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An analytical approach to compute conductivity of p-wave
holographic superconductors
Suchetana Pala∗, Diganta Paraib†, Sunandan Gangopadhyayc‡
a,bDepartment 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 have analytically deduced the frequency dependent expression of conductivity
and the band gap energy in AdS4Schwarzschild background for p-wave holographic supercon-
ductors considering Einstein-Yang-Mills theory. We also used the self consistent approach to
obtain the expressions of conductivity for different frequency ranges at low temperature. We
then compared the imaginary part of conductivity at low frequency region. The band gap energy
obtained from these two methods seem to agree very well.
1. Introduction
The AdS/CFT correspondence proposed by Maldacena [1,2,3,4,5] states that a weakly coupled
gravity theory in AdSn+1 spacetime is equivalent to a strongly coupled conformal field theory CF Tn
in one less dimension [6,7]. This conjecture derived from string theory has been used extensively
to understand the strongly coupled phenomena in field theories by looking at a weakly coupled dual
gravity theory. A very interesting application of this correspondence is the construction of holo-
graphic superconductors. The term “holographic” implies, by looking at a two (spatial) dimensional
superconductor one can identify a three-dimensional image that consists of a charged black hole with
non-trivial hair [8]. These holographic superconductors can successfully reproduce many important
properties of high Tcsuperconductors. The second-order superconducting phase transition below a
certain critical temperature can be understood by the condensation of a charged scalar field that
leads to U(1) symmetry breaking near the black hole horizon in the dual gravitational description.
∗suchetanapal92@gmail.com, sp15rs004@iiserkol.ac.in
†digantaparai007@gmail.com, dp16rs028@iiserkol.ac.in
‡sunandan.gangopadhyay@gmail.com, sunandan.gangopadhyay@bose.res.in
1
arXiv:2201.06908v1 [hep-th] 18 Jan 2022
There exist a large number of studies to describe the Meissner effect which is another important
feature of superconductors from the holographic superconductor point of view [9,10,11,12]. There
are also studies on effect of nonlinear electrodynamics and higher curvature correction on holo-
graphic superconductor [13,14]. Different mechanisms have been proposed where at a finite tem-
perature the system undergoes a spontaneous symmetry breaking and enters the superconducting
phase [15,16,17,18,19]. In this paper, we consider a p-wave holographic superconductor model as
considered in [15,19,20,21] and try to shed light on the optical conductivity at the superconducting
phase in an analytical approach.
Our analysis is based on the simplest example of p-wave holographic superconductor depicted by
SU (2) Einstein-Yang-Mills theory given by the following action
S=Zddx√−gh1
2(R−2Λ) −1
4Fa
µν Faµν i,(1.1)
where the negative cosmological constant Λ = −(d−1)(d−2)
2L2with L= 1 and Yang-Mills field strength
Fa
µν =∂µAa
ν−∂νAa
µ+qf abc Ab
µAc
ν. The gauge field can be written as A=Aa
µσadxµ, where σaare the
generators of the SU (2) group and (a, b, c) = (1,2,3) are the indices of the generators. The equation
of motion for the field variable Aµcoming from the above action is given by
1
√−g∂µ(√−gF aµν ) + qfabcAb
µFcµν = 0.(1.2)
For finite q, one must consider the effect of the gauge field on the metric but when qis large the back
reaction is negligible. This q→ ∞ limit is known as probe limit which simplifies the problem but
retains important properties of the system. In this paper our analysis is done in the probe limit.
In this paper we aim to obtain the frequency dependent expression for conductivity in AdS4
Schwarzschild background. The paper is organized in seven sections. We start with the basic formal-
ism in section 2 where field equations are discussed. In section 3 we provide the relationship between
critical temperature Tcand charge density ρ. In section 4 we analyse the system at low temperature
limit (T→0) and determine the behaviour of the charged field ψand gauge field φ, that is consistent
with the boundary conditions and also obtain the relationship between condensation operator hO1i
and critical temperature Tc. In section 5 we we discuss conductivity and compute band gap energy
for the case ψ0= 0. In section 6 we consider the case ψ1= 0 and compute conductivity and band
gap energy. In section 7 we draw conclusions from the findings.
