Noncommutative effects on holographic superconductors with power Maxwell electrodynamics

Abstract

The matching method is employed to analytically investigate the properties of holographic superconductors in higher dimensions in the framework of power Maxwell electrodynamics taking into account the effects of spacetime noncommutativity. The relationship between the critical temperature and the charge density and the value of the condensation operator is obtained first. The Meissner like effect is then studied. The analysis indicate that larger values of the noncommutative parameter and the parameter q appearing in the power Maxwell theory makes the condensate difficult to form. The critical magnetic field however increases with increase in the noncommutative parameter θ\theta.

Full text

Noncommutative effects on holographic
superconductors with power Maxwell
electrodynamics
Suchetana Pala∗
,Sunandan Gangopadhyaya,b†
aIndian Institute of Science Education and Research, Kolkata
Mohanpur, Nadia 741246, India
bVisiting Associate in Inter University Centre for Astronomy & Astrophysics,
Pune 411007, India
Abstract
The matching method is employed to analytically investigate the properties of holo-
graphic superconductors in higher dimensions in the framework of power Maxwell
electrodynamics taking into account the effects of spacetime noncommutativity. The
relationship between the critical temperature and the charge density and the value
of the condensation operator is obtained first. The Meissner like effect is then stud-
ied. The analysis indicate that larger values of the noncommutative parameter and
the parameter qappearing in the power Maxwell theory makes the condensate dif-
ficult to form. The critical magnetic field however increases with increase in the
noncommutative parameter θ.
1 Introduction
It is well known that the BCS theory of superconductivity [1] has proved to be an impor-
tant theoretical discovery which describes the properties of low temperature superconduc-
tors. However, there has been no successful theory yet to understand the superconductiv-
ity in a class of materials that exhibit the phenomenon at high temperatures. Examples
of such materials are the high Tccuprates. It is reasonably evident that these are strongly
coupled materials and hence lies the difficulty in developing a theory for them.
Recently, there has been an upsurge in the study of holographic superconductors.
The importance of these were realized as they reproduced some properties of high Tc
superconductors. The theoretical input that goes in the construction of these holographic
superconductor models is the gauge/gravity correspondence first discovered in the con-
text of string theory [2]-[4]. The physical mechanism behind this understanding involves
∗suchetanapal92@gmail.com, sp15rs004@iiserkol.ac.in
†sunandan.gangopadhyay@gmail.com, sunandan@iiserkol.ac.in, sunandan@associates.iucaa.in
1
arXiv:1708.06240v2 [hep-th] 24 Aug 2017

the demonstration of a spontaneous symmetry breaking in an Abelian Higgs model in a
black hole background that is asymptotically AdS [5]-[8]. Thereafter, several important
properties of these gravity models have been studied analytically [9]-[24].
An important aspect of superconductors is their response to an external magnetic field
known as the Meissner effect [25]. The response shows perfect diamagnetism as the tem-
perature is lowered below a critical temperature Tc. Quite a few number of investigations
have been carried out both numerically [26]-[28] as well as analytically [29]-[33]. However,
these studies were restricted mainly to the framework of Maxwell electrodynamics. In [30],
the conventional action for Maxwell electrodynamics was replaced by the power Maxwell
action [34]. The motivation for such a study comes from the question of investigating the
behaviour of the condensate in the presence of higher curvature corrections coming from
the power Maxwell theory.
Noncommutativity of spacetime is another important area of theoretical physics where
considerable research has been carried out. The idea which first came in 1947 [35] was
brought into forefront from studies in string theory [36]. Very recently, black hole back-
grounds have been provided incorporating the ideas of noncommutativity [37]-[39]. The
noncommutative effect gets introduced here by a smeared source of matter. This is then
used to solve Einstein’s equation of general relativity.
In this paper, we want to investigate the role of noncommutative spacetime on the
properties of holographic superconductors. Such an investigation has been carried out
earlier in [40] in the framework of Born-Infeld electrodynamics. Here, we shall carry out
such a study in power Maxwell electrodynamics using the matching method approach
[11] in the probe limit approximation which essentially means that the backreaction of
the spacetime has been neglected. Moreover, we shall also investigate the Meissner effect
in this framework and study the role played by spacetime noncommutativity.
The paper is organized as follows. In section 2, the basic formalism for the d-
dimensional holographic superconductor in noncommutative spacetime coupled to power
Maxwell electrodynamics is presented. In section 3, we obtain the relationship between
the critical temperature and the charge density and the value of the condensation opera-
tor using the matching method approach. In section 4, we investigate the Meissner like
effect using the same approach. We finally conclude in section 5.
2 Basic analytical set up
The analysis proceeds by setting up a gravitational dual for higher dimensional supercon-
ductors. We consider the metric of a d-dimensional planar noncommutative Schwarzschild-
AdS spacetime as the gravity dual of the holographic superconductor. The metric of such
a black hole spacetime reads
ds2=−f(r)dt2+1
f(r)dr2+r2dxidxi(1)
where f(r) is given by [37]
f(r) = r2
L2−2MGd
rd−3Γ(d−1
2)γd−1
2,r2
4θ+k(2)
2

