# One-dimensional backreacting holographic $p$-wave superconductors

Mahya Mohammadi  Physics Department and Biruni Observatory, Shiraz University,
Shiraz 71454, Iran  Ahmad Sheykhi<EMAIL_ADDRESS>Physics Department
and Biruni Observatory, Shiraz University, Shiraz 71454, Iran  Research
Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box:
55134-441, Maragha, Iran  Mahdi Kord Zangeneh<EMAIL_ADDRESS>Physics
Department, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz
61357-43135, Iran

###### 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 $T_{c}$ and the chemical potential $\mu$ for different values of
mass $m$ of charged spin-$1$ field $\rho_{\mu}$ 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 ${T_{c}}/{\mu}$. 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.

##  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

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{3}x\sqrt{-g}\left(R+\frac{2}{l^{2}}\right)+\int
d^{3}x\sqrt{-g}\mathcal{L}_{m},$ $\displaystyle\mathcal{L}_{m}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\rho_{\mu\nu}^{\dagger}\rho^{\mu\nu}-m^{2}\rho_{\mu}^{\dagger}\rho^{\mu}+iq\gamma\rho_{\mu}\rho_{\nu}^{\dagger}F^{\mu\nu},$
(1)

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

$\displaystyle\frac{1}{2\kappa^{2}}\left[R_{\mu\nu}-g_{\mu\nu}\left(\frac{R}{2}+\frac{1}{l^{2}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}F_{\mu\lambda}F_{\nu}{}^{\lambda}+\frac{1}{2}\mathcal{L}_{m}g_{\mu\nu}+\frac{1}{2}\left[\rho^{\dagger}{}_{\mu\lambda}\rho_{\nu}^{\lambda}+m^{2}\rho^{\dagger}{}_{\mu}\rho_{\nu}-i\gamma
qF_{\nu}^{\lambda}\left(\rho_{\mu}\rho^{\dagger}{}_{\lambda}-\rho^{\dagger}{}_{\mu}\rho_{\lambda}\right)+\mu\leftrightarrow\nu\right].$
$\nabla^{\nu}F_{\nu\mu}=iq\left(\rho^{\nu}\rho^{\dagger}{}_{\nu\mu}-\rho^{\nu\dagger}\rho_{\nu\mu}\right)+iq\gamma\nabla^{\nu}\left(\rho_{\nu}\rho^{\dagger}{}_{\mu}-\rho^{\dagger}{}_{\nu}\rho_{\mu}\right),$
(3) $D^{\nu}\rho_{\nu\mu}-m^{2}\rho_{\mu}+iq\gamma\rho^{\nu}F_{\nu\mu}=0,$ (4)

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,

${ds}^{2}=-f(r)e^{-\chi(r)}{dt}^{2}+\frac{{dr}^{2}}{f(r)}+r^{2}{dx}^{2},$ (5)

with the following choices for the vector and gauge fields,

$\rho_{\nu}dx^{\nu}=\rho_{x}(r)dx,\ \ \ A_{\nu}dx^{\nu}=\phi(r)dt.$ (6)

The Hawking temperature of this black hole is given by mahya

$T=\frac{e^{-\chi(r_{+})/2}f^{{}^{\prime}}(r_{+})}{4\pi}.$ (7)

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

$\phi^{\prime\prime}(r)+\left[\frac{1}{2}\chi^{\prime}(r)+\frac{1}{r}\right]\phi^{\prime}(r)-\frac{2q^{2}\text{$\rho_{x}$}(r)^{2}\phi(r)}{r^{2}f(r)}=0,$
(8)
$\rho^{\prime\prime}_{x}(r)+\left[\frac{f^{\prime}(r)}{f(r)}-\frac{1}{2}\chi^{\prime}(r)-\frac{1}{r}\right]\rho^{\prime}_{x}(r)+\left[\frac{q^{2}e^{\chi(r)}\phi(r)^{2}}{f(r)^{2}}-\frac{m^{2}}{f(r)}\right]\rho_{x}(r)=0,$
(9)

$f^{\prime}(r)-\frac{2r}{l^{2}}+\frac{2\kappa^{2}}{r}\left[\frac{q^{2}\text{$\rho_{x}$}(r)^{2}e^{\chi(r)}\phi(r)^{2}}{f(r)}+f(r)\rho^{\prime
2}_{x}+m^{2}\rho_{x}(r)^{2}+\frac{r^{2}}{2}e^{\chi(r)}\phi^{\prime
2}\right]=0,$ (10)
$\chi^{\prime}(r)+\frac{4\kappa^{2}}{r}\left[\frac{q^{2}\text{$\rho_{x}$}(r)^{2}e^{\chi(r)}\phi(r)^{2}}{f(r)^{2}}+{\rho^{\prime}_{x}}^{2}\right]=0.$
(11)