2. Discussion of the holographic model
For this part of our analysis we have chosen the fixed background of 3+ 1-dimensional Schwarzschild
AdS black hole, whose metric reads
ds2=−f(r)dt2+1
f(r)dr2+r2(dx2+dy2) (2.1)
where,
f(r) = r2g(r), g(r) = 1−r3
+
r3.(2.2)
2
Here r+is the horizon radius. The Hawking temperature is given by
T=3r+
4π.(2.3)
In order to investigate the metal/superconductor phase transition let us consider the following ansatz
A=φ(r)σ3dt +ψ(r)σ1dx . (2.4)
Here the gauge field A3
t=φ(r) is the U(1) subgroup of SU (2) and is associated with the chemical
potential in the boundary field theory. The charged field A1
x=ψ(r) is associated with the condensa-
tion operator hOi in the boundary field theory whose condensation is responsible for U(1) symmetry
breaking. Note that our analysis has been done in the probe limit, hence we did not consider the
back reaction of the gauge field on the metric eq.[2.2].
Plugging on the ansatz given by eq.[2.4] in eq.[1.2] we obtain the following equations of motion for
the field variables φ(z) and ψ(z) respectively
φ00(z)−ψ2(z)
r2
+g(z)φ(z)=0,(2.5)
ψ00(z) + g0(z)
g(z)ψ0(z) + φ2(z)ψ(z)
r2
+g2(z)= 0 .(2.6)
Note that here we considered the coordinate change z=r
r+for simplicity. At the horizon(r=r+)
z= 1 and at the boundary(r→ ∞)z→0.
Let us now discuss the boundary conditions. At the horizon φ(1) = 0 and ψ(1) is finite. At the
boundary, that is z→0 the behaviour of φ(z) and ψ(z) are as follows
φ(z) = µ−ρ
r+
z , (2.7)
ψ(z) = ψ0+ψ1
r+
z . (2.8)
According to the AdS/CFT dictionary, µand ρrespectively represent the dual to chemical potential
and charge density at the boundary. ψ0and ψ1are related to the source and the expectation value
of the condensation operator.
For now let us consider ψ0= 0. We will now discuss this case in detail. Keeping in mind the
behaviour at the boundary, we may write
ψ(z) = hO1i
√2r+
zF (z).(2.9)
In section 6, we also briefly discuss the case where we set ψ1= 0. In this case, we write
ψ(z) = hOi
√2F(z).(2.10)
For both of these cases F(z) obeys the following conditions
F(0) = 1, F 0(0) = 0 .(2.11)
3
3. Relation between critical temperature Tcand charge densityρ
In this section, our analysis will be mostly focused around T→Tcand we will develop the relationship
between Tcand ρ, which is necessary for later analysis and future constructions. But we will not be
discussing this in great detail as it is already available in the literature [21].
Using the fact that at the critical temperature Tc,ψ(z) = 0 from eq.[2.5] we obtain
φ(z) = λr+(c)(1 −z), λ =ρ
r2
+(c)
.(3.1)
By substituting φ(z) and ψ(z) from eq.[3.1] and eq.[2.9] respectively in eq.[2.6], we get
F00(z)−3z2
1−z3−2
zF0(z)−3z
1−z3F(z) + λ2
(1 + z+z2)2F(z) = 0 .(3.2)
This equation can be recast in Sturm-Liouville form corresponding to eigenvalue λ2, which minimizes
the following expression
λ2=R1
0dz[z2(1 −z3)F0(z)2+ 3z3F(z)2]
R1
0dz z2(1−z)
1+z+z2F(z)2.(3.3)
Here we choose the trial function Fβ(z)=1−βz2, which obeys the conditions given by eq.[2.11].
The minimum λ2is attained for β= 0.5078.
These findings yield [21]
Tc=3
4πr+(c)=3
4πrρ
λβ≈0.1239√ρ . (3.4)
4. Condensation operator hO1iat low temperature
As we are interested in low temperature limit (T→0), we consider the scaling z=s
bwhere b→ ∞.
We will determine blater on, and it will be clear that b→ ∞ corresponds to low temperature limit
[22,23]. Under this condition the dominant contribution comes from neighbouring region of the
boundary (z→0) and eq.[2.5] and eq.[2.6] takes the following forms respectively
φ00(s)−hO1i2
2r4
+b4s2F2(s)φ(s) = 0 ,(4.1)
F00(s) + 2
sF0(s) + φ2(s)
r2
+b2F(s) = 0 .(4.2)
We aim to obtain the solutions of the eq.[4.1] and eq.[4.2] iteratively, which are consistent with the
boundary conditions. To do that, we will start with the following form of F(s), which is essentially
the behavior of F(s) at s > 1 or z > 1
b
F(s)≈α
s.(4.3)
4
Here αis a constant to be determined later. Substituting F(s) from eq.[4.3] in from eq.[4.1], we get
φ(s) = C1e−s+C2es.(4.4)
Now we choose bas
b=phO1iα
21
4r+
.(4.5)
Using the condition φ(z) = 0 at the horizon (z→1) for all values of b, it is easy to show that C2= 0
and hence φ(s) can be written as
φ(s) = Cr+be−s(4.6)
where, C1=C√hO1iα
21
4.