and γ(s, x) is the lower incomplete gamma function given by
γ(s, x) = Zx
0
ts−1e−tdt (3)
and kdenotes the curvature. dxidxirepresents the line element of a (d−2)-dimensional
hypersurface with vanishing curvature. In this analysis, we shall set k= 0 since we require
a planar holographic superconductor. The metric therefore takes the form
f(r) = r2
L2−2MGd
rd−3Γd−1
2γd−1
2,r2
4θ.(4)
The horizon radius can be obtained by setting [f(r)]r=r+= 0 :
rd−1
+=2MGd
Γ(d−1
2)γd−1
2,r2
+
4θ(5)
where we have set L= 1 for convenience. Using eq.(5), eq.(4) can be recast as
f(r) = r2 1−rd−1
+
rd−1
γ(d−1
2,r2
4θ)
γ(d−1
2,r2
+
4θ)!.(6)
The Hawking temperature Tof the black hole is given by
T=f0(r+)
4π.(7)
By the gauge/gravity duality, this temperature is interpreted as the temperature of the
boundary field theory. Using eq.(s) (4) and (5), we get
T=1
4π"(d−1)r+−r2
+
γ0(d−1
2,r2
+
4θ)
γ(d−1
2,r2
+
4θ)#
=r+
4π
d−1−4MGd
Γ(d−1
2)
e−r2
+
4θ
(4θ)d−1
2
.(8)
In the limit θ→0, the coefficient of the t-component of the noncommutative metric
reduces to
f(r) = r21−rd−1
+
rd−1(9)
where r+= 2MGd. This metric is the planar Schwarzschild-AdS black hole in d-spacetime
dimensions.
The Hawking temperature for this black hole reads
T=(d−1)r+
4π.(10)
We now write down an appropriate action for the bulk which can explain the phase tran-
sition at the boundary. The action involves a gravity theory with a negative cosmological
3

