#### Holographic model of hybrid and coexisting s-wave and p-wave Josephson junction

Eur. Phys. J. C
Holographic model of hybrid and coexisting s-wave and p-wave Josephson junction
Shuai Liu 0
Yong-Qiang Wang 0
0 Institute of Theoretical Physics, Lanzhou University , Lanzhou 730000 , People's Republic of China
In this paper the holographic model for a hybrid and coexisting s-wave and p-wave Josephson junction is constructed by a triplet charged scalar field coupled with a nonAbelian SU (2) gauge field in (3+1)-dimensional AdS spacetime. Depending on the value of chemical potential μ, one can show that there are four types of junctions (s+p-N-s+p, s+p-N-s, s+p-N-p and s-N-p). We show that the DC currents of all the hybrid and coexisting s-wave and p-wave junctions are proportional to the sine of the phase difference across the junction. In addition, the maximum current and the total condensation decay with the width of junction exponentially, respectively. For the s+p-N-s and s-N-p junctions, the maximum current decreases with growing temperature. Moreover, we find that the maximum current increases with growing temperature for the s+p-N-s+p and s+p-N-p junctions, which is different from the behavior of the s+p-N-s and s-N-p junctions.
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Holographic model of hybrid and coexisting s-wave
and p-wave Josephson junction . . . . . . . . . . . .
2.1 The model setup . . . . . . . . . . . . . . . . . .
2.2 The scaling symmetry . . . . . . . . . . . . . . .
3 Numerical results . . . . . . . . . . . . . . . . . . . .
3.1 s+p-N-s+p Josephson junction . . . . . . . . . .
3.2 s+p-N-s Josephson junction . . . . . . . . . . . .
3.3 s+p-N-p Josephson junction . . . . . . . . . . .
3.4 s-N-p Josephson junction . . . . . . . . . . . . .
4 Conclusion and discussion . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
a e-mail:
b e-mail:
1 Introduction
The study of superconductivity has been at the forefront of
condensed matter physics. In particular, what the origin is of
high temperature superconductivity is still one of the major
unsolved problems of condensed matter theory. Over the past
decade, one of the most important results in string theory
is the AdS/CFT correspondence, which was first proposed
by Maldacena in [1,2] and states that the strong coupled
field living on the AdS boundary can be described with a
weakly gravity theory in one higher-dimensional AdS
spacetime. By applying the AdS/CFT correspondence, one has
first achieved success in the study of holographic QCD and
heavy ions collisions. In recent times, the AdS/CFT
correspondence also has provided insights into condensed
matter theory. In [3–5], the authors investigated the action of
a complex scalar field coupled to a U (1) gauge field in
a (3 + 1)-dimensional Schwarzschild–AdS black hole and
found that below some critical value of the temperature due
to the U (1) symmetry breaking, the scalar field which
condenses near the horizon could be interpreted as a Cooper
pair-like superconductor condensation. Moreover, analyzing
the optical conductivity of the superconducting state, the rate
of the width of the gap to the critical temperature is close to
the value of a high temperature superconductor. Thus, it is
hoped that the holographic model can match the properties of
the high temperature superconducting behavior. Soon, The
p-wave and d-wave holographic superconductors proposed
in [6–8], respectively. For reviews of holographic
superconductors, see [9–12].
It is well known that the Josephson junction is a device
made up of two superconductor materials coupled by weak
link barrier in [13]. The weak link can be a thin
normal conductor; then it is named a superconductor–normal–
superconductor junction (SNS), or a thin insulating barrier;
then we have a superconductor–insulator–superconductor
junction (SIS). Recently, by studying the space-dependent
solution of the action of a Maxwell field coupled with a
complex scalar field in a (3+1)-dimensional Schwarzschild–AdS
black hole background, Horowitz et al. [14] had constructed a
(2 + 1)-dimensional holographic model of a Josephson
junction and found a sine relation between the tuning current and
the phase difference of the condensation across the junction.
The extension to a 4-dimensional Josephson junction has
been discussed in [15,16]. The holographic mode of a
Josephson junction array has been constructed based on a designer
multigravity in [17]. With the SU (2) gauge field coupled
with gravity, the holographic p-wave Josephson junction
has also been discussed in [18]. In [19], a holographic
model of a (1 + 1)-dimensional superconductor–insulator–
superconductor (S-I-S) Josephson junction has been
investigated. In [20,21], with the action of the
Einstein–Maxwellcomplex scalar field, the authors studied a holographic model
of superconducting quantum interference device (SQUID).
In [22] a holographic model of a Josephson junction in
the non-relativistic case with a Lifshitz geometry was
constructed.
