# Holographic conductivity of holographic superconductors with higher-order corrections

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## Abstract

We analytically and numerically disclose the effects of the higher-order correction terms in the gravity and in the gauge field on the properties of *s*-wave holographic superconductors. On the gravity side, we consider the higher curvature Gauss–Bonnet corrections and on the gauge field side, we add a quadratic correction term to the Maxwell Lagrangian. We show that, for this system, one can still obtain an analytical relation between the critical temperature and the charge density. We also calculate the critical exponent and the condensation value both analytically and numerically. We use a variational method, based on the Sturm–Liouville eigenvalue problem for our analytical study, as well as a numerical shooting method in order to compare with our analytical results. For a fixed value of the Gauss–Bonnet parameter, we observe that the critical temperature decreases with increasing the nonlinearity of the gauge field. This implies that the nonlinear correction term to the Maxwell electrodynamics makes the condensation harder. We also study the holographic conductivity of the system and disclose the effects of the Gauss–Bonnet and nonlinear parameters \(\alpha \) and *b* on the superconducting gap. We observe that, for various values of \(\alpha \) and *b*, the real part of the conductivity is proportional to the frequency per temperature, \(\omega /T\), as the frequency is large enough. Besides, the conductivity has a minimum in the imaginary part which is shifted toward greater frequency with decreasing temperature.

## 1 Introduction

The correspondence between the gravity in a *d*-dimensional anti-de Sitter (AdS) spacetime and the conformal field theory (CFT) residing on the (\(d-1\))-dimensional boundary of this spacetime, known as the AdS/CFT correspondence, provides an established method for calculating correlation functions in a strongly interacting field theory using a dual classical gravity description [1, 2, 3]. It has been confirmed that this duality can be applied to solving the problem of high temperature superconductors in condensed matter physics [4]. This is due to the fact the high temperature superconductors are basically in a strong coupling regime, and thus one expects that the holographic method could give some insights into the pairing mechanism in these systems. Understanding the mechanism of high temperature superconductors has long been a mysteries problem in modern condensed matter physics. Recently, it was suggested that it is logical to understand the properties of high temperature superconductors on the boundary of spacetime by considering classical general relativity in one higher dimension. This idea is called holographic superconductors (HSCs) [5, 6, 7] and has got a lot of attention in the past decade. According to the HSC proposal, on the gravity side, a Maxwell field and a charged scalar field are introduced to describe the *U*(1) symmetry and the scalar operator in the dual field theory, respectively. This holographic model undergoes a phase transition from a black hole with no hair (normal phase/conductor phase) to the case with scalar hair at low temperatures (superconducting phase) [8].

Nowadays, the investigations on the HSC have attracted considerable attention and have become an active field of research. Let us review some work in this direction. In the background of Schwarzschild AdS black holes in Einstein gravity, the properties of HSCs have been explored in [9, 10, 11, 12, 13, 14, 15, 16]. The studies were also generalized to higher-order gravity theories such as Gauss–Bonnet gravity [17, 18, 19, 20, 21, 22, 23]. It was argued that the critical temperature of the HSC decreases with increasing the backreaction, although the effect of the Gauss–Bonnet coupling is more subtle: the critical temperature first decreases then increases as the coupling tends towards the Chern–Simons value in a backreaction dependent fashion [20]. It was confirmed that the critical exponent of the condensation in Gauss–Bonnet HSCs still obeys the mean field theory and has the value 1 / 2 [19]. Other studies on the holographic superconductor have been carried out (see for example [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49] and the references therein).

