One-dimensional backreacting holographic p-wave superconductors

TL;DR

This paper investigates one-dimensional holographic p-wave superconductors with backreaction, using analytical and numerical methods to analyze critical temperature, phase transition order, and the effects of backreaction and mass parameters.

Contribution

It provides a combined analytical and numerical study of backreacting 1D holographic p-wave superconductors, highlighting how backreaction and mass influence critical temperature and phase transition.

Findings

Critical temperature increases with backreaction and mass.

Phase transition is second order with a critical exponent of 1/2.

Analytical and numerical results are in good agreement.

Abstract

We analytically as well as numerically study the properties of one-dimensional holographic p- wave superconductors in the presence of backreaction. We employ the Sturm-Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the AdS3/CFT2 correspondence and determine the relation between the critical temperature Tc and the chemical potential \mu for different values of mass m of charged spin-1 field and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find out that increasing the backreaction as well as the mass parameter cause the greater values for Tc/ \mu. Therefore, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent \beta is 1/2 which is the same as in the mean field theory. Moreover, both methods confirm the exhibition of a second order phase transition.

Tables

Table 1. Table 1: Analytical and Numerical results of T c / μ subscript 𝑇 𝑐 𝜇 {T_{c}}/{\mu} for different values of backreaction and mass parameters.

	$m^2=\frac{1}{16}$		$m^2=\frac{1}{4}$		$m^2=1$
	Analytical	Numerical	Analytical	Numerical	Analytical	Numerical
${\kappa }^2=0$	$0.1424$	$0.1433$	$0.0860$	$0.0880$	$0.0478$	$0.0503$
${\kappa }^2=0.05$	$0.1363$	$0.1397$	$0.0814$	$0.0823$	$0.0443$	$0.0410$
${\kappa }^2=0.1$	$0.1356$	$0.1361$	$0.0803$	$0.0768$	$0.0424$	$0.0330$
${\kappa }^2=0.15$	$0.1346$	$0.1326$	$0.0786$	$0.0716$	$0.0394$	$0.0260$
${\kappa }^2=0.2$	$0.1332$	$0.1292$	$0.0763$	$0.0666$	$0.0353$	$0.0201$
${\kappa }^2=0.25$	$0.1313$	$0.1258$	$0.0733$	$0.0619$	$0.0302$	$0.0152$

Table 2. Table 2: Analytical and Numerical results of γ 𝛾 \gamma for different values of backreaction and mass parameters.

	$m^2=\frac{1}{16}$		$m^2=\frac{1}{4}$		$m^2=1$
	Analytical	Numerical	Analytical	Numerical	Analytical	Numerical
${\kappa }^2=0$	$23.5779$	$20.0224$	$48.1780$	$70.3273$	$188.5$	$1042.6100$
${\kappa }^2=0.05$	$23.7791$	$20.5590$	$49.1955$	$76.0058$	$199.517$	$1390.0400$
${\kappa }^2=0.1$	$23.9228$	$21.1162$	$50.2391$	$82.3205$	$220.382$	$1901.3500$
${\kappa }^2=0.15$	$24.1599$	$21.6954$	$51.9956$	$89.3687$	$260.302$	$2682.0100$
${\kappa }^2=0.2$	$24.4981$	$22.2974$	$54.6113$	$97.2622$	$333.956$	$3928.0200$
${\kappa }^2=0.25$	$24.9451$	$22.9237$	$58.2727$	$106.138$	$475.166$	$6028.6000$

Equations

$$S$$

$${\mathcal{L}}_m$$

$$\frac{1}{2{\kappa }^2}\left[R_{\mu \nu }-g_{\mu \nu }\left(\frac{R}{2}+\frac{1}{l^2}\right)\right]$$

$${\nabla }^{\nu }{F}_{\nu \mu }=iq\left({\rho }^{\nu }{\rho }^{†}{}_{\nu \mu }-{\rho }^{\nu †}{\rho }_{\nu \mu }\right)+iq\gamma {\nabla }^{\nu }\left({\rho }_{\nu }{\rho }^{†}{}_{\mu }-{\rho }^{†}{}_{\nu }{\rho }_{\mu }\right),$$

