Meissner like effect in holographic superconductors with back reaction

Abstract

In this article we employ the matching method to analytically investigate the properties of holographic superconductors in the framework of Maxwell electrodynamics taking into account the effects of back reaction on spacetime. The relationship between the critical temperature (TcT_{c}) and the charge density (ρ\rho) has been obtained first. The influence of back reaction on Meissner like effect in this holographic superconductor is then studied. The results for the critical temperature indicate that the condensation gets harder to form when we include the effect of back reaction. The expression for the critical magnetic field (BcB_{c}) above which the superconducting phase vanishes is next obtained. It is observed from our investigation that the ratio of BcB_{c} and Tc2T_{c}^{2} increases with the increase in the back reaction parameter. However, the critical magnetic field BcB_c decreases with increase in the back reaction parameter.

Full text

Meissner like effect in holographic superconductors with
back reaction
Suchetana Pala∗, Saumya Ghoshb†, Sunandan Gangopadhyayc‡
aDepartment of Physical Sciences,
Indian Institute of Science Education and Research Kolkata,
Mohanpur, Nadia, West Bengal, 741 246, India.
cDepartment of Theoretical Sciences,
S. N. Bose National Centre for Basic Sciences,
Block - JD, Sector - III, Salt Lake, Kolkata - 700 106, India.
Abstract
In this article we employ the matching method to analytically investigate the properties of
holographic superconductors in the framework of Maxwell electrodynamics taking into account
the effects of back reaction on spacetime. The relationship between the critical temperature
(Tc) and the charge density (ρ) has been obtained first. The influence of back reaction on
Meissner like effect in this holographic superconductor is then studied. The results for the
critical temperature indicate that the condensation gets harder to form when we include the
effect of back reaction. The expression for the critical magnetic field (Bc) above which the
superconducting phase vanishes is next obtained. It is observed from our investigation that the
ratio of Bcand T2
cincreases with the increase in the back reaction parameter. However, the
critical magnetic field Bcdecreases with increase in the back reaction parameter.
1. Introduction
The AdS/CFT correspondence which gives a sound realization of the holographic principle, funda-
mentally originated from superstring theory [1,2]. The duality claims the equivalence of a strongly
coupled D-dimensional gauge theory with a gravitational theory in (D+1)-dimensional anti-deSitter
(AdS) spacetime. In other words this duality states the equivalence between two theories one of
which is strongly coupled and the other is weakly coupled. For the past two decades many successful
connections between condensed matter physics and gravitational theories have been built using this
AdS/CFT duality, also known as the gauge/gravity correspondence in the literature. To describe
finite temperature field theories, the gravitational part is replaced by AdS black holes. More specifi-
cally, exploiting the gauge/gravity duality one can use general relativity as a tool to describe various
strongly correlated systems of condensed matter physics. One of the phenomena in condensed matter
system that is explained by this correspondence is the phenomena of superconductivity [3,4].
∗suchetanapal92@gmail.com, sp15rs004@iiserkol.ac.in
†sgsgsaumya@gmail.com, sg14ip041@iiserkol.ac.in
‡sunandan.gangopadhyay@gmail.com, sunandan.gangopadhyay@bose.res.in
1
arXiv:1904.07731v1 [hep-th] 15 Apr 2019

