Holographic superconductors in 4D Einstein-Gauss-Bonnet gravity

We investigate the neutral AdS black-hole solution in the novel four-dimensional Einstein-Gauss-Bonnet gravity proposed in [D. Glavan and C.S. Lin, Phys. Rev. Lett. {\bf 124}, 081301 (2020)] and construct the gravity duals of ($2+1$)-dimensional superconductors with Gauss-Bonnet corrections in the probe limit. We find that the curvature correction has a more subtle effect on the scalar condensates in the s-wave superconductor in ($2+1$)-dimensions, which is different from the finding in the higher-dimensional superconductors that the higher curvature correction makes the scalar hair more difficult to be developed in the full parameter space. However, in the p-wave case, we observe that the higher curvature correction always makes it harder for the vector condensates to form in various dimensions. Moreover, we note that the higher curvature correction results in the larger deviation from the expected relation in the gap frequency $\omega_g/T_c\approx 8$ in both ($2+1$)-dimensional s-wave and p-wave models.


I. INTRODUCTION
In condensed matter physics, the mechanism leading to superconductivity with high critical temperature, which can not be described by the conventional Bardeen-Cooper-Schrieffer (BCS) theory [1], is still unclear.
However, recently developed holographic superconductor models, have been used to give some insights into the pairing mechanism in the high-temperature superconductor systems [2][3][4][5]. According to the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, which relates strongly coupled systems to weakly coupled systems [6], Gubser presented the spontaneous symmetry breaking of the U (1) symmetry for an Abelian Higgs model coupled to the gravity theory with a negative cosmological constant [7], and Hartnoll et al. reproduced the properties of a (2 + 1)-dimensional s-wave superconductor in the (3 + 1)-dimensional holographic superconductor model based on the framework of usual Maxwell electrodynamics [8]. To go a step further, the p-wave holographic superconductivity was realized by introducing an SU (2) Yang-Mills field into the bulk [9] while the d-wave holographic superconductor was constructed by introducing a charged massive spin two field propagating in the bulk [10,11]. Especially, introducing a charged vector field into an Einstein-Maxwell theory [12], Cai et al. developed a novel p-wave holographic superconductor model which is a generalization of the SU (2) model with a general mass and gyromagnetic ratio [13]. Considering the application of the Mermin-Wagner theorem to the holographic superconductors, Gregory et al. constructed holographic superconductors in the five-dimensional Einstein-Gauss-Bonnet gravity in the probe limit where the backreaction of matter fields on the spacetime metric is neglected [14]. It was observed that the higher curvature correction makes the scalar condensates harder to form and causes the behavior of the claimed universal ratio ω/T c ≈ 8 [15] unstable. Note that this conclusion still holds in higher dimensions [16] and beyond the probe limit [17][18][19]. Other generalized investigations based on the effects of the curvature correction on the holographic dual models can be found, for example, in Refs. [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].
The plan of the work is the following. In Sec. II we will present the 4D neutral AdS black-hole solution in the Einstein-Gauss-Bonnet gravity by rescaling the Gauss-Bonnet coupling constant α → α/(d − 4) and taking the limit d → 4 as in [36], which can also be obtained from Ref. [38] by setting the charge Q = 0.
In Sec. III we will construct the s-wave holographic superconductors in the 4D Gauss-Bonnet gravity and investigate the effect of the curvature correction on the (2 + 1)-dimensional superconductors. In Sec. IV we will explore the p-wave cases. We will conclude in the last section with our main results.