$$D^{\nu }{\rho }_{\nu \mu }-m^2{\rho }_{\mu }+iq\gamma {\rho }^{\nu }{F}_{\nu \mu }=0,$$

$${ds}^2=-f(r)e^{-\chi (r)}{dt}^2+\frac{{dr}^2}{f(r)}+r^2{dx}^2,$$

$${\rho }_{\nu }d{x}^{\nu }={\rho }_x(r)dx,A_{\nu }d{x}^{\nu }=\varphi (r)dt.$$

$$T=\frac{e^{-\chi (r_{+})/2}{f}^{{}^{\prime }}(r_{+})}{4\pi }.$$

$${\varphi }^{\prime \prime }(r)+\left[\frac{1}{2}{\chi }^{\prime }(r)+\frac{1}{r}\right]{\varphi }^{\prime }(r)-\frac{2q^2\text{ρx}(r{)}^2\varphi (r)}{r^{2}f(r)}=0,$$

$${\rho }_x^{\prime \prime }(r)+\left[\frac{f^{\prime }(r)}{f(r)}-\frac{1}{2}{\chi }^{\prime }(r)-\frac{1}{r}\right]{\rho }_x^{\prime }(r)+\left[\frac{q^2{e}^{\chi (r)}\varphi (r{)}^2}{f(r{)}^2}-\frac{m^2}{f(r)}\right]{\rho }_x(r)=0,$$

$$f^{\prime }(r)-\frac{2r}{l^2}+\frac{2{\kappa }^2}{r}\left[\frac{q^2\text{ρx}(r{)}^2{e}^{\chi (r)}\varphi (r{)}^2}{f(r)}+f(r){\rho }_x^{\prime 2}+m^2{\rho }_x(r{)}^2+\frac{r^2}{2}{e}^{\chi (r)}{\varphi }^{\prime 2}\right]=0,$$

$${\chi }^{\prime }(r)+\frac{4{\kappa }^2}{r}\left[\frac{q^2\text{ρx}(r{)}^2{e}^{\chi (r)}\varphi (r{)}^2}{f(r{)}^2}+{{\rho }_x^{\prime }}^2\right]=0.$$

$$q\to q/a,\varphi \to a\varphi ,{\rho }_x\to a{\rho }_x,\kappa \to \kappa /a,$$

$$l\to al,r\to ar,q\to q/a,m\to m/a.$$

$$\varphi (r)\sim \rho +\mu \ln (r),f(r)\sim r^2,\chi (r)\to 0,{\rho }_x(r)\sim \frac{{\rho }_{x_{-}}}{r^{-m}}+\frac{{\rho }_{x_{+}}}{r^m}.$$

$$e^{\chi }\to a^2{e}^{\chi },t\to at,\varphi \to \varphi /a.$$

$${\varphi }^{\prime \prime }(z)+\left(\frac{{\chi }^{\prime }(z)}{2}+\frac{1}{z}\right){\varphi }^{\prime }(z)-\frac{2{\rho }_x(z{)}^2\varphi (z)}{z^{2}f(z)}=0,$$

$${\rho }_x^{\prime \prime }(z)+\left(-\frac{1}{2}{\chi }^{\prime }(z)+\frac{f^{\prime }(z)}{f(z)}+\frac{3}{z}\right){\rho }_x^{\prime }(z)+{\rho }_x(z)\left(\frac{r_{+}^2{e}^{\chi (z)}\varphi (z{)}^2}{z^{4}f(z{)}^2}-\frac{m^2{r}_{+}^2}{z^{4}f(z)}\right)=0,$$

$$f^{\prime }(z)+\frac{2r_{+}^2}{l^2{z}^3}-2{\kappa }^2\left(\frac{e^{\chi (z)}\varphi (z{)}^2{\rho }_x(z){}^2}{zf(z)}+\frac{z^{3}f(z){\rho }_x^{\prime 2}}{r_{+}^2}+\frac{m^2{\rho }_x(z){}^2}{z}+\frac{z}{2}{e}^{\chi (z)}{\varphi }^{\prime 2}\right)=0,$$