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

$\displaystyle q\rightarrow q/a,\text{ \ \ \ \ }\phi\rightarrow a\phi,\text{ \
\ \ \ }\rho_{x}\rightarrow a\rho_{x},\ \ \ \kappa\rightarrow\kappa/a,$ (12)
$\displaystyle l\rightarrow al,\text{ \ \ \ \ }r\rightarrow ar,\text{ \ \ \ \
}q\rightarrow q/a,\ \ \ m\rightarrow m/a.$ (13)

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

$\phi(r)\sim\rho+\mu\ln(r),\ \ f(r)\sim r^{2},\ \ \chi(r)\rightarrow 0,\ \
\rho_{x}(r)\sim\frac{\rho_{x_{-}}}{r^{-m}}+\frac{\rho_{x_{+}}}{r^{m}}.$ (14)

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,

$e^{\chi}\rightarrow a^{2}e^{\chi},\text{ \ \ \ \ }t\rightarrow at,\text{ \ \
\ \ }\phi\rightarrow\phi/a.$ (15)

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

$\phi^{\prime\prime}(z)+\left(\frac{\chi^{\prime}(z)}{2}+\frac{1}{z}\right)\phi^{\prime}(z)-\frac{2\rho_{x}(z)^{2}\phi(z)}{z^{2}f(z)}=0,$
(16)
$\rho_{x}^{\prime\prime}(z)+\left(-\frac{1}{2}\chi^{\prime}(z)+\frac{f^{\prime}(z)}{f(z)}+\frac{3}{z}\right)\rho_{x}^{\prime}(z)+\rho_{x}(z)\left(\frac{r_{+}^{2}e^{\chi(z)}\phi(z)^{2}}{z^{4}f(z)^{2}}-\frac{m^{2}r_{+}^{2}}{z^{4}f(z)}\right)=0,$
(17)
$f^{\prime}(z)+\frac{2r_{+}^{2}}{l^{2}z^{3}}-2\kappa^{2}\left(\frac{e^{\chi(z)}\phi(z)^{2}\rho_{x}(z){}^{2}}{zf(z)}+\frac{z^{3}f(z)\rho_{x}^{\prime
2}}{r_{+}^{2}}+\frac{m^{2}\rho_{x}(z){}^{2}}{z}+\frac{z}{2}e^{\chi(z)}\phi^{\prime
2}\right)=0,$ (18) $\chi^{\prime
2}\left(\frac{e^{\chi(z)}\phi(z)^{2}\rho_{x}(z){}^{2}}{zf(z)^{2}}+\frac{z^{3}\rho_{x}^{\prime
2}}{r_{+}^{2}}\right)=0.$ (19)

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

$\epsilon\equiv\left\langle J_{x}\right\rangle.$

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

$\displaystyle\rho_{x}\approx\epsilon\rho_{x_{1}}+\epsilon^{3}\rho_{x_{3}}+\epsilon^{5}\rho_{x_{5}}+\cdots,$
(20)
$\displaystyle\phi\approx\phi_{0}+\epsilon^{2}\phi_{2}+\epsilon^{4}\phi_{4}+\cdots,$
(21) $\displaystyle f\approx
f_{0}+\epsilon^{2}f_{2}+\epsilon^{4}f_{4}+\cdots,$ (22)
$\displaystyle\chi\approx\epsilon^{2}\chi_{2}+\epsilon^{4}\chi_{4}+\cdots.$
(23)

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

$\mu=\mu_{0}+\epsilon^{2}\delta\mu_{2}+...\rightarrow\epsilon\thickapprox\Bigg{(}\frac{\mu-\mu_{0}}{\delta\mu_{2}}\Bigg{)}^{1/2},$
(24)

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

$\phi^{\prime\prime}(z)+\frac{\phi^{\prime}(z)}{z}=0.$ (25)

The solutions of this equation reads

$\phi(z)=\lambda r_{+}\log(z),\ \ \ \lambda=\frac{\mu}{r_{+}}.$ (26)

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

$f^{\prime}(z)+\frac{2r_{+}^{2}}{z^{3}}-\frac{\kappa^{2}\lambda^{2}r_{+}^{2}}{z}=0,$
(27)

which has the solutions,

$f(z)=\frac{r_{+}^{2}g(z)}{z^{2}},\text{ \ \ \ \
}g(z)=1-z^{2}+\kappa^{2}\lambda^{2}z^{2}\log(z).$ (28)

