Analytical approach to compute conductivity of p -wave holographic superconductors

Abstract

In this article we have analytically derived the frequency dependent expression of conductivity and the band gap energy in AdS4 Schwarzschild background for p-wave holographic superconductors 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 the low frequency region. The band gap energy obtained from these two methods seem to agree very well. © 2022 authors. Published by the American Physical Society.published article's title, journal citation, and DOI. Funded by SCOAP3.

Full text

Analytical approach to compute conductivity of p-wave
holographic superconductors
Suchetana Pal,1,* Diganta Parai,1,†and Sunandan Gangopadhyay2,‡
1Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur, Nadia, West Bengal 741 246, India
2Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block—JD,
Sector—III, Salt Lake, Kolkata—700 106, India
(Received 19 January 2022; revised 22 February 2022; accepted 24 February 2022; published 28 March 2022)
In this article we have analytically derived the frequency dependent expression of conductivity and the
band gap energy in AdS4Schwarzschild background for p-wave holographic superconductors 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 the low frequency region. The band gap energy obtained from these two methods seem to
agree very well.
DOI: 10.1103/PhysRevD.105.065015
I. INTRODUCTION
The AdS=CFT correspondence proposed by Maldacena
[1–5] states that a weakly coupled gravity theory in AdSnþ1
spacetime is equivalent to a strongly coupled conformal
field theory CFTnin 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 correspon-
dence is the construction of holographic superconductors.
The term “holographic”implies, by looking at a two
(spatial) dimensional superconductor, taht one can identify
a three-dimensional image that consists of a charged black
hole with nontrivial hair [8]. These holographic super-
conductors can successfully reproduce many important
properties of high Tcsuperconductors (see [9,10] for good
reviews on AdS=CFT duality and its applications to
condensed matter physics). The second-order supercon-
ducting phase transition below a certain critical temperature
can be understood by the condensation of a charged scalar
field that leads to Uð1Þsymmetry breaking near the black
hole horizon in the dual gravitational description. There
exist a large number of studies to describe the Meissner
effect which is another important feature of superconduc-
tors from the holographic superconductor point of view
[11–14]. There are also studies on the effect of nonlinear
electrodynamics and the higher curvature correction on the
holographic superconductor [15,16]. Different mechanisms
have been proposed where at a finite temperature the
system undergoes a spontaneous symmetry breaking and
enters the superconducting phase [17–21]. Recent studies
have shown the explicit matching of the holographic model
with relativistic superfluid hydrodynamics [22]. In this
paper, we consider a p-wave holographic superconductor
model as considered in [17,21,23,24] 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 the
p-wave holographic superconductor depicted by SUð2Þ
Einstein-Yang-Mills theory given by the following action:
S¼Zddxffiffiffiffiffiffi
−g
p1
2ðR−2ΛÞ−
1
4Fa
μνFaμν ;ð1:1Þ
where the negative cosmological constant Λ¼−ðd−1Þðd−2Þ
2L2
with L¼1and Yang-Mills field strength Fa
μν ¼∂μAa
ν−
∂νAa
μþqfabcAb
μAc
ν. The gauge field can be written as
A¼Aa
μσadxμ, where σaare the generators of the SUð2Þ
group and ða; b; cÞ¼ð1;2;3Þare the indices of the
generators. The equation of motion for the field variable
Aμcoming from the above action is given by
1
ffiffiffiffiffiffi
−g
p∂μðffiffiffiffiffiffi
−g
pFaμνÞþqfabcAb
μFcμν ¼0:ð1:2Þ
*suchetanapal92@gmail.com, sp15rs004@iiserkol.ac.in
†digantaparai007@gmail.com, dp16rs028@iiserkol.ac.in
‡sunandan.gangopadhyay@gmail.com,
sunandan.gangopadhyay@bose.res.in
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 105, 065015 (2022)
2470-0010=2022=105(6)=065015(12) 065015-1 Published by the American Physical Society

For finite q, one must consider the effect of the gauge field
on the metric but when qis large the backreaction is
negligible. This q→∞limit is known as the probe limit,
which simplifies the problem but retains important proper-
ties 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 AdS4Schwarzschild back-
ground. The paper is organized into seven sections. We start
with the basic formalism in Sec. II where field equations are
discussed. In Sec. III we provide the relationship between
critical temperature Tcand charge density ρ. In Sec. IV we
analyze the system at low temperature limit (T→0) and
determine the behavior of the charged field ψand gauge
field ϕ, which is consistent with the boundary conditions
and we also obtain the relationship between condensation
operator hO1iand critical temperature Tc. In Sec. Vwe
discuss conductivity and compute band gap energy for the
case ψ0¼0. In Sec. VI we consider the case ψ1¼0and
compute conductivity and band gap energy. In Sec. VII we
draw conclusions from the findings.