$${\chi }^{\prime 2}\left(\frac{e^{\chi (z)}\varphi (z{)}^2{\rho }_x(z){}^2}{zf(z{)}^2}+\frac{z^3{\rho }_x^{\prime 2}}{r_{+}^2}\right)=0.$$

$$ϵ\equiv ⟨J_x⟩.$$

$${\rho }_x\approx ϵ{\rho }_{x_1}+{ϵ}^3{\rho }_{x_3}+{ϵ}^5{\rho }_{x_5}+\cdots ,$$

$$\varphi \approx {\varphi }_0+{ϵ}^2{\varphi }_2+{ϵ}^4{\varphi }_4+\cdots ,$$

$$f\approx f_0+{ϵ}^2{f}_2+{ϵ}^4{f}_4+\cdots ,$$

$$\chi \approx {ϵ}^2{\chi }_2+{ϵ}^4{\chi }_4+\cdots .$$

$$\mu=\mu_{0}+\epsilon^{2}\delta\mu_{2}+...\rightarrow\epsilon\thickapprox\Bigg{(}\frac{\mu-\mu_{0}}{\delta\mu_{2}}\Bigg{)}^{1/2},$$

$${\varphi }^{\prime \prime }(z)+\frac{{\varphi }^{\prime }(z)}{z}=0.$$

$$\varphi (z)=\lambda r_{+}\log (z),\lambda =\frac{\mu }{r_{+}}.$$

$$f^{\prime }(z)+\frac{2r_{+}^2}{z^3}-\frac{{\kappa }^2{\lambda }^2{r}_{+}^2}{z}=0,$$

$$f(z)=\frac{r_{+}^{2}g(z)}{z^2},g(z)=1-z^2+{\kappa }^2{\lambda }^2{z}^2\log (z).$$

$${\rho }_x(z)=\frac{⟨J_x⟩}{\sqrt{2}{r}_{+}^{\Delta }}{z}^{\Delta }F(z).$$

$$F^{\prime \prime }(z)+F^{\prime }(z)\left(\frac{g^{\prime }(z)}{g(z)}+\frac{2\Delta }{z}+\frac{1}{z}\right)+F(z)\left(\frac{\Delta g^{\prime }(z)}{zg(z)}-\frac{m^2}{z^{2}g(z)}+\frac{{\Delta }^2}{z^2}\right)+\frac{F(z)\left({\lambda }^2{\kappa }^2{\log }^2(z)\right)}{g(z{)}^2}=0.$$

$${\left[T(z)F^{\prime }(z)\right]}^{\prime }+P(z)T(z)F(z)+{\lambda }^{2}Q(z)T(z)F(z)=0,$$

Full text

I Introduction

In $1911$ Heike Kamerlingh Onnes discovered that the electrical resistance of mercury completely disappeared at temperatures a few degrees above absolute zero Dahi . The phenomenon became known as superconductivity. He was also awarded the the Nobel Prize in Physics $1913$ for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium. Since the discovery of Onnes, the studies on the superconductors have been became an active field of research and a lot of papers have been appeared in the literatures to explain the mechanism of superconductivity. The aim was to explain the zero resistance of the materials from microscopic point of view. The great step in this direction put forwarded in $1957$ by John Bardeen, Leon Neil Cooper and John Robert Schrieffer who described superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. They also awarded the Nobel Prize in physics $1972$ for their jointly developed theory of superconductivity, usually called the BCS-theory. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. In the BCS framework, superconductivity is a macroscopic effect which results from the condensation of Cooper pairs. It was the first widely-accepted theory that explained superconductivity at low temperatures. Based on this theory superconductivity occurs because of condensation of Cooper pairs (including electrons with different spins and momenta) at low temperature. According to angular momentum of Cooper pairs, we can classify superconductors as $s$-wave $(\ell=0)$, $p$-wave $(\ell=1)$, $d$-wave $(\ell=2)$ etc alkac . Since the Cooper pairs are decoupled at higher temperatures, the BCS theory has argued to be inadequate to fully explain the mechanism of high temperature superconductivity BCS57 . In order to shed some light on the problem of high temperature superconductivity the Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence was argued to taken into account Maldacena. (1998); H08 . AdS/CFT duality is a duality that relates the strong coupling conformal field theory living on the boundary in $d$-dimensions to a weak coupling gravity in ($d+1$ )-dimensional spacetime in the bulk. Through AdS/CFT, each quantity in the bulk has a dual on the boundary Maldacena. (1998); H08 ; Gubser et al. (1998); Witten et al. (1998); Horowitz et al. (2008); Ren. (2010). In $2008$, Hartnoll et al. proposed a holographic $s$-wave superconductor model based on the gauge/gravity duality H08 . In his holographic model, Hartnoll assumed that there is a phase transition from a black hole with no hair (normal phase) to a hairy one (superconducting phase) below the critical temperature. Through this process, the system faces with spontaneous $U(1)$ symmetry breaking. The studies on the holographic superconductors have arisen a lot of attentions in the past decade (see e.g. Hartnoll (2009); Herzog. (2009); Horowitz (2011); Gubser. (2009); HHH. (2008); Jing ,Chena (210); Salahi et al. (2016); Sheykhi et al. (2016); cai (2015); SHsh (17); Ge (2010, 2012); Kuang (2013); Pan (2011); Wang6 ; CAI (11); SHSH (16); shSh(16) ; Doa ; Afsoon ; cai (10); yao (2013); n3 ; n4 ; n5 ; n6 ; Gan1 and references therein).