Recently, the holographic approach has been applied to
the coexistence and competition order phenomena, in which
the phase diagram has a rich structure, such as the
competition of two scalar order parameters in the probe limit [23,24]
and with the backreaction of the scalars [25], the coexistence
of two p-wave order parameters [26], and the competition of
s-wave and d-wave order parameters [27]. Especially, there
is another interesting paper about competition and
coexistence of s-wave and p-wave order in [28,29]; the authors
studied a charged triplet scalar coupled with an SU (2) gauge
field in the (3 + 1)-dimensional spacetime and confirmed s+p
coexisting phases. Furthermore, the phase transitions of the
holographic s+p-wave superconductor with backreaction is
investigated in Ref. [30].
Since the holographic model of hybrid and s+p
coexisting superconductors has been constructed, it is natural to
set up a holographic model for hybrid and coexisting s-wave
and p-wave Josephson junction. We will study a non-Abelian
SU (2) Yang–Mills field and a scalar triplet charged under an
SU (2) gauge field in (3+1)-dimensional AdS spacetime and
construct the holographic model of a hybrid and coexisting
s-wave and p-wave Josephson junction, which will be related
to the s-N-s junction in [14] and the p-N-p junction in [18].
To construct the holographic model of a junction, ones need
to tune the value of the chemical potential.
The paper is organized as follows: in Sect. 2, we
consider a non-Abelian SU (2) gauge field coupled with a scalar
triplet charged field in (3 + 1)-dimensional AdS spacetime
and construct a gravity dual model for a (2 + 1)-dimensional
s+p coexisting Josephson junction. We show the numerical
results of the equations of motion and study the
characteristics of the (2 + 1)-dimensional s+p coexisting Josephson
junction in Sect. 3. The conclusion and discussion are in the
last section.
2 Holographic model of hybrid and coexisting s-wave and p-wave Josephson junction
2.1 The model setup
In [28], the author considered the action of a charged scalar
field coupled to an SU (2) gauge field, in which the charged
scalar field transform as a triplet under the gauge group
SU (2). This model can realize the competition and
coexistence of s-wave and p-wave order parameters. We will also
adopt the same form of the action in the (3 + 1)-dimensional
AdS spacetime:
d4x√−g(R − 2 )
where = −3/L2 is the negative cosmology constant, and
L is the radius of asymptotic AdS spacetime. a is a complex
scalar triplet charged under the SU (2) Yang–Mills field and
Fμν = ∂μ Aν − ∂ν Aμ + εabc Abμ Acν ,
a a a
where the one form A = Aμdx μ = Aaμτ a dx μ and the
superscript a is the index of generator τ a of SU (2) gauge field
with a = 1, 2, 3. gc is the coupling constant of non-Abelian
SU (2) Yang–Mills field. In this paper we will consider the
probe limit by taking gc → ∞ to ignore the backreaction of
the matter.
In the probe limit, we still choose a (3 + 1)-dimensional
planar Schwarzschild–AdS black hole solution as the
background geometry with metric
dr 2
ds2 = − f (r )dt 2 + f (r ) + r 2(dx 2 + dy2),
where x and y are the coordinates of a 2-dimensional
Euclidean space. The function f (r ) is
where rh is the radius of the black hole’s event horizon . The
Hawking temperature of the black hole is given by
The temperature T relates to the radius of the black hole’s
event horizon rh and the AdS radius L. In the rest of paper,
we will work in unit in which L = 1. T corresponds to the
temperature of the dual field theory on the AdS boundary.
The variation of the action (2.1) with respect to the scalar
field a and Aaμ lead to the equations of motion, respectively.
We have
∇μ(Dμ a ) + εabc Abμ Dμ c − m2 a = 0 ,
∇ν Fνaμ +εabc Abν Fνcμ − 2εabc b(∂μ
For the hybrid and coexisting s-wave and p-wave
Josephson junction, there are several different kinds of matter fields
ansatzes. Let us consider one of them:
A3x =
where the field functions 3, θ a , φ, x , Ar , and Ay are
dependent of the coordinates r and y. Thus the holographic
model of the Josephson junction would be along the y
direction. Without loss of generality, we will consider the SU (2)
gauge-invariant vector field Aaμ and the scalar field a :
With the black hole background (2.4) and the above ansatz
(2.9), the equations of the matter fields (2.7) and (2.8) can be
written as
m2 φ2 My2
∂r 3 + − f + f 2 − Mr2 − r 2 f
2 1 2 2 φ x2
∂r φ + r 2 f ∂y φ + r ∂r φ − r 2 f −
2
2φ 3
f
= 0,
∂y2 Mr − ∂r ∂y My − Mr x − 2r 2 Mr 3 = 0,
2 2
f
∂r2 My − ∂r ∂y Mr + f (∂r My − ∂y Mr )
2
My x
− r 2 f
= 0,
3 = 0,
x = 0,
Mr = 0,
where a prime denotes the derivative with respect to r . In our
paper, we will work with the case m2 ≥ −9/4 in order to
satisfy the BF bound [31]. Let us inspect Eqs. (2.11a)–(2.11g).