II. DISCUSSION OF THE HOLOGRAPHIC MODEL
For this part of our analysis we have chosen the fixed
background of a 3þ1-dimensional Schwarzschild AdS
black hole, whose metric reads
ds2¼−fðrÞdt2þ1
fðrÞdr2þr2ðdx2þdy2Þð2:1Þ
where
fðrÞ¼r2gðrÞ;gðrÞ¼1−r3
þ
r3:ð2:2Þ
Here rþis the horizon radius. The Hawking temperature is
given by
T¼3rþ
4π:ð2:3Þ
In order to investigate the “metal/superconductor”phase
transition1let us consider the following ansatz:
A¼ϕðrÞσ3dt þψðrÞσ1dx: ð2:4Þ
Here the gauge field A3
t¼ϕðrÞis the Uð1Þsubgroup of
SUð2Þand is associated with the chemical potential in the
boundary field theory. The charged field A1
x¼ψðrÞis
associated with the condensation operator in the boundary
field theory whose condensation is responsible for Uð1Þ
symmetry breaking. Note that our analysis has been done in
the probe limit; hence, we did not consider the backreaction
of the gauge field on the metric equation (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 ϕðzÞand ψðzÞ, respectively,
ϕ00ðzÞ−ψ2ðzÞ
r2
þgðzÞϕðzÞ¼0;ð2:5Þ
ψ00ðzÞþg0ðzÞ
gðzÞψ0ðzÞþϕ2ðzÞψðzÞ
r2
þg2ðzÞ¼0:ð2:6Þ
Note that here we considered the coordinate change z¼r
rþ
for simplicity. At the horizon(r¼rþ)z¼1and at the
boundary(r→∞)z→0.
Let us now discuss the boundary conditions. At the
horizon ϕð1Þ¼0and ψð1Þis finite. At the boundary,
which is z→0, the behavior of ϕðzÞand ψðzÞare as
follows:
ϕðzÞ¼μ−ρ
rþ
z; ð2:7Þ
ψðzÞ¼ψ0þψ1
rþ
z: ð2:8Þ
According to the AdS=CFT dictionary, μand ρ, respec-
tively, represent the dual to the 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 ψ0¼0. We will now discuss this
case in detail. Keeping in mind the behavior at the
boundary, we may write
ψðzÞ¼hO1i
ffiffiffi
2
prþ
zFðzÞ:ð2:9Þ
In Sec. VI, we also briefly discuss the case where we set
ψ1¼0. In this case, we write
ψðzÞ¼hOi
ffiffiffi
2
pFðzÞ:ð2:10Þ
For both of these cases FðzÞobeys the following con-
ditions:
Fð0Þ¼1;F
0ð0Þ¼0:ð2:11Þ
1It should be noted that since we are working in the probe limit,
the normal phase does not contain any momentum operator.
Hence, this phase does not correspond to a metal in the strict
condensed matter sense. A proper metallic ground state can be
obtained by including backreaction and the breaking of trans-
lations [25–27].
PAL, PARAI, and GANGOPADHYAY PHYS. REV. D 105, 065015 (2022)
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III. RELATION BETWEEN CRITICAL
TEMPERATURE TcAND CHARGE DENSITY ρ
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 [24].
Using the fact that at the critical temperature Tc,ψðzÞ¼0
from Eq. (2.5) we obtain
ϕðzÞ¼λrþðcÞð1−zÞ;λ¼ρ
r2
þðcÞ
;ð3:1Þ
where rþðcÞis the horizon radius at the critical temperature.
By substituting ϕðzÞand ψðzÞfrom Eq. (3.1) and Eq. (2.9),
respectively, in Eq. (2.6),weget
F00ðzÞ−3z2
1−z3−
2
zF0ðzÞ−3z
1−z3FðzÞ
þλ2
ð1þzþz2Þ2FðzÞ¼0:ð3:2Þ
This equation can be recast in Sturm-Liouville (SL) form
corresponding to eigenvalue λ2, which minimizes the fol-
lowing expression:
λ2¼R1
0dz½z2ð1−z3ÞF0ðzÞ2þ3z3FðzÞ2
R1
0dz z2ð1−zÞ
1þzþz2FðzÞ2:ð3:3Þ
Here we choose the trial function FβðzÞ¼1−βz2,which
obeys the conditions given by Eq. (2.11). The minimum λ2is
attained for β¼0.5078.
These findings yield [24]
Tc¼3
4πrþðcÞ¼3
4πffiffiffiffiffi
ρ
λβ
r≈0.1239 ffiffiffi
ρ
pð3:4Þ
which matches with the numerical finding Tc¼
0.1249 ffiffiffi
ρ
p[17].
Note that in case of the s-wave holographic super-
conductor, the relation between Tcand ρfor the same
conformal dimension reads [28]
Tc¼0.225 ffiffiffi
ρ
p:ð3:5Þ
Equations (3.4) and (3.5) indicate that the critical temper-
ature for the p-wave holographic superconductor is less
than s-wave holographic superconductor.
IV. CONDENSATION OPERATOR hO1i
AT LOW TEMPERATURE
As we are interested in the low temperature limit
(T→0), we consider the scaling z¼s
bwhere b→∞.
We will determine blater on, and it will be clear that b→
∞corresponds to the low temperature limit [28,29]. Under
this condition the dominant contribution comes from the
neighboring region of the boundary (z→0) and Eq. (2.5)
and Eq. (2.6) take the following forms, respectively,
ϕ00ðsÞ−hO1i2
2r4
þb4s2F2ðsÞϕðsÞ¼0;ð4:1Þ
F00ðsÞþ2
sF0ðsÞþϕ2ðsÞ
r2
þb2FðsÞ¼0:ð4:2Þ
We aim to obtain the solutions of Eq. (4.1) and Eq. (4.2)
iteratively, which are consistent with the boundary con-
ditions. To do that, we will start with the following form
of FðsÞ, which is essentially the behavior of FðsÞat s>1
or z>1
b:
FðsÞ≈α
s:ð4:3Þ
Here αis a constant to be determined later. Substituting
FðsÞfrom Eq. (4.3) in from Eq. (4.1), we get
ϕ00ðsÞ−hO1i2α2
2r4
þb4ϕðsÞ¼0:ð4:4Þ
Choosing bas
b¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
hO1iα
p21
4rþð4:5Þ
the solution of Eq. (4.4) reads
ϕðsÞ¼C1e−sþC2es:ð4:6Þ
Using the condition ϕðzÞ¼0at the horizon (z→1) for all
values of b, it is easy to show that C2¼0and hence ϕðsÞ
can be written as
ϕðsÞ¼Crþbe−sð4:7Þ
where C1¼Cffiffiffiffiffiffiffiffiffi
hO1iα
p
21
4
. Note that Eq. (4.5) shows that bgoes
as inverse of rþ. This implies that b→∞as rþ→0, and
hence from Eq. (2.3) it is evident that the large blimit
corresponds to the low temperature limit. The coefficient
ffiffiffiffiffi
α
ffiffi2
p
qin Eq. (4.5) has been taken for computational
convenience. Any other choice would give the same result
as can be checked easily. The factor ffiffiffiffiffiffiffiffiffiffi
hO1i
phas to be there
for dimensional consideration.
