# Holographic conductivity of holographic superconductors with higher order
corrections

Ahmad Sheykhi1,2, Afsoon Ghazanfari1, Amin Dehyadegari1
1 Physics Department and Biruni Observatory, College of Sciences,
Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
P.O. Box 55134-441, Maragha, Iran
<EMAIL_ADDRESS>
###### Abstract

We analytically as well as 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 make the condensation harder. We also
study the holographic conductivity of the system and disclose the effects of
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
conductivity is proportional to the frequency per temperature, $\omega/T$, as
frequency is enough large. Besides, the conductivity has a minimum in the
imaginary part which is shifted toward greater frequency with decreasing the
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, well-known as AdS/CFT
correspondence, provides an established method for calculating correlation
functions in a strongly interacting field theory using a dual classical
gravity description [1]. It has been confirmed that this duality can be
applied for solving the problem of high temperature superconductors in
condensed matter physics [2]. 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 temperatures
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 a classical general relativity in one higher dimensions. This idea
is called the holographic superconductors (HSC) [3, 4, 5] and has got a lot of
attentions in the past decade. According to the HSC proposal, in 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 black
hole with no hair (normal phase/conductor phase) to the case with scalar hair
at low temperatures (superconducting phase) [6].

Nowadays, the investigations on the HSC have attracted considerable attention
and become an active field of research. Let us review some works in this
direction. In the background of Schwarzschild AdS black holes in Einstein
gravity, the properties of HSC have been explored in [7, 8, 9, 10, 11, 12, 13,
14]. The studies were also generalized to higher order gravity theories such
as Gauss-Bonnet gravity [15, 16, 17, 18, 19]. 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 [18]. It was confirmed
that the critical exponent of the condensation in Gauss-Bonnet HSC still obeys
the mean field theory and has the value $1/2$ [17]. Other studies on the
holographic superconductor have been carried out in (see for example [20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45] and 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 were computed in [2, 4, 5]. In the presence of nonlinear
electrodynamics, the conductivity of HSC have been investigated in [46, 47].
Also, in the context of Born-Infeld nonlinear electrodynamics, the optical
properties of Lifshitz HSC has been explored in [48]. It was demonstrated that
this superconductor exhibits metamatrial property in low frequency of the
external electric field for certain region of nonlinear parameter. The effects
of the Weyle coupling parameter and Lifshitz dynamic exponent on the
conductivity of HSC have been explored in [49]. In Ref. [50], a rotating BTZ
black holes was considered as the gravity dual to $(1+1)$ dimensional
superconductor. In this case, depending of the angular momentum on the
conductivity has been investigated. Recently, the authors of [51] have
analytically computed the holographic conductivity of HSC in the presence of
Born-Infeld nonlinear electrodynamic by considering the backreaction of the
matter field on the bulk metric. Further investigations on the holographic
conductivity of HSC have been performed in [52, 53].

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 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 [54], which provides the most general models for the
$s$-wave HSC. Furthermore, in an effective action approach to the string
theory, the Gauss-Bonnet term corresponds to the leading order quantum
corrections to gravity, and its presence guarantees a ghost-free action[55].
The purpose of this work is to anallytically as well as numerically explore
the effects of these correction terms on the properties of $s$-wave HSC.

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

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

We consider the 5D Einstein-Gauss-Bonnet gravity in the background of AdS
spaces which is described by the action [56],

$S=\int
d^{5}x\sqrt{-g}\left[R-2\Lambda+\frac{\alpha}{2}\left(R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}\right)+L_{M}\right],$
(1)

where $\Lambda=-6/l^{2}$ is the cosmological constant of $5$-dimensional AdS
spacetime with radius $l$, $\alpha$ is the Gauss-Bonnet coefficient with
dimension $(\mathit{length})^{2}$, $R_{\mu\nu\rho\sigma}$, $R_{\mu\nu}$ 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

$L_{M}={L}_{NL}-|\nabla\psi-iqA\psi|^{2}-m^{2}|\psi|^{2}.$ (2)

where $\psi$ is a scalar field, $q$ and $m$ are, respectively, the charge and
the mass of the scalar field, and the Lagrangian density of the nonlinear
electrodynamics is given by [57, 58]

$\mathcal{L}_{\mathrm{NL}}=\mathcal{F}+b\mathcal{F}^{2}+O(\mathcal{F}^{4}),$
(3)

where $\mathcal{F}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ is the Maxwell Lagrangian
and $b$ is a parameter. The term $b\mathcal{F}^{2}$ is the first order leading
nonlinear correction term to the Maxwell field. There are several motivation
for choosing the nonlinear Lagrangian in the form of (3). First, the series
expansion of the three well-known Lagrangian of nonlinear electrodynamics such
as Born-Infeld, Logarithmic and Exponential nonlinear electrodynamics have the
form of (3) [59]. Second, calculating one-loop approximation of QED, it was
shown [60] that the effective Lagrangian is given by (3). Besides, if one
neglect all other gauge fields, one may arrive at the effective quadratic
order of $U(1)$ as $\mathcal{F}^{2}$ [61, 62]. Furthermore, considering the
next order correction terms in the heterotic string effective action one can
obtain the $\mathcal{F}^{2}$ term as a corrections to the bosonic sector of
supergravity, which has the same order as the Gauss-Bonnet term [61, 62, 63,
64],

$L_{\mathrm{cor}}=\beta\left[\alpha\left(R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}\right)+b(F^{\mu\nu}F_{\mu\nu})^{2}\right].$
(4)

The field equations can be obtained by varying action (1) with respect to the
metric $g_{\mu\nu}$, the scalar field $\psi$, and and the gauge field
$A_{\mu}$. We find