If x , Ar , and Ay are turned off, 3 and φ are only
dependent on r , and the remaining two equations will be the same
as the equations of the s-wave holographic
superconductivity. Similarly, if we turn off 3, Ar , and Ay , x and φ are
only dependent on r , and the two remaining equations will
be the equations which describe the p-wave condensation.
Next, we will consider the Josephson junction. If x is only
turned off, the remaining equations are the same equations
as of the pure s-wave Josephson junction. In a similar way, if
we only turn off 3, we will get the pure p-wave Josephson
junction. So we can get the so-called s+p coexisting phase
Josephson junction under this ansatz (2.9). It is obvious that
Eqs. (2.11a)–(2.11g) are coupled and nonlinear, so we need to
solve them numerically instead of solving them analytically.
Near the AdS boundary (r → ∞), the matter fields take
the asymptotic forms
3 =
3(−)(y)
r −
x =
x(−)(y) +
Mr = O
Here, the dimensions of the operations
± =
According to the AdS/CFT dictionary, (±) is considered as
3
the source of the scalar operation of s-wave condensation,
and 3(∓) is the corresponding expectation value of the
operator. Meanwhile, x(±) is the source of the vector operation of
p-wave condensation, and x(∓) is the corresponding
expectation value of the operator. μ(y), ρ(y), ν(y), and J are the
chemical potential, charge density, velocity of the superfluid,
and the constant current in the dual field, respectively [32–
39].
In order to solve Eqs. (2.11a)–(2.11g) numerically, we
need to impose the boundary conditions on them. First, we
impose the Dirichlet boundary conditions on 3 and x on
the AdS boundary. In this paper, we set
3(−) = 0.
x(−) = 0.
(+) is the expectation value of s-wave scalar operator,
3
So
3(+) = Os . Similarly, we impose the Dirichlet boundary
condition on x on the AdS boundary:
(+) is the expectation value of the p-wave vector operator,
x
x(+) = O p . Second, we impose the Dirichlet boundary
condition on φ at the horizon:
Now, we would like to introduce the critical temperature Tc of
the Josephson junction, which is proportional to the chemical
potential μ(+∞) or μ(−∞), and we set
φ is the t component of A1μ, and φ (rh ) = 0 is to avoid the
divergence of gμν A1μ Aν1. In addition, the matter field
functions are independent of y at the spatial coordinate y → ±∞.
So, the boundary conditions of Eqs. (2.11a)–(2.11g) are
determined by the value of μ and J . The phase difference
γ across the junction is γ = θ 1 − Ay dy. The phase
difference θ 1 can be eliminated under the SU (2)
gaugeinvariant, thus it is convenient to set
where μc ≈ 3.65. In order to describe a hybrid and coexisting
s-wave and p-wave Josephson junction, we should construct a
chemical potential which can make a phase transition
occurring along the direction y of the Josephson junction; thus the
chemical potential μ(y) is dependent on the spatial
coordinate y,
1 −
μ(y) = μ 1 − 2 tanh( 2σ )
y + 2
σ
where the chemical potential μ(y) is proportional to μ and
is the width of the Josephson junction. We use the parameter
, and σ and β control the steepness and the depth of the
junction, respectively.
2.2 The scaling symmetry
Analyzing the EoM, we found that there is a scaling
symmetry in Eqs. (2.11a)–(2.11g). These equations are invariant
under the following scale transformation:
r → br, rh → brh , y → y/b, φ → bφ,
x → b x , 3 → 3, f → b2 f, f → b f , Ay → b Ay .
Under this scaling symmetry, we can get the behavior of the
following physical quantities:
Because of this scaling symmetry, we can set the radius of the
black hole’s event horizon rh = 1. In addition, we have set
L = 1, so the temperature T and the background geometry
is fixed. From Eq. (2.6), we can see that the temperature
T changes to bT under the scaling transformation. So the
following quantities are invariant:
These invariant quantities can change with T / Tc.
