#### Entanglement from dissipation and holographic interpretation

Eur. Phys. J. C
Entanglement from dissipation and holographic interpretation
M. Botta Cantcheff 2
Alexandre L. Gadelha 1
Dáfni F. Z. Marchioro 0
Daniel Luiz Nedel 0
0 Universidade Federal da Integração Latino-Americana, Instituto Latino-Americano de Ciências da Vida e da Natureza , Av. Tancredo Neves 6731 bloco 06, Foz do Iguaçu, PR CEP: 85867-970 , Brazil
1 Instituto de Física, Universidade Federal da Bahia, Campus Universitário de Ondina , Salvador, BA CEP: 40210-340 , Brazil
2 IFLP-CONICET CC 67 , 1900 La Plata, Buenos Aires , Argentina
In this work we study a dissipative field theory where the dissipation process is manifestly related to dynamical entanglement and put it in the holographic context. Such endeavour is realized by further development of a canonical approach to study quantum dissipation, which consists of doubling the degrees of freedom of the original system by defining an auxiliary one. A time dependent entanglement entropy for the vacumm state is calculated and a geometrical interpretation of the auxiliary system and the entropy is given in the context of the AdS/CFT correspondence using the Ryu-Takayanagi formula. We show that the dissipative dynamics is controlled by the entanglement entropy and there are two distinct stages: in the early times the holographic interpretation requires some deviation from classical General Relativity; in the later times the quantum system is described as a wormhole, a solution of the Einstein's equations near to a maximally extended black hole with two asymptotically AdS boundaries. We focus our holographic analysis in this regime, and suggest a mechanism similar to teleportation protocol to exchange (quantum) information between the two CFTs on the boundaries (see Maldacena et al. in Fortschr Phys 65(5):1700034, arXiv:1704.05333 [hep-th], 2017).
1 Introduction
The AdS/CFT correspondence is the most successful
application of the holographic principle, and it plays a very important
role in the study of the non perturbative sector of a class of
Yang–Mills theories. It allows to calculate non-perturbative
vacuum expectation values of Yang–Mills operators via tree
level calculation in the perturbative sector of type IIB string
theory on Anti-de Sitter background. In fact, the
conjecture enables calculations of Yang–Mills expectation values in
terms of classical propagators on asymptotically AdS
geometries. On the other hand, since the AdS/CFT correspondence
is a duality, it would be possible to read the holographic
dictionary in another way and use the Yang–Mills computations
to infer classical properties of the bulk geometry. A more
challenging application is to use the holographic dictionary
to analyze quantum aspects of gravity. However, our
understanding on the basic principle of the holography, even in the
context of string theory, is still limited. It is well-known that,
to address fundamental issues in quantum gravity such as
black hole singularity or quantum cosmology, it is necessary
to develop a more general framework for the holography,
which allows us to go beyond the Anti-de Sitter spacetime
and explore non-trivial generalizations of the AdS/CFT
correspondence.
A key property of holography is that it allows to relate
geometric quantities with microscopic information of the
quantum field theory. The most famous example is the
Bekenstein–Hawking entropy, which relates the area of a
black hole’s event horizon to its entropy: the entropy is
directly computable from the quantum state of the system. In
an outstanding application of the AdS/CFT dictionary, Ryu
and Takayanagi (RT) proposed in [
1
] that the area of a
minimal surface in a holographic geometry should be related to
the entanglement between the degrees of freedom contained
within this region and those of its complement. In particular,
they proposed the formula
A
S = 4G N ,
where S is the entropy of a spatial region and A is the area
of the minimal surface in the bulk whose boundary is given
(1)
by ∂ . Their proposal has been checked in many ways and
this relation is proved in [
2
] for a spherical symmetry case
and for a more general case in [
3
]. Quantum corrections to
this area law formula are considered in [
4,5
] and the time
evolution of the entropy was studied in [
6,7
]. Also, a
holographic entanglement entropy in a higher derivative gravity
theory has been obtained in several works, in particular in
[
8–13
].
The main goal of the present work is to explore the RT
formula in the context of dissipative quantum field theories.
