Refractive index in generalized superconductors with Born–Infeld electrodynamics
Eur. Phys. J. C
Refractive index in generalized superconductors with Born-Infeld electrodynamics
Jun Cheng 0
Qiyuan Pan 0
Hongwei Yu 0
Jiliang Jing 0
0 Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, and Department of Physics, Hunan Normal University , Changsha 410081, Hunan , China
We investigate, in the probe limit, the negative refraction in the generalized superconductors with the BornInfeld electrodynamics. We observe that the system has a negative Depine-Lakhtakia index in the superconducting phase at small frequencies and the greater the Born-Infeld corrections the larger the range of frequencies or the range of temperatures for which the negative refraction occurs. Furthermore, we find that the tunable Born-Infeld parameter can be used to improve the propagation of light in the holographic setup. Our analysis indicates that the Born-Infeld electrodynamics plays an important role in determining the optical properties of the boundary theory.
-
1 Introduction
As one of the most significant developments in the
theoretical physics over the last decade, the anti-de Sitter/conformal
field theories (AdS/CFT) correspondence [
1–4
], which
provides an astonishing duality between the gravity in the
ddimensional spacetime and the gauge field theory living on
its (d − 1)-dimensional boundary, has been employed to
study the strong coupled systems in condensed matter physics
which are intractable by the traditional approaches [5]. In
particular, such a strong/weak duality might give some insights
into the pairing mechanism in the high Tc superconductors
[
6
]. It was first suggested by Gubser [
7
] that the spontaneous
U (1) symmetry breaking by bulk black holes can be used
While we were completing this work, a complementary paper [
71
] on the
optical properties of Born–Infeld–dilaton-Lifshitz holographic
superconductors appeared in arXiv.
* Corresponding author.
to construct gravitational duals of the transition from a
normal state to a superconducting state in the boundary theory,
which exhibit characteristic behaviors of superconductors
[
8
]. Along this line, there have been a lot of works
studying various gravity models with the property of the so-called
holographic superconductor, for reviews, see Refs. [
9–12
]
and references therein.
On the other hand, the refractive index, which reflects the
propagation property of light in an electromagnetic medium,
is one of the most important electromagnetic properties of
a medium. In 1968, Veselago first considered the case of a
medium which has both negative dielectric permittivity and
negative magnetic permeability at a given frequency and
proposed in theory that the refraction index might be negative
[
13
]. Around 2000, this exotic electromagnetic phenomenon
– negative refraction, which implies that the energy flux of
the electromagnetic wave flows in the opposite direction with
respect to the phase velocity, was experimentally realized in
a new class of artificial media commonly called
“metamaterials” [
14,15
]. Since then, the study on the negative refraction
has attracted intensive interest and a large number of the
negative refractive index materials have been introduced [16].
Interestingly, recent studies show that in the radio, microwave
and low-terahertz frequency ranges the superconductors can
behave as metamaterials and exhibit negative refraction [
17
].
Using the AdS/CFT correspondence, Gao and Zhang
investigated the optical properties in the s-wave superconductors
and found that the negative refraction does not appear in these
holographic models in the probe limit [
18
]. However, in the
fully backreacted spacetime, Amariti et al. showed that the
negative refraction is allowed in the superconducting phase
[
19
]. Extending the investigation to the generalized
holographic superconductors in which the spontaneous
breaking of a global U (1) symmetry occurs via the Stückelberg
mechanism [
20,21
], Mahapatra et al. observed that a
negative Depine–Lakhtakia index may appear at low frequencies
in the theory dual to the R-charged black hole, which
indicates that the system exhibits negative refraction even in the
probe limit [22]. Other investigations based on the
refractive index in the holographic dual models can be found, for
example, in Refs. [
23–32
].