Holographic superconductors have been studied extensively in the last few years and the results
obtained on the field theory side from the holographic superconducting gravitational models showed
considerable promise to explain some of the glaring features of high Tcsuperconductors. The basic
idea involved in holographic superconductors is the following. The local spontaneous U(1) symmetry
breaking of a charged black hole, minimally coupled to a complex scalar field, was first studied in
[5,6,7]. The gravitational theory in the bulk can be mapped into a dual field theory, residing on the
boundary of the AdS spacetime, using the AdS/CFT correspondence. This boundary theory suffers
a global U(1) symmetry breaking. In the last few years investigation on holographic superconductors
have been widely explored in the context of Maxwell electrodynamics [8]-[26] as well as non-linear
electrodynamics [27]-[31].
It is also well known that superconductors expel magnetic fields as the temperature is lowered
below a critical temperature (Tc). In the presence of an external magnetic field, ordinary supercon-
ductors can be classified into two classes, namely type I and type II. Using the AdS/CFT dictionary
it has been found that at low temperatures (T < Tc), s-wave condensate in holographic supercon-
ductors expels the magnetic field [32,33]. Such studies were first carried out in [34,35]. Thereafter
studies of Meissner like effect in holographic superconductors have also been carried out using the
matching method [36] in [37,38].
In this paper we have investigated the influence of back reaction on holographic superconductors
as well as the Meissner like effect in holographic superconductors. Our holographic superconducting
theory consists of a Einstein-Hilbert gravity theory along with a complex scalar field minimally
coupled to Maxwell field. The model also involves the effect of back reaction of matter fields on the
bulk spacetime. Hence, we are away from the probe limit through out the entire analysis. Studies
involving back reaction have been carried out earlier in [39,40].
The paper is organized as follows. In section 2, the basic formalism for the d-dimensional holo-
graphic superconductor considering the effect of back reaction in the spacetime geometry is presented.
In section 3, we obtain the relationship between the critical temperature and the charge density 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 formalism
The action of the gravitational dual able to describe the phase transition in the conformal field theory
living in the boundary reads
S=Zddxp|g|
2κ2hR−2Λ + 2κ2Lmi,Λ = −(d−1)(d−2)
2L2,(2.1)
Lm=−1
4Fµν Fµν −"(Dµψ)∗Dµψ+m2ψ∗ψ#
≡ L1− L2;µ, ν = 0,1,2, ...d . (2.2)
It is basically the Einstein-Hilbert gravity theory with a complex scalar field minimally coupled to
the Maxwell field. Where the symbols have their usual meanings and κ2= 8πGd, with Gdbeing the
ddimensional Newton’s gravitational constant. Lmdenotes the Lagrangian density of the matter
sector. The second term (L2) in the above equation (2.2) consists of the complex scalar field ψ, the
covariant derivative Dµis defined as Dµ≡∂µ−iqAµwhere Aµis the gauge field. On the other hand
2

the first term (L1) gives the dynamics of the gauge field. Next we obtain the equations of motion
by varying the action (2.1) with respect to the field variables. The equations of motion for the field
variables gµν, Aµand ψ, respectively, are given as follows
δgµν :"Rµν −1
2Rgµν + Λgµν #−2κ2"1
2L1gµν +1
2FµβFνβ−1
2L2gµν
+ (Dµψ)∗(Dνψ)#= 0 (2.3a)
δAµ:∂ν"p|g|Fνµ #−2p|g|q2Aµψ∗ψ= 0 (2.3b)
δψ:∂µhp|g|∂µψi−iqp|g|Aµ∂µψ−iq∂µhp|g|Aµψi−p|g|q2AνAνψ−p|g|m2ψ= 0 .
(2.3c)
The d-dimensional plane-symmetric black hole metric in AdS spacetime with back reaction reads
ds2=−f(r)e−χ(r)dt2+1
f(r)dr2+r2hij dxidxj(2.4)
where hijdxidxjis the metric on a (d−2)-dimensional hypersurface which has a flat geometry.
Now we consider a change of coordinate from r→z=r+
rwhere r+is the horizon radius
(f(r+) = 0) and make the following ansatz for the gauge field and scalar field
Aµ= (φ(r),0,0, ...), ψ =ψ(r).(2.5)
By putting µ=ν= 0 in eq.(2.3a), we obtain
f0(z)−d−3
zf(z)+(d−1)r2
+
L2z3−2k2
d−2zψ02(z)f(z)+r2
+
z3m2ψ(z)+r2
+φ2(z)ψ2(z)e−χ(z)
z2f(z)+1
2zφ02(z)= 0 .
(2.6)
Now setting µ=ν= 1 in eq.(2.3a) and then adding up with eq.(2.6) we obtain
χ0(z)−4κ2r2
+
(d−2)z3"z4
r2
+
ψ02(z) + φ2(z)ψ2(z)eχ(z)
f2(z)#= 0 .(2.7)
Also putting µ= 0 in eq.(2.3b), we obtain
φ00(z) + "χ0(z)
2−d−4
z#φ0(z)−2r2
+φ(z)ψ2(z)
f(z)z4= 0 .(2.8)
Finally rewriting eq.(2.3c) in terms of zgives
ψ00(z) + "f0(z)
f(z)−d−4
z−χ0(z)
2#ψ0(z) + r2
+
z4"φ2(z)eχ(z)
f2(z)−m2
f(z)#ψ(z)=0.(2.9)
3