constant together with a complex scalar field ψminimally coupled to the power Maxwell
field
S=Zddx√−gR−2Λ −β(Fµν Fµν )q−|5µψ−iAµψ|2−m2|ψ|2(11)
where βis the coupling constant of power Maxwell electrodynamics, qis the power pa-
rameter of the power Maxwell field, Λ = −(d−1)(d−2)
2is the cosmological constant.
In the subsequent discussion, we shall work in the probe limit which means that the effect
of back reaction of the matter fields on the metric is neglected.
The equations of motion for the Maxwell and the scalar field can be obtained by varying
the action. These read
4βq
√−g∂µ(√−g(FλσFλσ)q−1Fµν )−i(ψ∗∂νψ−ψ(∂νψ)∗)−2Aν|ψ|2= 0 (12)
∂µ(√−g∂µψ)−i√−gAµ∂µψ−i∂µ(√−gAµψ)−√−gAµAµψ−√−gm2ψ= 0 .(13)
To make further progress, we make the following ansatz
A=φ(r)dt , ψ =ψ(r).(14)
With this ansatz, eq.(s)(12) and (13) now take the form
∂2
rφ+1
rd−2
2q−1∂rφ−2φψ2(∂rφ)2(1−q)
(−1)q−12q+1βq(2q−1)f(r)= 0 (15)
∂2
rψ+f0
f+d−2
r∂rψ+φ2ψ
f2−m2ψ
f= 0 .(16)
Making a change of variable z=r+
r, the above equations take the form
∂2
zφ+1
z2−d−2
2q−1∂zφ+2φ(z)ψ2(z)r2q
+(∂zφ)2(1−q)
z4q2q+1(−1)3qβq(2q−1)f(z)= 0 (17)
∂2
zψ+f0(z)
f(z)−d−4
z∂zψ+φ2ψr2
+
z4f2(z)−m2ψr2
+
z4f(z)= 0 .(18)
In the limit z→0, the asymptotic behaviour of the fields φand ψcan be expressed as
φ(z) = µ−ρ1
2q−1
r
d−2
2q−1−1
+
zd−2
2q−1−1(19)
ψ(z) = ψ−
rλ−
+
zλ−+ψ+
rλ+
+
zλ+(20)
where
λ±=1
2hd−1±p(d−1)2+ 4m2i.(21)
The gauge/gravity duality interprets µand ρas the chemical potential and the charge
density. ψ+and ψ−are the vacuum expectation values of the dual operator O. For the
rest of our analysis, we shall choose ψ+=hO+iand ψ−= 0.
4

3 Critical temperature, charge density relation and
condensation operator
To begin with, we start by substituting z=r+
rin eq. (6). This yields
f(z) = r2
+
z2g0(z) (22)
where
g0(z) = 1−zd−1γ(d−1
2,r2
+
4θz2)
γ(d−1
2,r2
+
4θ)!.(23)
We now apply the matching method to proceed further. Near the horizon, that is z= 1,
we make Taylor series expansions of the fields φ(z) and ψ(z). These read
φ(z) = φ(1) −φ0(1)(1 −z) + 1
2φ00 (1)(1 −z)2+O((1 −z)3)
=−φ0(1)(1 −z) + 1
2φ00 (1)(1 −z)2+O((1 −z)3) (24)
ψ(z) = ψ(1) −ψ0(1)(1 −z) + 1
2ψ00 (1)(1 −z)2+O((1 −z)3) (25)
since φ(1) = 0.
To evaluate the expressions for φ00(1) and ψ0(1), ψ00 (1), we look at eq.(s) (17) and (18) for
z= 1. This yields
φ00 (1) = d−2
2q−1−2φ0(1) −2r2(q−1)
+ψ2(1)(φ0(1))3−2q
2q+1(−1)3q(βq)(2q−1)g0
0(1) (26)
ψ0(1) = m2
g0
0(1)ψ(1) (27)
ψ00 (1) = 1
2d−4−g00
0(1)
g0
0(1) +m2
g0
0(1)m2
g0
0(1)ψ(1) −φ02(1)ψ(1)
2r2
+g02
0(1) .(28)
Substituting these expressions in eq.(s) (24) and (25), we get
φ(z) = −φ0(1)(1 −z)
+1
2(1 −z)2"d−2
2q−1−2φ0(1) −2r2(q−1)
+ψ2(1)(φ0(1))3−2q
(−1)3q2q+1(βq)(2q−1)g0
0(1)#(29)
ψ(z) = ψ(1) −m2
g0
0(1)ψ(1)(1 −z)
+1
2(1 −z)21
2d−4−g00
0(1)
g0
0(1) +m2
g0
0(1)m2
g0
0(1)ψ(1) −φ02(1)ψ(1)
2r2
+g02
0(1) .(30)
5