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

$\rho_{x}(z)=\frac{\langle
J_{x}\rangle}{\sqrt{2}r_{+}^{\Delta}}z^{\Delta}F(z).$ (29)

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

$F^{\prime\prime}(z)+F^{\prime}(z)\left(\frac{g^{\prime}(z)}{g(z)}+\frac{2\Delta}{z}+\frac{1}{z}\right)+F(z)\left(\frac{\Delta
g^{\prime}(z)}{zg(z)}-\frac{m^{2}}{z^{2}g(z)}+\frac{\Delta^{2}}{z^{2}}\right)+\frac{F(z)\left(\lambda^{2}\kappa^{2}\log^{2}(z)\right)}{g(z)^{2}}=0.$
(30)

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

$\left[T(z)F^{\prime}(z)\right]^{\prime}+P(z)T(z)F(z)+\lambda^{2}Q(z)T(z)F(z)=0,$
(31)

where

$T(z)=z^{2\Delta+1}g(z),\text{ \ \ \ \ }P(z)=\left[\frac{\Delta
g^{\prime}(z)}{zg(z)}-\frac{m^{2}}{z^{2}g(z)}+\frac{\Delta^{2}}{z^{2}}\right],\text{
\ \ \ \ }Q(z)=\frac{\log^{2}(z)}{g(z)^{2}}.$ (32)

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:

$\lambda^{2}=\frac{\int_{0}^{1}T\left(F^{\prime
2}-PF^{2}\right)dz}{\int_{0}^{1}TQF^{2}dz},$ (33)

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

$\kappa_{n}=n\Delta\kappa,\ \ \ n=0,1,2,\cdots,\ \
\Delta\kappa=\kappa_{n+1}-\kappa_{n}.$ (34)

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

$\kappa^{2}\lambda^{2}={\kappa_{n}}^{2}\lambda^{2}={\kappa_{n}}^{2}(\lambda^{2}|_{\kappa_{n-1}})+O[(\Delta\kappa)^{4}],$
(35)

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).

$T_{c}=\frac{f^{\prime}\left(r_{+c}\right)}{4\pi}=r_{+c}\left(\frac{2-{\kappa}^{2}\lambda^{2}}{4\pi}\right)=\frac{\mu}{\lambda}\left(\frac{2-{\kappa}_{n}^{2}\lambda^{2}|_{\kappa_{n-1}}}{4\pi}\right).$
(36)

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

$\displaystyle f(z)=f_{1}\left(1-z\right)+f_{2}\left(1-z\right){}^{2}+\cdots,$
(37)
$\displaystyle\phi(z)=\phi_{1}\left(1-z\right)+\phi_{2}\left(1-z\right){}^{2}+\cdots,$
(38)
$\displaystyle\rho_{x}(z)=\rho_{x_{0}}+\rho_{x_{1}}\left(1-z\right)+\rho_{x_{2}}\left(1-z\right){}^{2}+\cdots,$
(39)
$\displaystyle\chi(z)=\chi_{0}+\chi_{1}\left(1-z\right)+\chi_{2}\left(1-z\right){}^{2}+\cdots.$
(40)

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

$r\rightarrow ar,\text{ \ \ \ \ }f\rightarrow a^{2}f,\text{ \ \ \ \
}\phi\rightarrow a\phi.$ (41)

(a) $m^{2}=\frac{1}{16}$ (b) $m^{2}=\frac{1}{4}$

(c) $m^{2}=1$

Figure 1: The behavior of condensation parameter as a function of temperature
for different values of backreaction.

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.

| $m^{2}=\dfrac{1}{16}$ | $m^{2}=\dfrac{1}{4}$ | $m^{2}=1$
---|---|---|---
| 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 1: Analytical and Numerical results of ${T_{c}}/{\mu}$ for different
values of 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

$\phi^{\prime\prime}(z)+\frac{\phi^{\prime}(z)}{z}-\frac{2\rho_{x}(z)^{2}\phi(z)}{z^{2}f(z)}=0.$
(42)

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

$\phi^{\prime\prime}(z)+\frac{\phi^{\prime}(z)}{z}=\frac{\langle
J_{x}\rangle^{2}z^{2\Delta}}{r_{+}^{2\Delta+2}g(z)}F(z)^{2}\phi(z),$ (43)