3. Critical temperature (Tc) in terms of charge density (ρ)
The Hawking temperature of a black hole is given by
TH=f0(r+)e−χ(r+)/2
4π.(3.1)
To solve the non-linear equations (2.6)−(2.9), we need to set the boundary conditions at the black
hole horizon r=r+and at spatial infinity r=∞. In this context we recall that f(r=r+) = 0 and
eχ(r=r+)is finite. We also set lim
r→∞ e−χ(r)→1. The matter fields obey [5,6]
φ(r) = µ−ρ
rd−3(3.2)
ψ(r) = ψ−
r∆−+ψ+
r∆+(3.3)
where
∆±=(d−1) ±p(d−1)2+ 4m2L2
2.(3.4)
Here µand ρare the duals to the chemical potential and charge density of the conformal field theory
part. We also choose ψ−= 0 so that ψ+is the expectation value of the condensation operator Jat
the boundary. For the matter field to be regular we require φ(r+) = 0 and ψ(r+) to be finite. In
terms of the coordinate variable zthis reads φ(z= 1) = 0 and ψ(z= 1) to be finite.
At the critical temperature T=Tc,ψ(z) = 0. Using this fact in eq.(2.7) we have
χ0(z) = 0 .(3.5)
With this relation and our previous argument we get
lim
z→0e−χ(z)→1 =⇒χ(z)=0.(3.6)
Using eq.(s)(3.5,3.6) in eq.(2.8), we get the following behavior for the field φat the critical temper-
ature
φ00(z)−d−4
zφ0(z) = 0 .(3.7)
Now using the boundary condition (3.2), we solve the above equation to get
φ(z) = λr+(1 −zd−3) (3.8)
where
λ=ρ
rd−2
+
.(3.9)
Now the field equation for f(z) (2.6) at the critical temperature T=Tcbecomes
f0(z)−d−3
zf(z) + (d−1)r2
+
L2z3−κ2z
d−2φ02(z)=0.(3.10)
Substituting the solution of φ(z) from eq.(3.8) in the above equation yields
f0(z)−d−3
zf(z) + (d−1)r2
+
L2z3−(d−3)2κ2λ2r2
+
d−2z2d−7= 0 .(3.11)
4

The solution of this equation (3.11) subject to the condition f(z= 1) = 0 reads
f(z) = r2
+"1
L2z2−1
L2+d−3
d−2κ2λ2zd−3+d−3
d−2κ2λ2z2(d−3)#.(3.12)
In the rest of the analysis we shall work with L= 1. With this value of Lthe form of f(z) reduces
to
f(z) = r2
+
z2g0(z) (3.13)
where
g0(z) = 1 −h1 + d−3
d−2κ2λ2izd−1+d−3
d−2κ2λ2z2(d−2) .(3.14)
Analysis by matching method
Now we proceed to find the relationship between the critical temperature Tcand charge density ρ
using the matching method [36]. For that we expand φ(z) and ψ(z) in Taylor’s series around z= 1
and equate it with the boundary condition mentioned in eq.(s)(3.2,3.3) at some point z=zm. This
yields
hµ−ρ
rd−3
+
zd−3iz=zm
=hφ(1) −(1 −z)φ0(1) + 1
2(1 −z)2φ00(1) − O((1 −z)3)iz=zm
.(3.15)
Now using the values of χ0(z) and χ(z) at the critical temperature given in eq.(3.5) and eq.(3.6), in
eq.(2.8), we end up with
φ00(z)−d−4
zφ0(z)−2r2
+φ(z)ψ2(z)
f(z)z4= 0 .(3.16)
From the above equation, we get
φ00(1) = "(d−4) + 2ψ2(1)
g0
0(1) #φ0(1) .(3.17)
With this result eq.(3.15) simplifies to the form
(µ−ρ
rd−3
+
zd−3=−(1 −z)φ0(1) + 1
2(1 −z)2 (d−4) + 2ψ2(1)
g0
0(1) !φ0(1))z=zm
.(3.18)
Taking derivative on both sides of the above relation and setting z=zmgives
(−(d−3) ρ
rd−3
+
zd−4=φ0(1) −(1 −z) (d−4) + 2ψ2(1)
g0
0(1) !φ0(1))z=zm
.(3.19)
Setting ψ(1) = α, φ0(1) = −vand ˜v=v
r+in the above equation, we arrive at
α2=g0
0(1)
2(1 −zm)h1−(1 −zm)(d−4)i"1−Tc
Td−2#(3.20)
where
5