We now proceed to implement the matching method. We match the above solutions with
the asymptotic solutions (19) and (20) at z=zm. This gives the following relations
µ−ρ1
2q−1z
d−2
2q−1−1
m
(r+)d−2
2q−1−1=
v(1 −zm) + 1
2(1 −zm)2"2−d−2
2q−1v−2r2(q−1)
+α2(−v)3−2q
(−1)3q2q+1(βq)(2q−1)(g0
0)#(31)
hO+izλ+
m
rλ+
+
=
α−(1−zm)αm2
g0
0(1)+1
2α(1−zm)21
2m2
g0
0(1)d−4−g00
0(1)
g0
0(1) +m2
g0
0(1)−˜v2
2g0
0(1)2.
(32)
where v=−φ0(1), α =ψ(1) and ˜v=v
r.
Taking derivative on both sides of eq.(31) and (32) yields
−ρ1
2q−1z
d−2
2q−1−2
m
(r+)d−2
2q−1−1d−2
2q−1−1=
−v−(1 −zm)"2−d−2
2q−1v−2r2(q−1)
+α2(−v)3−2q
(−1)3q2q+1(βq)(2q−1)g0
0(1)#(33)
λ+hO+izλ+−1
m
rλ+
+
=
αm2
g0
0(1)−α(1 −zm)1
2m2
g0
0(1)d−4−g00
0(1)
g0
0(1) +m2
g0
0(1)−˜v2
2g0
0(1)2.(34)
From the above set of equations together with eq.(8), it is simple to obtain
α2≡α2
NC =−(−1)5q−32q(βq)(2q−1)g0
0(1)
˜v2(1−q)
NC (1 −zm)
×1 + 2−d−2
2q−1(1 −zm)×(Tc)NC
Td−2
2q−1"1−T
(Tc)NC d−2
2q−1#(35)
where
(Tc)NC =ξNC ρ1
d−2(36)
ξNC =−z(d−2
2q−1−2)( 2q−1
d−2)
m
˜v
2q−1
d−2
NC g0
0(1)
4π(d−2
2q−1−1)2q−1
d−2
[1 + (2 −d−2
2q−1)(1 −zm)]2q−1
d−2
.(37)
6

Note that NC in the above equations stand for the noncommutative case. The above
results give the relation between the critical temperature and the charge density. It can
be observed from the analytical results that the critical temperature decreases with in-
crease in the noncommutative parameter θwhich clearly indicate that the condensate
gets harder to form as the spacetime noncommutativity increases. However, as the mass
of the black hole increases, the critical temperature for a particular value of θincreases
which tells that the effects of spacetime noncommutativity becomes prominent for lower
mass black holes. Further, we can infer from the Tables 3 and 4 (comparing the results
with Table 2) that the onset of power Maxwell electrodynamics (for a value of q6= 1)
makes the condensate difficult to form. However, in this case also the effect of the power
Maxwell theory on the formation of the condensate decreases with increase in the mass
of the black holes.
In the Tables 1, 2, 3 and 4, we present the analytical results for ξNC for different values
of Mand θ.
Table 1: Analytical values of ξNC for different values of Mand θ[q= 1, m2= 0, zm= 0.5
and d= 5]
θ ξNC
MGd= 10 MGd= 50 M Gd= 100
0.3 0.1507 0.16933 0.1702
0.5 0.1384 0.16058 0.1678
0.7 0.1395 0.1492 0.1608
0.9 0.1439 0.1418 0.1525
Table 2: Analytical values of ξNC for different values of Mand θ[q= 1, m2=−3,
zm= 0.5 and d= 5]
θ ξNC
MGd= 10 MGd= 50 M Gd= 100
0.3 0.1761 0.2003 0.2015
0.5 0.1649 0.18798 0.1977
0.7 0.1701 0.1744 0.1883
0.9 0.1767 0.1669 0.1782
Table 3: Analytical values of ξNC for different values of Mand θ[q= 5/4, m2=−3,
zm= 0.5 and d= 5]
θ ξNC
MGd= 10 MGd= 50 M Gd= 100
0.3 0.1015 0.1126 0.1134
0.5 0.0981 0.1067 0.1114
0.7 0.1020 0.1008 0.1069
0.9 0.1056 0.0980 0.1024
7