$\displaystyle
R_{\mu\nu}-\frac{(R-2\Lambda)}{2}g_{\mu\nu}-\frac{\alpha}{2}\left\\{\frac{1}{2}g_{\mu\nu}\left(R^{2}-4R^{\rho\sigma}R_{\rho\sigma}+R^{\kappa\lambda\rho\sigma}R_{\kappa\lambda\rho\sigma}\right)\right.$
$\displaystyle\left.-2RR_{\mu\nu}+4R_{\mu\lambda}R_{\
\nu}^{\lambda}+4R_{\mu\rho\nu\sigma}R^{\rho\sigma}-2R_{\mu}^{\
\rho\sigma\lambda}R_{\nu\rho\sigma\lambda}\right\\}=T_{\mu\nu},$ (5)
$(\nabla_{\mu}-iqA_{\mu})(\nabla^{\mu}-iqA^{\mu})\psi-m^{2}\psi=0,$ (6)
$\nabla_{\mu}\left[\left(1+2b\mathcal{F}\right)F^{\mu\nu}\right]=iq\left[\psi^{\ast}(\nabla^{\nu}-iqA^{\nu})\psi-\psi(\nabla^{\nu}+iqA^{\nu})\psi^{\ast}\right],$
(7)

where $T_{\mu\nu}$ is the matter-stress tensor

$\displaystyle
T_{\mu\nu}=\frac{1}{2}\left(\mathcal{F}+b\mathcal{F}^{2}\right)g_{\mu\nu}-2\left(1+2b\mathcal{F}\right)F_{\mu\rho}F_{\nu}^{\
\rho}-\frac{1}{2}m^{2}|\psi|^{2}g_{\mu\nu}-\frac{1}{2}g_{\mu\nu}|\nabla\psi-
iqA\psi|^{2}$
$\displaystyle+\frac{1}{2}\left[(\nabla^{\nu}-iqA^{\nu})\psi(\nabla^{\mu}+iqA^{\mu})\psi^{\ast}+\mu\leftrightarrow\nu\right].$
(8)

The metric of a planar Schwarzschild-AdS black hole in 5D is [65]

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(dx^{2}+dy^{2}+dz^{2}),$ (9)

with

$f(r)=\frac{r^{2}}{2\alpha}\left(1-\sqrt{1-4\alpha\left(1-\frac{r_{+}^{4}}{r^{4}}\right)}\right).$
(10)

The Hawking temperature at the horizon can be written in the form

$T=\frac{f^{{}^{\prime}}(r)}{4\pi}=\frac{r_{+}}{\pi}.$ (11)

It is worthwhile to note that in the limit $r\rightarrow\infty$, we can obtain

$f(r)\sim\frac{r^{2}}{2\alpha}\left[1-\sqrt{1-4\alpha}\right],$ (12)

so we can introduce the effective AdS radius as

$L_{\mathrm{eff}}^{2}=\frac{2\alpha}{1-\sqrt{1-4\alpha}}.$ (13)

We choose the gauge and the scalar fields in the form [2]

$A_{\mu}=(\phi(r),0,0,0,0),~{}\ \ \ ~{}~{}~{}~{}\psi=\psi(r),$ (14)

Inserting the metric (9) and the gauge and scalar fields (14) in the field
equations (6) and (7), we arrive at

$\partial_{r}^{2}\psi+\left(\frac{3}{r}+\frac{\partial_{r}f}{f}\right)\partial_{r}\psi+\left(\frac{\phi^{2}}{f^{2}}-\frac{m^{2}}{f}\right)\psi=0,$
(15)
$\partial_{r}^{2}\phi+\frac{3}{r}\left(1-2b(\partial_{r}\phi)^{2}\right)\partial_{r}\phi-\frac{2\psi^{2}\phi}{f}\left(1-3b(\partial_{r}\phi)^{2}\right)=0.$
(16)

The horizon radius is defined as the root of $f(r_{+})=0$. The regularity
condition for the gauge field $A_{t}$ on the horizon $r_{+}$, implies the
boundary condition $\phi(r_{+})=0$, which substituting in Eq. (15) yields

$\
~{}~{}~{}~{}~{}\psi(r_{+})=\frac{\partial_{r}f(r_{+})}{m^{2}}\partial_{r}\psi(r_{+}).$
(17)

Near the AdS boundary ($r\rightarrow\infty$) the asymptotic behaviors of the
solutions are given by

$\displaystyle\phi(r)$ $\displaystyle=$ $\displaystyle\mu-\frac{\rho}{r^{2}},$
(18) $\displaystyle\psi(r)$ $\displaystyle=$
$\displaystyle\frac{\psi_{-}}{r^{\Delta_{-}}}+\frac{\psi_{+}}{r^{\Delta_{+}}},$
(19)

where $\Delta_{\pm}=2\pm\sqrt{4+m^{2}L_{\mathrm{eff}}^{2}}$. It is clear that
we should have $m^{2}L_{\mathrm{eff}}^{2}\geq-4$. The value of $\Delta_{\pm}$
depend on $\tilde{m}^{2}=m^{2}L_{\mathrm{eff}}^{2}$. For example, setting
$\tilde{m}^{2}=-3$, we have $\Delta_{+}=3$ and $\Delta_{-}=1$. The
coefficients $\psi_{\pm}$ correspond to the vacuum expectation values of the
condensate operator, namely $\psi_{\pm}=<\mathcal{O_{\pm}}>$, where
$\mathcal{O_{\pm}}$ is the dual operator to the scalar field with the
conformal dimension $\Delta_{\pm}$. Following [2], we can impose the boundary
condition in which either $\psi_{-}$ or $\psi_{+}$ vanishes, so that the
theory is stable in the asymptotic AdS region. In what follow, we set
$\psi_{-}=0$ and take $\psi_{+}=\langle\mathcal{O_{+}}\rangle$ non zero. The
interpretation of the parameters $\mu$ and $\rho$, also comes from the
gauge/gravity dictionary and are, respectively, interpreted as the chemical
potential and charge density of the conformal field theory on the boundary.