Tc=ξρ 1
d−2(3.21)
ξ=−g0
0(1)
4π(d−3) 1
d−2z
d−4
d−2
m
˜v1
d−21−(1 −zm)(d−4)1
d−2
.(3.22)
Now we want to figure out the form of ˜vin terms of known parameters zm,d,m2. Here again we
use eq.(s) (3.5,3.6) in eq.(2.9) and end up getting
ψ00(z) + hf0(z)
f(z)−d−4
ziψ0(z) + r2
+
z4hφ2(z)
f2(z)−m2
f(z)iψ(z)=0.(3.23)
From this relation, we get
ψ0(1) = m2
g0
0(1)ψ(1) (3.24)
ψ00(1) = 1
2h(d−4) −g00
0(1)
g0
0(1) +m2
g0
0(1)im2
g0
0(1)ψ(1) −φ02(1)
2r2
+g0
0
2(1)ψ(1) .(3.25)
The Taylor’s series expansion of ψ(z) around z= 1 reads
ψ(z) = ψ(1) −(1 −z)ψ0(1) + (1 −z)2
2! ψ00(1) + .... . (3.26)
Following the matching technique as earlier, gives the expression for ˜vto be
˜v2=m4+m2g0
0(1)hd−4−g00
0
g0
0i−4m2g0
0∆+(1 −zm) + zm
(1 −zm)∆+(1 −zm+ 2zm)+4g0
0
2∆+
(1 −zm)∆+(1 −zm)+2zm.(3.27)
The values of g0
0(1), g00
0(1) and ξhave been calculated from eq.(3.14) and eq.(3.22) respectively.
For that we have taken different values of λobtained using Sturm-Liouville eigen value method
corresponding to different back-reaction parameter κ[40] . The values of λand ξare displayed in
Table [1] for different back-reaction parameter κ. In the calculations, we have chosen m2=−3 and
d= 5.
Table 1: Values of λand ξfor different κ[m2=−3, d= 5 ]
κ λ ξ
0 18.23 0.2016
0.05 18.11 0.1997
0.10 17.75 0.1938
0.15 17.16 0.1839
In Fig[1], we have plotted the results of ξvs. κ. The plot shows that the critical temperature
decreases with increase in the back-reaction parameter.
6

Figure 1: ξvs κplot : zm= 0.5, m2=−3, d = 5
4. Condensate with magnetic field
In this section we shall explore the behaviour of the condensate solution in the presence of a magnetic
field. To make progress we proceed with the ansatz
Aµ= (φ(r),0, Bx, 0,0, ...), ψ ≡ψ(r, x).(4.1)
With this choice in hand, we rewrite eq.(2.3c) as
r2f(r)"∂2ψ(r, x)
∂r2+d−2
r+f0(r)
f(r)−χ0(r)
2∂ψ(r, x)
∂r +q2eχ(r)
f2(r)φ2−m2
f(r)ψ(r, x)#
=−∂2ψ(r, x)
∂x2+q2B2x2ψ(r, x).(4.2)
Using the separation of variables method, we have
ψ(r, x) = R(r)X(x).(4.3)
Eq.(4.2) now decouples into two differential equations which has the following forms
1
X(x)
d2X(x)
dx2−q2B2x2=−K(4.4a)
r2f(r)
R(r)"∂2R(r)
∂r2+d−2
r+f0(r)
f(r)−χ0(r)
2∂R(r)
∂r +q2eχ(r)
f2(r)φ2−m2
f(r)R(r)#=K(4.4b)
where Kis the constant of separation. Eq.(4.4a) is of the form of Schrodinger’s equation for a
simple harmonic oscillator. The eigen values are K=λnqB with λn= 2n+ 1. The corresponding
eigenfunctions can be given in terms of Hermite polynomials, Hn(√2qBx). As explained in [34], for
7