Table 4: Analytical values of ξNC for different values of Mand θ[q= 7/4, m2=−3,
zm= 0.5 and d= 5]
θ ξNC
MGd= 10 MGd= 50 M Gd= 100
0.3 0.0167 0.0177 0.0178
0.5 0.0172 0.0171 0.0176
0.7 0.0183 0.0167 0.0171
0.9 0.0187 0.0168 0.0168
From eq.(s).(32) and (34), we obtain the expression for ˜vN C
˜v≡˜v2
NC =m4+m2g0
0(1) d−4−g00
0(1)
g0
0(1) −4m2g0
0(1)(zm+λ+(1 −zm))
(1 −zm)(2zm+λ+(1 −zm))
+4g02
0(1)λ+
(1 −zm)(2zm+λ+(1 −zm)).(38)
Now we have all the expressions in hand required to compute the condensation operator
for this problem. The expression for hO+ican be obtained by substituting ˜vNC from eq.
(38) in eq.(33). This gives
hO+iNC =
rλ+
+(1 −m2(1−zm)
2g0
0(1) )
zλ+
m(1 + λ+(1−zm)
2zm)pANC ×(Tc)NC
Td−2
2(2q−1) s1−T
(Tc)NC d−2
2q−1
(39)
where
ANC =(−1)5q−32q(βq)(2q−1)(−g0
0(1))
˜v2(1−q)
NC (1 −zm)×1 + 2−d−2
2q−1(1 −zm).(40)
Now we proceed to take the θ→0 limit of the above findings. The expression for
α≡αC=ψ(1) now reduces to1
α2
C=(−1)5q−12q(βq)(2q−1)(d−1)
˜v2(1−q)(1 −zm)
×1 + 2−d−2
2q−1(1 −zm)×(Tc)C
Td−2
2q−1"1−T
(Tc)Cd−2
2q−1#(41)
where
(Tc)C=ξCρ1
d−2(42)
1It is to be noted that our expression differs from that in [30] since an algebraic error was made in
that paper.
8

with ξCgiven by
ξC=z(d−2
2q−1−2)( 2q−1
d−2)
m
˜v
2q−1
d−2
Cd−1
4π(d−2
2q−1−1)2q−1
d−2
[1 + (2 −d−2
2q−1)(1 −zm)]2q−1
d−2
.(43)
The expression for ˜v≡˜vCand the condensation operator in the commutative case take
the form
˜vC=m4+ 2m2(d−1) 2(zm+λ+(1 −zm))
(1 −zm)(2zm+λ+(1 −zm)) + 1+4λ+(d−1)2
(1 −zm)(2zm+λ+(1 −zm))1
2
(44)
hO+iC=rλ+
+(1 + m2(1−zm)
2(d−1) )
zλ+
m(1 + λ+(1−zm)
2zm)pAC×(Tc)C
Td−2
2(2q−1) s1−T
(Tc)Cd−2
2q−1
(45)
where
AC=(−1)5q−32q(βq)(2q−1)(d−1)
˜v2(1−q)
C(1 −zm)×1 + 2−d−2
2q−1(1 −zm).(46)
In Table 5, we display the analytical results for ξCfor the commutative case. In Figure
Table 5: Analytical values of ξCfor zm= 0.5 and d= 5
m2ξC
q= 1 q= 5/4q= 7/4
0 0.17028 0.0880 0.0117
-3 0.2017 0.1135 0.0179
5, we use the analytical results in Tables 2, 3 and 4 to plot ξvs θfor different values of
the power Maxwell parameter.
Figure 1: ξvs θplot : zm= 0.5, M Gd= 100, m2=−3, d = 5
9