##  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 as well as the gauge field is
taken into account. We shall continue our study both analytically and
numerically and compare the two method with each other.

###  3.1 Analytical method

Here, we analytically obtain the relation between the critical temperature and
the charge density of Gauss-Bonnet HSC. To do this, we first transform the
coordinate $r$ to $z$, such that $z={r_{+}}/{r}$. Using this new coordinates,
the equations of motion (15) and (16) can be rewritten as

$\partial_{z}^{2}\phi-\frac{1}{z}\partial_{z}\phi+\frac{6bz^{3}}{r_{+}^{2}}(\partial_{z}\phi)^{3}-\frac{2\psi^{2}\phi
r_{+}^{2}}{fz^{4}}+\frac{6b(\partial_{z}\phi)^{2}\phi\psi^{2}}{f}=0,$ (20)
$\partial_{z}^{2}\psi+(\frac{\partial_{z}f}{f}-\frac{1}{z})\partial_{z}\psi+\frac{r_{+}^{2}}{z^{4}}(\frac{\phi^{2}}{f^{2}}-\frac{m^{2}}{f})\psi=0.$
(21)

Near the critical temperature ($T=T_{c}$) we have $\psi=0$, and thus equation
(20) reduces to

$\partial_{z}^{2}\phi-\frac{1}{z}\partial_{z}\phi+\frac{6bz^{3}}{r_{+}^{2}}(\partial_{z}\phi)^{3}=0.$
(22)

Solving the above equation for the small value of nonlinear parameter $b$, we
find

$\phi(z)=\lambda
r_{+c}(1-z^{2})\left[1-\frac{b\lambda^{2}}{2}\xi(z)\right]+O(b^{2}),$ (23)

where

$\xi(z)=(1+z^{2})(1+z^{4}),~{}~{}~{}\ \ \
~{}~{}~{}\lambda=\frac{\rho}{r_{+c}^{3}}.$ (24)

Next, we consider the boundary conditions for $\psi$ near the critical point
($T\rightarrow T_{c}$). We assume $\psi$ has the following form [66]

$\psi|_{z\rightarrow
0}\sim\frac{\langle\mathcal{O_{+}}\rangle}{r_{+}^{3}}z^{3}F(z),$ (25)

where $F(z)$ 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

$F^{{}^{\prime\prime}}+p(z)F^{{}^{\prime}}+q(z)F+\lambda^{2}w(z)F=0,$ (26)

where the prime now indicates the derivative with respect to $z$, and $p(z)$,
$q(z)$ and $w(z)$ read

$\displaystyle p(z)$ $\displaystyle=$
$\displaystyle\frac{3(1-\sqrt{1-4\alpha+4\alpha z^{4}})-12\alpha+20\alpha
z^{4}}{z[1-4\alpha+4\alpha z^{4}-\sqrt{1-4\alpha+4\alpha z^{4}}]},$ (27)
$\displaystyle q(z)$ $\displaystyle=$
$\displaystyle\frac{1}{z^{2}}\left[\frac{3(1-4\alpha-4\alpha
z^{4}-\sqrt{1-4\alpha+4\alpha z^{4}})}{\sqrt{1-4\alpha+4\alpha
z^{4}}-1+4\alpha-4\alpha z^{4}}+\frac{2m^{2}\alpha}{\sqrt{1-4\alpha+4\alpha
z^{4}}-1}\right],$ (28) $\displaystyle w(z)$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}(1-z^{2})^{2}(1-\frac{b}{2}\lambda^{2}\xi(z))^{2}}{(1-\sqrt{1-4\alpha+4\alpha
z^{4}})^{2}}.$ (29)

It is a matter of calculations to convert Eq. (26) to the standard form of the
Sturm-Liouville equation

$(T(z)F^{{}^{\prime}}(z))^{{}^{\prime}}-Q(z)F(z)+\lambda^{2}P(z)F(z)=0,$ (30)

where,

$\displaystyle T(z)$ $\displaystyle=$
$\displaystyle\frac{z^{3}}{2\sqrt{\alpha}}(\sqrt{1-4\alpha+4\alpha
z^{4}}-1)\approx z^{3}\sqrt{\alpha}(z^{4}-1)[1-\alpha(z^{4}-1)],$ (31)
$\displaystyle Q(z)$ $\displaystyle=$
$\displaystyle-T(z)q(z)\approx-3z\sqrt{\alpha}(3z^{4}+6\alpha z^{4}-7\alpha
z^{8}),$ (32) $\displaystyle P(z)$ $\displaystyle=$ $\displaystyle
T(z)w(z)\approx\frac{\sqrt{\alpha}z^{3}(z^{2}-1)(1+\alpha(z^{4}-1))(1-\frac{b}{2}\lambda^{2}\xi(z))^{2}}{z^{2}+1}.$
(33)

In the above equations we have only kept the terms up to order $\alpha^{3/2}$.
Next, we perform a perturbative expansion $b\lambda^{2}$ and retain only the
terms that are linear in $b$ such that

$b\lambda^{2}=b(\lambda^{2}|_{b=0})+O(b^{2}),$ (34)

where $\lambda^{2}|_{b=0}$ is the value of $\lambda^{2}$ for $b=0$. Thus we
can rewrite Eq. (33) as

$P(z)\approx\frac{\sqrt{\alpha}z^{3}(z^{2}-1)(1+\alpha(z^{4}-1))(1-b(\lambda^{2}|_{b=0})\xi(z))}{z^{2}+1}.$
(35)

Employing the Sturm-Liouville eigenvalues problem, the eigenvalues of Eq. (30)
can be obtained by varying the following function

