Analytical study on holographic superfluid in AdS soliton background

We analytically study the holographic superfluid phase transition in the AdS soliton background by using the variational method for the Sturm-Liouville eigenvalue problem. By investigating the holographic s-wave and p-wave superfluid models in the probe limit, we observe that the spatial component of the gauge field will hinder the phase transition. Moreover, we note that, different from the AdS black hole spacetime, in the AdS soliton background the holographic superfluid phase transition always belongs to the second order and the critical exponent of the system takes the mean-field value in both s-wave and p-wave models. Our analytical results are found to be in good agreement with the numerical findings.


I. INTRODUCTION
As we know, the phenomenology of conventional superconductors is extremely well explained by Bardeen-Cooper-Schrieffer (BCS) theory [1] and its extensions [2]. However, these theories fail to describe the core mechanism governing the high-temperature superconductor systems which is one of the unsolved mysteries in modern condensed matter physics. Interestingly, the anti-de Sitter/conformal field theories (AdS/CFT) correspondence [3][4][5], which can map strongly coupled non-gravitational physics to a weakly coupled perturbative gravitational problem, might provide some meaningful theoretical insights to understand the physics of high T c superconductors from the gravitational dual [6][7][8][9]. The main idea is that the spontaneous U (1) symmetry breaking by bulk black holes can be used to construct gravitational duals of the transition from normal state to superconducting state in the boundary theory, which exhibits the behavior of the superconductor [10,11].
In additional to the bulk AdS black hole spacetime, it was found that a holographic model can be constructed in the bulk AdS soliton background to describe the insulator and superconductor phase transition [12].
In general, the studies on the gravitational dual models of the superconductorlike transition focus on the vanishing spatial components of the U (1) gauge field on the AdS boundary. Considering that the supercurrent in superconducting materials is a well studied phenomenon in condensed matter systems, the authors of Refs. [13,14] constructed a holographic superfluid solution by performing a deformation of the superconducting black hole, i.e., turning on a spatial component of the gauge field that only depends on the radial coordinate.
It was found that the second-order superfluid phase transition can change to the first order when the velocity of the superfluid component increases relative to the normal component. Interestingly, the holographic superfluid phase transition remains second order for all allowed fractions of superfluid density in the strongly-backreacted regime at low charge q [15]. However, in the case of the fixed supercurrent, the superfluid phase transition is always of the first order for any nonzero supercurrent [16][17][18]. In Ref. [19], the effect of the scalar field mass on the superfluid phase transition was investigated and it was observed that the Cave of Winds exists for some special mass in the superfluid model. In order to explore the effect of the vector field on the superfluid phase transition, a holographic p-wave superfluid model in the AdS black holes coupled to a Maxwell complex vector field was introduced [20,21] and it was revealed that the translating superfluid velocity from second order to first order increases with the increase of the mass squared of the vector field. On the other hand, from the perspective of the QNM analysis, the question of stability of holographic superfluids with finite superfluid velocity was revisited and it was suggested that there might exist a spatially modulated phase slightly beyond the critical temperature [22,23].
The aforementioned works on the holographic superfluid models concentrated on the AdS black hole configuration. More recently, the authors of Refs. [24,25] extended the investigation to the soliton spacetime and investigated numerically the holographic s-wave superfluid model in the AdS soliton background. It was found that, in the probe limit, the first-order phase transition cannot be brought by introducing the spatial component of the vector potential of the gauge field in the AdS soliton background, which is different from the black hole spacetime [25]. In order to back up numerical results and further reveal the properties of the holographic superfluid model in the probe limit, in this work we will use the analytical Sturm-Liouville (S-L) method, which was first proposed in [26,27] and later generalized to study holographic insulator/superconductor phase transition in [28], to analytically investigate the holographic s-wave superfluid model in the AdS soliton background. Considering that the increasing interest in study of the Maxwell complex vector field model [29][30][31][32][33][34][35][36][37][38][39][40], we will also extend the investigation to the holographic p-wave superfluid model in the AdS soliton background, which has not been constructed as far as we know. Besides to be used to check numerical computation, the analytical study can clearly disclose some general features for the effects of the spatial component of the gauge field on the holographic superfluid model in the AdS soliton background.
The structure of this work is as follows. In Sec. II we will investigate the holographic s-wave superfluid model in the AdS soliton background. In particular, we calculate the critical chemical potential of the system as well as the relations of condensed values of operators and the charge density with respect to (µ − µ c ). In Sec. III we extend the discussion to the p-wave case which has not been constructed as far as we know. We will conclude in the last section with our main results.