It is also interesting to investigate the electrical conductivity of HSC in the dual CFT as a function of frequency. In the AdS/CFT correspondence, the electrical conductivity can be computed by looking at the linear response of the system to fluctuations of the fields \(A_{x}\) and \(g_{tx}\) in the bulk. These fluctuations are dual to the electric current \(J_{x}\) and energy current \(T_{tx}\) operators in the CFT. In the context of linear Maxwell field, the conductivity of HSC was computed in [4, 6, 7]. In the presence of nonlinear electrodynamics, the conductivity of HSC has been investigated in [50, 51]. Also, in the context of Born–Infeld nonlinear electrodynamics, the optical properties of Lifshitz HSC have been explored in [52]. It was demonstrated that this superconductor exhibits a metamaterial property for low frequency of the external electric field for a certain region of the nonlinear parameter. The effects of the Weyl coupling parameter and the Lifshitz dynamic exponent on the conductivity of HSC have been explored in [53]. In Ref. [54], rotating BTZ black holes were considered as the gravity dual to a \((1+1)\) dimensional superconductor. In this case, the dependence on the angular momentum on the conductivity has been investigated. Recently, the authors of [55] have analytically computed the holographic conductivity of HSCs in the presence of Born–Infeld nonlinear electrodynamics by considering the backreaction of the matter field on the bulk metric. Further investigations on the holographic conductivity of HSCs have been performed in [56, 57].

In this work, we will address the effects of the higher-order corrections on the holographic conductivity of the *s*-wave HSC. On the gravity side, we will consider the Gauss–Bonnet curvature correction terms which is the most general action in the 5D spacetime and on the gauge field side we add the quadratic nonlinear gauge term. We shall investigate the effects of these correction terms on the imaginary and real parts of the electrical conductivity of the system. With these correction terms, especially including a Gauss–Bonnet correction to the 5D action, we have the most general action with second-order field equations in 5D [58], which provides the most general model for a *s*-wave HSC. Furthermore, in an effective action approach to string theory, the Gauss–Bonnet term corresponds to the leading-order quantum corrections to gravity, and its presence guarantees a ghost-free action [59]. The purpose of this work is to analytically and numerically explore the effects of these correction terms on the properties of the *s*-wave HSC.

The plan of the work is as follows. In Sect. 2, we set up our model of the HSC in Gauss–Bonnet gravity with nonlinear electrodynamics in the probe limit and derive the equations of motion. In Sect. 3, we analytically and numerically compute the relationship between the critical temperature and the charge density of Gauss–Bonnet HSC. In Sect. 4, we study condensation operator near the critical temperature using analytical and numerical method. In Sect. 5 we investigate the electrical conductivity of the HSC in Gauss–Bonnet gravity with a nonlinear correction term to the Maxwell field. In particular, we shall find the ratio of the gap frequency in conductivity to the critical temperature. Finally, we summarize and discuss our results in Sect. 6.

## 2 HSC in Gauss–Bonnet gravity with nonlinear electrodynamics

*l*, \(\alpha \) is the Gauss–Bonnet coefficient with dimension \((\mathrm{length})^2\), \(R_{\mu \nu \rho \sigma }\), \(R_{\mu \nu } \) and

*R*are the Riemann curvature tensor, Ricci tensor, and the Ricci scalar, respectively. For convenience, hereafter we set the AdS radius \(l=1\). We consider the Lagrangian density of the matter field, \(L_{M}\), as

*q*and

*m*are, respectively, the charge and the mass of the scalar field, and the Lagrangian density of the nonlinear electrodynamics is given by [61, 62]

*b*is a parameter. The term \(b\mathcal {F}^{2}\) is the first-order leading nonlinear correction term to the Maxwell field. There are several motivations for choosing the nonlinear Lagrangian in the form of (3). First, the series expansion of the three well-known Lagrangians of nonlinear electrodynamics such as Born–Infeld, logarithmic and exponential nonlinear electrodynamics have the form of (3) [63]. Second, calculating a one-loop approximation of QED, it was shown [64] that the effective Lagrangian is given by (3). Besides, if one neglects all other gauge fields, one may arrive at the effective quadratic order of

*U*(1) as \(\mathcal {F}^{2}\) [65, 66]. Furthermore, considering the next-order correction terms in the heterotic string effective action one can obtain the \(\mathcal {F}^{2}\) term as a correction to the bosonic sector of supergravity, which has the same order as the Gauss–Bonnet term [65, 66, 67, 68],

## 3 Relation between critical temperature and charge density

In this section, we are going to study the critical temperature of HSC, when the higher-order corrections to the gravity side and the gauge field are taken into account. We shall continue our study both analytically and numerically and compare the results of the two methods.