II. 4D GAUSS-BONNET-ADS BLACK HOLES
Rescaling the Gauss-Bonnet coupling constant α → α/(d − 4), we can rewrite the Einstein-Hilbert action with a Gauss-Bonnet term in d dimensions [90] as where Λ = −(d − 1)(d − 2)/(2L 2 ) is the negative cosmological constant. Taking the metric ansatz for the d-dimensional topological black hole where h ij dx i dx j denotes the line element of a (d − 2)-dimensional hypersurface with constant curvature (d − 2)(d − 3)k and volume Σ k , we obtain the simple form of Einstein-Hilbert action [91] where ϕ = r −2 (k − e −2λ ) is the function of r only, and the prime denotes the derivative with respect to r.
Thus, from this action we have Obviously, we can get the exact solution for the limit d → 4 where k = 1, −1 or 0 without loss of the generality. Obviously, it coincides with the black-hole solution in the regularized 4D Einstein-Gauss-Bonnet gravity proposed in Refs. [79,82]. It should be noted that the charged AdS solution in the 4D Einstein-Gauss-Bonnet gravity, which has the same form with that obtained in a conformal anomaly gravity [39], was obtained in [37,38]. Since we can recover the Schwarzschild-AdS black hole in the limit α → 0, we are more interested in the physically interesting solution with the minus sign.
In order to construct a superconductor dual to an AdS black hole configuration in the probe limit, in this work we consider the background of the 4D planar Gauss-Bonnet-AdS black hole where the black hole horizon r + is related to the mass M through r 3 + = 8πGM L 2 /Σ 0 with the volume of the Ricci flat space Σ 0 . It should be noted that in the asymptotic region, i.e., r → ∞, we find f (r) ∼ where the so-called Chern-Simons limit α = L 2 /4 [92] is the upper bound of the Gauss-Bonnet parameter. For simplicity, in the following we will consider the Gauss-Bonnet parameter in the range −L 2 /10 ≤ α ≤ L 2 /4.
We can easily obtain the Hawking temperature of this 4D Gauss-Bonnet-AdS black hole which will be interpreted as the temperature of the CFT. For convenience, we will scale L = 1 in the following calculation.

III. S-WAVE HOLOGRAPHIC SUPERCONDUCTORS
We want to know the influence of the curvature correction on the (2 + 1)-dimensional superconductors since we have rescaled the Gauss-Bonnet coupling constant α → α/(d − 4). In this section, we first introduce the s-wave holographic superconductor models in the 4D Gauss-Bonnet-AdS black hole.

A. Condensates of the scalar field
We consider a Maxwell field and a charged complex scalar field coupled via the action Adopting the ansatz for the matter fields ψ = |ψ|, A t = φ where ψ, φ are both real functions of r only, we arrive at the equations of motion where the prime denotes differentiation in r.
To get the solutions in the superconducting phase, i.e., ψ(r) = 0, we have to count on the appropriate boundary conditions. At the horizon r = r + , the regularity leads to the boundary conditions ψ(r + ) = f ′ (r + )ψ ′ (r + )/m 2 and φ(r + ) = 0. Near the AdS boundary r → ∞, the asymptotic behaviors of the solutions with the characteristic exponents where µ and ρ are interpreted as the chemical potential and charge density in the dual field theory, respectively.
Considering the stability of the scalar field, we observe that the mass should be above the Breitenlohner-Freedman bound m 2 BF = −9/(4L 2 eff ) [93], which depends on the Gauss-Bonnet parameter α. From the AdS/CFT correspondence, provided λ − is larger than the unitarity bound, both ψ − and ψ + can be normalizable and correspond to the vacuum expectation values O − = ψ − , O + = ψ + of an operator O dual to the scalar field, respectively. We will impose boundary condition that either ψ + or ψ − vanishes since imposing boundary conditions in which both ψ − and ψ + are non-zero makes the asymptotic AdS theory unstable [8].
It is interesting to note that, from Eqs. (10) and (11), we can get the useful scaling symmetries and the transformation of the relevant quantities where β is a real positive number. We will make use of these properties to set r + = 1 and q = 1 when performing numerical calculations.

