#### Holographic insulator/superconductor transition with exponential nonlinear electrodynamics probed by entanglement entropy

Eur. Phys. J. C
Holographic insulator/superconductor transition with exponential nonlinear electrodynamics probed by entanglement entropy
Weiping Yao 1
Chaohui Yang 1
Jiliang Jing 0
0 Department of Physics Key Laboratory of Low Dimensional Quantum Structures, Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University , Changsha 410081, Hunan , People's Republic of China
1 Department of electrical engineering, Liupanshui Normal University , Liupanshui 553004, Guizhou , People's Republic of China
From the viewpoint of holography, we study the behaviors of the entanglement entropy in insulator/superconductor transition with exponential nonlinear electrodynamics (ENE). We find that the entanglement entropy is a good probe to the properties of the holographic phase transition. Both in the half space and the belt space, the nonmonotonic behavior of the entanglement entropy in superconducting phase versus the chemical potential is general in this model. Furthermore, the behavior of the entanglement entropy for the strip geometry shows that the confinement/deconfinement phase transition appears in both insulator and superconductor phases. And the critical width of the confinement/deconfinement phase transition depends on the chemical potential and the exponential coupling term. More interestingly, the behaviors of the entanglement entropy in their corresponding insulator phases are independent of the exponential coupling factor but depends on the width of the subsystem A.
1 Introduction
As a strong-week duality, the anti-de Sitter/conformal field
theories (AdS/CFT) correspondence [
1–3
] establishes a dual
relationship between the (d − 1) dimensional strongly
interacting theories on the boundary and the d dimensional weekly
coupled gravity theories in the bulk. Based on this novel
idea, the AdS/CFT correspondence have received
considerable interest in modeling strongly coupled physics, in
particular the construction of the holographic superconductor,
might shed some light on the problem of understanding the
mechanism of the high temperature superconductors in
condensed matter physics [
4–9
]. Such holographic
superconductor models are interesting since they exhibit many
characteristic properties shared by real superconductor. In recent
years, the studies on the holographic superconductors in AdS
spacetime have received a lot of attentions [
10–24
].
In addition, the entanglement entropy is expected to be a
useful tool to keep track of the degrees of freedom of strongly
coupled systems while other traditional methods might not be
available. In the spirit of AdS/CFT correspondence, a
holographic method for calculating the entanglement entropy has
been proposed by Ryu and Takayanagi [
25,26
]. Presently,
consider a subsystem A of the total boundary system, the
entanglement entropy for a region A of the boundary system
is obtained from gravity side as the area of the minimal
surface γA in the bulk which ends at ∂A. Then the entanglement
entropy of A with its complement is given by
SA =
Area(γA) ,
4G N
(1)
where G N is the Newton’s constant in the bulk. With this
elegant and executable approach, the holographic
entanglement entropy has recently been applied to disclose properties
of phase transitions in various holographic superconductor
models [
27–39
]. It turns out that the entanglement entropy
is a good probe to the critical phase transition points and
the order of holographic phase transition [
40–45
]. However,
most studies on the holographic entanglement entropy are
focused on the cases where the gauge field is in the form of
the Maxwell field. When thinking about the higher
derivative correction to the gauge field, the Refs. [
46,47
] studied the
holographic entanglement entropy in superconductor
transition with Born-Infeld electrodynamics. Then, the behaviors
of holographic entanglement entropy in the time-dependent
background with nonlinear electrodynamics has been present
in [
48
].
In 1930’s Born and Infeld [
49
] introduced the theory of
nonlinear electrodynamics to avoid the infinite self
energies for charged point particles arising in Maxwell theory.
The ENE theory, as a extended Born–Infeld-like
nonlinear electrodynamics, was introduced by Hendi [
50,51
]. It’s
Lagrangian density is L = 41b2 e−b2 F2 − 1 with F 2 =
F μν Fμν . When the ENE factor b → 0, the Lagrangian
will reduce to the Maxwell case. Compared to the Born–
Infeld nonlinear electrodynamics (BINE) [
52–55
], the ENE
displays different effect on the electric potential and
temperature for the same parameters and its singularity is much
weaker than the Einstein–Maxwell theory [
56–58
]. Recently,
this theory has applications in several branches of physics
being particularly interesting in systems where the ENE is
minimally coupled with gravitation as in the cases of charged
black holes [
59–64
] and cosmology [
65–67
].