3 Numerical results
In this section we will solve the coupled and nonlinear
equations (2.11a)–(2.11g) numerically. Before we solve these
equations, we will have coordinate transformations, z = 1/r ,
y = tanh( 4yσ ). It is more convenient to impose boundary
conditions at z = 1, z = 0, and y = ±1, rather than at r = 1,
r = +∞, and y = ±∞. There are four kinds of Josephson
junctions.
(i) The s+p-N-s+p Josephson junction consists of the s+p
coexisting phase in the two leads, and the normal phase
between them.
(ii) The s+p-N-s Josephson junction consists of the s+p
coexisting phase in the left lead, the conventional s-wave
phase in the right lead, and the normal phase between
them.
(iii) The s+p-N-s Josephson junction consists of the s+p
coexisting phase in the left lead, the conventional p-wave
phase in the right lead, and the normal phase between
them.
(iv) The s-N-p Josephson junction consists of the
conventional s-wave phase in the left lead, the conventional
s-wave phase in the right lead, and the normal phase
between them.
We can tune the chemical potential μ(y) to realize the four
cases. In [28], when the value of + or m2 is in a special
region, as the temperature decreases, the p-wave
condensation will appear first and increase; the s-wave condensation
will not appear. When the temperature continues to decrease
and reaches the critical temperature Tcs1p, the s-wave
condensation will appear and increase, and the p-wave condensation
will decrease. When the temperature reaches another critical
temperature Tcs2p, the p-wave condensation will disappear.
So when the temperature is in the region Tcs2p ∼ Tcs1p, the
swave phase and p-wave will coexist. When the temperature
is higher than Tcs1p, there is only a p-wave phase. When the
temperature is below Tcs2p, there is only an s-wave phase. In
order to construct the junction of the s+p coexisting phase, we
can write the region of the chemical potential as μc1 ∼ μc2
in [28].
3.1 s+p-N-s+p Josephson junction
In this subsection, in order to obtain the model of a s+p-N-s+p
Josephson junction, we need to tune the value of chemical
potential μ(y) at y = ±∞ such that it is in the s+p coexisting
region μc1 ∼ μc2. Because the superconductor phase in the
two leads are symmetrical, the phase difference γ can be
obtained by (2.21); we have
The profiles of At1 and A1y are shown in Fig. 1.
We show the relationship between the DC current J and
the phase difference γ across the junction on the left panel
of Fig. 2. From the figure, we can see that J is proportional
to the sine of γ . We have
J / Tc2 ≈
Note that we only obtain the phase difference in the interval
(−π/2, π/2), and we can see that the points which represent
our numerical data fit the sine line very well. We can see
that when the value of m2 increases, the maximum current
Jmax will grow. The dependence of Jmax on the temperature
T is shown on the right panel of Fig. 2. The graph shows
that Jmax/ Tc2 increases with growing T / Tc; however, in the
s-N-s or p-N-p Josephson junction [14,18], Jmax/ Tc2 decays
with increasing T / Tc. The reason is that in the s-wave and
pwave coexisting region [28], the condensation of the s-wave
decreases and the condensation of the p-wave increases with
growing temperature, respectively. But the total condensation
would decrease when the temperature drops. So, the Jmax
decreases with rising T .
The relationship of between Jmax/ Tc2 and is shown on
the left panel of Fig. 3. Jmax/ Tc2 will decay with
exponentially and the change will be larger when m2 becomes larger.
The total condensation in y = 0 when the current J = 0 is
the sum of s-wave and p-wave condensation. Here, we define
O(0) J =0 = Os (0) J =0/ Tc + O p(0) J =0/ Tc2.
We plot the O(0) J =0 on the right panel in Fig. 3. O(0) J =0
also decays with growing exponentially and O(0) J =0
becomes larger with increasing m2. We have
Fig. 1 We represent the components μ(y) and Ay of Yang–Mills fields. Left The profile of μ(y). Right The figure of Ay. The parameters are
m2 = −33/16, μ = 7.6, = 0.25, σ = 0.5, = 3, and β = 1. We can see that these two figures have symmetry
Fig. 3 Left The maximum current Jmax as exponential function of .
Right The total condensate O(0) J=0 as exponential function of . Our
numerical results are the points with m2 = −33/16 (red), −1031/500
O(0) J =0 ≈ 22.03e− /(2×0.7534).
We can obtain the coherence length (0.7505, 0.7534) from
the above two equations, respectively. The error of the two
values is about 0.4 %. We have
for m2 = −33/16,
O(0) J =0 ≈ 20.66e− /(2×0.8295).
The coherence length (0.7856, 0.8295) is obtained from the
above two equations, respectively. The error of the two values
is about 5.6 %.
for m2 = −1031/500,
(blue). In all the plots, we use μ = 7.6, = 0.25, σ = 0.5, and β = 1.