The holographic $p$-wave superconductors can be studied by condensation of a charged vector field in the bulk which is the dual of a vector order parameter in the boundary which can also be considered as the condensation of a $2$-form field in the boundary. For this type of holographic superconductor, the formation of vector hair below the critical temperature is observed. Various models of holographic $p$-wave superconductors have been proposed. In Gubser a $p$-wave superconductors proposed by using an SU(2) Yang-Mills field in the bulk and one of the gauge degrees of freedom which is dual to spin-$1$ order parameter in the field theory. Also, the $p$-wave type of superconductivity may arisen by the condensation of a $2$-form field Donos and a massive spin-$1$ vector in the bulk Caip ; cai13p . The holographic $p$-wave superconductors have been widely investigated in the literatures (e.g.Roberts8 ; zeng11 ; cai11p ; pando12 ; momeni12p ; gangopadhyay12 ; chaturverdip15 ).

on the other side, holographic superconductors have also been explored when the bulk spacetime is a three dimensional black hole. The Einstein field equations admit a three dimensional solution known as BTZ (Bandos-Teitelboim-Zanelli) black holes. BTZ black holes have a crucial effect on the several improvement in string theory Car1 ; Ash ; Sar ; Wit1 ; Car2 . The corresponding superconductor living on the boundary of BTZ black hole is one dimensional. Using the probe brane construction, the holographic $p$-wave superconductors were investigated in Bu . One dimensional holographic $p$-wave superconductors coupled to a massive complex vector field and in the probe limit were explored in alkac . It was argued that below a certain critical temperature, there is a formation of a vector hair around the black hole alkac . It is worth noting that in order to analyze one-dimensional holographic superconductor on the boundary of the three dimensional spacetime, one needs to apply the AdS3/CFT2 Wit2 . One-dimensional holographic $s$-wave and $p$-wave superconductors were investigated analytically as well as numerically from different point of view (see e.g. Wang ; chaturvedi ; Li (2012); momeni ; peng17 ; lashkari ; hua ; yanyan ; yan ; kord ; bina ; mahya ). All investigations on the ($1+1$)-dimensional holographic $p$-wave superconductors are restricted to the case where the vector and gauge fields do not back react on the background geometry. In the present work, we would like to extend the study on the holographic $p$-wave superconductors by considering the effects of the vector and gauge fields on the background of spacetime and disclose the effects of the backreaction on the properties of superconductor.

We shall employ the Sturm-Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. For each method, the relation between critical temperature and chemical potential as well as critical exponent are investigated. We shall also compare the analytical results with the numerical data.

This paper is outlined as fallow. In section II, we present the basic field equations and the boundary conditions of the ($1+1$)-dimensional backreacting holographic $p$-wave superconductors. In section III, by using the Sturm-Liouville variational method, we obtain a relation between the critical temperature and the chemical potential. We also apply the shooting method and study the problem numerically and confirm that the analytical results are compatible with the numerical data. In section IV, we calculate the critical exponent both analytically and numerically. The last section is devoted to closing remarks.