Now we proceed to estimate a more accurate behavior of
FðsÞthat is consistent with the conditions FðzÞ¼1and
F0ðzÞ¼0at the boundary z¼0. By substituting ϕðsÞfrom
Eq. (4.7) in (4.2), we obtain
ANALYTICAL APPROACH TO COMPUTE CONDUCTIVITY OF …PHYS. REV. D 105, 065015 (2022)
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F00ðsÞþ2
sF0ðsÞþC2e−2sFðsÞ¼0:ð4:8Þ
The above equation can be solved using the SL approach in
the interval ð0;∞Þ. The corresponding eigenvalue C2is
given by
C2¼R∞
0s2F0ðsÞ2ds
R∞
0s2e−2sF2ðsÞds :ð4:9Þ
Here we choose the following trial function as the eigen-
function FðsÞthat minimizes Eq. (4.9) and is consistent
with the boundary conditions
FðsÞ¼α
stanh s
α:ð4:10Þ
We obtain the minimum for α¼αS:L ¼0.8179 and that
corresponds to C¼CS:L ¼2.4065.
Next we aim to solve Eq. (4.1) once again perturbatively
by substituting FðsÞfrom Eq. (4.10) and considering ϕðsÞ
from Eq. (4.7) as the zeroth order solution. We obtain
ϕðzÞ¼c1−c2
bz
αþbrþCe−bz1−2α
αþ22
e−2bz
α3F2
×2;1þα
2;1þα
2;2þα
2;2þα
2;−e−2bz
α:
ð4:11Þ
Using the fact that at z¼1,ϕðzÞ¼0and b→∞,wemay
show that c1¼c2¼0. Finally, we obtain
ϕðzÞ¼brþCe−bz1−2α
αþ22
e−2bz
α3F2
×2;1þα
2;1þα
2;2þα
2;2þα
2;−e−2bz
α:
ð4:12Þ
By comparing the coefficient of zfrom the above equation
with the boundary behavior of ϕðzÞgiven by Eq. (2.7),we
obtain
ρ
rþ¼0.4911b2rþC: ð4:13Þ
Now we substitute ρfrom Eq. (3.4) and bfrom Eq. (4.5) in
Eq. (4.13) and get
ffiffiffiffiffiffiffiffiffiffi
hO1i
p¼ξTcð4:14Þ
where the dimensionless parameter ξ¼9.7624.
Restoring the zcoordinate now, we may write FðzÞans
ψðzÞas follows:
FðzÞ¼ α
bz tanh bz
α;ð4:15Þ
ψðzÞ¼hO1i
ffiffiffi
2
prþ
α
btanh bz
α:ð4:16Þ
Interestingly we can determine the constant Cand αby direct
analytical approach as Eq. (4.8) is analytically solvable.
Using the condition Fð0Þ¼1from Eq. (4.8), we get
FðsÞ¼π
2s½Y0ðCÞJ0ðCe−sÞ−J0ðCÞY0ðCe−sÞ:ð4:17Þ
To compute the constant Cwe will use the condition
FðsÞ→0as s→∞.FromEq.(4.17),wethusobtain
J0ðCÞ¼0:ð4:18Þ
The above equation implies C¼Cdirect ¼2.4048,whichis
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 SL approach. Now FðsÞmay be written
as follows:
FðsÞ¼π
2s½Y0ðCÞJ0ðCe−sÞ:ð4:19Þ
Now let us compare the behavior of the FðsÞfrom the above
equation (4.19) with Eq. (4.3) for s>1or z>1
bto find out α.
This gives
FðsÞ¼πY0ðCÞ
2sð4:20Þ
α¼πY0ðCÞ
2¼0.8009:ð4:21Þ
We see that α¼αdirect ¼0.8009 is in good agreement with
the previous estimate of αS:L.
Following the earlier steps using FðsÞgiven by
Eq. (4.19), and considering ϕðsÞfrom Eq. (4.7) as the
zeroth order solution, we compute ϕðzÞfrom Eq. (4.1) and
the relationship between hO1iand Tc.
FIG. 1. FðsÞvs s.
PAL, PARAI, and GANGOPADHYAY PHYS. REV. D 105, 065015 (2022)
065015-4

ϕðzÞ¼brþCe−bz3F41
2;1
2;1
2;1;1;3
2;3
2;−C2e−2bz
ð4:22Þ
ffiffiffiffiffiffiffiffiffiffi
hO1i
p¼10.0518Tc:ð4:23Þ
In this case we may write FðzÞand ψðzÞas follows:
FðzÞ¼ π
2bz ½Y0ðCÞJ0ðCe−bzÞ;ð4:24Þ
ψðzÞ¼hO1i
ffiffiffi
2
prþ
π
2b½Y0ðCÞJ0ðCe−bzÞ:ð4:25Þ
It is worth noting that for both the cases given by Eq. (4.15)
and Eq. (4.24), it is possible to write FðzÞ¼1þOðz2Þas
expected.