Eq.[4.5] shows that b→ ∞ as r+→0, and hence from eq.[2.3] it is easy to note that it corresponds
to low temperature limit as we claimed earlier.
Now we proceed to estimate a more accurate behaviour of F(s) that is consistent with the conditions
F(z) = 1 and F0(z) = 0 at the boundary z= 0. By substituting φ(s) from eq.[4.6] in [4.2], we obtain
F00(s) + 2
sF0(s) + C2e−2sF(s) = 0 .(4.7)
The above equation can be solved using the Sturm-Liouville approach in the interval (0,∞). The
corresponding eigenvalue C2is given by
C2=R∞
0s2F0(s)2ds
R∞
0s2e−2sF2(s)ds .(4.8)
Here we choose the following trial function as the eigen function F(s) that minimizes eq.[4.8] and is
consistent with the boundary conditions
F(s) = α
stanh s
α.(4.9)
We obtain the minimum for α=αS.L = 0.8179 and that corresponds to C=CS.L = 2.4065.
Next we aim to solve eq.[4.1] once again perturbatively by substituting F(s) from eq.[4.9] and con-
sidering φ(s) from eq.[4.6] as the zeroth order solution. We obtain
φ(z) = c1−c2
bz
α+br+Ce−bz"1−2α
α+ 22
e−2bz
α3F2n2,1 + α
2,1 + α
2; 2 + α
2,2 + α
2;−e−2bz
αo#.
(4.10)
Using the fact that at z= 1, φ(z) = 0 and b→ ∞, we may show that c1=c2= 0. Finally, we obtain
φ(z) = br+Ce−bz"1−2α
α+ 22
e−2bz
α3F2n2,1 + α
2,1 + α
2; 2 + α
2,2 + α
2;−e−2bz
αo#.(4.11)
5
By comparing the coefficient of zfrom the above equation with the boundary behaviour of φ(z) given
by eq.[2.7], we obtain ρ
r+
= 0.4911b2r+C . (4.12)
Now we substitute ρfrom eq.[3.4] and bfrom eq.[4.5] in eq.(4.12) and get
phO1i= 9.7624Tc≡ξTc.(4.13)
Restoring the zcoordinate now, we may write F(z) ans ψ(z) as following
F(z) = α
bz tanh bz
α,(4.14)
ψ(z) = hO1i
√2r+
α
btanh bz
α.(4.15)
Interestingly we can determine the constant Cand αby direct analytical approach as eq.[4.7] is
analytically solvable. Using the condition F(0) = 1 from eq.[4.7], we get
F(s) = π
2sY0(C)J0(Ce−s)−J0(C)Y0(Ce−s).(4.16)
To compute the constant Cwe will use the condition F(s)→0 as s→ ∞. From eq.[4.16], we thus
obtain
J0(C)=0.(4.17)
The above equation implies C=Cdirect = 2.4048, which is the first root of the Bessel function J0.
We can see that this result agrees well with our previous estimate of the constant Cobtained using
the Sturm-Liouville approach. Now F(s) may be written as follows
F(s) = π
2sY0(C)J0(Ce−s).(4.18)
Now let us compare the behaviour of the F(s) from the above eq.[4.18] with eq.[4.3] for s > 1 or,
z > 1
bto find out α. This gives
F(s) = πY0(C)
2s(4.19)
α=πY0(C)
2= 0.8009 .(4.20)
We see that α=αdirect = 0.8009 is in good agreement with the previous estimate of αS.L .
Following the earlier steps using F(s) given by eq.[4.18], and considering φ(s) from eq.[4.6] as the
zeroth order solution we compute φ(z) from eq.[4.1] and the relationship between hO1iand Tc.
φ(z) = br+Ce−bz3F41
2,1
2,1
2; 1,1,3
2,3
2;−C2e−2bz(4.21)
6
phO1i= 10.0518Tc.(4.22)
In this case we may write F(z) and ψ(z) as following
F(z) = π
2bz Y0(C)J0(Ce−bz),(4.23)
ψ(z) = hO1i
√2r+
π
2bY0(C)J0(Ce−bz).(4.24)
It is worth noting that for both the cases given by eq.[4.14] and eq.[4.23], it is possible to write
F(z) = 1 + O(z2) as expected.