Dissipative quantum field theories have been studied in the
last years for several reasons, and one of these reasons is the
experimental results that came out of the Relativistic Heavy
Ion Collider (RHIC). The experimental results suggest that
the quark gluon plasma is “strongly coupled” with a very
small value of η/s, where η is the shear viscosity and s the
thermodynamical entropy density. This result motivates the
development of finite temperature holographic techniques to
calculate transport coefficients, and a universal result for the
shear viscosity, η/s = 1/(4π ), was derived in [
14
]. On the
other hand, using the Bañados–Teitelboim–Zanelli (BTZ)
black hole [
15
] as its holographic dual, the Brownian motion
of a heavy quark in a finite temperature Conformal Field
Theory (CFT) was studied in [
16
] in the context of Langevin
equation. The holographic Schwinger–Keldysh method was
developed for this case in [
17
] and the Thermo Field
Dynamics (TFD) in [
18
]. One of the interesting results is that there
is a drag force on the fluctuating external quark even at zero
temperature, owing to radiation. This was studied in detail in
[
19
], where the same result for the zero temperature term is
obtained for pure AdS in all dimensions. This suggests that,
even at zero temperature, important dissipative effects can be
read from holographic techniques.
A close relation between dissipation and entanglement
was pointed out in [
20
], where it was reported an
experiment where dissipation generates continuously entanglement
between two macroscopic objects. Concerning the
quantum treatment of dissipative systems, the quantum theory
of damped linear harmonic oscillators can lead to a vacuum
state that is in fact an entangled state [
21
]. By putting
forward the Feshbach–Ticochinski approach in [
22
], it was
realized a relationship between the canonical quantization of the
damped harmonic oscillator and the TFD formalism, where
the damped harmonic oscillator was interpreted as the
thermal vacuum and the TFD entropy operator appears naturally
as the entanglement entropy operator, in a suitable
extension to quantum fields. Following these ideas, we are going
to use in this work the RT formula to give a holographic
interpretation of the relationship between entanglement and
dissipation, concerning conformal field theories at zero and
finite temperature.
In order to handle dissipation using a canonical approach,
one needs settings which are wider than those of usual
applications of quantum field theory and the AdS/CFT
correspondence. In particular, if one introduces dissipation from the
outset in the usual settings, two immediate problems come
to light. The first one is the Lorentz covariance breaking:
dissipation is a typical process that breaks the invariance under
boost transformations. Just because there is a deep relation
between dissipation and the “arrow of time”, a dissipative
process defines a natural preferred frame. Although it is
possible to write a covariant action, the expectation values break
the Lorentz covariance [
23–25
]. Also, as thermal effects are
taken into account, the thermal bath’s frame works as the
preferred frame. The second problem is the loss of unitarity.
However, this problem can be circumvented and it is possible
to preserve the unitarity and make usual quantum
mechanics in a finite volume limit. In order to do such endeavour,
we shall work therefore with Hamiltonian theories in which
dissipative effects are caused by interactions with additional
degrees of freedom [
22–24
]. In this case, the AdS/CFT
correspondence allows an elegant interpretation of the auxiliary
degrees of freedom. The explanation follows the same spirit
of Israel’s work in black holes using TFD [
26
]. In TFD, in
order to take care of thermal expectation values, the
original system is duplicated and the thermal vacuum is defined
via a Bogoliubov transformation, which actually entangles
the system and its copy. In maximally extended black hole
solutions, the auxiliary degrees of freedom are interpreted
as fields living behind the horizon; for AdS black holes,
this defines two asymptotic conformal theories. In references
[
27,28
], a mechanism to send a signal between the
boundaries is presented. From a teleportation protocol, an
effective potential reproduces an “attractive force” that brings the
boundaries closer. The potencial is set up with the product
of two hermitian operators, one on each side. In our work,
we obtain a similar potential that couples the two
boundaries.
We consider a particular coupling between two systems
that will drive dissipation in one of the boundaries. Note
that what defines the system and the auxiliary system is
just the geometric region where the measures are taken. We
show that, for one class of asymptotic observers, the vacuum
state is an entangled state and we calculate the
entanglement entropy in a “canonical way”. Actually, we show that
the entanglement entropy can be calculated via expectation
value of an operator, defined as the entanglement entropy
operator, which is in fact a representation for the modular
Hamiltonian [
29
], playing a dynamical role in this system.
In fact, it will be seen that the time evolution of the vacuum
entangled state is generated by the time derivative of that
operator. So, after establishing a geometric description for
the dissipative conformal field theory (DCFT), we can use
the RT formula to infer that the time evolution of the minimal
surface is controlled by the time dependence of the entangled
state.
We observe at least two distinct holographic stages: for
the first one, at very early times, some type of deviation from
classical Einstein’s gravity should be considered in order to
describe changes of the spacetime topology [
30
]; in the
second one, for later times, we can study the leading large-N
effects by considering a classical solution of the Einstein’s
equations as the holographic dual. The final sections of the
present paper will be devoted to study the second holographic
stage.
In order to find out the geometric picture of the
dissipative scenario presented here, we look for an asymptotically
AdS gravitational system that reproduces the same
dissipative dynamics on the boundary. It is known for a long time
that scalar field coupled to Friedmann–Robertson–Walker
(FRW) metric has a kind of dissipative behavior [
31–35
].