II. HOLOGRAPHIC S-WAVE SUPERFLUID MODEL
We start with the five-dimensional Schwarzschild-AdS soliton in the form where f (r) = r 2 (1 − r 4 s /r 4 ) with the tip of the soliton r s which is a conical singularity in this solution. We can remove the singularity by imposing a period β = π/r s for the coordinate ϕ. As a matter of fact, this soliton can be obtained from a five-dimensional AdS Schwarzschild black hole by making use of two Wick rotations.
In order to construct the holographic s-wave model of superfluidity in the AdS soliton background, we consider a Maxwell field and a charged complex scalar field coupled via the action where q and m represent the charge and mass of the scalar field ψ respectively. Taking the ansatz of the matter fields as where both a time component A t and a spatial component A ϕ of the vector potential have been introduced in order to consider the possibility of DC supercurrent, we can get the equations of motion in the probe limit where the prime denotes the derivative with respect to r. From the equation of motion for ψ, we can obtain the effective mass of the scalar field which implies that the increasing m 2 and A ϕ or decreasing A t will hinder the s-wave superfluid phase transition.
We will get the consistent result in the following calculation.
In order to solve above equations, we have to impose the appropriate boundary conditions at the tip r = r s and the boundary r → ∞. At the tip r = r s , the fields behave as whereψ i ,Ã ti andÃ ϕi (i = 0, 1, 2, · · · andÃ ϕ0 = 0) are the integration constants, and we have imposed the Neumann-like boundary conditions to render the physical quantities finite [12]. Obviously, we can find a constant nonzero gauge field A t (r s ) at r = r s , which is in strong contrast to that of the holographic superfluid model in the AdS black hole background where A t (r + ) = 0 at the horizon [13,14,25].
At the asymptotic AdS boundary r → ∞, we have asymptotic behaviors where ∆ ± = 2 ± √ 4 + m 2 is the conformal dimension of the scalar operator dual to the bulk scalar field, µ and S ϕ are the chemical potential and superfluid velocity, while ρ and J ϕ are the charge density and current in the dual field theory, respectively. Note that, provided ∆ − is larger than the unitarity bound, both ψ − and ψ + can be normalizable and they will be used to define operators in the dual field theory according to the We can impose boundary conditions that either ψ − or ψ + vanishes [11,41].
Interestingly, from Eq. (4) we can get the useful scaling symmetries where λ is a real positive number. Using these symmetries, we can obtain the transformation of the relevant with i = + or i = −. We can use them to set q = 1 and r s = 1 when performing numerical calculations and check the analytical expressions in this section.
Applying the S-L method to analytically study the properties of the holographic s-wave model of superfluidity in AdS soliton background, we will introduce a new variable z = r s /r and rewrite Eq. (4) into Here and hereafter in this section the prime denotes the derivative with respect to z.