The aforementioned works on the refractive index in the
holographic superconductor models are based on the usual
Maxwell electrodynamics. Considering the Maxwell theory
as only a special case of or a leading order approximation to
nonlinear electrodynamics, we may extend the investigation
to the nonlinear electrodynamics which essentially implies
higher derivative corrections of the gauge field [
33
]. As a
first attempt to investigate how the nonlinear
electrodynamics affects the properties of the holographic dual model, Jing
and Chen introduced the holographic superconductors in the
Born–Infeld electrodynamics and observed that the nonlinear
Born–Infeld corrections make it harder for the scalar
condensation to form [
34
]. Later the holographic dual models with
the Power-Maxwell electrodynamics [
35
], the Maxwell field
strength corrections [
36
], the logarithmic form of nonlinear
electrodynamics [
37
] and the exponential form of
nonlinear electrodynamics [
38
] were realized and some interesting
properties were disclosed. Considering the increasing
interest in the study of the holographic dual models with the
nonlinear electrodynamics [
39–66
], in this paper, we are going
to examine the influence of the Born–Infeld
electrodynamics on the optical properties of the generalized holographic
superconductors in the probe limit. We will show that the
Born–Infeld corrections affect not only the range of
frequencies or the range of temperatures for which negative
refraction occurs but also the dissipation in the system, and this is
helpful for us to understand the influences of the 1/N (N is
the color quantum number) or 1/λ (λ is the ’t Hooft coupling)
corrections on the holographic superconductor models and
their optical properties.
The structure of this work is as follows. In Sect. 2 we
will construct the generalized holographic superconductors
with the Born–Infeld electrodynamics in the probe limit and
analyze the effect of the Born–Infeld electrodynamics on the
condensate of the system. In Sect. 3 we will consider the
effect of the Born–Infeld electrodynamics on the negative
refraction in generalized superconductors. We will
summarize our results in the last section.
2 Generalized holographic superconductors with
Born–Infeld electrodynamics
In order to construct the generalized holographic
superconductors with the Born–Infeld electrodynamics in the probe
limit, we consider a four-dimensional planar
SchwarzschildAdS black hole background
1
ds2 = − f (r ) dt 2 + f (r ) dr 2 + r 2 d x 2 + d y2 ,
(1)
(2)
(3)
(4)
(5)
(6)
(7)
with
f (r ) = r 2 1 − rrh33
,
where rh is the radius of the event horizon. The Hawking
temperature of the black hole, which will be interpreted as
the temperature of the CFT, is determined by T = 3rh /4π .
We then introduce the Born–Infeld electrodynamics and a
charged scalar field coupled via a generalized Lagrangian
S =
4 √
d x
−
where both the charged scalar field and the phase α are
real, and the local U (1) gauge symmetry in this theory is
given by Aμ → Aμ + ∂μλ together with α → α + λ. When
the Born–Infeld parameter b → 0, the model (3) reduces to
the generalized holographic superconductors with the usual
Maxwell electrodynamics investigated in [
20–22
].
Using the gauge symmetry to fix the phase α = 0 and
taking the ansatz = (r ), A = (r )dt , we can get the
equations of motion
2 f
r + f
+
2
+ r (1 − b
2)
m2
− f
2G( )
− f
2 dG( )
+ f 2 d
= 0,
(1 − b
2)3/2
= 0,
where a prime denotes the derivative with respect to r .
Imposing the appropriate boundary conditions, we can solve Eqs.
(4) and (5) numerically by doing integration from the horizon
out to the infinity [
6
]. At the horizon r = rh , we require the
regularity conditions
(rh ) = 0,
(rh ) =
m2 (rh ) .
f (rh )
ρ
= μ − r ,
− + ,
= r − + r +
At infinity r → ∞, we have asymptotic behaviors
with ± = (3 ± √9 + 4m2)/2. According to the AdS/CFT
correspondence, μ and ρ are interpreted as the chemical
potential and the charge density in the dual field theory,
respectively. Note that, provided − is larger than the
unitarity bound, both − and + can be normalizable and be
used to define operators on the dual field theory, − = O− ,
+ = O+ , respectively [
6
]. In this work, we impose the
boundary condition − = 0 since we concentrate on the
condensate for the operator O+ . Considering that the choices of
from left to right correspond to increasing Born–Infeld parameter, i.e.,
b = 0.00 (red), 0.05 (blue), 0.10 (green) and 0.15 (black), respectively.