In particular, the equation of motion has the same damping
term of the DCFT studied here. In [
36
] it was shown that,
using a particular coordinate transformation, it is possible to
have FRW metric on the boundary from a BTZ black hole
on the bulk. So, this appears to be the geometric system that
we are looking for. However, it will be shown that the time
dependent entanglement entropy derived here demands that
the original metric is not BTZ, but a sort of Vaidya solution in
the adiabatic approximation; that is, a BTZ black hole with a
slowly time dependent mass. Keeping this scenario in mind,
the RT formula allows to find a natural relationship between
dissipation, entanglement, thermodynamics and black hole
physics. However, as it will be discussed in Sect. 7, for Ad S3
the holographic picture based on the FRW boundary is just
an approximation of the DCFT. In the asymptotic limit, the
entangled state derived here can be seen as an approximation
of the TFD vacuum.
This work is organized as follows: in Sect. 2 we present
the dissipative model; in Sects. 3 and 4 the time
dependent entropy is canonically computed; in Sect. 5 we show
how we deal with the time dependence of the entropy
in holographic computations; in Sect. 6 the holographic
model is constructed; and Sect. 7 is devoted to the
conclusions.
2 The dissipative model
In this section we are going to develop the main idea of this
work. We start with a system A, whose degrees of freedom
are described by a free conformal field φ living on a
compact space ⊂ d . The usual canonical approach to study
dissipative systems consists of putting the system A inside
another system A¯ (bath, medium, or environment), whose
degrees of freedom are unknown. The system A¯ interacts
with the system A such that, from the φ-field perspective,
there is a dissipative process which can be described by the
equation
where γ is the damping coefficient that will describe the
coupling between the original fields and the medium (the A¯
system). Here we study the problem from a different
perspective. We think of A and A¯ as two physical systems living
in different geometric regions, corresponding to two
asymptotic conformal theories. In a certain moment an interaction
is suddenly switched on, allowing energy exchange between
the two theories.1 The Eq. (2) is just the resulting dynamics
as seen by the A system.
If we represent the A¯ system by ψ fields, we can write the
following Lagrangian that reproduces the Eq. (2)
L =
dd x ψ (∂t2 − ∇2)φ + γ ∂t φ .
Integrating (3) by parts and neglecting the boundary terms,
we can write it in the symmetric form
L =
dd x ∂μφ∂μψ + γ2 (φ∂t ψ − ψ ∂t φ) .
The variation of (4) with respect to ψ gives (2), while
minimizing it with respect to φ leads to the equation of motion
for the ψ field
(∂t2 − ∇2)ψ − γ ∂t ψ = 0.
Equation (5) shows that ψ describes an identical copy of the
physical field φ with the time direction reversed. The
interaction between ψ and φ describes the dissipative behavior
precisely [
22–24
].
We want to draw attention for the similarity of this
construction with the rules of TFD: the field ψ can be considered
the TFD’s double of the system, and it can be interpreted as
a derivation of the TFD picture. In this context, the field ψ
evolves in the inverse time direction. In the usual
interpretation, if the coupling is switched off (γ → 0), we recover two
non-interacting (free) CFT’s whose states describe
spacetimes with two locally asymptotically AdS regions [
30
]. In
particular, the ground state |0 >φ ⊗|0 >ψ represents two
disconnected copies of the exact AdS spacetime [
38
], and
the thermal vacuum2 is the Kruskal extension of an
eternal AdS black hole3 [
40
]. Furthermore, the AdS/CFT
dictionary implies that, in this example, the gravity dual theory
is very stringy, and therefore quantum gravity interprets it
as quantizing strings on AdS such that these metrics shall
be recovered in the proper large N-limit. In the following
1 This may be viewed as a sort of quantum quenching [
37
], where an
interaction is suddenly turned on.
2 In a TFD description.
3 For further applications of TFD in string theory, see [
18
] and [
39
].
(2)
(3)
(4)
(5)
sections, the behavior of the entropy will refer us to two
different scenarios. We will focus on the large γ t limit, where it
will be possible to make an approximation compatible with
the large N limit, and the holographic picture can be
understood in terms of Vaidya black holes, written in a particular
coordinate system. Also note that the Lagrangian (4) is
different from the ones studied in [
41,42
] just because we are
interested in the dissipative process and its relation with the
entanglement entropy. As the dissipative process imposes
a privileged direction of time, Lorentz invariance is broken
[
23–25
]. However, the entanglement entropy for models with
broken Lorentz symmetry has the same behavior as the
relativistic ones [43].