Consequently, it is of great interest to investigate the
holographic entanglement entropy in AdS spacetime by
considering the exponential form of nonlinear electrodynamics. In
our previous work [
68
], we have investigated the effects of
the ENE sector on the holographic entanglement entropy
in metal/superconductor phase transition. As a further step
along this line, in this paper, we will further study the
properties of phase transitions by calculating the behaviors of the
scalar operator and the entanglement entropy in holographic
insulator/superconductor model with ENE.
The paper is organized as follows. In the next section, we
will derive the equations of motions and give the boundary
conditions of the holographic model in AdS soliton
spacetime. Then in Sect. 3, we will study the properties of
holographic phase transition by examining the scalar operator.
In Sect. 4, we will calculate the holographic entanglement
entropy in insulator/superconductor transition with ENE.
Finally, Sect. 5 is devoted to conclusions.
2 Equations of motion and boundary conditions
The action for a ENE field coupling with a charged scalar field
with a negative cosmological constant in five-dimensional
spacetime reads
(2)
(3)
S =
5 √
d x
12
−g R + L2 − |∇
1
+ 4b2
e−b2 F − 1
,
− i q A |2 − m2| |2
where g is the determinant of the metric, L is the radius
of AdS spacetime, q and m are respectively the charge and
the mass of the scalar field, F = Fμν F μν here Fμν is the
electromagnetic field tensor. The Einstein equation derived
from the above action becomes
1 6 1
Rμν − 2 gμν R − L2 gμν = 2 Tμν ,
where the energy-momentum tensor Tμν is
Tμν = e−b2 F FμλFμλ + 2∇μψ ∇ν ψ + 2q2 Aμ Aμψ 2
1
+ gμν 4b2
φ 3
= 0 ,
− m2 ψ = 0 ,
3 A B C
r + 2 + B − 2
φ
−
A =
r 2 B
2r 2C + r 2C 2 + 4r C + 4r 2ψ 2 − 2e−C+2b2e−C Bφ 2 φ 2
r (6 + r C )
C + 21 C 2
+
5 A B
r + 2 + B
C
e−c
− r 2
e2b2e−C Bφ 2 φ 2 + 2q2φ2ψ2
r 2 B
= 0,
2 −eC(r)dt2 + d x2 + d y2 + eA(r) B(r )dχ 2 ,
dr 2
ds2 = r 2 B(r ) + r
At = φ(r ), ψ = ψ(r ).
Without lose of generality, we set L = 1 in this paper. In order
to get a smooth geometry at the tip r0 satisfying B(r0) = 0,
χ should be made with an identification
χ = χ + , wi t h
=
4π e−A(r0)/2
r02 B (r0)
.
The independent equations of motion under the above ansatz
can be obtained as follows
B
3 C
r − 2
+B ψ 2
1
− 2 A C +
e−C+2b2e−C Bφ 2 φ 2
where the prime denotes the derivative with respect to r . For
the sake of integrating the field equations from the tip of the
soliton out to the infinity for this system, we need to specify
the asymptotic behavior both at the tip and the infinity. At
the tip (r = r0), the above equations can be Taylor expand
in the form [
12
]
ψ (r ) = ψ0 + ψ1(r − r0) + · · · ,
φ (r ) = φ0 + φ1(r − r0) + · · · ,
A(r ) = A0 + A1(r − r0) + · · · ,
B(r ) = B0(r − r0) + B1(r − r0)2 + · · · ,
C (r ) = C0 + C1(r − r0) + · · · .
The boundary conditions near the AdS boundary where r →
∞ are
ψ ∼ rψ−− + rψ++ , φ ∼ μ − rρ2 ,
A4 B4 C4
A ∼ r 4 + · · · , B ∼ 1 + r 4 + · · · , C ∼ r 4 + · · · .
2W±he√re4th+emco2n,fμoramnadlρdicmanenbseioinntseorpfrtehteedopaesrtahteorcsoarrreespo±nd=ing chemical potential and charge density in the dual field
theory respectively. According to the AdS/CFT
correspondence, both ψ− and ψ+ can be normalizable and they
correspond to the vacuum expectation values ψ− =< O− >,
ψ+ =< O+ > of an operator O dual to the scalar field [
4,5
].