To get a clear idea about the behavior of FðsÞ
obtained from two different methods, in Fig. 1we
have plotted FðsÞvs s. The blue, orange, and green
curves represent Eqs. (4.3),(4.10),and(4.19), respec-
tively. By looking at the blue curve representing
Eq. (4.3) we can easily say that indeed it is the
behavior of FðsÞat s>1or z>1
b, and by looking
at the orange and green curve it is clear that the
behavior of FðsÞpredicted by Eqs. (4.10) and (4.19)
is very similar as expected.
In Fig. 2, we have plotted ϕðsÞ
ffiffiffiffiffiffiffi
hO1i
pvs s. The blue and
orange curves, respectively, represent Eqs. (4.12) and
(4.22), which suggests the behavior of ϕdepicted by these
equations is very similar. Note that for the rest of the
analysis we will use α¼αS:L ¼0.8179.
In Fig. 3we have plotted ffiffiffiffiffiffiffi
hO1i
p
Tcvs T
Tcnumerically and
observe that in the low temperature region hO1iis almost
constant. We get hO1i¼9.51Tcfor T
Tc¼0.09, which
matches nicely with the analytical results presented in
Table I.
V. CONDUCTIVITY AT LOW TEMPERATURE
To study the conductivity at the boundary, we will
consider an electromagnetic perturbation in the bulk by
applying a nonzero gauge field in the ydirection. Let us
consider the following ansatz:
Ay¼AðrÞe−iωtσ3:ð5:1Þ
Note that as our analysis is done on the probe limit, we do
not consider the backreaction 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
FIG. 2. ϕðsÞ
ffiffiffiffiffiffiffi
hO1i
pvs s.
FIG. 3. ffiffiffiffiffiffiffi
hO1i
p
Tcvs T
Tc.
TABLE I. Comparison for ξobtained analytically and numeri-
cally.
Approach ξ
SL 10.0518
Direct 9.7624
Numerical 9.51
ANALYTICAL APPROACH TO COMPUTE CONDUCTIVITY OF …PHYS. REV. D 105, 065015 (2022)
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A00ðrÞþf0ðrÞ
fðrÞA0ðrÞþω2
f2ðrÞ−ψ2ðrÞ
r2fðrÞAðrÞ¼0:ð5:2Þ
Switching to zcoordinate, we obtain
A00ðzÞþg0ðzÞ
gðzÞA0ðzÞþ 1
r2
þω2
g2ðzÞ−ψ2ðzÞ
gðzÞAðzÞ¼0:ð5:3Þ
At the boundary, the behavior of the gauge field AðzÞcan be
found from Eq. (5.2) given by
AðzÞ¼A0þA1
rþ
z: ð5:4Þ
The conductivity σyy for our system is given as follows
(refer to the Appendix) [30–32]:
σyy ¼−iA1
ωA0¼−irþA0ð0Þ
ωAð0Þ:ð5:5Þ
Let us now switch to the tortoise coordinate (as z→1,
r→−∞) defined as follows, where the integration con-
stant is chosen such that at the boundary (z¼0), r¼0
r¼Zdr
fðrÞ¼1
6rþ2lnð1−zÞ−lnð1þzþz2Þ
−2ffiffiffi
3
ptan−1ffiffiffi
3
pz
2þz:ð5:6Þ
Now Eq. (5.3) takes the following form:
A00ðrÞþω2AðrÞ¼VAðrÞ;VðrÞ¼ψ2ðrÞfðrÞ
r2:
ð5:7Þ
Note that at the horizon (r¼rþ), V¼0. Taking into
account the ingoing boundary condition [28] at the horizon,
solving the above equation, we get
A∼e−iωr∼ð1−zÞ−iω
3rþ:ð5:8Þ
Near the horizon (z¼1), the main contribution in rcomes
from the first term as given in Eq. (5.6). In order to obtain
an expression for AðzÞ, Eq. (5.3) ought to be solved by
taking into account the boundary behavior. We may now
write
AðzÞ¼ð1−zÞ−iω
3rþGðzÞ:ð5:9Þ
Substituting AðzÞfrom Eq. (5.9) in Eq. (5.3), we get
3ð1−z3ÞG00ðzÞ−9z2−2ð1þzþz2Þiω
rþG0ðzÞ
−3ψ2ðzÞ
r2
þ
−ð1þ2zÞiω
rþ
−ð2þzÞð4þzþz2Þ
3ð1þzþz2Þ
ω2
r2
þ
×GðzÞ¼0:ð5:10Þ
At the horizon (z¼1), from the above equation we deduce
3−
2iω
rþG0ð1Þþψ2ð1Þ
r2
þ
−iω
rþ
−
2ω2
3r2
þGð1Þ¼0:ð5:11Þ
Turning on the low temperature limit, Eq. (5.10) may be
approximated as
G00ðzÞþ2iω
3rþ
G0ðzÞ−ψ2ðzÞ
r2
þ
−iω
3rþþ8ω2
9r2
þGðzÞ¼0:
ð5:12Þ
Substituting ψðzÞfrom Eq. (4.16), we exactly solve the
above equation and obtain
GðzÞ¼e−iω
3rþz"cþP
αffiffiffiffiffiffiffiffiffiffiffi
1−ffiffi2
pω2
αhO1i
q
1
2ð−1þffiffiffiffiffiffiffiffiffiffi
1þ4α2
pÞtanh bz
α
þc−P