For concreteness, we have set the mass of the scalar field m2 = −2
0.5
0.6
the scalar field mass will not qualitatively modify our results,
we will set m2 = −2 for concreteness. Thus, the scalar
condensate is now described by the operator O2 = 2.
We will investigate the effect of the Born–Infeld
electrodynamics on the condensate of the system. For simplicity,
we consider a particular form of G( ), i.e.,
G( ) =
2 + ξ
8,
(8)
with the model parameter ξ introduced in Ref. [
22
]. We solve
the equations of motion numerically, and plot in Fig. 1 the
condensate around the critical region for chosen values of ξ
and various Born–Infeld parameters. In the case of ξ = 0 or
small ξ , for example ξ = 0.2, the transition is of the second
order and the condensate approaches zero as O2 ∼ (μ −
μc)1/2 for all values of b considered here. However, the story
is different if we increase the model parameter ξ . Focusing
on the case of ξ = 0.5 in Fig. 1, as an example, we observe
that O2 becomes multivalued near the critical point and the
first-order phase transition appears. This behavior keeps for
all values of b except for b = 0.00 and increasing b makes the
behavior more distinct, which supports the findings in Ref.
[
39
] and indicates that the greater the Born–Infeld corrections
the easier it is for the first-order phase transition to emerge.
Thus, we find that not only the model parameter ξ but also
the Born–Infeld parameter b can manipulate the order of the
phase transition. The Born–Infeld electrodynamics provides
richer physics in terms of the phase transition. On the other
hand, we can see clearly from Fig. 1 that the critical chemical
potential μc increases with the increase of the Born–Infeld
parameter for a fixed model parameter ξ , which indicates
that large Born–Infeld electrodynamics corrections hinder
the formation of the condensation. This agrees well with the
findings in the first holographic superconductor model with
the Born–Infeld electrodynamics introduced in [
34
].
3 Negative refraction in generalized superconductors
with Born–Infeld electrodynamics
In the preceding section we have constructed the generalized
holographic superconductors with the Born–Infeld
electrodynamics. Now we are in a position to discuss the optical
properties of these holographic systems and reveal the effect
of the Born–Infeld electrodynamics on the negative
refraction in generalized superconductors. Just as the standard
assumption in the usual Maxwell electrodynamics [
23,26
],
we will assume that the boundary theory is weakly coupled
to a dynamical electromagnetic field since the Born–Infeld
electrodynamics is a correction to the usual Maxwell
electrodynamics and calculate the refractive index of the system
perturbatively.
3.1 Holographic setup
Using the linear response theory (for more details see Ref.
[
23
]), we can find that the electric permittivity and the
effective magnetic permeability for isotropic media are
determined by the frequency dependent transverse current
correlators as follows
4π
(ω) = 1 + ω2 Ce2m G0T (ω) ,
1
μ (ω) = 1 − 4π Ce2m G2T (ω) ,
where Cem is the electromagnetic coupling which will be
set to unity when performing numerical calculations, and
G0T (ω) and G2T (ω) are the expansion coefficients of the
retarded correlators [
67
] in the spatial momentum k, i.e.,
GT (ω, k) = G0T (ω) + k2G2T (ω) + · · · .
(9)
So the refractive index can be given by
n2(ω) = (ω)μ (ω) .
Generally, the existence of a negative refractive index can be
predicted by using the Depine–Lakhtakia (DL) index n DL
expressed as [
68
]
n DL = Re[ (ω)]|μ(ω)| + Re[μ(ω)]| (ω)|,
where the negativity of the DL index indicates that the phase
velocity in the medium is opposite to the direction of the
energy flow, i.e., the system has negative refractive index.