$\lambda^{2}=\frac{\int_{0}^{1}dz(T(z)(F^{{}^{\prime}}(z))^{2}+Q(z)F^{2}(z))}{\int_{0}^{1}dzP(z)F^{2}(z)},$
(36)

where we also choose $F(z)=1-az^{2}$ and $m^{2}=-3/L_{\mathrm{eff}}^{2}$ to
appraise this expression. At last, using Eqs. (11) and (24), for $T\sim
T_{c}$, one can obtain

$T_{c}=\zeta\rho^{1/3},$ (37)

where $\zeta=\frac{1}{\pi\lambda_{\mathrm{min}}^{1/3}}$ and
$\lambda_{\mathrm{min}}$ is the minimum eigenvalue which can be obtained by
variation of Eq. (36). Our strategy, in the analytical method, for calculating
the critical temperature for condensation is to minimize the function (36)
with respect to the coefficient $a$ 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 relation (37).
As an example, we bring the details of our calculation for $\alpha=0.01$ and
$b=0.01$. In this case Eq. (36) reduces to

$\lambda^{2}=\frac{0.150900-0.226000a+0.117119a^{2}}{0.003844-0.003350a+0.000911a^{2}},$
(38)

whose minimum is $\lambda_{\mathrm{min}}=25.6427$ at $a=0.747087$. And thus
according to Eq. (37), the critical temperature becomes
$T_{c}=0.185363\rho^{1/3}$. In tables 1, 2 and 3, we summarize our results for
$\lambda_{\mathrm{min}}$ and $\zeta$ for different values of the parameters
$\alpha$, $a$ 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 results is in a very good agreement with the numerical results.

###  3.2 Numerical method

Now, we numerically investigate the critical behavior of the HSC in Gauss-
Bonnet gravity with quadratic correction term to the gauge field. For the
numerical study we employ the shooting method [67]. For simplicity we assume
$r_{+}=1$, and thus Eqs. (20) and (21)for $\phi$ and $\psi$ reduces to

$\partial_{z}^{2}\phi-\frac{1}{z}\partial_{z}\phi+6bz^{3}(\partial_{z}\phi)^{3}-\frac{2\psi^{2}\phi}{fz^{4}}+\frac{6b(\partial_{z}\phi)^{2}\phi\psi^{2}}{f}=0,$
(39)
$\partial_{z}^{2}\psi+\left(\frac{\partial_{z}f}{f}-\frac{1}{z}\right)\partial_{z}\psi+\frac{1}{z^{4}}\left(\frac{\phi^{2}}{f^{2}}-\frac{m^{2}}{f}\right)\psi=0.$
(40)

Near the horizon ($z=1$), we can expand $\phi$ and $\psi$ as

$\phi\approx\phi^{{}^{\prime}}(1)(1-z)+\frac{\phi^{{}^{\prime\prime}}(1)}{2}(1-z)^{2}+\dots,$
(41)
$\psi\approx\psi(1)+\psi^{{}^{\prime}}(1)(1-z)+\frac{\psi^{{}^{\prime\prime}}(1)}{2}(1-z)^{2}+\dots,$
(42)

while near the AdS boundary ($z=0$), they behave like

$\phi\approx\mu-\rho z^{2},$ (43)
$\psi\approx\psi_{-}z^{\Delta_{-}}+\psi_{+}z^{\Delta_{-}}.$ (44)

(a) $\alpha=0.0001,b=0.01$

(b) $\alpha=0.01,b=0$

(c) $\alpha=0.01,b=0.01$

(d) $\alpha=0.1,b=0$

Figure 1: The behavior of $\psi(z)$ versus $z$ for Gauss-Bonnet HSC and for $\tilde{m}^{2}=-3$ and different $\alpha$ and $b$. $b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\zeta_{SL}\left(=\frac{1}{\pi\lambda_{\mathrm{min}}^{1/3}}\right)$ | $\zeta_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 1: Comparison of analytical and numerical values of $\zeta=T_{c}/\rho^{1/3}$ for $\alpha=0.0001$. $b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\zeta_{SL}\left(=\frac{1}{\pi\lambda_{\mathrm{min}}^{1/3}}\right)$ | $\zeta_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 2: Comparison of analytical and numerical values of $\zeta=T_{c}/\rho^{1/3}$ for $\alpha=0.01$. $b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\zeta_{SL}\left(=\frac{1}{\pi\lambda_{\mathrm{min}}^{1/3}}\right)$ | $\zeta_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 3: Comparison of analytical and numerical values of
$\zeta=T_{c}/\rho^{1/3}$ for $\alpha=0.1$.

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 specific value of the reduced scalar
field mass $\tilde{m}^{2}$, we can perform 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 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 Gauss-
Bonnet coefficient $\alpha$ and nonlinear parameter $b$. In the absence of
quadratic correction term ($b=0$), our results exactly coincide with those
presented in [17, 20]. The acceptable diagram for us is the red one in each
plot since there is nothing in the bulk to effect on speed of the wave, so the
diagram of $\psi$ will be stable. From tables $1-3$, it is evident that when
$b$ becomes larger the condensation gets harder. 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

We would like to obtain the critical exponent and the condensation values of
the condensation operator near the critical temperature using the analytical
method. Inserting Eq. (25) into Eq. (20), we get

$\partial_{z}^{2}\phi-\frac{1}{z}\partial_{z}\phi+\frac{6bz^{3}}{r_{+}^{2}}(\partial_{z}\phi)^{3}=\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{4}}\mathcal{B}\phi,$
(45)

and,

$\mathcal{B}=\frac{2z^{2}}{f}\left(1-\frac{3bz^{4}(\partial_{z}\phi)^{2}}{r_{+}^{2}}\right)F^{2}(z).$
(46)