−αffiffiffiffiffiffiffiffiffiffiffi
1−ffiffi2
pω2
αhO1i
q
1
2ð−1þffiffiffiffiffiffiffiffiffiffi
1þ4α2
pÞtanh bz
α#;ð5:13Þ
where Pμ
νare the fractional Legendre functions. Finally, we
may write AðzÞfor the low frequency (ω≪hO1i) region as
AðzÞ¼ð1−zÞ−iω
3rþe−iω
3rþzcþPα
1
2ð−1þffiffiffiffiffiffiffiffiffiffi
1þ4α2
pÞtanh bz
α
þc−P−α
1
2ð−1þffiffiffiffiffiffiffiffiffiffi
1þ4α2
pÞtanh bz
α:ð5:14Þ
Using the definition of conductivity from Eq. (5.5) for low
temperature and low frequency, we write
σðωÞ¼0.4616iffiffiffiffiffiffiffiffiffiffi
hO1i
pω1−1.3911 cþ
c−
1−0.4085 cþ
c−:ð5:15Þ
Next, we aim to determine the ratio cþ
c−. Note that at z→1,
tanhðbz
αÞ≈1, and under this condition we may approximate
Pα
1
2ð−1þffiffiffiffiffiffiffiffiffiffi
1þ4α2
pÞtanhbz
α¼2α
2
Γð1∓αÞ1−tanhbz
α∓α
2þ:
ð5:16Þ
Now for the low frequency region from Eq. (5.13), we also
get
PAL, PARAI, and GANGOPADHYAY PHYS. REV. D 105, 065015 (2022)
065015-6

Gð1Þ¼cþ
Γð1−αÞebþc−
Γð1þαÞe−be−iω
3rþ;
G0ð1Þ¼cþðb−iω
3rþÞ
Γð1−αÞebþc−ðbþiω
3rþÞ
Γð1þαÞe−be−iω
3rþ:ð5:17Þ
Using Eq. (5.11) and Eq. (5.17) the ratio cþ
c−becomes
cþ
c−¼−e−2bΓð1−αÞ
Γð1þαÞb−3
bþ3þ4ðb2−3Þ
bðbþ3Þ2
iω
rþþOðω2Þ:
ð5:18Þ
Substituting the above ratio in Eq. (5.15), we obtain σðωÞat
low frequency (ω→0). This yields the following equations:
Im½σðωÞ ≈0.4616 ffiffiffiffiffiffiffiffiffiffi
hO1i
pω;ð5:19Þ
Re½σðω¼0Þ ∼e−2b½1þOð1=bÞ ≡e
−Eg
T;ð5:20Þ
Eg¼3ffiffiffiffiffiffiffiffiffiffiffiffiffi
αhO1i
p25
4π
≈0.3631 ffiffiffiffiffiffiffiffiffiffi
hO1i
p:ð5:21Þ
We have used α¼αSL ¼0.8179 obtained from the SL
method in Sec. IV in the above equation. In Fig. 4we have
plotted ReðσÞvs ω
ffiffiffiffiffiffiffi
hO1i
pat T¼0.09Tcnumerically. From
there we can observe that Eg¼0.37 ffiffiffiffiffiffiffiffiffiffi
hO1i
p.
In the probe limit, the gap frequency ωg¼2Eg[20,33],
where Egis the energy gap. Now using Eq. (5.21) and
Eq. (4.14), we get
ωg
Tc¼2Eg
Tc¼7.0894:ð5:22Þ
Next we evaluate the following ratio using Eqs. (4.14),
(5.19), and (3.4)
lim
ω→0
ω
ffiffiffi
ρ
pIm½σðωÞ ¼ 0.5583:ð5:23Þ
This agrees exactly with the numerical result given in [17].
We would like to make a comment here. Relations of the
form given by Eqs. (5.19),(5.20), and (5.21) also hold in
the s-wave case. In this case, the divergence in the linear
part of the conductivity can be determined explicitly from
hydrodynamics and it coincides with ρs
μ, where ρsis the
superfluid density and μis the chemical potential [22].
It would therefore be interesting to see whether this holds
even in the p-wave case. From Eq. (4.14), we find that
ffiffiffiffiffiffi
O1
p∼Tc∼rþðcÞ∼ρ=μwhere we have used the fact that
rþðcÞ¼ρ=μ. Hence, Im½σ½ω ∼ρ
μ
1
ω. One can, in principle,
also try to get this rigorously from a p-wave hydrodynamic
model which is, however, beyond the scope of this
present work.
In this article, we also compute the expression of
conductivity in a 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
AðrÞ¼e−iffiffiffiffiffiffiffiffiffiffiffi
ω2−hVi
pr;ð5:24Þ
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
σðωÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−hVi
ω2
rð5:25Þ
where
hVi¼R0
−∞ VjAðrÞj2dr
R0
−∞ jAðrÞj2dr
:ð5:26Þ
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¼−
1
rþzþz4
4þz7
7þz10
10 þ:¼−
˜z
rþ
:ð5:27Þ
Now Eq. (5.26) can be rewritten as follows:
hVi¼R∞
0Vð˜zÞe
−2ffiffiffiffiffi
hVi
pffiffiffiffiffiffiffiffi
1−ω2
hVi
q˜z
rþd˜z
R∞
0e
−2ffiffiffiffiffi
hVi
pffiffiffiffiffiffiffiffi
1−ω2
hVi
q˜z
rþd˜z
;
VðzÞ¼hO1iα
ffiffiffi
2
pð1−z3Þtanh2bz
α:ð5:28Þ
FIG. 4. ReðσÞvs ω
ffiffiffiffiffiffiffi
hO1i
pat T¼0.09Tc.