II Basic Field Equations and boundary conditions

As we mentioned our study is based on the AdS3/CFT2 duality. Due to this model, we have a spontaneous local/global $U(1)$ symmetry breaking in the bulk/at the boundary. The action which can describe a charged massive spin-$1$ field $\rho_{\mu}$ with charge $q$ and mass $m$ into ($2+1$)-dimensional Einstein-Maxwell theory with a negative cosmological constant is given by

[TABLE]

where $g$, $R$ and $l$ are the metric determinant, Ricci scalar and AdS radius, respectively. $\kappa^{2}=8\pi G_{3}$ in which $G_{3}$ characterizes the $3$-dimensional Newton gravitation constant in the bulk. Also, by considering $A_{\mu}$ as the vector potential, the strength of Maxwell field reads $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$. In addition, $\rho_{\mu\nu}=D_{\mu}\rho_{\nu}-D_{\nu}\rho_{\mu}$ where $D_{\mu}=\nabla_{\mu}-iqA_{\mu}$. A nonlinear interaction between $\rho_{\mu}$ with $\gamma$ (the magnetic moment) and $A_{\mu}$ is described by the last term in the above action. Since we consider the case without external magnetic field, this term plays no role.

We obtain the equations of motion for matter and gravitational fields by varying the action (1) with respect to the metric $g_{\mu\nu}$, the gauge field $A_{\mu}$ and the vector field $\rho_{\mu}$. We find

[TABLE]

[TABLE]

[TABLE]

The boundary value of $\rho_{\mu}$ is the origin of a charged vector operator which its expectation value plays the role of order parameter in the boundary theory. When the temperature decreases below the critical value, the normal phase becomes unstable and the vector hair which corresponds to superconducting phase appears.

In order to study the one-dimensional holographic $p$-wave superconductor in the presence of backreaction, we take the following metric for the background geometry,

[TABLE]

with the following choices for the vector and gauge fields,

[TABLE]

The Hawking temperature of this black hole is given by mahya

[TABLE]

Substituting metric (5) and relation (6) in the field equations (LABEL:Eein) and (3), we arrive at

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here, the prime denotes derivative with respect to $r$. If we consider the probe limit ($\kappa\rightarrow 0$), the equations of motion (8) and (9) reduce to the corresponding equations in alkac . In the following, we set $q$ and $l$ equal to unity by using the symmetries

[TABLE]

The asymptotic behavior $(r\rightarrow\infty)$ of the solutions are given by

[TABLE]

in which $\mu$ and $\rho$ are chemical potential and charge density, respectively. Note that in (14), the value of $\chi$ has been set to zero by virtue of symmetry,

[TABLE]

The asymptotic behavior of the vector field $\rho_{x}(r)$ is in agreement with the result of wen18 . Here, the Breitenlohner-Freedman (BF) bound is $m^{2}\geq 0$. In this limit, $\rho_{x_{-}}$ plays the role of the source and $\rho_{x_{+}}$ known as $x$-component of the expectation value of the order parameter $\langle J_{x}\rangle$. In the next sections, we will analyze the properties of one-dimensional backreacting holographic $p$-wave superconductor analytically as well as numerically.

III Superconductivity phase transition

In this section, we are going to investigate the phase transition and critical temperature of ($1+1$)-dimensional backreacting holographic $p$-wave superconductors. We address the relation between critical temperature $T_{c}$ and chemical potential $\mu$ as well as the effect of backreaction parameter on $T_{c}$ in the vicinity of transition point.

III.1 Analytical approach

For the analytical approach, we employ the Sturm-Liouville eigenvalue problem. To do this we use a coordinate transformation as $z=r_{+}/r$ where $0\leqslant z\leqslant 1$. In the new coordinates, the field equations (8)-(11) turn to

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here, the prime indicates the derivative with respect to $z$. Near the critical temperature, the expectation value of $\langle J_{x}\rangle$ is small so we can take it as an expansion parameter

[TABLE]

Since in the vicinity of critical temperature $\epsilon\ll 1$, we focus on solutions for small values of condensation parameter $\epsilon$. Therefore, we can expand the model functions as

[TABLE]

Furthermore, we have a similar expression for the chemical potential which can be expressed as

[TABLE]

where $\delta\mu_{2}>0$. Thus near the phase transition point ($\mu_{c}=\mu_{0}$), the order parameter $\epsilon$ vanishes. In addition, we obtain the mean field value of the critical exponent as $\beta=1/2$.