### 3.1 Analytical method

*r*to

*z*, such that \(z={r_{+}}/{r}\). Using these new coordinates, the equations of motion (15) and (16) can be rewritten as

*b*, we find

*F*(

*z*) is the trial function near the boundary \(z=0\), which satisfies the boundary conditions \(F(0)=1\) and \(F^{\prime }(0)=0\). Substituting Eqs. (23) and (25) in Eq. (21) one arrives at

*z*, and

*p*(

*z*),

*q*(

*z*) and

*w*(

*z*) read

*b*such that

*a*by fixing other parameters of the model such as

*b*and \(\alpha \). Then we obtain \(\lambda _{{\mathrm {min}}}\) and hence the maximum value of \(T_{c}/\rho ^{1/3}\) can be deduced through Eq. (37). As an example, we present the details of our calculation for \(\alpha =0.01\) and \(b=0.01\). In this case Eq. (36) reduces to

*a*and

*b*. This table shows that, for a small and fixed value of \(\alpha \), with increasing the nonlinear parameter

*b*, the value of \(\zeta =T_{c}/\rho ^{1/3}\) decreases as well. As we shall see in the next section this result is in a very good agreement with the numerical results.

Comparison of analytical and numerical values of \(\zeta =T_{c}/\rho ^{1/3}\) for \(\alpha =0.0001\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\zeta _{SL}\left( =\frac{1}{\pi \lambda _{{\mathrm {min}}}^{1/3}}\right) \) | \(\zeta _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.721772 | 18.2331 | 0.196204 | 0.197957 |

0.01 | 0.747560 | 25.0682 | 0.186064 | 0.181057 |

0.02 | 0.777479 | 36.1392 | 0.175060 | 0.165642 |

0.03 | 0.809805 | 55.2025 | 0.163126 | 0.151671 |

Comparison of analytical and numerical values of \(\zeta =T_{c}/\rho ^{1/3}\) for \(\alpha =0.01\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\zeta _{SL}\left( =\frac{1}{\pi \lambda _{{\mathrm {min}}}^{1/3}}\right) \) | \(\zeta _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.720561 | 18.5392 | 0.195660 | 0.196843 |

0.01 | 0.747087 | 25.6427 | 0.185363 | 0.179433 |

0.02 | 0.777900 | 37.2577 | 0.174173 | 0.163592 |

0.03 | 0.811046 | 57.4907 | 0.162026 | 0.149279 |

Comparison of analytical and numerical values of \(\zeta =T_{c}/\rho ^{1/3}\) for \(\alpha =0.1\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\zeta _{SL}\left( =\frac{1}{\pi \lambda _{{\mathrm {min}}}^{1/3}}\right) \) | \(\zeta _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.709061 | 21.5679 | 0.190787 | 0.186114 |

0.01 | 0.743348 | 31.6982 | 0.178928 | 0.162657 |

0.02 | 0.783517 | 49.9342 | 0.165876 | 0.141983 |

0.03 | 0.823574 | 85.6808 | 0.151601 | 0.124005 |

### 3.2 Numerical method

We calculate \(\phi ^{^{\prime \prime }}(1)\), \(\psi ^{^{\prime }}(1)\) and \(\psi ^{^{\prime \prime }}(1)\) in the term of \(\psi (1)\) and \(\phi ^{^{\prime }}(1)\) by using the equations of motion for \(\phi \) and \(\psi \), namely Eqs. (39) and (40), respectively. Since near the critical point \(\psi \) is very small, thus we choose \(\psi (1)=0.0001\). Our strategy for using the shooting method is as follows. For a specific value of the reduced scalar field mass \(\tilde{m}^{2}\), we can perform a numerical calculation near the horizon boundary with one shooting parameter \(\phi ^{^{\prime }}(1)\) to get proper solutions at the infinite boundary. For specific values of \(\phi ^{^{\prime }}(1)\), we impose the boundary condition \(\psi _{-}=0\). We also calculate the analytical values and numerical values of \(\zeta \) for different *b*. We compare our numerical results with analytical results in Tables 1, 2 and 3.