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Notice that at low temperature as rþ→0, the main
contribution to the integral comes when ˜z→0and this
condition implies z¼˜zor rðzÞ¼−z
rþ, which is essen-
tially the region near the boundary. Hence, we put
Vð˜zÞ¼hO1iα
ffiffiffi
2
pð1−˜z3Þtanh2b˜z
α:ð5:29Þ
As ˜z→0, we may write
hVi¼hO1iα
ffiffiffi
2
pR∞
0tanh2ðb˜z
αÞe
−2ffiffiffiffiffi
hVi
pffiffiffiffiffiffiffiffi
1−ω2
hVi
q˜z
rþd˜z
R∞
0e
−2ffiffiffiffiffi
hVi
pffiffiffiffiffiffiffiffi
1−ω2
hVi
q˜z
rþd˜z
:ð5:30Þ
After integrating we deduce
ffiffiffi
2
pˆ
V¼1þ25
4αffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
V−ˆ
ω2
pþ2ffiffiffi
2
pα2ðˆ
V−ˆ
ω2Þ
×ψα
23
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
V−ˆ
ω2
p−ψ1
2þα
23
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
V−ˆ
ω2
p
ð5:31Þ
where
ˆ
V¼hVi
hO1iα;ˆ
ω2¼ω2
hO1iα:ð5:32Þ
For low frequency, which is ω→0, from Eq. (5.31) we
obtain
ˆ
V¼0.2924. Hence, for low temperature and
low frequency, conductivity given by Eq. (5.25) may be
written as
σðωÞ¼0.489iffiffiffiffiffiffiffiffiffiffi
hO1i
pω:ð5:33Þ
By comparing the above equation (5.33) with the imaginary
part of conductivity obtained in Eq. (5.19), we see that they
are in good agreement.
At high frequencies, which is ω→∞from Eq. (5.31),
we obtain
ˆ
V¼−1
4α2ˆ
ω2. Hence, at low temperature and
high frequency, conductivity given by Eq. (5.25) may be
written as
σðωÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þhO1i2
4ω4
s:ð5:34Þ
When
ˆ
Vis comparable with ˆ
ω2, which is
ˆ
V¼ˆ
ω2, from
Eq. (5.31) we obtain hVi¼αhO1i
ffiffi2
pand
σðωÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−αhO1i
ffiffiffi
2
pω2
s:ð5:35Þ
Interestingly, there is another way to solve Eq. (5.7)
by treating δV¼V−αhO1i
ffiffi2
pas the perturbation and
AðrÞ¼e
−iffiffiffiffiffiffiffiffiffiffiffiffiffi
ω2−αhO1i
ffiffi2
p
qras zeroth order solution [28].To
see this, we rewrite Eq. (5.7) as follows:
A00ðrÞþω2−αhO1i
ffiffiffi
2
pAðrÞ¼V−αhO1i
ffiffiffi
2
pAðrÞ:
ð5:36Þ
By solving the above equation we obtain
AðrÞ¼e
−iffiffiffiffiffiffiffiffiffiffiffiffiffi
ω2−αhO1i
ffiffi2
p
qr1þα2
2β−α2π
sin πβ e2iffiffiffiffiffiffiffiffiffiffiffiffiffi
ω2−αhO1i
ffiffi2
p
qr
þα2
2β2F11;β;1þβ;−e
−2ffiffiffiffiffiffi
hO1i
ffiffi2
pα
qr
−α2
2ð1þβÞe
−2ffiffiffiffiffiffi
hO1i
ffiffi2
pα
qr2F1
×1;1þβ;2þβ;−e
−2ffiffiffiffiffiffi
hO1i
ffiffi2
pα
qr ð5:37Þ
where β¼iαffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffi2
pω2
αhO1i−1
q. Note note that to determine the
integration constants we use the ingoing boundary con-
dition at the horizon and we also use the relation
2F1ð1;β;1þβ;xÞ¼eiπβ
xβ
πβ
sin πβ when jxj→∞.
The expression of conductivity given by Eq. (5.5) can be
rewritten now as
σðωÞ¼i
ωdAðrÞ
dr
AðrÞr¼0
:ð5:38Þ
Using Eq. (5.37) and Eq. (4.14) in Eq. (5.38), we deduce
σðωÞ¼ iξβ
21
4ffiffiffi
α
pðω
TcÞ1−
2ð1þα2
2βÞ
1−πα2
sin πβ þα2
2fψð1þβ
2Þ−ψðβ
2Þg;
β¼iαffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffi
2
p
αξ2ω
Tc2
−1
s:ð5:39Þ
Figures 5and 6depict the dependency of Re½σðωÞ and
Im½σðωÞ on ω
Tc. The green curves are obtained using
Eq. (5.39) analytically and the blue curves are plotted
numerically for T
Tc¼0.09. In Table II we compare ωg
Tc,
Im½σðωÞ, and ω→0
ω
ffiffiρ
pIm½σðωÞ obtained from different
approaches.
From Fig. 5, we find that at T→0,Re½σðωÞ vanishes
for ω<ωgand a gap appears as expected. From the plot we
obtain ωg
Tc¼7.4242 [34] which is consistent with our
previous estimate given by Eq. (5.22).