The numerical data fit the exponential curves well
3.2 s+p-N-s Josephson junction
In this subsection, the model of the s+p-N-s Josephson
junction will be constructed. We tune the chemical potential μ(y)
such that μ(−∞) is in the s+p coexisting region μc1 ∼ μc2
and μ(+∞) > μc2 is in the pure s-wave phase region.
Because the superconductor condensations in the two leads
are not symmetrical, the phase difference γ can be calculated
by (2.21)
Fig. 4 We represent the components φ and Ar of the Yang–Mills field. Left The profile of φ. Right The figure of Ar . The parameters are
m2 = −33/16, μ = 8.3, = 0.3, σ = 0.5, = 3, and β = 1.286. We can see that these two figures are not symmetrical
Fig. 5 Left The current J/Tc2 as sine function of phase difference γ . Right The Jmax/Tc2 decays with growing T /Tc. The parameters are
m2 = −33/16, μ = 8.3, = 0.3, σ = 0.5, = 3, and β = 1.286. The numerical data fit the exponential curves well
First, we show the profiles of At1 and A1y in Fig. 4, with the
parameters m2 = −33/16, μ = 8.3, = 0.3, σ = 0.5,
= 3, and β = 1.286.
The dependence of the current J on the phase difference
γ across the junction is shown on the left panel of Fig. 5. The
figure shows that J is proportional to the sine of γ .
J / Tc2 ≈ 0.4095sinγ , for m2 = −33/16.
From the above result, we can see our numerical data which
is drawn with the red points fits sine line very well. The
dependence of Jmax on the temperature T is shown on the
right panel of Fig. 5. The graph shows that Jmax/ Tc2 decreases
with growing T / Tc.
The dependence of Jmax on the width of the gap is shown
on the left panel of Fig. 6. The figure shows that Jmax/ Tc2
decays with growing exponentially. The dependence of
O(0) J =0 on the width of the gap is shown on the right
panel of Fig. 6. The figure predicts that O(0) J =0 decays
with growing exponentially. We have
O(0) J =0 ≈ 26.91e− /(2×0.6567).
The coherence length (0.7138, 0.6567) can be obtained from
these equations. The error of these two values is about 8 %.
3.3 s+p-N-p Josephson junction
In this subsection, let us to continue to study the
s+p-Np Josephson junction. We tune the chemical potential μ(y)
such that μ(−∞) is in the s+p coexisting region μc1 ∼ μc2
for m2 = −33/16,
Fig. 6 Left The maximum current Jmax/Tc2 as an exponential function of the width . Right The total condensate O(0) J=0 as an exponential
function of the width . In all the plots, we use m2 = −33/16, μ = 8.3, = 0.3, σ = 0.5, and β = 1.286. The numerical data fit exponential
curves well
Fig. 7 We represent the components φ and Ay of the Yang–Mills field. Left The profile of φ. Right The figure of Ay. The parameters are
m2 = −33/16, μ = 6.3, = 0.0, σ = 0.5, = 3, and β = 0.6. We can see that these two figures are not symmetrical
and μ(+∞) < μc1 is in the pure p-wave phase region. We
still calculate the phase difference γ from Eq. (3.3). The
profiles of At1 and A1y are shown in Fig. 7, with the parameters
m2 = −33/16, μ = 6.3, = 0.0, σ = 0.5, = 3, and
β = 0.6.
The dots are determined by the DC current J and the phase
difference γ across the junction is shown on the left panel of
Fig. 8. With data fitting, we can see that J is proportional to
the sine of γ . We have
J / Tc2 ≈
From the figure, it is shown that when the value of m2
increases the maximum current Jmax will grow. The
dependence of Jmax on the temperature T is shown on the right
panel of Fig. 8. The graph shows that Jmax/ Tc2 increases
with growing T / Tc, which is for the same reason as the case
of the s+p-N-s+p junction.
In the left panel of Fig. 3, the dependence of Jmax/ Tc2 on
is shown. We can see that Jmax/ Tc2 decays with
exponentially and the change will be larger when m2 becomes
larger. Furthermore, we show the dependence of O(0) J =0
on in the right panel in Fig. 9. O(0) J =0 also decays with
growing exponentially and O(0) J =0 becomes larger with
increasing m2. We have
O(0) J =0 ≈ 15.61e− /(2×0.6021).
for m2 = −33/16,
Fig. 8 Left The current J/Tc2 as a sine function of the difference phase γ . Right Jmax/Tc2 increases with growing T /Tc. The black lines are sine
curves and the points with m2 = −33/16 (red), −1031/500 (blue) are numerical results. The parameters are μ = 6.3, = 0.0, σ = 0.5, = 3,
and β = 0.6.