Let us now move on to the strategy to calculate these
quantities in our holographic superconducting system with the
Born–Infeld electrodynamics. Considering the gauge field
perturbation
δ Ax = Ax (r )e−iωt+iky ,
we obtain the equation of motion for Ax (r ) as
Ax +
f b
f + 1 − b
ω2 k2
+ f 2 − r 2 f −
We can numerically solve this equation with the appropriate
boundary conditions, i.e., the ingoing wave boundary
condition at the event horizon
Ax ∝ f − 3irωh ,
A(x1)
AA((xx01)) .
and the asymptotic behavior at the asymptotic AdS boundary
Ax = A(0)
x +
+ · · · .
r
Thus, according to the AdS/CFT correspondence, the retarded
correlator has the form [
69
]
GT (ω, k) =
In order to calculate G0T (ω) and G2T (ω), we will expand
Ax (r ) in powers of k just as in Eq. (10) for GT (ω, k)
Ax (r ) = Ax0(r ) + k2 Ax2(r ) + · · · ,
which leads to the equations of motion
Ax0 +
Ax2 +
ω2
+ f 2 −
ω2
+ f 2 −
f b
f + 1 − b
√
2 1 − b
f b
f + 1 − b
√
2 1 − b
f
f
Ax0 = 0,
Ax0
Ax2 − r 2 f = 0.
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
With the asymptotic forms of Ax0 and Ax2 found from Eqs.
(19) and (20), i.e.,
Ax0 = A(x00) +
+· · · ,
AA((xx0100)) ,
G2T (ω) =
3.2 Numerical results and discussion
Using the shooting method, we can solve the equations of
motion (19) and (20) numerically and then examine the effect
of the Born–Infeld electrodynamics on the negative
refraction. For concreteness, we will set ξ = 0.2 for the fixed mass
of the scalar field m2 = −2. It should be noted that other
choices of the model parameter ξ will not qualitatively
modify our results. On the other hand, we will investigate our
system below the critical temperature, i.e., T < Tc since we
concentrate on the superconducting phase.
In Figs. 2 and 3 , we plot the permittivity and
permeability μ as a function of ω/ T with the fixed temperatures
T = 0.81Tc (left) and T = 0.86Tc (right) for different
values of the Born–Infeld parameter b, i.e., b = 0.00 (red), 0.05
(blue), 0.10 (green) and 0.15 (black), respectively.
Regardless of the fixed temperature and Born–Infeld parameter, we
observe that, for the permittivity, Re( ) is negative at low
frequencies and Im( ) is always positive and has a pole at
the zero frequency, but for the permeability, Re(μ) is always
positive and Im(μ) is negative. Obviously, Re( ) and Re(μ)
are not simultaneously negative, which may be a signal of
the negative refraction [
23
]. However, just as pointed out
in [
19,22,29
], Im(μ) < 0 could imply some problem in
the − μ approach, although we have defined an effective
magnetic permeability that is not an observable. We will not
comment on this further since this delicate issue is beyond
the scope and the purpose of our work.
Our main purpose here is to calculate the DL index n DL
and its dependence on the Born–Infeld parameter b. In Fig. 4,
we present n DL as a function of ω/ T with the fixed
temperatures T = 0.81Tc (left) and T = 0.86Tc (right) for
different values of the Born–Infeld parameter b, i.e., b = 0.00
(red), 0.05 (blue), 0.10 (green) and 0.15 (black), respectively.
From this Figure, we can see the emergence of the negative
DL index, below a certain value of ω/ T except in the case
of b = 0 with T = 0.86Tc. Fixing the temperature of the
system, we find that the value of ω/ T , below which the
negative n DL appears, increases as the Born–Infeld parameter
increases, which indicates that the geater the Born–Infeld
corrections, the larger the range of frequencies for which
negative refraction occurs. On the other hand, we note that the
negative n DL appears in the case of the Born–Infeld
parameter b = 0 at the temperature T = 0.81Tc and disappears
at T = 0.86Tc, but the negative n DL always appears in the
case of the Born–Infeld parameter b = 0.05 (b = 0.10 or
b = 0.15) at the temperatures T = 0.81Tc and T = 0.86Tc,
which shows that greater Born–Infeld corrections also make
the range of temperatures larger for which negative
refraction occurs. Interestingly, the Born–Infeld electrodynamics
can play an important role in determining the appearance of
negative refraction.
In order to ensure the validity of the expansion used in Eq.