12345
s
0.5
1.0
1.5
F(s)
s
sTanh[s
]
sJ0(Ce-s)
Figure 1: F(s) vs. s
To get a clear idea about the behaviour of F(s) obtained from two different methods, in Figure[1]
we have plotted F(s) vs. s. The blue, orange and green curves represent eq.[4.3], [4.9] and [4.18]
respectively. By looking at blue curve representing eq.[4.3] we can easily say that indeed it is the
behaviour of F(s) at s > 1 or, z > 1
band by looking at the orange and green curve it is clear that
the behaviour of F(s) predicted by eq.[4.9] and [4.18] are very similar as expected.
12345
s
0.5
1.0
1.5
ϕ
(
s)
1〉
αC
2
1
4
e-s[1- ( 2α
α+2)2e-2s
α3F2(2,1+α
2,1+α
2; 2+α
2,2+α
2;-e-2s
α)]
αC
2
1
4
e-s3F4(1
2,1
2,1
2; 1,1, 3
2,3
2;-C2e-2s)]
Figure 2: φ(s)
√hO1ivs. s
7
In Figure[2], we have plotted φ(s)
√hO1ivs. s. The blue and orange curves respectively represents eq.[4.11]
and [4.21] which suggests the behaviour of φdepicted by these equations are very similar. Note that
for the rest of the analysis we will use α=αS.L = 0.8179 .
Table 1: α,Cand phO1iobtained from Sturm-liouville approach and direct analytical approach:
Approach α C phO1i
S.L 0.8179 2.4065 10.0518 Tc
Direct 0.8009 2.4048 9.7624 Tc
5. Conductivity at low temperature
To study the conductivity at the boundary, we will consider an electromagnetic perturbation in the
bulk by applying a non-zero gauge field in the y- direction. Let us consider the following ansatz
Ay=A(r)e−iωtσ3.(5.1)
Note that, as our analysis is done on probe limit we do not consider the back reaction of this newly
introduced component (Ay) on the metric or the other components of the gauge field (At,Ax).
Plugging in the above ansatz in the equation of motion given by eq.[1.2], we get
A00(r) + f0(r)
f(r)A0(r) + ω2
f2(r)−ψ2(r)
r2f(r)A(r) = 0 .(5.2)
Switching to zcoordinate, we obtain
A00(z) + g0(z)
g(z)A0(z) + 1
r2
+hω2
g2(z)−ψ2(z)
g(z)iA(z)=0.(5.3)
At the boundary the behaviour of the gauge field A(z) can be found from eq.(5.2) given by
A(z) = A0+A1
r+
z. (5.4)
The conductivity σyy for our system is given as following (refer Appendix) [24,25,26]
σyy =−iA1
ωA0
=−ir+A0(0)
ωA(0) .(5.5)
Let us now switch to the tortoise coordinate (as z→1, r∗→ −∞) defined as follows, where the
integration constant is chosen such that at the boundary (z= 0), r∗= 0
r∗=Zdr
f(r)=1
6r+
[2 ln(1 −z)−ln(1 + z+z2)−2√3 tan−1√3z
2 + z].(5.6)
8
Now eq.[5.3] takes the following form
A00(r∗) + ω2A(r∗) = V A(r∗), V (r) = ψ2(r)f(r)
r2.(5.7)
Note that at the horizon (r=r+), V= 0. Taking into account the ingoing boundary condition [22]
at the horizon, solving the above equation, we get
A∼e−iωr∗∼(1 −z)−iω
3r+.(5.8)
Near the horizon (z= 1), the main contribution in r∗comes from the first term as given in eq.[5.6].