Further, the above equations of motion have useful scaling
symmetries [9]
r → αr, (χ , x , y, t ) → (χ , x , y, t )/α, φ → αφ,
(17)
Using the scaling symmetries (17), we can take r0 = 1. And
the useful quantities can be scaled as
1
→ α , μ → αμ, ρ → α3ρ ,
Oˆ +
5
→ α 2 Oˆ + . (18)
Therefore, we will use the following dimensionless quantities
in next section
μ , ρ 3,
Oˆ + 25 ,
(14)
(15)
(16)
3 Insulator/superconductor phase transition
In this section, we want to study of the phase transition in the
five-dimensional AdS soliton background with ENE field.
In order to obtain the solutions in the complicated model
and ensure the validity of the results, we here concentrate
on the case in the weak effects of ENE field and study its
influences on the properties of the holographic phase
transition. From above discussion, for given m2, q, ψ (r0), we can
solve the equations of motion by choosing φ (r0) as a
shooting parameter. Considering the BF bound [
69,70
], we choose
m2 = − 145 , q = 2 in this paper. Then, ψ− can either be
identified as an expectation value or a source of the operator O of
the dual superconductor. In the following calculation, we will
consider ψ− as the source of the operator and use the ψ+ =<
O+ > to describe the phase transition in the dual CFT.
Here, we plot pictures to display the explicit dependence
of the chemical potential for operator Oˆ + and charge
density ρ on the ENE factor b. It can be seen from the Fig. 1 that
there is a phase transition at the critical chemical potential
μc and its value is independent of the ENE factor b which is
shown in the right-hand panel. That is to say, the ENE has
no effect on the critical potential of the holographic phase
transition for this physical model. When μ < μc, the system
is described by the AdS soliton solution itself which
indicates a insulator phase turns on. When the chemical potential
is bigger than the critical value μc the condensation of the
operator emerges, which means the AdS soliton reaches a
superconductor phase. It is interesting to find that the effect
of the ENE factor b on the operator in the condensate phase
is not trivial. With the increase of the strength of the ENE,
the value of the scalar operator becomes bigger. In Fig. 2, we
note that the charge density ρ in the superconductor phase
drops when the ENE parameter becomes lower and the
insulator/superconductor phase transition here is typically the
second order in this case.
4 Holographic entanglement entropy
In this section, we will study the behavior of holographic
entanglement entropy in this holographic model and discuss
the effect of the ENE factor b on the entanglement entropy.
Since the choice of the subsystem A is arbitrary, we can
define infinite entanglement entropy correspondingly. For
concreteness, we investigate the holographic entanglement
entropy of dual field with a half space and a belt geometry in
the AdS boundary, respectively.
4.1 Holographic entanglement entropy for half geometry
(19)
We first consider the subsystem A with a half space defined
by x > 0, − R2 < y < R2 (R → ∞), 0 ≤ χ ≤ .
Accord5.03
5.0
4.2 Holographic entanglement entropy for strip geometry
In the Following calculation, we are interested in a more
nontrivial geometry which is a strip shape for region A. We
assume that the strip shape with a finite width along the x
direction, along the η direction with a period , but infinitely
extending in y direction. The holographic dual surface γA
6
5
4
3
2
1
0
0.0
0.1
0.3
0.4
0.2
b
while the second term s is a finite term, so it is physically
important. According to the scaling symmetry (17), we here
choose the following scale invariants to explore physics in
the entanglement entropy s.
In Fig. 3, we plot the behavior of the entanglement entropy
s with respect of chemical potential μ and the ENE factor
b in the half geometry. It can be seen from the figure that
the entanglement entropy is continuous but its slop has a
discontinuous change at the critical phase transition point μc.
Which indicates some kind of new degree of freedom like the
Cooper pair would emerge after the condensation.
Furthermore, the discontinuous change of the entanglement entropy
at μc signals that the phase transition here is the second order
transition. With the increase of the ENE factor, the value of
μc dose not change. Which means the ENE parameter has no
effect on the critical point of the phase transition. Before the
phase transition, the entanglement entropy is a constant as
we change the parameters b and μ which can be interpreted
as the insulator phase. After the phase transition, for a given
b, the entanglement entropy in the superconductor phase first
increases and then decreases monotonously for larger μ. And
the value of the entanglement entropy becomes lager as we
choose a lager ENE parameter for a given μ. When the factor
b → 0, the ENE field will reduce to the Maxwell field and
our results are consistent with the one discussed in Ref. [
34
].
ing to the proposal (1), the entanglement entropy can be
expressed as
Shal f
A
R
= 4G N
1
r0
A(r) Rπ
r e 2 dr = 8G N
where r = 1 is the UV cutoff. Note that the first term is
divergent as → 0 and will not change after the operator
condensation [
34
]. The second term s is independent of the
UV cutoff and s = −1 corresponds to the pure AdS soliton.