Fig. 9 Left The maximum current Jmax as an exponential function of
. Right The total condensate O(0) J=0 as an exponential function
of . Our numerical results are the points with m2 = −33/16 (red),
From the above result, we can obtain the coherence length
(0.7376, 0.6021) from the above two equations, respectively.
The error of the two values is about 18.4 %. We have
O(0) J=0 ≈ 14.68e− /(2×0.6337).
Similarly, the coherence length (0.7558, 0.6337) is obtained
from the above two equations, respectively. The error of the
two values is about 16.2 %.
3.4 s-N-p Josephson junction
In this subsection, we will construct the hybrid model of
the s-wave and p-wave Josephson junction, namely, the
sfor m2 = −1031/500,
−1031/500 (blue). In all the plots, we use μ = 6.3, = 0.0, σ = 0.5,
and β = 0.6. The numerical data fit the exponential curves well
N-p Josephson junction. We tune the value of the chemical
potential μ(y) such that μ(−∞) > μc2 is in the pure
swave region and μ(+∞) < μc1 is in the pure p-wave phase
region. The phase difference γ can be obtained in Eq. (3.3).
The profiles of At1 and A1y are shown in Fig. 10, with the
parameters m2 = −33/16, μ = 7, = 0.0, σ = 0.5, = 3,
and β = 0.6.
We show the dependence of the current J on the phase
difference γ across the junction on the left side of Fig. 11, in
which it is shown that J is proportional to the sine of γ . We
have
J / Tc2 ≈ 0.1166sinγ , for m2 = −33/16.
Fig. 10 We represent the components φ and Ay of the Yang–Mills field. (Left) The profile of φ. (Right) The figure of Ay. The parameters are
m2 = −33/16, μ = 7, = 0.0, σ = 0.5, = 3, and β = 0.6. We can see that these two figures are not symmetrical
Fig. 11 Left The current J/Tc2 as a sine function of γ . Right Jmax/Tc2 decays with growing T /Tc. In all the plots, we use m2 = −33/16, μ = 7,
= 0.0, σ = 0.5, = 3, and β = 0.6. The numerical data fit the exponential curves well
The dependence of Jmax on the temperature T is shown
on the right panel of Fig. 11, which shows that Jmax/ Tc2
decreases with growing T / Tc.
The dependence of Jmax on the width of the gap is shown
on the left panel of Fig. 12, which shows that Jmax/ Tc2 decays
with growing exponentially. The dependence of O(0) J =0
on the width of the gap is shown on the right panel of Fig.
12, which predicts that O(0) J =0 decays with growing
exponentially. We have
O(0) J =0 ≈ 12.95e− /(2×0.6362).
The coherence length (0.6798, 0.6362) can be obtained
from these equations. The error of these two values is about
6.4 %.
4 Conclusion and discussion
In this paper, we set up a holographic model for a hybrid
swave and p-wave DC Josephson junction with a scalar triplet
charged under the SU (2) gauge field in the background of a
(3 + 1)-dimensional AdS black hole. We get a set of partial
differential equations of fields that are nonlinear and coupled
and solve them numerically. We construct a new chemical
Fig. 12 Left The maximum current Jmax/ Tc2 as an exponential function of . Right The total condensate O(0) J =0 as exponential function of .
In all the plots, we use m2 = −33/16, μ = 7, = 0.0, σ = 0.5, and β = 0.6. The numerical data fit the exponential curves well
potential μ(y) and tune the parameters in it, so the
s+p-Ns+p junction, s+p-N-s junction, s+p-N-p junction, and s-N-p
junction can be obtained, respectively. For the four kinds of
junctions, we find that the DC is proportional to the sine
of the phase difference across the junction and the
coherence lengths are different. We also study the relationship
between Jmax/ Tc2 and , the total condensation O(0) J =0
and , Jmax/ Tc2 and T / Tc, respectively.
The reason we take m2 = −33/16 is that when m2 <
−33/16, the region of s+p coexistence is too small, when
m2 > −33/16, the value in the region of s+p coexistence is
too large for the junction, the numerical results are not good.