The equation of motion for the gauge field (16) at zeroth order of $\epsilon$ is given by

[TABLE]

The solutions of this equation reads

[TABLE]

Combining the solutions (26) with Eq. (18), the equation for $f(z)$, at zeroth order of $\epsilon$, can be obtained as

[TABLE]

which has the solutions,

[TABLE]

Near the boundary, the vector field can be defined as:

[TABLE]

Inserting Eqs. (28) and (29) in Eq. (17) yields to

[TABLE]

If we define some new functions as below, we can rewrite Eq. (30) in the Sturm-Liouville form as

[TABLE]

where

[TABLE]

Next, we define a trial function $F(z)=1-\alpha z^{2}$ which is satisfied in the boundary conditions $F(0)=1$ and $F^{{}^{\prime}}(0)=0$. By minimizing the following expression with respect to $\alpha$, equation (31) will be solved:

[TABLE]

With the help of iteration method, definition of backreacting parameter is LPJW2015

[TABLE]

Here, $\Delta\kappa=0.05$. In addition, we have

[TABLE]

where $\kappa_{-1}=0$ and $\lambda^{2}|_{\kappa_{-1}}=0$. At the critical point, at zeroth order with respect to $\epsilon$, critical temperature is defined as111Note that $\chi$ tends to zero near the critical point according to (23).

[TABLE]

The analytical results of $T_{c}/\mu$ for different values of mass and backreaction parameters are shown in table 1. According to these results, enlarging the values of mass have the same effect as increasing backreaction parameter on $T_{c}/\mu$ and makes it smaller. So, it causes condensation harder to form.

III.2 Numerical Method

We employ the shooting method Hartnoll (2009) to numerically investigate the properties of ($1+1$)-dimensional holographic $p$-wave superconductor developed in BTZ black hole background, when the gauge and vector fields backreact on the background geometry. For this purpose, we must know the behavior of the model functions both at horizon and boundary. By using Taylor expansion around horizon we arrive at

[TABLE]

We impose the boundary condition $\phi(z=1)$ which is motivated from the fact that the gauge field $A_{\nu}$ has a finite norm at the horizon. In this method, all coefficients will be defined in terms of $\phi_{1}$, $\rho_{x_{0}}$ and $\chi_{0}$. Our desirable state is $\rho_{x_{-}}(\infty)=\chi(\infty)=0$. This will be achieved by varying $\phi_{1}$, $\rho_{x_{0}}$ and $\chi_{0}$ at the horizon. Furthermore, we can set $r_{+}=1$ by virtue of the equations of motion’s symmetry

[TABLE]

The consequence of this method is finding the values of $T_{c}/\mu$ for different masses and backreaction parameters. In order to compare the numerical and analytical results, data are given in table 1. The results of Sturm-Liouville method are confirmed by numerical data. The effects of mass and backreaction parameters on the behaviour of condensation are shown in Fig. 1. We see that all curves follow the same behaviour. As it is clear from Fig. 1, enhancing the values of mass and backreaction parameter causes the gap in curves larger and so it makes the formation of condensation harder. As a result, the critical temperature decreases with increasing the backreaction and mass parameters.

IV Critical exponents

In this section we are going to calculate the expectation value of $\langle J_{x}\rangle$ in the boundary theory near the critical temperature for one-dimensional holographic $p$-wave superconductor in the presence of backreaction. Furthermore, we compute the values of the critical exponents both analytically and numerically.

