. Now if we take the ψ x (r) as the order parameter of the p-wave superconductivity as it was suggested in the original model [23], there is no angular dependence in the gap, which is a contradiction. Our analysis in the present setup is telling us that the gap function of the p-wave model is ψ u , not the ψ x . Notice that ψ x can not be the order parameter of p-wave superconductivity because it does not have any angular dependence, and ψ θ can not be the one either, because it shows the vanishing gap at u = 0, which is a coordinate singularity not the real nature.
There is nothing wrong here but what we got is not really what we would expect in the usual tensor analysis. All the oddities come from the assumption that only ψ x = 0 while ψ i = 0, i = y, t, r which is very unusual gauge choice from rotation tensor point of view. The better ansatz for the gap structure should be the following one: Then by a simple calculation, we can get the identification A = Ψ(r) where Ψ(r) is the function we met before. Here we can regards any of A i as the order parameter. Then, the order parameter for the gap structure p x + ip y can be naturally introduced as which has not been possible so far. This is simple but one of the main points of this paper.

The critical temperature and field solutions
We now proceed to solve the radial part of the matter field. Using the vortex free condition, eq.(s)(4.9,4.10) can be written as Substitute eq.(4.16) in eq.(4.23) and eq.(4.24), we get a single equation for Ψ(r) which takes form as where we have substitute the zeroth order of gauge field part from the gauge field ex- t (r, u, θ) near the critical temperature. Substitute ψ u (r, u, θ) and ψ θ (r, u, θ) for l p = 1, the zeroth order gauge field equation becomes The above two field equations are same with the field equations for vector field model in literature. Therefore, the critical temperature and the temperature dependence condensation operator value will be unchanged. To get the solution of the fields, we need to know the asymptotic behavior of the fields. At the asymptotic limit, we consider f (r) ≈ r 2 and O(r 4 ) = ∞. Using this, we obtain the field equation near boundary Using the gauge/gravity duality, the asymptotic behavior of the field reads is the scaling dimension, µ and ρ is the chemical potential and the charge density respectively and Ψ ± maps to temperature dependent condensation operator value of the boundary theory. The Breitenlohner-Freedman mass bound [51], [52] for this holographic set-up is m 2 BF ≥ − 1 4 . Under the coordinate transformation z = r h r , the metric field reads In the z-coordinate, the field equations become At T = T c , the matter field Ψ(z) = 0 which leads to the zeroth order gauge field eq.(4.32) Using the aymptotic behavior of fields (4.29), the solution of the above equation reads . We can write the radial part of the matter field in following form where F (z) is the trail function for Sturm-Liouville eigenvalue method, δ(= δ ± ) is the scaling dimension and C p is unknown constant which need to be determined. We now substitute this in the matter field equation which yields whereλ = qλ. The above equation can be written in the Sturm-Liouville form (4.38) The above identification enables us to write down an equation for the eigenvalueλ 2 which minimizes the expressioñ where the Sturm-Liouville eigenvalueλ is estimated from eq.(4.39). For m 2 = − 3 16 , the scaling dimensions are δ = δ − = 5 4 and δ = δ + = 7 4 . We have shown the critical temperatures in the Table 1. The critical temperature for δ = 2 (setting m 2 = 0) is √ ρ which matches with the result from the non-abelian model [23].  We now move to calculate the constant C p from the gauge field equation near T c . Substituting eq.(4.35) in eq.(4.32), we get where B(z) = 2q 2 z 2δ−2 F 2 (z) g(z) . We may now expand A (0) t (z) in the small parameter with χ(1) = 0 = χ (1). From eq.(4.42), we get the asymptotic behavior (near z = 0) of the gauge field. Comparing the both equations of the gauge field about z = 0, we obtain Comparing the coefficient of z on both sides of eq.(4.43), we obtain We now need to find out the χ (0) by substituting eq. Using the boundary condition of χ(z), we integrate (4.45) between the limits z = 0 and z = 1 which gives . Using eq.(4.46) and eq.(4.44), we obtain where the definition of λ is used. Using the expression for the critical temperature eq.(4.47), we get (4.48) Using the fact that T ≈ T c , we can write T 2δ Tc Using this, we finally obtain the constant which gives the temperature dependence condensation value in following form Near the critical temperature, the radial part of the scalar field solution now takes form