4 Meissner like effect
In this section we introduce an external magnetic field Bin the bulk theory and observe
how the condensation behaves at low temperature for noncommutative black hole back-
ground in the bulk. The intention is to find a critical magnetic filed Bcabove which the
condensation vanishes. We therefore make the following ansatz
At=φ(z), Ay=Bx , ψ =ψ(x, z).(47)
The equation of motion for the complex scalar field ψthat follows from the above ansatz
reads
∂2
zψ(x, z) + f0(z)
f(z)−d−4
z∂zψ(x, z)
+φ2(z)ψ(x, z)r2
+
z4f2(z)−m2r2
+ψ(x, z)
z4f(z)+1
z2f(z)(∂2
xψ−B2x2ψ)=0.(48)
For solving the above equation, we write ψ(x, z) as
ψ(x, z) = X(x)R(z).(49)
Substituting eq.(49) in eq.(48), we arrive at the following expression
z2f(z)
R(z)∂2
zR(z) + f0(z)
f(z)−d−4
z∂zR(z)+φ2(z)r2
+
z2f(z)−m2r2
+
z2
−1
X(x)−∂2
xX(x) + B2x2X(x)= 0 .(50)
This equation implies that X(x) satisfies a 1-dimensional simple harmonic oscillator equa-
tion with frequency B
−X00 (x) + B2x2X(x) = λnBX (x) (51)
where the separation constant is given by λn= 2n+ 1. For the rest of our analysis we
shall set n= 0, since this corresponds to the most stable mode. The equation for R(z)
has the following form
R00 (z) + f0(z)
f(z)−d−4
zR0(z) + φ2(z)r2
+R(z)
z4f2(z)−m2r2
+R(z)
z4f(z)=BR(z)
z2f(z).(52)
Now we shall expand R(z) in a Talylor series around z= 1 and equate it with the
asymptotic solution of R(z) at some point z=zm.
The Taylor series expansion of R(z) around z= 1 reads
R(z) = R(1) −R0(1)(1 −z) + 1
2R00 (1)(1 −z)2+O(1 −z)3.(53)
Further the asymptotic form for R(z) reads
R(z) = hOi+
rλ+
+
zλ+.(54)
10

Equating these at z=zmyields
"hOi+
rλ+
+
zλ+#z=zm
=R(1) −R0(1)(1 −z) + 1
2R00 (1)(1 −z)2+O(1 −z)3z=zm
.(55)
Differentiating eq.(s)(53) and (54) with respect to zand evaluating at z=zmyields
"λ+hOi+
rλ+
+
zλ+−1#z=zm
=hR0(1) −R00(1)(1 −z) + O(1 −z)3iz=zm
.(56)
Now for the noncommutative black hole spacetime (6), we have from eq.(52)
R0(1) = m2
g0
0(1) +B
r2
+g0
0(1)R(1) (57)
R00 (1) = 1
2d−4 + m2
g0
0(1) +B
r2
+g0
0(1) −g00
0(1)
g0
0(1)  m2
g0
0(1) +B
r2
+g0
0(1)R(1)
+BR(1)
r2
+g0
0(1) −φ02(1)R(1)
2r2
+g02
0(1) .(58)
Substituting R0(1) and R00 (1) in eq.(s)(55) and (56), we have
"hOi+
rλ+
+
zλ+
m#=R(1) −m2
g0
0(1) +B
r2
+g0
0(1)(1 −zm)R(1)
+1
2(1 −zm)2[1
2d−4 + m2
g0
0(1) +B
r2
+g0
0(1) −g00
0(1)
g0
0(1)  m2
g0
0(1) +B
r2
+g0
0(1)R(1)
+BR(1)
r2
+g0
0(1) −φ02(1)R(1)
2r2
+g02
0(1) ] (59)
"λ+hOi+
rλ+
+
zλ+−1
m#=m2
g0
0(1) +B
r2
+g0
0(1)R(1)
−(1 −zm)[1
2d−4 + m2
g0
0(1) +B
r2
+g0
0(1) −g00
0(1)
g0
0(1)  m2
g0
0(1) +B
r2
+g0
0(1)R(1)
+BR(1)
r2
+g0
0(1) −φ02(1)R(1)
2r2
+g02
0(1) ].(60)
Eq.(s)(59) and (60) yields a quadratic equation for B. This reads
B2+pr2
+B+nr4
+−φ02(1)r2
+= 0 (61)
where
p= 2m2+d−4−g00
0(1)
g0
0(1) g0
0(1) + 2g0
0(1) −4g0
0(1)(λ+(1 −zm) + zm)
(1 −zm)(λ+(1 −zm)+2zm)(62)
11