Scalar operator O+
Using the shooting method [8], we can solve numerically the equations of motion (10) and (11)  just as in the cases of d ≥ 5 [14,16]. Obviously, we have to reinvestigate the issue related to the different choices of the mass of the scalar field for the (2 + 1)-dimensional Gauss-Bonnet superconductors.
In Fig. 1, we present the condensates of the scalar operator O + as a function of temperature with various Gauss-Bonnet parameters α for the fixed masses of the scalar field m 2 L 2 = −2 and m 2 L 2 eff = −2. It is found that, the curves have similar behavior to the BCS theory for different α, where the condensate goes to a constant at zero temperature, which indicates that the s-wave holographic superconductors still exist even we consider Gauss-Bonnet correction terms to the standard (2 + 1)-dimensional holographic superconductor model built in [8]. Fitting these curves for small condensate, we observe that there is a square root behavior 1/2 for all values of α. Furthermore, the left panel of Fig. 1 shows that the higher correction term α makes the condensation gap larger for the operator O + , which implies the higher Gauss-Bonnet correction will make the scalar hair more difficult to be developed. However, fixing the scalar field mass by m 2 L 2 eff = −2, we can see clearly from the right panel of Fig. 1 that there is an abnormal behavior in the Chern-Simons limit α = 0.25, which is completely different from that of the (3 + 1)-dimensional Gauss-Bonnet superconductors [14,16]. This means that the different choices of the mass of the scalar field will modify the effect of α on the behavior of the condensates even for the operator O + .  We give in Fig. 3 the condensates of the scalar operator O − as a function of temperature with various Gauss-Bonnet parameters α for the fixed masses m 2 L 2 = −2 and m 2 L 2 eff = −2, which will diverge at low temperatures, similar to that for the standard holographic superconductor model in the probe limit neglecting backreaction of the spacetime [8]. More importantly, from this figure, we find that the condensate of the operator O − for different choices of the mass of the scalar field has completely different behavior as the Gauss-Bonnet parameter is changing, which is in agreement with the result obtained in the Fig. 4, where the Gauss-Bonnet term has completely different effects on the critical temperature T c for m 2 L 2 = −2 and m 2 L 2 eff = −2.
Considering the correct consistent influence due to the Gauss-Bonnet parameter in various condensates for all dimensions [16,30] and the pure effect of the curvature correction on the critical temperature T c for the fixed mass m 2 L 2 = m 2 L 2 eff = 0 in Fig. 2, i.e., the increase of α results in the decrease of T c , we argue that it is more appropriate to choose the mass of the scalar field by selecting the values of m 2 L 2 eff for the Gauss-Bonnet superconductors even in (2 + 1)-dimensions. Therefore, the higher curvature corrections make it harder for the condensate of the scalar operator O − to form.

B. Conductivity
In order to calculate the conductivity, we consider the perturbed Maxwell field δA x = A x (r)e −iωt dx, which results in the equation of motion With the ingoing wave boundary condition near the horizon A x (r) ∼ f (r) −iω/(3r+) , and the behavior in the asymptotic AdS region from Ohm's law we can obtain the conductivity of the dual superconductor [8] We will focus on the case of the scalar operator O + for the fixed scalar field masses m 2 L 2 eff = −2 and 0, and study the effect of the Gauss-Bonnet correction term on the conductivity by solving Eq. (15) numerically. frequency ω g which can be defined as the frequency minimizing Im[σ(ω)] for all cases considered here [15].
For the same mass of the scalar field, we observe that with the increase of the Gauss-Bonnet parameter α, the gap frequency ω g becomes larger, just as in the (3 + 1)-dimensional Gauss-Bonnet superconductors [14,16].
Also, for increasing absolute value of α, we have larger deviations from the value ω g /T c ≈ 8, especially in the case of the Chern-Simons limit α = 0.25 where ω g /T c ≈ 13.0 for m 2 L 2 eff = −2 and ω g /T c ≈ 14.5 for m 2 L 2 eff = 0. This shows that the higher curvature corrections really change the universal relation in the gap frequency ω g /T c ≈ 8 [15] for (2 + 1)-dimensional superconductors.

IV. P-WAVE HOLOGRAPHIC SUPERCONDUCTORS
In the previous section, we have investigated the influence of the curvature correction on the (2 + 1)dimensional s-wave superconductors, which shows that the Gauss-Bonnet terms have more subtle effects on the condensates of the scalar field and change the ratio in the gap frequency ω g /T c ≈ 8. In this section, we will construct the p-wave holographic superconductors in the 4D Gauss-Bonnet-AdS black hole and analyze the influence of the curvature correction on these models.