As the aim of requiring the lower bound of the integral is still
equal to unit, we define a useful dimensionless coordinate in
the form
r0
z = r .
Shal f
A
= −
Then, the entanglement entropy for a half space can be
rewritten as
1
r0 r 2e A2(z)
0
z3
1
d z = 2
(22)
defined as a codimension three surface is t = 0, x =
x (r ), − R2 < y < R2 (R → ∞), 0 ≤ χ ≤ .
Considering the surface is smooth, we suppose that the
holographic surface γA starts from x = 2 at r = 1 , extends
into the bulk until it reaches r = r∗, then returns back to the
AdS boundary r = 1 at x = − 2 . The entanglement entropy
of the belt geometry with connected surface in z coordinate
is given by
R
Sconnect [x ] = − 2G N z
A
∗
r0 r 2 A(z)
0 e 2
z3
1+r02 B(z)(d x /d z)2d z,
where z∗ = r0/r∗ is the turning point. As the physics model
is a static situation, the above Eq. (24) dose not depend on
the time slice. And we could consider Sconnect [x ] as the
inteA
gral of the Lagrangian with x direction thought of as time.
Because the translations of x direction is symmetry, the
corresponding Hamiltonian is conserved. Therefore, the
equation of motion for the minimal surface from Eq. (24) can be
deduced as following
A(r)
r04 B(z)(d x /d z)e 2
z3 1 + r02 B(z)(d x /d z)2
=
r 3√B(zs )e 2
A(zs)
0
z3
s
,
where zs is a constant. And the width and the entanglement
entropy S can be easily calculated in the form
2 = −
r0
zs∗ r0
1
B(z)( z6zB6 (Bz(∗z))eeAA((zz∗))) − 1)
∗
d zs ,
(26)
(24)
(25)
Here, we show in Fig. 4 the results for the entanglement
entropy s versus the width of the subsystem A and the ENE
factor b with the dimensionless quantities s 2, μ , −1 and
b. We find that the discontinuous solutions represented the
horizontal dotted lines in the figure is independent of the
width but its value of the entanglement entropy with a
smaller b is smaller. The connected configuration denoted
by the solid lines has two solutions. Specifically, the
socalled confinement/deconfinement phase transition [
29–31
]
emerges as we change the width and the critical value c
indicated by the vertical dotted lines becomes bigger with the
increase of the parameter b. For a fixed b, in the
deconfinement phase where < c, considering the physical entropy
determined by the choice of the lowest one, the entanglement
entropy comes from the connected surface and the lowest
branch in the figure is finally favored. However, the physical
entanglement entropy in confinement phase where > c
is dominated by the discontinuous surface and has
nothing to do with the factor . Thus, there exists four phases
in the dual boundary field theory, including the insulator
phase, superconductor phase and their corresponding
confinement/deconfinement phases. When the parameter is
fixed, we observe that the entanglement entropy increases
as we choose a bigger ENE factor both in the confinement
and deconfinement superconducting phases.
Interestingly, the entanglement entropy with respect to the
chemical potential μ as one fixes the ENE factor b or the
width is presented in Fig. 5. At the insulator/superconductor
phase transition point μ = μc, we also find that the jump of
the slop of the entanglement entropy indicates that the
system undergoes the second order phase transition. Both in
the confinement and deconfinement superconducting phases
where μ > μc, we can see that the behavior of the
entanglement entropy as a function of the chemical potential is
non-monotonic and similar to the case in the half geometry
which we have discussed above. As the chemical potential is
fixed, the value the entanglement entropy becomes smaller
when the factors b and become lower. More specifically,
2
s 8
6
7
9
Fig. 4 The entanglement entropy as a function of the strip width with
different parameter b for μ = 6.9165. The black curve is for |b| = 0
and the blue curve is for |b| = 0.2. The red curve is for |b| = 0.3
and the green curve is for |b| = 0.4. The corresponding critical widths
are c −1 = 0.2109, c −1 = 0.2122, c −1 = 0.2139, c −1 =
0.2164, respectively
the effect of the ENE factor on the entanglement entropy is
weaker than the width of the subsystem A. In their
corresponding insulator phases where μ < μc, we observe that
the value of the entanglement entropy does not change as we
alter the parameter b for a given μ. On the other hand, with
the increase of the width the entanglement entropy increases.