and
n=m4+m2g0
0(1) d−4−g00
0(1)
g0
0(1) −4(zm+λ+(1 −zm))
(1 −zm)(2zm+λ+(1 −zm))
+4λ+g02
0(1)
(1 −zm)(2zm+λ+(1 −zm)) .(63)
Now when B=Bc, the condensate vanishes and hence we can take ψ= 0 and eq.(17)
now takes the form
∂2
zφ+1
z2−d−2
2q−1∂zφ= 0 .(64)
Solving this, we get
φ(z) = ρ
rd−2
+1
2q−1
r+(1 −zd−2
2q−1−1) (65)
⇒φ02(1)r2
+=ρ
rd−2
+2
2q−1
r4
+d−2
2q−1−12
.(66)
Using eq.(66) in eq.(61) we get the expression for the critical magnetic field Bc:
(Bc)NC =(−g0
0(1)) d−2
2q−1−2
2(4π)d−2
2q−1−2ξ
d−2
2q−1
NC
(Tc)2
NC ×"ΩNC (d, q, m)−p−4πξN C
g0
0(1) d−2
2q−1T
(Tc)NC d−2
2q−1#
(67)
where
ΩN C (d, q, m) = "4( d−2
2q−1−1)2−(4n−p2)−4πξN C
g0
0(1) 2(d−1)
2q−1T
(Tc)NC 2(d−1)
2q−1#
1
2
.(68)
Once again we take the θ→0 limit of the above results. This gives the critical magnetic
field in the commutative case :
(Bc)C=(d−1) d−2
2q−1−2
2(4π)d−2
2q−1−2ξ
d−2
2q−1
c
(Tc)2
C×"ΩC(d, q, m)−p4πξC
d−1d−2
2q−1T
(Tc)Cd−2
2q−1#(69)
where
ΩC(d, q, m) = "4( d−2
2q−1−1)2−(4n−p2)4πξc
d−12(d−1)
2q−1T
(Tc)C2(d−1)
2q−1#
1
2
.(70)
The above findings are displayed in Figures 2 and 3. It is evident from these figures that
there exists a critical magnetic field as well as a critical temperature above which the
superconducting phase vanishes. In Fig.2, we present our results for Bc−Tfor two sets
of values, namely, m2=−3, q= 1 and d= 5 and m2= 0, q= 1 and d= 5 for different
values of the noncommutative parameter θ. In Fig.3, the plots are made for q= 5/4 with
m2=−3 and d= 5 for different values of the noncommutative parameter θ.
It is interesting to note that the critical magnetic field above which the condensate vanishes
increases with increase in the noncommutative parameter θ.
12

Figure 2: Bc/T 2
cvs T/Tcplot : zm= 0.5, M Gd= 100, d = 5
Figure 3: Bc/T 2
cvs T/Tcplot : zm= 0.5, M Gd= 100, d = 5
5 Conclusions
In this paper, we have explored the role of noncommutativity of spacetime in holographic
superconductors in the framework of power Maxwell electrodynamics using the match-
13

ing method technique. In our study, the relation between the critical temperature and
the charge density has been obtained in d-dimensions. It is observed that the critical
temperature not only depends on the charge density but also on the noncommutative
parameter θ, mass of the black hole and the parameter qappearing in the power Maxwell
theory. We have presented the analytical results for the ratio of the critical temperature
and charge density for d= 5. We have also analytically obtained the expression for the
condensation operator in d-dimensions. Our analytical results indicate that the conden-
sation gets harder to form in the presence of the power Maxwell parameter qand the
noncommutative parameter θ. However, with increase in the mass of the black hole, the
critical temperature for a particular value of θincreases which reveals that the effects of
spacetime noncommutativity becomes weaker for black holes with mass much larger in
comparison to the noncommutative parameter θ. We also conclude from our results that
the onset of power Maxwell electrodynamics (for a value of q6= 1) makes the conden-
sate harder to form. However, once again the effect of the power Maxwell parameter on
the formation of the condensate decreases with increase in the mass of the black holes.
We then study the Meissner like effect by introducing an external magnetic field in our
model. The critical magnetic field above which the condensate vanishes is obtained by
the matching method and is observed to increase with increase in the noncommutative
parameter θ.
Acknowledgments
SP wants to thank the Council of Scientific and Industrial Research (CSIR), Govt. of
India for financial support. SG acknowledges the support by DST SERB under Start Up
Research Grant (Young Scientist), File No.YSS/2014/000180.
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