Gap structure in vector field model
We would like to mention the general prescription for the mapping between the order parameter (gap function) and the matter field by the near boundary behavior of the bulk field ψ: 51) in the limit z → 0. Hereψ(z, k, θ) = d 2 xe i k· x ψ(z, u, θ) is the Fourier transform and k = k 2 x + k 2 y . Since the radial coordinate in gravity theory is associated with the energy in boundary theory, the temperature dependence is solely coming from the radial part of the field Ψ(z) and angle dependence is from the solution R(u, θ) so that we can see that these dependencies are factorized as in the eq. (4.13). From this, the angle dependent condensation operator can be written as (4.52) where H(u, θ) is R(u, θ) upto a constant and I(k x , k y ) is the Fourier transformation of H(u, θ). From eq.(4.50), we can write the solution near boundary for ground state as where The angle dependence part is separated by the separation constant l in which equation takes form The solution can be chosen such that Θ(θ) = e ilθ , where l can be identified as angular quantum number. Using the separation constant l, the equation (4.58) now becomes which is nothing but the Bessel equation whose solution takes form for the finiteness at the origin u = 0. The value of u runs from 0 to system size a. The R(u) should vanish at the boundary of the system for which we have to set α = α l1 a where α l1 is the first zero of the J l polynomial. The solution now reads (4.62) We can finally express the solution R(u, θ) in the following form where k = k 2 x + k 2 y , k x = k cos φ, k y = k sin φ and φ is the angle in momentum space. After some calculation, we obtain This result is very crucial for understanding the different wave state structures in the scalar field model. We can identify the angles in coordinate space and that in momentum space, φ = θ as it is well known, the states in momentum space become The α l1 values are α 01 = 0, α 11 = 3.8317, α 21 = 5.1356 for s, p, d-wave state respectively.
Using real part of I(k, θ) (4.66), the density plot of s, d-wave states in momentum space are presented in Figure 3, where k x and k y is expressed in inverse unit of a.

The critical temperatures
We already know that the ground state in scalar model is in s-wave state and the ground state in vector field model is in p-wave state. In this subsection, we will discuss the critical temperature of the ground state in both models and compare them for each value of scaling dimension δ. For example, at the same value of the scaling dimension δ = 2, the critical temperature for vector field is T c = 0.124 √ ρ while that of the scalar field model if we recast eq.(A.23) in Sturm-Liouville form. If we denote q(z) as q v (z) (from eq.(4.38)) for vector field model, then the difference which is responsible for the simple result of eq.(5.2). Before we finish this subsection, we mention that in the ref. [30], the competition between the s-and p-wave condensations were studied. However, the authors compared p-wave and s-wave such that the p-wave model has fixed conformal weight 2 while the s-wave model has varying weights. In contrast, we compared s-wave and p-wave at the same weight for various values of weight.
In the presence of the condensate and the charge density, it is not necessary to respect the Lorentz invariance and density operator and current operator may have different weights.