IV.1 Analytical approach

We focus on the behavior of the gauge field in the vicinity of the critical temperature. In this limit, the field equation (16) turns to

[TABLE]

Because of nonzero value of the condensation in the vicinity of the critical temperature, we have an extra term in the above equation in comparison with the field equation in the previous section. Inserting Eqs. (28) and (29) in Eq. (42) we have

[TABLE]

Using the fact that the value of $\frac{\langle J_{x}\rangle^{2}}{r_{+}^{2\Delta+2}}$ is small in $T\sim T_{c}$ limit, we assume that Eq. (43) has the following answer

[TABLE]

Since at the horizon $\phi(z=1)=0$, thus we have $\eta(1)=0$. Substituting the above equation in Eq. (43) up to $\frac{\langle J_{x}\rangle^{2}}{r_{+}^{2\Delta+2}}$ order, we arrive at

[TABLE]

Multiplying the both sides of Eq. (45) by $z$ and integrating from $z=0$ to $z=1$, we get

[TABLE]

where

[TABLE]

Combining Eqs. (14) and (44) and taking into account the fact that the first term on the rhs of Eq. (44) is the solution of $\phi(z)$ at the critical point, and the second term is a correction term, we can write near the critical point,

[TABLE]

Now, we use a coordinate transformation $z\rightarrow Z+1$, then by expanding the resulting equation around $Z=0$ we get

[TABLE]

Comparing the coefficients $Z$ on both sides of Eq. (49) and using Eq. (46) we find

[TABLE]

Near the critical point we have $T\sim T_{c}$, and thus using relation (36), we can find the equation of $r_{+}$ as below

[TABLE]

Inserting Eqs. (36) and (51) in Eq. (50) and taking the absolute values of the resulting equation, we arrive at

[TABLE]

where

[TABLE]

Based on the above equation, it is obvious that the critical exponent $\beta=1/2$ is in a perfect agreement with the mean field theory. Since the value of $\beta$ is independent of the effect of backreaction, we have the second order phase transition for all values of the backreaction parameter. The analytical results are shown in table 2. Increasing the values of the mass and backreaction parameters, causes the larger values of the condensation parameter. Therefore, the larger values of the mass as well as the backreacting of the gauge and vector fields on the background geometry makes the condensation harder to form.

IV.2 Numerical Method

Based on the behavior of condensation near the critical temperature which is obtained by using analytical approach (i.e. Eq. (52)) we have

[TABLE]

In Fig 2, the behavior of $\log\langle J_{x}\rangle/T_{c}^{\Delta+1}$ as a function of $\log\left(1-T/T_{c}\right)$ for different values of the backreaction and mass parameters was shown. The slope of curves is $1/2$ which is in agreement with mean field theory and shows that we face with a second order phase transition as same as the analytical approach. In addition, this value of critical exponent is independent of the backreaction parameter. Using Eq. (54), it is obvious that the intercept of curves represent the values of $\log\gamma$, numerically. In order to compare the analytical and numerical values of $\gamma$, the results are listed in table 2. The most agreement between the values of $\gamma$ from these two approaches appears in $m^{2}=1/16$ and for larger values of mass we observe less match. In addition, the values of $\gamma$ increase for larger values of backreaction parameter. Same results are obtained in analytical method, too.

V Closing remarks

In this paper, we analyzed a holographic $p$-wave superconductor model in a three-dimensional Einstein-Maxwell theory in the presence of negative cosmological constant and a vector field when the gauge and vector fields backreact on the background geometry. In order to study the problem analytically, we employ the Sturm-Liouville eigenvalue problem while the numerical data were achieved with help of shooting method. We analytically calculated the relation between the critical temperature and chemical potential for different values of the mass and backreaction parameters. These data were confirmed by numerical results. We found out that increasing the values of the mass and backreaction parameters makes the condensation harder to form and thus the critical temperature decreases. In addition, critical exponent of this system have also obtained both analytically and numerically. Based on these investigations, it was expressed that we face a second order phase transition. Furthermore, the obtained critical exponent value $\beta={1}/{2}$ follows the mean field theory value. Since the nonlinear electrodynamics give more information in comparison with Maxwell case, it is worthwhile to consider the effect of nonlinearity on the physical properties of holographic $p$-wave superconductors. We leave this issue for future investigations.

Acknowledgements.

We thank Shiraz University Research Council. The work of AS has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran. MKZ thanks Shahid Chamran University of Ahvaz for supporting this work.