Near the critical temperature,
$\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{4}}$ is a very small and thus
we can expand $\phi(z)$ as

$\frac{\phi(z)}{r_{+}}=\lambda(1-z^{2})\left[1-\frac{b\lambda^{2}}{2}\xi(z)\right]+\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{4}}\chi(z).$
(47)

where $\chi$ satisfies the following boundary condition

$\displaystyle\chi(1)=\chi^{{}^{\prime}}(1)=0.$ (48)

With the help of Eq. (47), and comparing the coefficient of
$\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{4}}$ on both sides, Eq. (45)
leads to

$\chi^{{}^{\prime\prime}}(z)-\frac{\chi^{{}^{\prime}}}{z}+72b\lambda^{2}z^{5}\chi^{{}^{\prime}}=\lambda\mathcal{B}(1-z^{2})\left(1-\frac{b}{2}\lambda^{2}\xi(z)\right).$
(49)

From Eq. (49) we figure out, in the limit $z\rightarrow 0$, the following
equation

$\chi^{{}^{\prime\prime}}(0)=\frac{\chi^{{}^{\prime}}(z)}{z}|_{z\rightarrow
0}.$ (50)

It is a matter of calculations to shaw that Eq. (49) can be written

$\frac{d}{dz}\left(e^{12b\lambda^{2}z^{6}}\frac{\chi^{{}^{\prime}}}{z}\right)=\lambda\frac{2z^{3}}{r_{+}^{2}}\frac{e^{12b\lambda^{2}z^{6}}(1-\frac{b}{2}\lambda^{2}\Gamma(z))}{(1+z^{2})(1+\alpha(1-z^{4}))}F^{2}(z),$
(51)

where $\Gamma(z)=1+z^{2}+z^{4}+25z^{6}$. Integrating both sides of the above
equation in the interval $[0,1]$ and using the boundary condition (48), we
arrive at

$\displaystyle\frac{\chi^{{}^{\prime}}}{z}|_{z\rightarrow
0}=-\frac{\lambda}{r_{+}^{2}}\mathcal{A},$ (52)

where

$\displaystyle\mathcal{A}$ $\displaystyle\approx$
$\displaystyle\int_{0}^{1}\frac{2z^{3}F^{2}(z)\left[1-\frac{b}{2}\lambda^{2}(1+z^{2}+z^{4}+z^{6})\right][1-\alpha(1-z^{4})]}{1+z^{2}}dz.$

Combining Eqs. (18) and (47), we achieve

$\frac{\mu}{r_{+}}-\frac{\rho}{r_{+}^{3}}z^{2}=\lambda(1-z^{2})\left[1-\frac{b\lambda^{2}}{2}\xi(z)\right]+\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{4}}(\chi(0)+z\chi^{{}^{\prime}}(0)+\frac{z^{2}}{2}\chi^{{}^{\prime\prime}}(0)+...),$
(53)

where in the last step we have expanded $\chi(z)$ around $z=0$. Equating the
coefficients of $z^{2}$ on both sides of Eq. (53), we find

$\displaystyle\frac{\rho}{r_{+}^{3}}=\lambda\left(1+\frac{\langle\mathcal{O_{+}}\rangle^{2}}{r_{+}^{6}}\mathcal{A}\right).$
(54)

Using the fact that $\lambda={\rho}/{r^{3}_{+c}}$ as well as definition (11),
we can obtain the order parameter $\langle\mathcal{O_{+}}\rangle$ near the
critical temperature $T_{c}$ as

$\displaystyle\langle\mathcal{O_{+}}\rangle=\gamma\pi^{3}T_{c}^{3}\sqrt{1-\frac{T}{T_{c}}},$
(55)

where

$\displaystyle\gamma=\sqrt{\frac{6}{\mathcal{A}}}.$ (56)

From Eq. (55) we observe that the critical exponent has the mean field value
$1/2$, which is independent of the nonlinear parameter $b$ 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 back to
calculation the condensation value $\gamma$ in the next subsection.

(a) $\alpha=0.0001$

(b) $\alpha=0.01$

(c) $\alpha=0.1$

Figure 2: The condensate operator $<\mathcal{O}_{+}>$ as a function of
temperature for different values of $b$ and various values of $\alpha$, where
we have set $<\mathcal{O}_{-}>=0$ and $\tilde{m}^{2}=-3$.

###  4.2 Numerical method

We use the numerical method to explore the behaviour 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 rises quickly as the system is cooled and finally
goes to a constant for sufficiently low temperatures. This behaviour is
qualitatively similar to that obtained in BCS theory and observed in many
materials.

$b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\gamma_{SL}$ | $\gamma_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 4: The analytical and numerical results for the condensation operator for $\alpha=0.0001.$ $b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\gamma_{SL}$ | $\gamma_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 5: The analytical and numerical results for the condensation operator for $\alpha=0.01.$ $b$ | $a$ | $\lambda_{\mathrm{min}}^{2}$ | $\gamma_{SL}$ | $\gamma_{\mathrm{Numerical}}$ |  |
---|---|---|---|---|---|---
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 |  |
Table 6: The analytical and numerical results for the condensation operator
for $\alpha=0.1.$

Now we are going to study the condensation operator
$\langle\mathcal{O}_{+}\rangle$ in the close neighborhood of the
superconductor critical temperature to compute the critical exponents and the
condensation value $\gamma$ of the Gauss-Bonnet HSC with quadratic nonlinear
electromagnetic. For this purpose, we first take the logarithmic of Eq. (55).
We arrive at

$\mathcal{\log}\left(\frac{\langle\mathcal{O_{+}}\rangle}{T_{c}^{3}}\right)=\mathcal{\log}\left(\pi^{3}\gamma\right)+\frac{1}{2}\mathcal{\log}\left(1-\frac{T}{T_{c}}\right).$
(57)

(a) $\alpha=0.0001$ (b) $\alpha=0.01$

(c) $\alpha=0.1$

Figure 3: These figures show the behavior of
$\log{\left(<\mathcal{O}_{+}>/T_{c}^{3}\right)}$ versus
$\log\left(1-T/Tc\right)$ near the critical temperature $Tc$ for different
values of $b$ and $\alpha$. The numerical results are highlighted in the
filled squares.