That is to say, the behavior of the entanglement entropy in
the corresponding insulator phases is independent of the ENE
factor but depends on the width of the subsystem A.
4.3 Phase diagram
Finally, according to our calculation for entanglement entropy
in holographic insulator/superconductor transition with ENE
field, we use a picture to display the phase diagram of
entanglement entropy with a straight geometry.
5
10
15
Fig. 6 The phase diagram of the entanglement entropy for a strap
geometry in the holographic insulator/superconductor transition with
ENE field. The black curve is for |b| = 0 and the blue curve is for
|b| = 0.2. The red curve is for |b| = 0.3 and the green curve is for
|b| = 0.4
In Fig. 6, the insulator phase and the superconductor
phase are separated by the green vertical dashed line and
the phase boundary between the confinement phase and
the deconfinement phase is separated by the red
horizontal dashed line and the solid curve. Therefore, the phases
characterized by the parameters μ and contain the
insulator phase, superconductor phase and their corresponding
confinement/deconfinement phases. It can be clearly seen
from the figure that the critical width c of the
confinement/deconfinement phase transition in the insulator phase
is independent of the ENE parameter. In the
superconductor phase, however, the critical width c increases with the
increase of the ENE factor. To further study, we observe that
the critical width c has a non-monotonic change as the
chemical potential becomes bigger. Concretely, the entanglement
entropy first increases beyond the cusp at the certain
chemi1 0.165
b 0.0.
b 0.2
b 0.3
b 0.4
0
5
10
15
5
10
15
Fig. 5 The entanglement entropy s as a function of the chemical
potential μ for the various factors,i.e., the ENE factor b and the belt width
. The horizontal dotted lines denote the entropy in the insulator phase,
the green vertical dashed line denotes the critical phase transition point
where μc = 2.9662, and the solid curves denote the entropy in the
superconductor phase. The left-hand figure corresponds to −1 = 1
and different b: black curve for |b| = 0 , blue curve for |b| = 0.2, red
curve for |b| = 0.3 and green curve for |b| = 0.4. The right-hand figure
corresponds to |b| = 0.3 and different : green curve for −1 = ∞,
red curve for −1 = 0.181 and blue one for −1 = 0.165
cal potential, reaches to a maximum and decreases to a
minimum, and then approaches a plateau at very large μ.
5 Summary
We have studied the properties of phase transitions by
calculating the behaviors of the scalar operator and the
entanglement entropy in holographic insulator/superconductor model
with ENE. On the basis of the behaviors of the scalar
operator in this holographic model, we find that there is
a insulator/superconductor transition at the critical
chemical potential point and the effect of the ENE factor on the
scalar condensation is quite different from those observed in
the holographic metal/superconductor transition with ENE
field model [
68
]. Specifically, in the holographic
insulator/superconductor system the ENE factor does not have
any effect on the critical chemical potential of the transition.
These conclusions can also be understood from the behavior
of the entanglement entropy. From the Fig. 3, the
discontinuity of the slop of the entanglement entropy in half space at
the critical chemical potential point signals some kind of new
degree of freedom like the Cooper pair would emerge after
the condensation and indicates the order of associated phase
transition in the system. In Fig. 5, we observed the behavior
of the entanglement entropy with respect chemical potential
in strip geometry at the insulator/superconductor transition
point is similar to the half case. That is to say, the
entanglement entropy is indeed a good tool to search for the phase
transition point.
In the superconducting phase, compared to the
phenomenon observed in the scalar operator, the entanglement
entropy versus the chemical potential displays more rich
behaviors. Both in the half space and the belt space, the
nonmonotonic behavior of the entanglement entropy versus the
chemical potential is general in this model as the ENE
parameter is fixed. For a given chemical potential, the value the
entanglement entropy becomes smaller when the ENE factor
or the with becomes lower. In the insulator phase, however,
the behavior of entanglement entropy is independent of the
ENE parameter.
Interestingly, considering the effect of the belt width on
the entanglement entropy, we obtained that the
confinement/deconfinement phase transition appears in both
insulator and superconductor phases and the complete phase
diagram of the entanglement entropy with a straight geometry
is presented in Fig. 6. It is shown that the critical width of
the confinement/deconfinement phase transition depends on
the chemical potential and the ENE term.
Acknowledgements This work was supported by the National Natural
Science Foundation of China under Grant Nos. 11665015, 11475061;
Guizhou Provincial Science and Technology Planning Project of China
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