Comparing p-wave states in vector and scalar models
The p-wave state in the scalar field model is in the excited state of the system while that state in the vector field model is the ground state. Nevertheless they can be the same since they are states in different models. Therefore the question here is how much they are different if they are different. Before we compare these, we would like to mention that the condensation to p-wave state in the scalar field model is possible only under the constraint such that s-wave condensation is forbidden for some reason. In such situation, the p-wave state is the ground state in the scalar field model. Then one may ask whether the p-wave gap structure in scalar field model and in vector field model are similar or not.
We only need to focus on the momentum dependent part I(k, θ) of the gap function here.
From the scalar field model, the excited p-wave state in momentum space is represented by (from eq.(4.66)) The momentum dependence part of the gap energy in vector field model reads −a 2 k 2 4 cos θ and I θ (k, θ) = 2πa 2 iJ 2 (ak) k sin θ (5.5) Since I u (k, θ) and I θ (k, θ) are related and ψ u is the measure of the order parameter, we now discuss only about the I u (k, θ) part from the gap energy ∆ (u) k . From the Fig.(1) and Fig.(5), we observe that the density plot of the imaginary part of I(k, θ) is very similar to the density plot of I u (k, θ). We now take the ratio between this two function Im. (I(k, θ)) which is independent of θ. We can now plot this ratio function (5.6) in Figure 6 for fixed value of a, which is almost constant function for ka << 1 4 . Notice , however, that this ratio function diverges at the roots of the generalized hypergeometric function. The first of which is ka = 5.8843.
(a) (b) Figure 6: The plot of the ratio function

Discussion
In this paper, we have investigated the angle dependent gap structures in the vector field models. In order to understand the necessity of the vector field model for p-wave superconductors, we have started with the vector field model. We showed that the normalizable ground states of this system is given by the p-wave state while the state with l p = 0 is not normalizable. Therefore, the p-wave ground state can be achieved only from the vector field model. We have found that the order parameter for vector field model is represented by ψ u , not the ψ x since ψ x does not have any angular dependency.
We then explore the angular dependence in the scalar field model, where all l-wave states are available. Here the ground state of the system is from the s-wave state. For l > 0, they are excited states of the system. We have then compare the momentum dependent part I k,θ of gap function from the scalar field model and the vector field model.
We observe that the structure of both gap energy is almost same for small momentum 4 If we consider ka < 1, then we can write this ratio function as Im. (I(k,θ)) ≈ 0.32 + 0.006k 2 a 2 . In this limit ka < 1, we can neglect the higher order terms in k 2 a 2 .
range. The point is that p-and d-wave gap structures can be explained through the scalar field model if we assume that the states for lower value of angular momentum number (l = 0, 1) is forbidden in the scalar field model.
We also studied the critical temperature in the probe approximation of the gravity background using matrix-eigenvalue algorithm method and Sturm-Liouville's eigenvalue method for different scaling dimensions. Another interesting point is that the critical temperature for the ground state of vector field model is higher than the ground state of the scalar field model for same value of the scaling dimensions.
We would now like to mention the drawbacks and future works. The Fermi surface is not easily demonstrated in our set-up, where fermions are not included at all. The appearance of Fermi arc and Fermi surface is only possible when one consider the interaction between fermion and tensor field in holographic set-up [38]. We will come back to this issue in the future work.
Factoring out the behavior near the boundary z = 0 and the horizon, we define Then, F is normalized as F (0) = 1. We now substitute this in the matter field equation (A.23) which yields Notice that this is the generalized Heun's equation that has five regular singular points at z = 0, 1, −1± √ 3i 2 , ∞. Substituting y(z) = ∞ n=0 d n z n into (A.25), we obtain the following four term recurrence relation: The first four d n 's are given by α 0 d 1 + β 0 d 0 = 0, α 1 d 2 + β 1 d 1 + γ 1 d 0 = 0, d −1 = 0 and d −2 = 0. The series y(z) = ∞ n=0 d n z n is absolutely convergent for |z| < 1. The condition for convergence at |z| = 1 involves parameters of the equation. The convergence of the series y(z) = ∞ n=0 d n z n can be analyzed by studying asymptotic behaviour of the linear difference equation eq. (A.26) as n → ∞. One finds that eq. (A.26) possesses three linearly independent asymptotic solution of the form From this expression, we can identify the temperature dependent part of the condensation operator as Õ = C s 1 − T Tc . To determine the integration constant C s in the above