In Fig. 1, we plot \(\psi \) versus *z* for three first boundary condition \(\phi ^{^{\prime }}(1)\), \(\tilde{m}^{2}=-3\) and different values of the Gauss–Bonnet coefficient \(\alpha \) and nonlinear parameter *b*. In the absence of a quadratic correction term (\(b=0\)), our results exactly coincide with those presented in [19, 24]. The acceptable diagram for us is the red one in each plot, since there is nothing in the bulk to have effect on the speed of the wave, so the diagram of \(\psi \) will be stable. From Tables 1, 2 and 3, it is evident that when *b* becomes larger the condensation gets harder. A similar behavior can be seen for the fixed value of *b* and different values of \(\alpha \), namely the critical temperature reduces and condensation becomes harder when the Gauss–Bonnet coupling parameter \(\alpha \) gets larger.

## 4 Critical exponent and condensation values

In this section, our aim is to calculate the critical exponent of HSC with first-order correction terms in gravity and gauge field. Again, we continue our studying both analytically and numerically.

### 4.1 Analytical method

*b*and Gauss–Bonnet parameter \(\alpha \). It is worth noting that \(\langle \mathcal {O} \rangle \) is zero at \(T = T_{c}\) and condensation occurs for \(T < T_{c}\). We shall come back to calculation of the condensation value \(\gamma \) in the next subsection.

### 4.2 Numerical method

We use the numerical method to explore the behavior of the condensate operator \(\langle \mathcal {O}_{+}\rangle \) in terms of temperature for different values of \(\alpha \) and *b* (see Fig. 2). These curves are obtained by the shooting method which we described in the previous section. As one can see from this figure there is a critical temperature \(T_{c}\) below which the condensate appears, then it increases quickly as the system is cooled and finally goes to a constant for sufficiently low temperatures. This behavior is qualitatively similar to that obtained in BCS theory and observed in many materials.

*b*and \(\alpha \). We see that, for a fixed value of \(\alpha \), the condensation operator \(\gamma \) increases with increasing

*b*, while for a fixed value of

*b*, it decreases with increasing \(\alpha \).

## 5 Holographic conductivity

The analytical and numerical results for the condensation operator for \(\alpha =0.0001\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\gamma _{SL}\) | \(\gamma _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.721772 | 18.2331 | 7.70525 | 12.4879 |

0.01 | 0.747560 | 25.0682 | 9.05565 | 17.4138 |

0.02 | 0.777479 | 36.1392 | 14.4103 | 24.5663 |

The analytical and numerical results for the condensation operator for \(\alpha =0.01\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\gamma _{SL}\) | \(\gamma _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.720561 | 18.5392 | 7.72419 | 11.2411 |

0.01 | 0.747087 | 25.6427 | 9.44846 | 15.9088 |

0.02 | 0.777900 | 37.2577 | 15.6715 | 23.5892 |

*K*is also a constant parameter with length dimension which is considered so as to have a dimensionless logarithmic argument.

The analytical and numerical results for the condensation operator for \(\alpha =0.1\)

| | \(\lambda _{{\mathrm {min}}}^{2} \) | \(\gamma _{SL}\) | \(\gamma _{{\mathrm {Numerical}}}\) |
---|---|---|---|---|

0 | 0.709061 | 21.5679 | 7.90303 | 4.05071 |

0.01 | 0.743348 | 31.6982 | 9.81189 | 10.0376 |

0.02 | 0.783517 | 49.9342 | 37.6568 | 17.7555 |

*T*is the Hawking temperature and

The numerical results for holographic conductivity associated with the condensation operator \(<\mathcal {O}_{+}>\) are plotted in Figs. 4, 5, 6 and 7. The real and imaginary parts of the electrical conductivity versus frequency are illustrated at different temperatures below \(T_c\) in Figs. 4 and 5, respectively. As one can see from Fig. 4, the superconducting gap has opened below the critical temperature, which became deeper with decreasing the temperature. Besides, for various values of \(\alpha \) and *b*, the real part of the conductivity is proportional to the frequency per temperature, \(\omega /T\), as the frequency is large enough. According to Fig. 5 the divergence in the imaginary part, at \(\omega =0\), points out a delta function in the real part, \(Re[\sigma ]\), at \(\omega =0\), which is not displayed in Fig. 4. As one can see from Fig. 5, the holographic conductivity of the HSC has a minimum in the imaginary part. Thus with decreasing the temperature the minimum in the imaginary part shifts toward greater frequency for various values of the Gauss–Bonnet and nonlinear parameters.