In order to obtain an expression for A(z), eq.[5.3] ought to be solved taking into account the boundary
behaviour. We may now write
A(z) = (1 −z)−iω
3r+G(z).(5.9)
Substituting A(z) from eq.[5.9] in eq.[5.3], we get
3(1 −z3)G00(z)−9z2−2(1 + z+z2)iω
r+G0(z)
−3ψ2(z)
r2
+−(1 + 2z)iω
r+−(2 + z)(4 + z+z2)
3(1 + z+z2)
ω2
r2
+G(z)=0.(5.10)
At the horizon (z= 1), from above equation we deduce
3−2iω
r+G0(1) + ψ2(1)
r2
+−iω
r+−2ω2
3r2
+G(1) = 0 .(5.11)
Turning on the low temperature limit, eq.[5.10] may be approximated as
G00(z) + 2iω
3r+
G0(z)−ψ2(z)
r2
+−iω
3r+
+8ω2
9r2
+G(z) = 0.(5.12)
Substituting ψ(z) from eq.[4.15], we exactly solve the above equation and obtain
G(z) = e−iω
3r+zc+P
αr1−√2ω2
αhO1i
1
2(−1+√1+4α2)tanh bz
α+c−P−αr1−√2ω2
αhO1i
1
2(−1+√1+4α2)tanh bz
α,(5.13)
where Pµ
νare the fractional Legendre functions. Finally we may write A(z) for low frequency (ω <<
hO1i) region as
A(z) = (1 −z)−iω
3r+e−iω
3r+zc+Pα
1
2(−1+√1+4α2)tanh bz
α+c−P−α
1
2(−1+√1+4α2)tanh bz
α,(5.14)
Using the definition of conductivity from eq.[5.5] for low temperature and low frequency, we write
9
σ(ω)=0.4616iphO1i
ω1−1.3911c+
c−
1−0.4085c+
c−!.(5.15)
Next, we aim to determine the ratio c+
c−. Note that at z→1, tanh(bz
α)≈1, and under this condition
we may approximate
P±α
1
2(−1+√1+4α2)tanh bz
α=2±α
2
Γ(1 ∓α)1−tanh bz
α∓α
2
+... . (5.16)
Now for low frequency region from eq.[5.13], we also get
G(1) = c+
Γ(1 −α)eb+c−
Γ(1 + α)e−be−iω
3r+, G0(1) = c+(b−iω
3r+)
Γ(1 −α)eb+c−(b+iω
3r+)
Γ(1 + α)e−be−iω
3r+.(5.17)
Using eq.[5.11] and eq.[5.17] the ratio c+
c−becomes
c+
c−
=−e−2bΓ(1 −α)
Γ(1 + α)b−3
b+ 3 +4(b2−3)
b(b+ 3)2
iω
r+
+O(ω2).(5.18)
Substituting the above ratio in eq.[5.15], we obtain σ(ω) at low frequency (ω→0). This yields the
following equations
Im[σ(ω)] ≈0.4616phO1i
ω,(5.19)
Re[σ(ω= 0)] ∼e−2b1 + O(1/b)≡e−Eg
T,(5.20)
Eg=3pαhO1i
25
4π≈0.3631phO1i.(5.21)
We have used α=αSL = 0.8179 obtained from the SL method in section 4 in the above equation.
Eq.[5.20] depicts that the zero frequency limit of Re[σ(ω)] is governed by thermal fluctuations,
where Egis the energy gap. In the probe limit gap frequency ωg= 2Eg[27]. Now using eq.[5.21]
and eq.[4.13], we get
ωg
Tc
=2Eg
Tc
= 7.0894 .(5.22)
Next we evaluate the following ratio using eq.[4.13], eq.[5.19], eq.[3.4]
lim
ω→0
ω
√ρIm[σ(ω)] = 0.5583 .(5.23)
This agrees exactly with the numerical result given in [15].
In this article we also compute the expression of conductivity in self consistent approach and compare
the results with our previous estimates. To do that we will be replacing Vwith its average hViin a
self consistent manner. From eq.[5.7], we write
10
A(r∗) = e−i√ω2−hVir∗.(5.24)
which is consistent with the ingoing boundary condition at the horizon as mentioned earlier.
From eq.[5.5], the expression of conductivity in this case is given by
σ(ω) = r1−hVi
ω2(5.25)
where,
hVi=R0
−∞ V|A(r∗)|2dr∗
R0
−∞ |A(r∗)|2dr∗
.(5.26)
We evaluate the integral considering ωhas an imaginary part, which we will set zero at the end of
the calculation.
Let us now consider a change in variable for better understanding. From eq.[5.6], we may write
r∗=−1
r+z+z4
4+z7
7+z10
10 +....=−˜z
r+
.(5.27)
Now eq.[5.26] can be rewritten as following
hVi=R∞
0V(˜z)e−2√hViq1−ω2
hVi
˜z
r+d˜z
R∞
0e−2√hViq1−ω2
hVi
˜z
r+d˜z
, V (z) = hO1iα
√2(1 −z3) tanh2bz
α.(5.28)
Notice that at low temperature as r+→0, the main contribution to the integral comes when ˜z→0
and this condition implies z= ˜zor r∗(z) = −z
r+, which is essentially the region near boundary.