We have plotted the behaviour of the above function in Fig. (3). From this
figure, we observe that the numerical results are fitted to the above analytic
form in the vicinity of the critical temperature. We summarize our results in
Fig. (3) and also tables $4-6$ for different values of $b$ 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

In this section, we study the energy gap in the holographic superconductor
phase which is constructed on the boundary of the background spacetime. In
particular, we investigate the influence of the Gauss-Bonnet and nonlinear
parameters on the superconducting gap. In order to do this, we must compute
the electrical conductivity of holographic superconductor by turning on a
small perturbation $\delta A_{x}=A_{x}(r)\exp(-i\omega t)$ to the gauge field
in the bulk where $\omega$ is the frequency. At linearized order in
perturbation ($\delta A_{x}$), the equation of motion for the gauge field
$A_{x}(r)$, which obeys Eq. (7), is

$\partial_{r}^{2}A_{x}+\left[\frac{1}{r}\left(1-6b\partial_{r}\phi^{2}\right)+\frac{\partial_{r}f}{f}+\frac{4b\phi\partial_{r}\phi\psi^{2}}{f}\right]\partial_{r}A_{x}-\left[\frac{2\psi^{2}}{f}\left(1-b\partial_{r}\phi^{2}\right)-\frac{\omega^{2}}{f^{2}}\right]A_{x}=0.$
(58)

In the absence of the nonlinear correction ($b=0$), this differential equation
reduces to the Maxwell case as presented in [2, 3]. To determine the
conductivity, we need the asymptotic ($r\rightarrow\infty$) form of the second
order differential equation (58), which may be obtained as

$\partial_{r}^{2}A_{x}+\frac{3}{r}\partial_{r}A_{x}+\frac{\omega^{2}L_{\mathrm{eff}}^{4}}{r^{4}}A_{x}+\ldots=0,$
(59)

which admits the following solution near the boundary

$A_{x}=A_{x}^{(0)}+\frac{A_{x}^{(1)}}{r^{2}}+\frac{\omega^{2}L_{\mathrm{eff}}^{4}\mathrm{ln}(Kr)}{2r^{2}}A_{x}^{(0)}+\cdots,$
(60)

where $A_{x}^{(0)}$, $A_{x}^{(1)}$ are two constant and $K$ is also a constant
parameter with length dimension which is considered for a dimensionless
logarithmic argument.

According to AdS/CFT correspondence, the two point correlation function of the
current operators in a system is given by its on shell action where the action
is evaluated on the equations of motion. Here, the on shell action is

$S_{o.s.}\equiv\int_{r_{+}}^{r_{\infty}}dr\int d^{4}x\sqrt{-g}\mathcal{L},$
(61)

which, in the quadratic approximation for the gauge field perturbation becomes

$S_{o.s.}=\int
d^{4}x\int_{r_{+}}^{r_{\infty}}dr\Bigg{\\{}-\frac{1}{2}r\left[\left(2\psi(r)^{2}-\frac{\omega^{2}}{f(r)}-\frac{b\omega^{2}\partial_{r}\phi^{2}}{f(r)}\right)A_{x}(r)^{2}+f(r)\left(1+b\partial_{r}\phi^{2}\right)\partial_{r}A_{x}(r)^{2}\right]\Bigg{\\}}.$
(62)

After performing an integration by parts and using Eq. (58), we get

$S_{o.s.}=\int
d^{4}x\left[-\frac{1}{2}rf(r)\left(1+b\partial_{r}\phi^{2}\right)\partial_{r}A_{x}(r)A_{x}(r)\right]\Bigg{|}_{r=r_{\infty}}.$
(63)

Substituting Eqs. (12), (18) and (60) in the above expression, one arrives at

$\displaystyle S_{o.s.}=\int
d^{4}x\left[\frac{A_{x}^{(0)}A_{x}^{(1)}}{L_{\mathrm{eff}}^{2}}-\frac{\omega^{2}L_{\mathrm{eff}}^{2}A_{x}^{(0)2}}{4}+\frac{1}{2}\omega^{2}L_{\mathrm{eff}}^{2}\mathrm{ln}(Kr)A_{x}^{(0)2}+\frac{A_{x}^{(1)2}}{L_{\mathrm{eff}}^{2}r^{2}}\right.$
$\displaystyle-\frac{\omega^{2}L_{\mathrm{eff}}^{2}A_{x}^{(0)}A_{x}^{(1)}}{4r^{2}}-\frac{\omega^{2}L_{\mathrm{eff}}^{6}\mathrm{ln}(Kr)A_{x}^{(0)2}}{8r^{2}}+\frac{\omega^{2}L_{\mathrm{eff}}^{2}\mathrm{ln}(Kr)A_{x}^{(0)}A_{x}^{(1)}}{r^{2}}$
$\displaystyle\left.+\frac{\omega^{4}L_{\mathrm{eff}}^{6}\mathrm{ln}(Kr)A_{x}^{(0)2}}{4r^{2}}+\mathcal{O}\left(\frac{1}{r^{3}}\right)\right]\Bigg{|}_{r=r_{\infty}},$
(64)

thus we can obtain $S_{o.s.}$ as follows

$S_{o.s.}=\int
d^{4}x\left[\frac{A_{x}^{(0)}A_{x}^{(1)}}{L_{\mathrm{eff}}^{2}}-\frac{\omega^{2}L_{\mathrm{eff}}^{2}A_{x}^{(0)2}}{4}+\frac{1}{2}\omega^{2}L_{\mathrm{eff}}^{2}\mathrm{ln}(Kr)A_{x}^{(0)2}\right],$
(65)

in which logarithmic divergences appears. In order to cancel out this
divergency, we obtain the boundary counterterm as described in Appendix by
using Skenderis’s method of holographic renormalization [68]. Therefore, the
finite on shell action may be written as