(10), we require a constraint
k2G2T (ω)
G0T (ω)
=
G2T (ω) n2 ω2
G0T (ω)
(23)
Thus, our − μ analysis is valid only for the frequencies
for which this constraint is not violated. The behaviors of
| GG20TT ((ωω)) n2|ω2 as a function of ω/ T with the fixed
temperatures T = 0.81Tc (left) and T = 0.86Tc (right) for
different values of the Born–Infeld parameter b are given in
Fig. 5, which shows that, within the plotted frequency range,
the constraint (23) is marginally satisfied in the frequency
region where n DL is negative for all values of b considered
here. Of course, Eq. (23) is not very strictly satisfied and the
caveat may be related to the appearance of a negative
imaginary part of the magnetic permeability, just as pointed out in
[
22, 29
]. Though Markel proposed that Im(μ) (the loss term
in the permeability) can in fact be negative for diamagnetic
materials [70], it is worthwhile to have a better
understanding of this point in our holographic approach. Interestingly,
the inclusion of the backreaction can make Im(μ) positive
[
19, 29, 31
]. We will leave this subject for future research.
Finally, we consider the ratio Re(n)/I m(n) and discuss
the dissipation effects in our system. In Fig. 6, we plot the
ratio Re(n)/I m(n) as a function of ω/ T with the fixed
temperatures T = 0.81Tc (left) and T = 0.86Tc (right) for
different values of the Born–Infeld parameter b, i.e., b = 0.00
(red), 0.05 (blue), 0.10 (green) and 0.15 (black), respectively.
For the fixed temperature, we observe that the ratio decreases
with increasing values of b for the fixed ω/ T , and the
magb 0.15
b 0.10
b 0.05
b 0.00
nitude of Re(n)/I m(n) is small within the negative
refraction frequency range, which implies large dissipation in the
system. However, we can use the Born–Infeld corrections to
reduce the dissipation since the negative refraction frequency
range depends on the Born–Infeld parameter b. For example,
the ratio Re(n)/I m(n) is about 0.10 when ω/ T = 2.0 with
b = 0.00 but is about 0.13 when ω/ T = 2.6 with b = 0.05
for the case of T = 0.81Tc. Obviously, the Born–Infeld
electrodynamics can be used to improve the propagation in the
holographic setup.
ω T
1.5
Re n
Im n 0.2
Fig. 5 | GG02TT ((ωω)) n2|ω2 as a function of ω/ T with the fixed temperatures
T = 0.81Tc (left) and T = 0.86Tc (right) for different values of the
Born–Infeld parameter b. The four lines in each panel from top to
bottom correspond to increasing Born–Infeld parameter, i.e., b = 0.00
(red), 0.05 (blue), 0.10 (green) and 0.15 (black), respectively
1
2
ω T
We have constructed the generalized superconductors with
the Born–Infeld electrodynamics and studied their negative
refraction in the probe limit, which may help us to
understand the influences of the 1/N or 1/λ corrections on the
holographic superconductor models and their optical
properties. Varying the Born–Infeld parameter as well as the
temperature, we calculated in details the electric permittivity, the
effective magnetic permeability, the refractive index, and the
Depine–Lakhtakia (DL) index of our system and observed
the existence of negative refraction in the superconducting
phase at small frequencies. Interestingly, we found that the
greater the Born–Infeld corrections the larger the range of
frequencies or the range of temperatures for which a
negative DL index occurs, which indicates that the Born–Infeld
electrodynamics facilitates the appearance of negative
refraction. Furthermore, we analyzed the dissipation effects in our
system and found that the tunable Born–Infeld parameter can
be used to improve the propagation in the holographic setup.
Thus, we concluded that the Born–Infeld electrodynamics
can play an important role in determining the optical
properties of the boundary theory. The extension of this work to
the fully backreacted spacetime would be interesting since
the backreaction provides richer physics in the generalized
holographic superconductors and significantly affects their
optical properties [
29, 31
]. We will leave this for future study.
Acknowledgements This work was supported by the National Natural
Science Foundation of China under Grant nos. 11775076, 11690034
and 11475061; Hunan Provincial Natural Science Foundation of China
under Grant no. 2016JJ1012.
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