Hence we put
V(˜z) = hO1iα
√2(1 −˜z3) tanh2b˜z
α.(5.29)
As ˜z→0, we may write
hVi=hO1iα
√2R∞
0tanh2(b˜z
α)e−2√hViq1−ω2
hVi
˜z
r+d˜z
R∞
0e−2√hViq1−ω2
hVi
˜z
r+d˜z
.(5.30)
After integrating we deduce
√2ˆ
V= 1 + 2 5
4αqˆ
V−ˆω2+ 2√2α2(ˆ
V−ˆω2)ψα
23
4qˆ
V−ˆω2−ψ1
2+α
23
4qˆ
V−ˆω2(5.31)
where
ˆ
V=hVi
hO1iα,ˆω2=ω2
hO1iα.(5.32)
For low frequency that is ω→0, from eq.[5.31] we obtain ˆ
V= 0.2924. Hence for low temperature
and low frequency, conductivity given by eq.[5.25] may be written as
11
σ(ω) = 0.489iphO1i
ω.(5.33)
By comparing the above eq.[5.33] with the imaginary part of conductivity obtained in eq.[5.19], we
see that they are in good agreement.
At high frequencies that is ω→ ∞ from eq.(5.31), we obtain ˆ
V=−1
4α2ˆω2. Hence at low temperature
and high frequency, conductivity given by eq.(5.25) may be written as
σ(ω) = r1 + hO1i2
4ω4.(5.34)
When ˆ
Vis comparable with ˆω2that is ˆ
V= ˆω2, from eq.[5.31] we obtain hVi=αhO1i
√2and
σ(ω) = s1−αhO1i
√2ω2.(5.35)
Interestingly there is another way to solve eq.[5.7] by treating δV =V−αhO1i
√2as the perturbation
and A(r∗) = e−irω2−αhO1i
√2r∗as zero-th order solution [22]. To see this we rewrite eq.[5.7] as following
A00(r∗) + ω2−αhO1i
√2A(r∗) = V−αhO1i
√2A(r∗).(5.36)
By solving the above equation we obtain
A(r∗) = e−irω2−αhO1i
√2r∗"1 + α2
2β−α2π
sin πβ e2irω2−αhO1i
√2r∗+α2
2β2F1 1, β; 1 + β;−e−2rhO1i
√2αr∗!
−α2
2(1 + β)e−2rhO1i
√2αr∗2F1 1,1 + β; 2 + β;−e−2rhO1i
√2αr∗!#
(5.37)
where β=iαq√2ω2
αhO1i−1. Note note that to determine the integration constants we use the ingoing
boundary condition at the horizon and we also use the relation 2F11, β; 1 + β;x=eiπβ
xβ
πβ
sin πβ when
|x|→ ∞.
The expression of conductivity given by eq.(5.5) can be rewritten now as
σ(ω) = i
ω"dA(r∗)
dr∗
A(r∗)#r∗=0
.(5.38)
Using eq.(5.37) and eq.(4.13) in eq.[5.38], we deduce
σ(ω) = i ξβ
21
4√αω
Tc
1−
21 + α2
2β
1−πα2
sin πβ +α2
2ψ1+β
2−ψβ
2
, β =iαs√2
αξ2ω
Tc2
−1.(5.39)
12
ωg
Tc
5 10 15 20
ω
Tc
0.5
0.5
1.0
Re(σ
Figure 3: Re(σ) Vs. ω
Tcat low temperature
ωg
Tc
5 10 15 20
ω
Tc
1.5
1.0
0.5
0.5
1.0
1.5
Im(σ
Figure 4: Im(σ) Vs. ω
Tcat low temperature
We have plotted Figure [3] and Figure[4] using eq.(5.39) which depicts the dependency of Re[σ(ω)]
and Im[σ(ω)] on ω
Tc.
From Figure[3], we find that at T→0, Re[σ(ω)] vanishes for ω < ωgand a gap appears as expected.
From the plot we obtain ωg
Tc= 7.4242 [28] which is consistent with our previous estimate given by
eq.[5.22].