$S=S_{o.s.}+S_{c.t.},$ (66)

(a) $\alpha=0.0001,b=0$ (b) $\alpha=0.0001,b=0.01$ (c) $\alpha=0.0001,b=0.02$
(d) $\alpha=0.01,b=0$ (e) $\alpha=0.01,b=0.01$

(f) $\alpha=0.1,b=0$

(g) $\alpha=0.1,b=0.01$

Figure 4: The real part of the conductivity as a function of frequency for
different values of $b$ and $\alpha$. Each figure is plotted for different
temperature $T/Tc$.

(a) $\alpha=0.0001,b=0$ (b) $\alpha=0.0001,b=0.01$

(c) $\alpha=0.0001,b=0.02$

(d) $\alpha=0.01,b=0$ (e) $\alpha=0.01,b=0.01$ (f) $\alpha=0.1,b=0$ (g)
$\alpha=0.1,b=0.01$

Figure 5: The imaginary part of the conductivity as a function of frequency
for different values of $b$ and $\alpha$. Each figure is plotted for different
temperature $T/Tc$.

where the gauge invariant counterterm $S_{c.t.}$ is given by Eq. (79). Now, we
can obtain the current operator in the boundary field theory [2, 3] as

$\left\langle J_{x}\right\rangle=\frac{\delta S}{\delta
A_{x}^{(0)}}=\frac{2}{L_{\mathrm{eff}}^{2}}A_{x}^{(1)}-\frac{\omega^{2}L_{\mathrm{eff}}^{2}}{2}A_{x}^{(0)}\text{
}.$ (67)

According to Ohm’s law, the electrical conductivity can be expressed as

$\sigma\left(\omega\right)=\frac{\left\langle J_{x}\right\rangle}{E_{x}},$
(68)

where $E_{x}=-\partial_{t}\delta A_{x}$. Hence, using the current (67), the
holographic conductivity is given by

$\sigma=-\frac{2iA_{x}^{(1)}}{\omega
L_{\mathrm{eff}}^{2}A_{x}^{(0)}}+\frac{i\omega L_{\mathrm{eff}}^{2}}{2}.$ (69)

Consequently, the holography conductivity is calculated by solving numerically
a differential equation (58) such that the infalling boundary condition is
imposed at the event horizon

$A_{x}(r)=\mathrm{exp}\left(-\frac{i\omega}{4\pi T}\right)S(r),$ (70)

in which $T$ is the Hawking temperature and

$S(r)=1+a_{1}(r-r_{+})+a_{2}(r-r_{+})^{2}+\ldots,$ (71)

where $a_{1},a_{2},\ldots$ are calculated by Taylor series expansion of
equation (58) around the horizon.

(a) $\alpha=0.0001$

(b) $\alpha=0.01$

(c) $\alpha=0.1$

(d) $b=0$

(e) $b=0.005$

(f) $b=0.01$

(g) $b=0.02$

Figure 6: The real part of the conductivity as a function of
$\omega/<\mathcal{O_{+}}>^{1/3}$, at low temperatures, around $T\approx
0.1Tc$.

(a) $\alpha=0.0001$

(b) $\alpha=0.01$

(c) $\alpha=0.1$ (d) $b=0$ (e) $b=0.005$ (f) $b=0.01$ (g) $b=0.02$

Figure 7: The imaginary part of the conductivity as a function of
$\omega/<\mathcal{O_{+}}>^{1/3}$, at low temperatures, around $T\approx
0.1Tc$.

The numerical results for holographic conductivity associated with the
condensation operator $<\mathcal{O_{+}}>$ are plotted in Figs. 4-7. The real
and imaginary parts of electrical conductivity versus frequency are
illustrated at different temperature below $Tc$ in Figs.4 and 5, respectively.
As one can see from Fig. 4, the superconducting gap is opened below the
critical temperature which became deeper with decreasing the temperature.
Besides, for various values of $\alpha$ and $b$, the real part of conductivity
is proportional to the frequency per temperature, $\omega/T$, as frequency is
enough large. According to the Fig. 5 the divergence in imaginary part, at
$\omega=0$, points out a delta function in the real part, $Re[\sigma]$, at
$\omega=0$ which is not played in the Fig. 4. As one can see from Fig. 5, the
holographic conductivity of 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 formation of the superconducting gap with changing $\alpha$ and $b$
at low temperature, e.g., $T\approx 0.1Tc$, the real and 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 Gauss-Bonnet
coefficient $\alpha$, the energy gap ($\omega/{<\mathcal{O_{+}}>}^{1/3}$)
enlarges with increasing the nonlinear parameter $b$. It is evident from Figs.
6 and 7, that the energy gap of HSC for various $\alpha$ exhibits different
behavior based on the nonlinear correction $b$. In case of the Maxwell field
($b=0$), the superconducting energy gap decreases as $\alpha$ increases at low
temperature (see Fig. 6(d)). When we take into account the nonlinear
correction $b$, the energy gap of HSC increases with increasing $\alpha$ (see
Figs. 6(d)-6(g)). From Fig. 7, we see that for a fixed value of $\alpha$, the
minimum of imaginary part of conductivity goes to the larger value of
$\omega/<\mathcal{O_{+}}>^{1/3}$ when $b$ increases. In the absence of
correction ($b=0$), it decreases with enhancing the Gauss-Bonnet coefficient
(Fig. 6(d)). 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 increase 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
to the gravity side as well as 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
Lagrangian has a series expansion which their first two terms are exactly in
the above form.