A. Critical chemical potential
It has been shown numerically that [12,42,43], adding the chemical potential to the AdS soliton, the solution is unstable to develop a hair for the chemical potential bigger than a critical value, i.e., µ > µ c . For lower chemical potential µ < µ c , the scalar field is zero and it can be interpreted as the insulator phase since in this model the normal phase is described by an AdS soliton where the system exhibits a mass gap. Therefore, there is a phase transition when µ → µ c and the AdS soliton reaches the superconductor (or superfluid) phase for larger µ.
Before going further, we would like to discuss the phase transition between the AdS soliton and AdS black holes at high chemical potential without the scalar (or vector) field since it is very important for us to understand the phase structure of the holographic dual model in the backgrounds of AdS soliton [12,42].
Considering that the Gibbs Euclidean action of AdS soliton coincides with that of the AdS charged black hole in the grand canonical ensemble, we find that the phase boundary between the AdS black hole and the AdS soliton at zero temperature will be at a chemical potential µ d = 2 1/2 3 1/4 ≃ 1.861 assuming r s = 1, which has been discussed in Refs. [12,42]. Obviously, the AdS soliton solution should be replaced with the AdS black hole at µ = µ c and the superconductor (or superfluid) phase transition gets unphysical if µ c > µ d .
Employing the analysis of the string theory embedding found in [44], the authors of [12] avoided this problem in an explicit string theory setup. In the following discussion, we will accept this way if we were in a similar situation.
At the critical chemical potential µ c , the scalar field ψ = 0. Thus, below the critical point Eq. (11) reduces to which leads to a general solution where c 1 is an integration constant. Obviously, the second term is divergent at the tip z = 1 if c 1 = 0.
Considering the Neumann-like boundary condition (6) for the gauge field A t at the tip z = 1, we have to set c 1 = 0 to keep A t finite, i.e., in this case A t will be a constant. Thus, we can get the physical solution Similarly, from Eq. (12) we have which results in a solution which is consistent with the boundary condition A ϕ (1) = 0 given in (6).
As µ → µ c from below the critical point, the scalar field equation (10) becomes With the boundary condition (7), we assume ψ takes the form where the trial function F (z) obeys the boundary conditions F (0) = 1 and F ′ (0) = 0. From Eq. (17), we arrive at where we have defined According to the Sturm-Liouville eigenvalue problem [45], the minimum eigenvalue of Λ = qµ/r s can be obtained from variation of the following functional where we will assume the trial function to be F (z) = 1 − az 2 with a constant a. When k = 0, Eq.  Table I is impressive. We see that, from Table I and Figs. 1 and 2, the critical chemical potential Λ c = qµ c /r s increases as the dimensionless parameter k = S ϕ /µ increases for the fixed mass of the scalar field, i.e., the critical chemical potential becomes larger with the increase of the superfluid velocity, which indicates that the spatial component of the gauge field to modeling the superfluid hinders the phase transition. This result is consistent with the observation obtained from the effective mass of the scalar field in Eq. (5), which implies that the increasing A ϕ will hinder the s-wave superfluid phase transition.

B. Critical phenomena
Now we are in a position to study the critical phenomena of this holographic s-wave superfluid system.
Considering that the condensation of the scalar operator O i is so small near the critical point, we can where we have introduced the boundary condition χ(1) = 0 at the tip. Defining a function ξ(z) as we obtain the equation of motion for ξ(z) with According to the asymptotic behavior in Eq. (7) and Eq. (23), we will expand A t when z → 0 as From the coefficients of the z 0 term in both sides of the above formula, we have with where c 2 and c 3 are the integration constants which can be determined by the boundary condition of χ(z). Comparing the coefficients of the z 1 term in Eq. (26), we observe that ξ ′ (0) → 0, which agrees well with the following relation by making integration of both sides of Eq. (24) Considering the coefficients of the z 2 term in Eq. (26), we get with a prefactor On the other hand, near the critical point Eq. (12) becomes Thus, we finally arrive at which obeys the boundary condition A ϕ (1) = 0 presented in (6)

III. HOLOGRAPHIC P-WAVE SUPERFLUID MODEL
Since the S-L method is effective to obtain the properties of the holographic s-wave model of superfluidity in the AdS soliton background, we will use it to investigate analytically the holographic p-wave model of superfluidity in the AdS soliton background which has not been constructed as far as we know.
Considering the Maxwell complex vector field model which was first proposed in [29,30], we will build the holographic p-wave model of superfluidity in the AdS soliton background via the action where the tensor ρ µν is defined by ρ µν = D µ ρ ν − D ν ρ µ with the covariant derivative D µ = ∇ µ − iqA µ , q and m are the charge and mass of the vector field ρ µ , respectively. Since we consider the case without external magnetic field in this work, the parameter γ, which describes the interaction between the vector field ρ µ and the gauge field A µ , will not play any role.
As in Refs. [20,21], we take the same ansatz for the gauge field A µ just as in Eq. (3) and assume the condensate to pick out the x direction as special where we can set ρ x (r) to be real by using the U (1) gauge symmetry. Thus, in the soliton background (1), we can obtain the equations of motion for the holographic p-wave superfluid model where the prime denotes the derivative with respect to r. Obviously, we find that the effective mass of the vector field has the same expression just as in (5), which means that the increasing m 2 and A ϕ or decreasing A t will hinder the p-wave superfluid phase transition.
Analyzing the boundary conditions of the matter fields, we observe that A t and A ϕ have the same boundary conditions just as Eq. (6) for the tip r = r s and Eq. (7) for the boundary r → ∞. But for the vector field ρ x , we find that at the tip with the integration constantρ xi (i = 0, 1, 2, · · · ), and at the asymptotic AdS boundary Defining a trial function F (z) which matches the boundary behavior (38) for ρ x [26] ρ with the boundary conditions F (0) = 1 and F ′ (0) = 0, from Eq. (44) we can get the equation of motion for with where V (z) and W (z) have been introduced in (20). Following the S-L eigenvalue problem [45], we deduce the eigenvalue Λ = qµ/r s minimizes the expression where we still assume the trial function to be F (z) = 1−az 2 with a constant a. When the dimensionless parameter k = 0, Eq. (48) reduces to the case considered in [37] for the holographic p-wave insulator/superconductor phase transition, where the spatial component A ϕ has been turned off.
For different values of k and m 2 , we can get the minimum eigenvalue of Λ 2 and the corresponding value of a, for example, Λ 2 min = 7.879 and a = 0.3716 for k = 0.25 with m 2 = 5/4, which leads to the critical chemical potential Λ c = Λ min = 2.807. In Table II, we present the critical chemical potential Λ c = qµ c /r s for chosen k.
In order to compare with numerical results given in Fig. 3, we fix the mass of the vector field by m 2 = 5/4.
Obviously, the analytical results derived from S-L method are in very good agreement with the numerical computations. From Table II, we observe that, for the fixed mass of the vector field, the critical chemical potential Λ c = qµ c /r s becomes larger with the increase of k = S ϕ /µ, i.e., the critical chemical potential increases with the increase of the superfluid velocity. The fact implies that the spatial component of the gauge field to modeling the superfluid hinders the phase transition, which supports the observation obtained from the effective mass of the vector field in Eq. (36).