Approach ωg
Tc
Direct 7.0894
Perturbative 7.424
Approach Im[σ(ω)]
Direct 0.4616√hO1i
ω
Self consistent 0.489 √hO1i
ω
Approach ω→0ω
√ρIm[σ(ω)]
Analytical 0.5583
Numerical 0.55
Table 2: Comparisons for ωg
Tc,Im[σ(ω)] and ω→0ω
√ρIm[σ(ω)] obtained from different approaches
It is worth mentioning that for Bi2Sr2CaCu2O8+δsample, which is a high Tcsuperconductor, this
ratio is found to be 7.9±0.5 [29]. On the other hand for weakly coupled low Tcsuperconductors
13
described by BCS theory [30], the value of this ratio is 3.5. Another interesting observation is that
at ω= 0, there is also a delta function in the Re[σ(ω)] for all T < Tc. Although we cannot detect
it by analytical or numerical computation as it gives us only the continuous part of σ(ω), this delta
function can be revealed by looking at the pole in Im[σ(ω)] at ω= 0. A general argument for such
a conclusion comes from the Kramers–Kronig relations. Recall that these relations relate the real
and imaginary parts of any causal quantity when expressed in frequency space. For conductivity this
relation gives us
Im[σ(ω)] = −1
πPZ∞
−∞
Re[σ(ω0)]dω0
ω0−ω.(5.40)
From the above relation [5.40], we conclude that the real part of the conductivity contains a delta
function, if and only if the imaginary part has a pole. It is clear from the Figure[4] and from eq.(5.19)
there is indeed a pole in Im[σ(ω)] at ω= 0 at low temperature. In the probe limit at T→0, this
delta function in the real part of conductivity implies infinite DC conductivity of superconducting
phase. Turning our attention to the gap that appears for the frequencies ω < ωg, we conclude there
exists a gap in the charge spectrum corresponding to the frequencies ω < ωgand the conduction is
non-dissipative. The finite conductivity for ω > ωgindicates dissipation. As ω→ ∞, conductivity
of the superconducting phase appears to be like the conductivity of the normal phase which in turn
implies that the degrees of freedom that contribute to the conductivity at high frequency corresponds
to the normal phase.
Note that the behaviour of σ(ω) depicted by Fig[3], [4] for p-wave holographic superconductor is
qualitatively similar to the ψ1= 0 case of s-wave holographic superconductor. The relations [5.19]
and [5.21] in our analysis corresponds to Im[σ(ω)] = 0.55√hO2i
ωand Eg= 0.43phO2ifor s-wave
holographic superconductor [22].
6. Conductivity for the case ψ1= 0
Now let us look into the case where we set ψ1= 0, and ψ(z) is given by eq.[2.10]. Here we simply
choose F(z) = 1 which is consistent with the condition given by eq.[2.11]. In order to deduce
conductivity let us substitute eq.(2.10) in eq.(5.10) and consider the low temperature rescaleing
z=s
band by letting b→ ∞, we obtain the following equation
G00(s) + 2iω
3br+
G0(s)−hOi2
2r2
+b2−iω
3r+b2−8ω2
9r2
+b2G(s) = 0 .(6.1)
Next we choose b=hOi
√2r+. Eq.(6.1) gives the following approximate solution that is valid for low
temperature and low frequency (ω << hOi) region as
G(s) = c+es+c−e−s
⇒A(z)≈e
iωz
3r+c+ehOi
√2r+
z+c−e−hOi
√2r+
z.(6.2)
Using the definition of conductivity given by eq.(5.5), we obtain
14
σ(ω)≈ihOi
√2ω
1−c+
c−
1 + c+
c−
.(6.3)
The ratio c+
c−can be found from boundary condition given by eq.(5.11), where we substitute ψ(1) ≈
hOi
√2. This gives
c+
c−
=−e−2bb−3
b+ 3 +2(2b2−3)
b(b+ 3)2
iω
r+
+O(ω2).(6.4)
By plugging the above ratio in eq.[6.3] leads to
σ(ω) = ihOi
√2ω1+2e−2bb−3
b+ 3 +2(2b2−3)
b(b+ 3)2
iω
r+
+O(ω2) .(6.5)
For the case of low temperature (T→0) and low frequency (ω→0), this yields the following
equations
Im[σ(ω)] ≈hOi
√2ω, Re[σ(ω= 0)] ∼e−2b1+O(1/b)=e−Eg
T⇒Eg=3
2√2πhOi ≈ 0.3376hOi .(6.6)
Next we deduce expression of conductivity using self consistent approach as earlier which is valid for
the entire frequency range at low temperature. Here we substitute V(z) = hOi2
2(1 −z3) in eq.(5.28)
and obtain
hVi=hOi2
2.(6.7)
0.5
1.0
1.5
2.0
2.5
3.0
Ω
XO\
0.2
0.4
0.6
0.8
Re Σ
Figure 5: Re(σ) Vs. ω
hOi at low temperature
By substituting eq.(6.7) in eq.(5.25), we get
σ(ω) = r1−hOi2
2ω2.(6.8)
15
For low frequency region, the above expression leads to
Im[σ(ω)] = hOi
√2ω.(6.9)
This matches perfectly with our previous estimate given by eq.[6.6]. In Figure[5], we have also plotted
Re[σ(ω)] vs. ω
hOi using eq.[6.8]. Note that the above analysis ( ψ1= 0 case of p-wave holographic
superconductor) is qualitatively similar to that of ψ2= 0 case of s-wave holographic superconductor
and eq.[6.6] for p-wave corresponds to Im[σ(ω)] = hO1i
ωand Eg= 0.48hO1ifor s-wave holographic
superconductor [22].