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It is worth mentioning that for the Bi2Sr2CaCu2O8þδ
sample, which is a high Tcsuperconductor, this ratio is
found to be 7.90.5[35]. On the other hand, for weakly
coupled low Tcsuperconductors described by Bardeen-
Cooper-Schrieffer (BCS) theory [36], the value of this ratio
is 3.5. Another interesting observation is that at ω¼0,
there is also a delta function in the Re½σðωÞ for all T<T
c.
Although we cannot detect it by analytical or numerical
computation as it gives us only the continuous part of σðωÞ,
this delta function can be revealed by looking at the pole in
Im½σðωÞ at ω¼0. A general argument for such a con-
clusion 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
Im½σðωÞ ¼ −
1
π
PZ∞
−∞
Re½σðω0Þdω0
ω0−ω:ð5:40Þ
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 Fig. 6
and from Eq. (5.19) that there is indeed a pole in Im½σðωÞ
at ω¼0at low temperature. In the probe limit at T→0,
this delta function in the real part of conductivity implies
infinite DC conductivity of the 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 nondissipative. 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. In Fig. 7we have plotted ReðσÞvs ω
Tcnumerically.
The curves are plotted for successively lower temperatures
from the left to right. The horizontal line describes the
constant conductivity at T¼Tcand the right most curve is
plotted for T
TC¼0.09. These plots clearly indicate that as we
FIG. 6. ImðσÞvs ω
Tcat low temperature.
FIG. 7. ReðσÞvs ω
Tcfor T
Tc¼1, 0.711, 0.507, 0.308, 0.208, 0.09
(from left to right).
TABLE II. Comparisons for ωg
Tc,Im½σðωÞ, and ω→0
ω
ffiffiρ
pIm½σðωÞ
obtained from different approaches.
Approach ωg
Tc
Direct 7.0894
Perturbative 7.424
Numerical 7.3
Approach Im½σðωÞ
Direct 0.4616 ffiffiffiffiffiffiffi
hO1i
p
ω
Self consistent 0.489 ffiffiffiffiffiffiffi
hO1i
p
ω
Numerical 0.4625 ffiffiffiffiffiffiffi
hO1i
p
ω
Approach ω→0
ω
ffiffiρ
pIm½σðωÞ
Analytical 0.5583
Numerical 0.55
FIG. 5. ReðσÞvs ω
Tcat low temperature.
ANALYTICAL APPROACH TO COMPUTE CONDUCTIVITY OF …PHYS. REV. D 105, 065015 (2022)
065015-9

lower the temperature below the critical temperature a gap
starts to form. Note that T¼0.09Tcis the lowest temper-
ature we could achieve numerically and at this temperature
the numerical results are in very good agreement with the
analytical results obtained in the low temperature limit.
From the nature of the numerics we can argue that if we can
further lower the temperature the numerical results will be
closer to the analytical results. From Figs. 5and 7we infer
that the analytical results starts to deviate from the numerics
above T
Tc¼0.09.
Note that the behavior of σðωÞdepicted by Figs. 5and 6
for p-wave holographic superconductor is qualitatively
similar to the ψ1¼0case of s-wave holographic super-
conductor. The relations (5.19) and (5.21) in our analysis
correspond to Im½σðωÞ ¼ 0.55 ffiffiffiffiffiffiffi
hO2i
p
ωand Eg¼0.43 ffiffiffiffiffiffiffiffiffiffi
hO2i
p
for the s-wave holographic superconductor [28].
VI. CONDUCTIVITY FOR THE CASE ψ1=0
Now let us look into the case where we set ψ1¼0,
and ψðzÞis given by Eq. (2.10). Here we simply choose
FðzÞ¼1which 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
rescaling z¼s
band by letting b→∞, we obtain the
following equation:
G00ðsÞþ 2iω
3brþ
G0ðsÞ−hOi2
2r2
þb2−iω
3rþb2−
8ω2
9r2
þb2GðsÞ¼0:
ð6:1Þ
Next we choose b¼hOi
ffiffi2
prþ
. Equation (6.1) gives the follow-
ing approximate solution that is valid for low temperature
and low frequency ðω≪hOiÞ region as
GðsÞ¼cþesþc−e−s
⇒AðzÞ≈eiωz
3rþcþehOi
ffiffi2
prþzþc−e−hOi
ffiffi2
prþz:ð6:2Þ
Using the definition of conductivity given by Eq. (5.5),we
obtain
σðωÞ≈ihOi
ffiffiffi
2
pω
1−cþ
c−
1þcþ
c−
:ð6:3Þ
The ratio cþ
c−can be found from the boundary condition
given by Eq. (5.11), where we substitute ψð1Þ≈hOi
ffiffi2
p. This
gives
cþ
c−¼−e−2bb−3
bþ3þ2ð2b2−3Þ
bðbþ3Þ2
iω
rþþOðω2Þ:ð6:4Þ
By plugging the above, the ratio in Eq. (6.3) leads to
σðωÞ¼ihOi
ffiffiffi
2
pω1þ2e−2bb−3
bþ3þ2ð2b2−3Þ
bðbþ3Þ2
iω
rþþOðω2Þ:
ð6:5Þ
For the case of low temperature (T→0) and low frequency
(ω→0), this yields the following equations:
Im½σðωÞ ≈hOi
ffiffiffi
2
pω;
Re½σðω¼0Þ ∼e−2b½1þOð1=bÞ ¼ e
−Eg
T⇒Eg
¼3
2ffiffiffi
2
pπhOi≈0.3376hOi:ð6:6Þ
Next we deduce the expression of conductivity using a self-
consistent approach at low temperature. Here we substitute
VðzÞ¼hOi2
2ð1−z3Þin Eq. (5.28) and obtain hVifor the
low frequency and high frequency regions as
hVi¼hOi2
2:ð6:7Þ
By substituting Eq. (6.7) in Eq. (5.25), we get
σðωÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−hOi2
2ω2
s:ð6:8Þ
The above expression is valid for the low frequency and
high frequency regions. For the low frequency region, the
above expression leads to
Im½σðωÞ ¼ hOi
ffiffiffi
2
pω:ð6:9Þ
This matches perfectly with our previous estimate given by
Eq. (6.6). The numerical analysis also agrees exactly with
the analytical result, Eq. (6.9). Note that the above analysis
(ψ1¼0case of p-wave holographic superconductor) is
qualitatively similar to that of the ψ2¼0case of the
s-wave holographic superconductor and Eq. (6.6) for the
p-wave corresponds to Im½σðωÞ¼hO1i
ωand Eg¼0.48hO1i
for the s-wave holographic superconductor [28].