First, we have analytically as well as numerically investigated the relation
between critical temperature of 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 out that for a
fixed value of $\alpha$, with increasing the nonlinear parameter $b$, the
value of $T_{c}/\rho^{1/3}$ decreases as well. This implies that when $b$
becomes larger the condensation gets harder. 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 this results
are in a very good agreement with our numerical results. Then, we obtained the
critical exponent of the Gauss-Bonnet HSC with 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 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 electrical
conductivity versus $\omega/T$ and $\omega/<\mathcal{O_{+}}>^{1/3}$ for
$T<Tc$. We observed that the superconducting gap is opened below the critical
temperature which became deeper with decreasing the temperature. Interstingly
enough, we found that for different values of $\alpha$ and $b$, and for large
frequency, the real part of 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 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
increase with increasing $b$.

## Acknowledgments

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

## Appendix: Holographic renormalization

To construct the boundary counterterm action, we utilize a holographic
renormalization method of Skenderis which was presented in [68, 69, 70]. To
apply this method, the spacetime metric takes the form

$ds^{2}=G_{ab}d\xi^{a}d\xi^{b}=-\frac{L_{\mathrm{eff}}^{2}}{\varrho}\mathcal{X}(\varrho)d\tau^{2}+\frac{L_{\mathrm{eff}}^{2}d\varrho^{2}}{4\varrho\mathcal{X}(\varrho)}+\frac{L_{\mathrm{eff}}^{2}}{\varrho}(dx^{2}+dy^{2}+dz^{2}),$
(72)

which relates to the metric (9) via $\tau=t/L_{\mathrm{eff}}$,
$\varrho=L_{\mathrm{eff}}^{2}/r$ and the asymptotic ($\varrho\rightarrow 0$)
metric function is $\mathcal{X}(\varrho)_{\varrho\rightarrow 0}=1$. Hence,
asymptotically metric becomes

$ds^{2}=\frac{L_{\mathrm{eff}}^{2}d\varrho^{2}}{4\varrho\mathcal{X}(\varrho)}+h_{\mu\nu}dx^{\mu}dx^{\nu},$
(73)

where

$h_{\mu\nu}=\frac{L_{\mathrm{eff}}^{2}}{\varrho}\gamma_{\mu\nu}^{0}.$ (74)

According to the electromagnetic contribution, one can evaluate the on-shell
action as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}d^{5}\xi\sqrt{-G}\left[\mathcal{F}+b\mathcal{F}^{2}\right]$
(75) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}d^{5}\xi\frac{\sqrt{-G}}{2}A_{b}\nabla_{a}\left[\left(1+2b\mathcal{F}\right)F^{ab}\right]-\int_{\partial\mathcal{M}}d^{4}x\frac{\sqrt{-h}}{2}A_{\mu}F^{\varrho\mu}\left(1+2b\mathcal{F}\right)$
$\displaystyle=$ $\displaystyle-
L_{\mathrm{eff}}\int_{\varrho=\epsilon}d^{4}x\sqrt{-\gamma^{0}}A_{\mu}\partial_{\varrho}A_{\nu}\gamma^{0\mu\nu}\left(1+2b\mathcal{F}\right),$

where $\epsilon$ is a small constant parameter. On the new coordinates (72),
the gauge field equation of bulk motion near boundary is given by

$\varrho\frac{d^{2}A_{i}}{d\varrho^{2}}+\frac{1}{4}\partial_{0}^{2}A_{i}+\mathcal{O}(\varrho^{2})=0,$
(76)

where $\partial_{0}^{2}$ points out to the wave operator of boundary metric
$\gamma_{\mu\nu}^{0}$ and the general solution of this equation is

$A_{i}=A_{i}^{0}+A_{i}^{1}\varrho+\psi\varrho\mathrm{ln}(\varrho),$ (77)

where $\psi=-1/4\partial_{0}^{2}A_{i}^{0}$. With $A_{i}$ at hand, the on-shell
electromagnetic action can be written as

$S=-L_{\mathrm{eff}}\int_{\varrho=\epsilon}d^{4}x\sqrt{-\gamma^{0}}A_{i}^{0}\left(A_{i}^{1}-\frac{1}{4}\partial_{0}^{2}A_{i}^{0}-\frac{1}{4}\mathrm{ln}(\epsilon)\partial_{0}^{2}A_{i}^{0}\right),$
(78)

which is logarithmically divergent. Now, according to Ref.[68], in order to
determine the counterterm action, we first invert the solution (77) to give
$A_{i}^{0}=A_{i}+\mathcal{O}(\epsilon)$. Thus, the counterterm action is
obtained as

$\displaystyle S_{c.t.}$ $\displaystyle=$
$\displaystyle-\frac{L_{\mathrm{eff}}}{4}\mathrm{ln}(\epsilon)\int_{\varrho=\epsilon}d^{4}x\sqrt{-\gamma^{0}}A_{i}\partial_{0}^{2}A_{i}$
(79) $\displaystyle=$
$\displaystyle-\frac{L_{\mathrm{eff}}}{4}\mathrm{ln}(\epsilon)\int_{\varrho=\epsilon}d^{4}x\frac{\sqrt{-h}}{2}F_{\mu\nu}F^{\mu\nu}.$

It is notable to mention that since we consider
$A_{i}=A_{i}(\varrho)\exp(-i\omega L_{\mathrm{eff}}\tau)$, one can calculate

$\partial_{0}^{2}A_{i}=\omega^{2}L_{\mathrm{eff}}^{2}A_{i}.$ (80)

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