7. Conclusions
This paper mostly focuses on the analytical computation of the conductivity of p-wave holographic
superconductors described by Einstein-Yang-Mills theory in the probe limit. In section 3 the system
was analysed around critical temperature (Tc) above which the condensate vanishes. We obtained the
behavior of the field variables φand ψ. Established the relationship between critical temperature and
charge density (ρ). In section 4 we have discussed the approximate behavior of field variables ψand
φby solving the coupled field equations analytically by two different approaches at low temperature
limit. We have also provided the relationship between condensation operator hO1iand Tcat low
temperature.
In section 5 we have discussed the conductivity for the case where ψ0is set to zero. First we
derived the expression of conductivity at low frequency and low temperature and established the
fact that at low frequency limit (ω→0) the real part of σ(ω) is governed by thermal fluctuations as
limω→0Re[σ(ω)] ∼e−Eg
Tand computed the value of the ratio ωg
Tc. We also obtained the expression of
conductivity for the entire frequency range using self-consistent approach. Then using perturbation
techniques the field equation [5.7] for the gauge field Awas solved. In Figures [3] and [4], we showed
the dependency of the real and imaginary part of conductivity on frequency at low temperature limit
and also obtained the ratio ωg
Tcfrom the plots which is consistent with the previously obtained result.
Next in section 6, we have computed the expression for conductivity for the case where ψ1is
set to zero. Another interesting observation we made is that at low temperature the gap energy
Egis proportional to phO1iif we consider the conformal dimension one, that is ψ0= 0 case, and
proportional to hOi if we consider the conformal dimension zero that is, ψ1= 0 case [27].
16
Appendix
Here we will derive the expression for holographic conductivity eq.(5.5). Matter Lagrangian is given
by
Sm=−1
4Zd4x√−gF a
µν Faµν
=−1
4Zd4x√−g∇µAa
ν− ∇νAa
µ+qf abc Ab
µAc
νFaµν
=−1
2Zd4x√−g∇µ(Faµν Aa
ν) + 1
2Zd4x√−g∇µ(Faµν )Aa
ν−q
4Zd4x√−gf abc Ab
µAc
νFaµν .
(7.1)
Now using eq.(1.2), the on shell action is given by
So.s =−1
2Zd4x√−g∇µ(Faµν Aa
ν) + q
4Zd4x√−gf abc Ab
µAc
νFaµν
=−1
2Z∂M
d3x√−hnµFaµν Aa
ν+q
4Zd4x√−gf abc Ab
µAc
νFaµν .(7.2)
Using the ansatz A3
t=φ(r), A1
x=ψ(r) and A3
y=δAyin eq.(7.2), we will get the on shell action to
be
So.s =−1
2Zd3xf(r)A3
y∂rA3
y+f(r)ψ(r)ψ0(r)−r2φ(r)φ0(r)r→∞ −q2
2Zd4xψ2(r)φ2(r)
r2f(r)
+q2
2Zd4xψ2(r)
r4(δAy)2.(7.3)
We can neglect the last term of eq.(7.3) since it contains the perturbation square term. Asymptotic
behaviour of the perturbation field is (from eq.(5.4))
A3
y=A(0) +A(1)
r+... . (7.4)
According to the AdS/CFT correspondence, the electrical current based on the on-shell bulk action
So.s reads
Jy=δSo.s
δA(0) =A(1).(7.5)
In the last line to compute variation of So.s with respect to A(0), we use the fact that A(1) is pro-
portional to A(0)[27]. Electric field at boundary is given by Ey=−[∂t(δAy)]r→∞. So conductivity is
given by
σyy =Jy
Ey
=−iA1
ωA0
.(7.6)
17
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