To study the formation of the superconducting gap with changing \(\alpha \) and *b* at low temperature, e.g., \(T\approx 0.1T_c\), the real and the imaginary parts of holographic conductivity as a function of \(\omega /{<\mathcal {O}_{+}>}^{1/3}\) are plotted in Figs. 6 and 7, respectively. For a fixed value of the Gauss–Bonnet coefficient \(\alpha \), the energy gap (\(\omega /{<\mathcal {O}_{+}>}^{1/3}\)) enlarges with increasing nonlinear parameter *b*. It is evident from Figs. 6 and 7 that the energy gap of HSC for various \(\alpha \) exhibits different behaviors based on the nonlinear correction *b*. In the case of the Maxwell field (\(b=0\)), the superconducting energy gap decreases as \(\alpha \) increases at low temperature (see Fig. 6d). When we take into account the nonlinear correction *b*, the energy gap of the HSC increases with increasing \(\alpha \) (see Fig. 6d, g). From Fig. 7, we see that, for a fixed value of \(\alpha \), the minimum of the imaginary part of the conductivity goes to the larger value of \(\omega /<\mathcal {O}_{+}>^{1/3}\) when *b* increases. In the absence of a correction (\(b=0\)), it decreases with enhancing Gauss–Bonnet coefficient (Fig. 6d). Besides, for a fixed value of *b*, the minimum of \(\mathrm {Im[\sigma ]}\) increases with increasing \(\alpha \), while for a fixed value of \(\alpha \), it increases with increasing *b*.

## 6 Conclusions

In this paper, we continue the studies on the *s*-wave holographic superconductors (HSC) by taking into account the higher correction terms both on the gravity side and on the gauge field side of the system. We considered the Gauss–Bonnet HSC when the Maxwell Lagrangian has a nonlinear correction term and is written in the form \(\mathcal {L}=\mathcal {F}+b \mathcal {F}^2\), where \(\mathcal {F}\) is the Maxwell Lagrangian. We have provided several motivations for choosing this kind of Lagrangian for the gauge field. For example, all well-known nonlinear Lagrangians have a series expansion of which the first two terms are exactly in the above form.

First, we have analytically and numerically investigated the relation between the critical temperature of the phase transition and charge density which depends on both the Gauss–Bonnet parameter \(\alpha \) and the nonlinear parameter *b*. For this purpose, we employed the analytical Sturm–Liouville eigenvalue problem and the numerical shooting method. We find that, for a fixed value of \(\alpha \), with increasing nonlinear parameter *b*, the value of \(T_{c}/\rho ^{1/3}\) decreases. This implies that when *b* becomes larger, the condensation gets harder. A similar behavior can be seen for the fixed value of *b* and different values of \(\alpha \), namely the critical temperature decreases and the condensation becomes harder when the Gauss–Bonnet coupling parameter \(\alpha \) gets larger. We confirmed that these results are in a very good agreement with our numerical results. Then we obtained the critical exponent of the Gauss–Bonnet HSC with a nonlinear gauge field. We observed that the critical exponent has the mean field value 1 / 2, which is independent of the nonlinear parameter *b* and the Gauss–Bonnet parameter \(\alpha \).

Then we explored, numerically, the holographic conductivity of the system. For this purpose, we plotted the real and imaginary parts of the electrical conductivity versus \(\omega /T\) and \(\omega /<\mathcal {O}_{+}>^{1/3}\) for \(T<T_c\). We observed that the superconducting gap has opened below the critical temperature, which became deeper with decreasing the temperature. Interestingly enough, we found that, for different values of \(\alpha \) and *b*, and for large frequency, the real part of the conductivity is proportional to \(\omega /T\). We observed that the holographic conductivity of HSC has a minimum in the imaginary part. Besides, with decreasing the temperature the minimum in the imaginary part shifts toward greater frequency for various values of the Gauss–Bonnet parameter \(\alpha \) and the nonlinear gauge field parameter *b*. Furthermore, for a fixed value of *b*, the minimum of \(\mathrm {Im[\sigma ]}\) increases with increasing \(\alpha \), while for a fixed value of \(\alpha \), it increases with increasing *b*.

## Notes

### Acknowledgements

We thank the Research Council of Shiraz University. The work of AS has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran.

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