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

$\frac{\phi(z)}{r_{+}}=\lambda\log(z)+\frac{\langle
J_{x}\rangle^{2}}{r_{+}^{2\Delta+2}}\eta(z),\ \ \ \lambda=\frac{\mu}{r_{+}}.$
(44)

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

$\eta^{\prime\prime}(z)+\frac{\eta^{\prime}(z)}{z}=\frac{\lambda
z^{2\Delta}\log(z)}{g(z)}F(z)^{2}.$ (45)

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

$\int_{0}^{1}d\left(z\eta^{\prime}(z)\right)=\eta^{\prime}(1)=\lambda\int_{0}^{1}\frac{z^{2\Delta+1}\log(z)}{1-z^{2}+\kappa^{2}\lambda^{2}z^{2}\log(z)}F(z)^{2}\,dz=\lambda\mathcal{A},$
(46)

where

$\mathcal{A}\equiv\int_{0}^{1}\frac{z^{2\Delta+1}\log(z)}{1-z^{2}+\kappa^{2}\lambda^{2}z^{2}\log(z)}F(z)^{2}\,dz.$
(47)

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,

$\frac{\rho}{r_{+}}+\frac{\mu}{r_{+}}\log(z)=+\frac{\mu}{r_{+c}}\log(z)+\frac{\langle
J_{x}\rangle^{2}}{r_{+c}^{2\Delta+2}}\eta(z),$ (48)

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

$\frac{\rho}{r_{+}}+\frac{\mu}{r_{+}}\left(Z-\frac{Z^{2}}{2}+...\right)=+\frac{\mu}{r_{+c}}\left(Z-\frac{Z^{2}}{2}+...\right)+\frac{\langle
J_{x}\rangle^{2}}{r_{+c}^{2\Delta+2}}\left(\eta(1)+Z\eta^{\prime}(1)+...\right).$
(49)

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

$\frac{\mu}{r_{+}}=\frac{\mu}{r_{+c}}\left(1+\frac{\langle
J_{x}\rangle^{2}}{r_{+c}^{2\Delta+2}}\mathcal{A}\right),$ (50)

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

$r_{+}=\frac{4\pi T}{\left(2-\kappa^{2}\lambda^{2}\right)},$ (51)

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

$\langle J_{x}\rangle=\gamma T_{c}^{\Delta+1}\sqrt{1-\frac{T}{T_{c}}},$ (52)

where

$\gamma=\frac{1}{\sqrt{\left|\mathcal{A}\right|}}\left(\frac{4\pi}{2-\kappa^{2}\lambda^{2}}\right)^{\Delta+1}.$
(53)

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

$\log\left(\frac{\langle
J_{x}\rangle}{T_{c}^{\Delta+1}}\right)=\log(\gamma)+\frac{1}{2}\log\left(1-\frac{T}{T_{c}}\right).$
(54)

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.

| $m^{2}=\dfrac{1}{16}$ | $m^{2}=\dfrac{1}{4}$ | $m^{2}=1$
---|---|---|---
| Analytical | Numerical | Analytical | Numerical | Analytical | Numerical
$\kappa^{2}=0$ | $23.5779$ | $20.0224$ | $48.1780$ | $70.3273$ | $188.5$ | $1042.6100$
$\kappa^{2}=0.05$ | $23.7791$ | $20.5590$ | $49.1955$ | $76.0058$ | $199.517$ | $1390.0400$
$\kappa^{2}=0.1$ | $23.9228$ | $21.1162$ | $50.2391$ | $82.3205$ | $220.382$ | $1901.3500$
$\kappa^{2}=0.15$ | $24.1599$ | $21.6954$ | $51.9956$ | $89.3687$ | $260.302$ | $2682.0100$
$\kappa^{2}=0.2$ | $24.4981$ | $22.2974$ | $54.6113$ | $97.2622$ | $333.956$ | $3928.0200$
$\kappa^{2}=0.25$ | $24.9451$ | $22.9237$ | $58.2727$ | $106.138$ | $475.166$ | $6028.6000$
Table 2: Analytical and Numerical results of $\gamma$ for different values of
backreaction and mass parameters.

(a) $m^{2}=\frac{1}{16}$ (b) $m^{2}=\frac{1}{4}$

(c) $m^{2}=1$

Figure 2: The behavior of $\log\langle J_{x}\rangle/T_{c}^{\Delta+1}$ as a
function of $\log\left(1-T/T_{c}\right)$ with slope of $1/2$ for different
values of mass and backreaction parameters.

##  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.

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