B. Critical phenomena
Since the condensation of the vector operator O x is so small when µ → µ c , we can expand A t (z) in small with the boundary condition χ(1) = 0 at the tip. Introducing a function ξ(z) as we get the equation of motion for ξ(z) where Q(z) has been defined in (25).
Considering the asymptotic behavior of A t and Eq. (50), near z → 0 we will expand A t as According to the coefficients of the z 0 term in both sides of the above formula, we obtain with where c 2 and c 3 are the integration constants which can be determined by the boundary condition of χ(z). From the coefficients of the z 1 term in Eq. (52), we find that ξ ′ (0) → 0, which is consistent with the following relation by making integration of both sides of Eq. (51) Comparing the coefficients of the z 2 term in Eq. (52), we arrive at with Γ(k, m) = 1 2ξ(0) which is a function of the parameter k and vector field mass m 2 . For the case of k = 0.25 with m 2 = 5/4, as an example, we can obtain ρ = 1.013 (µ − µ c ) when a = 0.3716 (we have scaled q = 1 and r s = 1 for simplicity), which is in good agreement with the result shown in the right panel of Fig. 3. Note that the parameter k and mass of the vector field m 2 will not alter Eq. (56), we can obtain the linear relation between the charge density and chemical potential near µ c , i.e., ρ ∼ (µ − µ c ), which can be used to back up the numerical result presented in the right panel of Fig. 3.
Similarly, considering Eq. (43) near the phase transition point, i.e., we can solve it and get which is consistent with the boundary condition A ϕ (1) = 0 at the critical point. For example, for the case of k = 0.25 with m 2 = 5/4, we have A ϕ = S ϕ [(1 − z 2 ) − 0.02450 O x 2 z 2 + · · ·] when a = 0.3716 (we have scaled q = 1 and r s = 1 for simplicity), which supports our numerical computation.

IV. CONCLUSIONS
We have applied the S-L method to study analytically the properties of the holographic superfluid models in the AdS soliton background in order to understand the influence of the spatial component of the gauge field on the superfluid phase transition. By investigating the s-wave (the scalar field) and p-wave (the vector field) models in the probe limit, we obtained analytically the critical chemical potentials which are perfectly in agreement with those obtained from numerical computations. We observed that the critical chemical potential i.e., the holographic superfluid phase transition always belongs to the second order and the critical exponent of the system takes the mean-field value 1/2 in both s-wave and p-wave models. The analytical results can be used to back up the numerical findings in both holographic s-wave [25] and p-wave superfluid models in the AdS soliton background. Since the superfluid velocity provides richer physics in the superfluid phase transition in the AdS black hole background [13,14], it would be of interest to generalize our study to the AdS black hole configuration and analytically discuss the effect of the spatial component of the gauge field on the system. We will leave it for further study.