the rest of our analysis we shall work with the lowest mode (n= 0) solution, which implies λ0= 1.
Eq.(4.4b) can now be written as (introducing new coordinate variable z=r+
r)
R00(z) + "f0(z)
f(z)−d−4
z−χ0(z)
2#R0(z) + "eχ(z)r2
+φ2(z)
z4f2(z)−m2r2
+
z4f(z)−B
z2f(z)#R(z) = 0 .(4.5)
We shall employ the matching method once again, discussed in the previous section, to find out the
relation between the critical magnetic field (Bc) and critical temperature (Tc). For that, first we
make a Taylor’s series expansion of R(z) around z= 1 which reads
R(z) = R(1) −R0(1)(1 −z) + 1
2R00 (1)(1 −z)2+O(1 −z)3.(4.6)
Further the asymptotic form for R(z) reads
R(z) = hOi+
rλ+
+
zλ+.(4.7)
At any interior point z=zmwe match these solutions. Hence, 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
.(4.8)
Differentiating eq.(s)(4.6,4.7) 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
.(4.9)
As was discussed in the earlier section, near the critical temperature we put χ0(z) = χ(z) = 0 in
eq.(4.5). This simplifies the form of eq.(4.5) and it reads
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).(4.10)
From this equation, we obtain
R0(1) = m2
g0
0(1) +B
r2
+g0
0(1)R(1) (4.11)
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) .(4.12)
Substituting R0(1) and R00 (1) in eq.(s)(4.8,4.9), we have
8

"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) ] (4.13)
"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) ].(4.14)
Eq.(s)(4.13) and (4.14) yields a quadratic equation for B. This reads
B2+pr2
+B+nr4
+−φ02(1)r2
+= 0 (4.15)
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)(4.16)
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)) .(4.17)
Since at T=Tc, the scalar field vanishes (ψ(z) = 0), this gives φ0(1) from eq.(3.8) and reads
φ0(1) = −λr+(d−3) .(4.18)
Substituting this in eq.(4.15) gives the critical magnetic field to be
Bc=1
2−g0
0(1)
4πd−41
ξd−2Ω(d, m, zm)−p−4πξ
g0
0(1)d−2T
Tcd−2(4.19)
where
Ω(d, m, zm) = 4(d−3)2−(4n−p2)−4πξ
g0
0(1)(2d−4)T
Tc(2d−4).(4.20)
9

Figure 2: Bc/T 2
cvs T/Tcplot : zm= 0.5, m2=−3, d = 5
In Fig.[2], we have plotted Bc/T 2
cagainst T/Tcfor different back-reaction parameters κ, with zm=
0.5, m2=−3, d= 5. It is evident from the figure that there exists a certain critical temperature Tc
and a critical magnetic field Bcabove which the superconducting phase vanishes.
From Fig.[2], it is to be noted that the ratio of Bc
T2
cat T= 0 increases with increase in the back
reaction parameter. The value of Bccan be estimated from the ratio Bc
T2
cand using eq.(3.21). These
values have been displayed in Table [2].
Table 2: Values of Bcfor different κ[m2=−3, d= 5, T= 0 ]
κBc
T2
cBc
0 77.5879 3.1533 ρ2
3
0.05 78.6834 3.1379 ρ2
3
0.10 82.0542 3.0818ρ2
3
0.15 89.1723 3.0157ρ2
3
From Table [2], it can be observed that the critical magnetic field Bcdecreases with increase in the
back reaction parameter κ. This indicates that the presence of the back reaction parameter destroys
the superconducting phase earlier, that is, for a smaller value of the critical magnetic field Bc.
5. Conclusion
In this paper we have studied the influence of back reaction on holographic superconductor in the
framework of Maxwell electrodynamics using the matching method. The holographic superconductor
model that we have considered in our analysis consists of Einstein-Hilbert gravity theory along with a
complex scalar field minimally coupled to Maxwell field. Further we have investigated the effect of an
external magnetic field in our holographic superconductor model. From our analysis we can conclude
from the critical temperature and charge density relationship that the critical temperature depends
on both charge density and the back reaction parameter. We have presented the analytical results for
10

the ratio of the critical temperature and charge density for d= 5 and m2=−3. Our results indicate
that condensation gets harder to form as we include the effect of back reaction. With these findings,
we then obtain the expression for the critical magnetic field above which the superconducting phase
vanishes. This is obtained using matching method once again. We observe that the ratio of Bcand
T2
cat T= 0 increases with the increase in back reaction parameter κ. However, we have found
that the critical magnetic field Bcdecreases with increase in the back reaction parameter κ. This
clearly tells that the presence of the back reaction parameter destroys the superconducting phase for
a smaller value of the critical magnetic field Bc.
Acknowledgment
S. Pal wants to thank the Council of Scientific and Industrial Research (CSIR), Govt. of India for
financial support. S. Gangopadhyay acknowledges the support of IUCAA, Pune for the Visiting
Associateship.
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