It is well known that the Josephson period is 2π in the py
wave s-N-p junction, so our ansatz just corresponds to the
py wave junction. To our surprise, the periods of the currents
are also 2π in the remaining three kinds of junctions. For
the s+p-N-s+p junction and the s+p-N-p junction, we take
different values of m2 = −33/16, −1031/500 and find that
when the value becomes larger, J , Jmax/ Tc2, and O(0) J =0
will become larger. The reason we take another value of
m2 = −1031/500 is that the region of s+p coexistence is
small, we should take the other value of m2 as it approaches
the m2 = −33/16. The maximum current increases with the
growing temperature in the s+p-N-s+p and the s+p-N-p
junction. Except the s+p-N-s+p junction, the phase difference
should be obtained by γ = − −c∞ d y[ν(y) − ν(−∞)] −
c+∞ d y[ν(y) − ν(+∞)] without loss of generality. When
c = 0, the relationship between the current and the phase
difference is J / Tc2 = ( Jmax/ Tc2)sin(γ + φ), where φ is the
origin’s phase difference and φ = 0. So it is more convenient
to set c = 0 to make φ = 0.
Note that our model also can describe the s-wave
superconductor, the p-wave superconductor, the s+p coexistence
superconductor, the s-wave junction, and the p-wave
junction. The p-wave contains a px wave and a py wave, the
Josephson periods are π and 2π , respectively. In the present
study, this ansatz just describes the py wave. So, it should be
of great interest to construct an ansatz which can describe a
px wave and a py wave, respectively.
Finally, we have studied the hybrid and coexisting s-wave
and p-wave junctions with the probe limit, and we would like
to study these kinds of junctions with the gravity backreaction
in the future. So far, we have studied the hybrid and coexisting
s-wave and p-wave DC Josephson junctions in the AdS black
hole background. It is intended to study these DC junctions
by taking an AdS soliton as the geometry background in our
further work.
Acknowledgments YQW would like to thank Li Li and Hai-Qing
Zhang for very helpful discussion. SL and YQW were supported by the
National Natural Science Foundation of China.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. J.M. Maldacena , The large N limit of superconformal field theories and supergravity . Int. J. Theor. Phys . 38 , 1113 ( 1999 )
2. J.M. Maldacena , Adv. Theor. Math. Phys. 2 , 231 ( 1998 ) arXiv:hep-th/9711200
3. S.S. Gubser , Breaking an Abelian gauge symmetry near a black hole horizon . Phys. Rev. D 78 , 065034 ( 2008 ). arXiv:0801.2977 [hep-th]
4. S.A. Hartnoll , C.P. Herzog , G.T. Horowitz , Building a holographic superconductor . Phys. Rev. Lett . 101 , 031601 ( 2008 ). arXiv:0803.3295 [hep-th]
5. S.A. Hartnoll , C.P. Herzog , G.T. Horowitz , Holographic superconductors . JHEP 0812 , 015 ( 2008 ). arXiv:0810.1563 [hep-th]
6. S.S. Gubser , S.S. Pufu , JHEP 0811 , 033 ( 2008 ). arXiv:0805.2960 [hep-th]
7. J.W. Chen , Y.J. Kao , D. Maity , W.Y. Wen , C.P. Yeh , Phys. Rev. D 81 , 106008 ( 2010 ). arXiv:1003.2991 [hep-th]
8. F. Benini , C.P. Herzog , R. Rahman , A. Yarom , JHEP 1011 , 137 ( 2010 ). arXiv:1007.1981 [hep-th]
9. S.A. Hartnoll , Lectures on holographic methods for condensed matter physics. Class. Quant. Grav . 26 , 224002 ( 2009 ). arXiv:0903.3246 [hep-th]
10. C.P. Herzog, Lectures on holographic superfluidity and superconductivity. J. Phys. A 42 , 343001 ( 2009 ). arXiv:0904.1975 [hep-th]
11. G.T. Horowitz , Introduction to holographic superconductors . Lect. Notes Phys. 828 , 313 ( 2011 ). arXiv:1002.1722 [hep-th]
12. R.G. Cai , L. Li , L.F. Li , R.Q. Yang , Sci. China Phys. Mech. Astron . 58 , 060401 ( 2015 ). arXiv:1502.00437 [hep-th]