VII. 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 Sec. III the system was analyzed around a critical
temperature (Tc) above which the condensate vanishes.
We obtained the behavior of the field variables ϕand ψ.
We then established the relationship between critical
temperature and charge density (ρ). In Sec. IV we have
discussed the approximate behavior of field variables ψand
PAL, PARAI, and GANGOPADHYAY PHYS. REV. D 105, 065015 (2022)
065015-10

ϕby solving the coupled field equations analytically by
two different approaches at the low temperature limit. We
have also provided the relationship between the condensa-
tion operator hO1iand Tcat low temperature.
In Sec. Vwe 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 the low frequency limit (ω→0)
the real part of σðωÞis governed by thermal fluctuations as
limω→0Re½σðωÞ ∼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 Figs. 5and 6, we showed
the dependency of the real and imaginary parts of con-
ductivity on the frequency at the low temperature limit and
also obtained the ratio ωg
Tcfrom the plots which is consistent
with the previously obtained result.
Next in Sec. VI, 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 ffiffiffiffiffiffiffiffiffiffi
hO1i
pif we consider
the conformal dimension one, which is the ψ0¼0case,
and proportional to hOiif we consider the conformal
dimension zero, which is the ψ1¼0case [33].
We have also numerically verified all the analytical
results obtained at T→0and observed that they are in
good agreement when we choose T¼0.09Tcnumerically.
This is available online Ref. [37].
ACKNOWLEDGMENTS
The authors would like to thank the referee for useful
comments. We would also like to mention that the
Mathematica code prepared by us for doing the numerical
work was based on the one developed by C. P. Herzog.
APPENDIX: DERIVATION OF HOLOGRAPHIC
CONDUCTIVITY EQUATION
Here we will derive the expression for the holographic
conductivity equation (5.5). The matter Lagrangian is
given by
Sm¼−
1
4Zd4xffiffiffiffiffiffi
−g
pFa
μνFaμν
¼−
1
4Zd4xffiffiffiffiffiffi
−g
p½∇μAa
ν−∇νAa
μþqfabcAb
μAc
νFaμν
¼−
1
2Zd4xffiffiffiffiffiffi
−g
p∇μðFaμνAa
νÞþ1
2Zd4xffiffiffiffiffiffi
−g
p∇μðFaμνÞAa
ν−q
4Zd4xffiffiffiffiffiffi
−g
pfabcAb
μAc
νFaμν:ðA1Þ
Now using Eq. (1.2), the on-shell action is given by
So:s¼−
1
2Zd4xffiffiffiffiffiffi
−g
p∇μðFaμνAa
νÞþq
4Zd4xffiffiffiffiffiffi
−g
pfabcAb
μAc
νFaμν
¼−
1
2Z∂M
d3xffiffiffiffiffiffi
−h
pnμFaμνAa
νþq
4Zd4xffiffiffiffiffiffi
−g
pfabcAb
μAc
νFaμν:ðA2Þ
Using the ansatz A3
t¼ϕðrÞ,A1
x¼ψðrÞ, and A3
y¼δAyin Eq. (A2), we will get the on-shell action to be
So:s¼−
1
2Zd3x½fðrÞA3
y∂rA3
yþfðrÞψðrÞψ0ðrÞ−r2ϕðrÞϕ0ðrÞr→∞−q2
2Zd4xψ2ðrÞϕ2ðrÞ
r2fðrÞþq2
2Zd4xψ2ðrÞ
r4ðδAyÞ2:ðA3Þ
We can neglect the last term of Eq. (A3) since it contains
the perturbation square term. Asymptotic behavior of the
perturbation field is [from Eq. (5.4)]
A3
y¼Að0ÞþAð1Þ
rþ:ðA4Þ
According to the AdS=CFT correspondence, the
electrical current based on the on-shell bulk action So:s
reads
Jy¼δSo:s
δAð0Þ¼Að1Þ:ðA5Þ
In the last line, to compute a variation of So:swith respect to
Að0Þ, we use the fact that Að1Þis proportional to Að0Þ[33].
The electric field at the boundary is given by Ey¼
−½∂tðδAyÞr→∞. So conductivity is given by
σyy ¼Jy
Ey¼−iA1
ωA0
:ðA6Þ
ANALYTICAL APPROACH TO COMPUTE CONDUCTIVITY OF …PHYS. REV. D 105, 065015 (2022)
065015-11

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