A. Condensates of the vector field
We start with the action of the Einstein-Maxwell-complex vector model [12,13] where m and q represent the mass and charge of the vector field ρ µ , and D µ = ∇ µ − iqA µ is the covariant derivative. The last term, which describes the interaction between the vector field ρ µ and gauge field A µ , will not play any role since we will only consider the case without an external magnetic field. Taking the ansatz for the matter fields ρ ν dx ν = ρ x (r)dx and A ν dx ν = A t (r)dt [12,13], we obtain the following equations of where the prime denotes the derivative with respect to r.
We impose the appropriate boundary conditions for ρ x (r) and A t (r) to get the solutions in the superconducting phase. At the horizon, we can easily get the boundary conditions ρ x (r + ) = f ′ (r + )ρ ′ x (r + )/m 2 and A t (r + ) = 0. At the asymptotic boundary, we have with the characteristic exponents where ρ x− and ρ x+ are interpreted as the source and the vacuum expectation value of the vector operator O x in the dual field theory according to the AdS/CFT correspondence, respectively. It should be noted that the mass has a lower bound m 2 BF = −1/(4L 2 eff ) with λ + = λ − = 1/2. Since we require that the condensate appears spontaneously, we will impose the boundary condition ρ x− = 0 and use λ to denote λ + in the following analysis.
Considering that Eqs. (19) and (20) are invariant with respect to the following scaling transformations with a positive number β. In what follows, we will use them to set r + = 1 and q = 1, just as in the previous section. well with the result given in the Fig. 7, where the critical temperature T c decreases as α increases for all vector field masses chosen by fixing m 2 L 2 and m 2 L 2 eff . This means that the higher curvature corrections make it harder for the vector field to condense in the (2 + 1)-dimensional superconductors, which is quite different from the findings obtained in Fig. 1 for the s-wave superconductors where the different choices of the mass of the scalar field will modify the effect of the Gauss-Bonnet corrections on the behavior of the condensates.

B. Conductivity
Considering the perturbed Maxwell field δA y = A y (r)e −iωt dy, we obtain the equation of motion which has the ingoing wave boundary condition near the horizon A y (r) ∼ f (r) −iω/(3r+) , and the general behavior near the asymptotic AdS boundary Using the AdS/CFT dictionary, we get the conductivity [12] For clarity, we choose the masses of the vector field m 2 L 2 eff = 3/4 and 0 in our calculation. 5 and 8, our study implies that the higher curvature correction results in the larger deviation from the expected universal relation in the gap frequency ω g /T c ≈ 8 [15] in both (2 + 1)-dimensional s-wave and p-wave superconductor models.

V. CONCLUSIONS
In this work, we first investigated the 4D neutral AdS black-hole solution in the Einstein-Gauss-Bonnet gravity by rescaling the Gauss-Bonnet coupling constant α → α/(d − 4) and taking the limit d → 4 just as in Ref. [36], which can also be obtained from Ref. [38] by setting the charge Q = 0. Then, we constructed the s-wave and p-wave holographic superconductors in this Gauss-Bonnet black hole background and analyzed the effect of the curvature correction on the (2 + 1)-dimensional superconductors, which may help to understand the influences of the 1/N or 1/λ (with λ being the 't Hooft coupling) corrections on the holographic superconductor models. In the probe limit, we found that it is more appropriate to choose the mass of the field by selecting the value of m 2 L 2 eff , since this choice can disclose the correct consistent influence due to the Gauss-Bonnet parameter in various condensates for all dimensions. In the s-wave model, although the underlying mechanism remains mysterious, the effect of the curvature correction is more subtle: the critical temperature first decreases then increases as the Gauss-Bonnet parameter tends towards the Chern-Simons value in a scalar mass dependent fashion for the scalar operator O + , but always decreases for the scalar operator O − , which exhibits a very interesting and different feature when compared to the higher-dimensional Gauss-Bonnet superconductors. In the p-wave model, we noted that the critical temperature decreases as the Gauss-Bonnet parameter increases, which tells us that the higher curvature correction makes it harder for the vector hair to form in the full parameter space. On the other hand, for all cases considered here, we pointed out that the curvature correction will not modify the critical phenomena, i.e., the holographic superconductor phase transition belongs to the second order and the critical exponent of the system takes the mean-field value. Moreover, we observed that the higher curvature correction results in the larger deviation from the expected universal relation in the gap frequency ω g /T c ≈ 8 in both (2 + 1)-dimensional s-wave and p-wave superconductors, which is the same to the finding obtained in (3 + 1)-dimensions.