13. B.D. Josephson, Possible new effects in superconductive tunnelling . Phys. Lett . 1 , 251 ( 1962 )
14. G.T. Horowitz , J.E. Santos , B. Way , A holographic Josephson junction . Phys. Rev. Lett . 106 , 221601 ( 2011 ). arXiv: 1101 .3326 [hepth]
15. Y.Q. Wang , Y.X. Liu , Z.H. Zhao , Holographic Josephson junction in 3+1 dimensions . arXiv: 1104 .4303 [hep-th]
16. M. Siani , On inhomogeneous holographic superconductors . arXiv:1104 .4463 [hep-th]
17. E. Kiritsis , V. Niarchos , Josephson junctions and AdS/CFT networks . JHEP 1107 , 112 ( 2011 ). arXiv:1105.6100 [hep-th] . (Erratum-ibid . 1110 , 095 ( 2011 ))
18. Y.Q. Wang , Y.X. Liu , Z.H. Zhao , Holographic p-wave Josephson junction . arXiv:1109 .4426 [hep-th]
19. Y.Q. Wang , Y.X. Liu , R.G. Cai , S. Takeuchi , H.Q. Zhang , Holographic SIS Josephson junction . JHEP 1209 , 058 ( 2012 ). arXiv:1205.4406 [hep-th]
20. R.G. Cai , Y.Q. Wang , H.Q. Zhang , JHEP 1401 , 039 ( 2014 ). arXiv:1308.5088 [hep-th]
21. S. Takeuchi , Int. J. Mod. Phys. A 30(09) , 1550040 ( 2015 ). arXiv:1309.5641 [hep-th]
22. H.F. Li , L. Li , Y.Q. Wang , H.Q. Zhang , JHEP 1412 , 099 ( 2014 ). arXiv:1410.5578 [hep-th]
23. P. Basu , J. He , A. Mukherjee , M. Rozali , H.H. Shieh , Competing holographic orders . JHEP 1010 , 092 ( 2010 ). arXiv:1007.3480 [hep-th]
24. D. Musso , JHEP 1306 , 083 ( 2013 ). arXiv:1302.7205 [hep-th]
25. R.G. Cai , L. Li , L.F. Li , Y.Q. Wang , Competition and coexistence of order parameters in holographic multi-band superconductors . JHEP 1309 , 074 ( 2013 ). arXiv:1307.2768 [hep-th]
26. A. Amoretti , A. Braggio , N. Maggiore , N. Magnoli , D. Musso , JHEP 1401 , 054 ( 2014 ). arXiv:1309.5093 [hep-th]
27. L.F. Li , R.G. Cai , L. Li , Y.Q. Wang , Competition between s-wave order and d-wave order in holographic superconductors . JHEP 1408 , 164 ( 2014 ). arXiv:1405.0382 [hep-th]
28. Z.Y. Nie , R.G. Cai , X. Gao , H. Zeng , Competition between the swave and p-wave superconductivity phases in a holographic model . JHEP 1311 , 087 ( 2013 ). arXiv:1309.2204 [hep-th]
29. I. Amado , D. Arean , A. Jimenez-Alba , L. Melgar , I. Salazar Landea , Phys. Rev . D 89 ( 2 ), 026009 ( 2014 ). arXiv:1309.5086 [hep-th]
30. Z.Y. Nie , R.G. Cai , X. Gao , L. Li , H. Zeng . arXiv: 1501 .00004 [hep-th]
31. P. Breitenlohner , D.Z. Freedman , Positive energy in anti-De Sitter backgrounds and gauged extended supergravity . Phys. Lett. B 115 , 197 ( 1982 )
32. P. Basu , A. Mukherjee , H.-H. Shieh , Supercurrent: vector hair for an AdS black hole . Phys. Rev. D 79 , 045010 ( 2009 ). arXiv:0809.4494 [hep-th]
33. C.P. Herzog , P.K. Kovtun , D.T. Son , Holographic model of superfluidity . Phys. Rev. D 79 , 066002 ( 2009 ). arXiv:0809.4870 [hep-th]
34. D. Arean , M. Bertolini , J. Evslin , T. Prochazka , On holographic superconductors with DC current . JHEP 1007 , 060 ( 2010 ). arXiv:1003.5661 [hep-th]
35. J. Sonner , B. Withers , A gravity derivation of the TiszaLandau model in AdS/CFT . Phys . Rev . D 82 , 026001 ( 2010 ). arXiv:1004.2707 [hep-th]
36. G.T. Horowitz , M.M. Roberts , Holographic superconductors with various condensates . Phys. Rev. D 78 , 126008 ( 2008 ). arXiv:0810.1077 [hep-th]
37. D. Arean , P. Basu , C. Krishnan , The many phases of holographic superfluids . JHEP 1010 , 006 ( 2010 ). arXiv:1006.5165 [hep-th]
38. H.B. Zeng , W.M. Sun , H.S. Zong , Supercurrent in p-wave holographic superconductor . Phys. Rev. D 83 , 046010 ( 2011 ). arXiv:1010.5039 [hep-th]
39. D. Arean , M. Bertolini , C. Krishnan , T. Prochazka , JHEP 1103 , 008 ( 2011 ). arXiv:1010.5777 [hep-th]