The canonical conjugate momenta of the fields are
∂ L γ
πφ = ∂φ˙ = ψ˙ − 2 ψ
∂ L γ
πψ = ∂ψ˙ = φ˙ + 2 φ,
and the Hamiltonian can be written as
H = πφ πψ + ∂i φ∂i ψ − 21 γ (φπφ − ψ πψ ) − γ42 ψ φ. (7)
The ansatz for the solution of the equations of motion is
γ
φ = φ e− 2 t ,
where φ is the plane wave solution with frequency ωk =
± k2 − γ42 . Note that the system has real frequencies since
an IR cut off is defined by γ .
The general solution can be expanded in terms of
quasi-normal frequencies (frequencies with imaginary part),
namely
φ = φ (x )ei t ,
= w + i .
Replacing this into the equation of motion and demanding
that k is real,4 we have that = γ2 and ωk = ± k2 − γ42 .
In summary, the general solution is plane waves damped by
γ
a decaying factor e− 2 t . In contrast, the solution for the field
γ
ψ has a growing factor e 2 t ; however, by taking the physical
time parameter −t , it has the correct damping behavior.
It is easier to quantize the theory by making the
redefinition
=
φ√+2ψ ,
=
φ√−2ψ .
4 Otherwise we would have that depends on k.
In terms of the new fields the Lagrangian is
where
H
H
where
¯
=
In momentum space, the Hamiltonian is
H =
dd k ¯ (k) ¯ (−k) − ¯ (k) ¯ (−k)
1 2
+ 2 k ( (k) (−k) −
(k) (−k)) ,
γ
− 2
¯
=
γ
+ 2 .
(6)
(8)
(9)
(10)
,
(11)
(12)
(13)
(14)
(15)
(16)
(17)
γ
− 2
γ
+ 2
2
2
+ 21 (∂i )2
As the total system is canonical, the fields satisfy the usual
canonical commutation relations and they can be replaced by
expressions in terms of the creation/annihilation operators.
As usual, the free fields can be written as
(k) =
(k) =
†
Ak + A−k ,
√2 k
†
Bk + B−k ,
√2 k
(k) =
(k) =
√
√
† )
2 k ( Ak − A−k ,
2i
† )
2 k (Bk − B−k ,
2i
where [ A(k), A†( p)] = [B(k), B†( p)] = δ(k − p) and k =
√k2. The momentum space Hamiltonian becomes
H =
k
γ † B†)
− 2i ( A−k Bk − A−k k
( Ak† Ak − Bk Bk )
†
H = H0 + Hint ,
with
H0 =
Hint =
k
k
.
It is easy to see that the Hamiltonian in (17) can be presented
as
the γ must not be independent of modes and the asymptotic
limit for the vacuum state cannot be (20).
As [H0, Hint ] = 0, the vacuum state time evolution is given
by 5
|0(t ) = exp(−i H t ) |0
= exp
k
δkk
= k cosh γ t exp tanh
2
† B†
A−k Bk − A−k k
|0
γ t
2
† B†
A−k k |0 . (19)
Note that the dissipative interaction of the two CFTs produces
an entangled state. In the next section we are going to show
how the entanglement entropy appears naturally in this
scenario. Also note that this state becomes stable for t 4/γ
[
46
],
|0(t ) ∼ N exp A†−k Bk† |0 ≡ |B ,
where |B is a boundary state in the field theory. In particular
this is the solution for the equation
[ (x , t ) −
(x , t )]|B
In terms of the original fields, this translates into φ |B =
ψ |B = 0, as we should expect in a dissipative theory if
we do not take into account thermal effects; this final state is
the maximum entanglement state for each mode. In fact, the
classical behavior of the fields for later times translates into
a (quantum) boundary condition that reads
φ |0(t ) ≈ 0,
φ |B
= 0.
for 4/γ t , while the state |B is actually defined by the
exact condition
However, thermal effects are always present in dissipative
theories [
47
]. The Brownian motion does not allow a
condition like (22) and (20). This implies that, at least in later times,
5 Actually, |0 is a SU (1, 1) rotated vacuum. See [
21,22,44,45
] for
details.
(18)
(20)
(22)
(23)
k
n
k
k
3 Entanglement entropy and dissipative dynamics
In order to analyze the result of Eq. (19) in a more general
context, let us consider for a moment a possible dependence of
the dissipative coefficient on the particles’ momenta, namely
γ → k . Following reference [
22
], the entanglement entropy
can arise in the scenario introduced above observing that the
vacuum at a finite time t , in a finite volume, can also be
obtained as
where we have introduced the following operators
Many interesting properties are related to these operators.6
First of all, considering the result of their expectation value
in the time evolved vacuum,
†
0 (t ) Ak Ak 0 (t ) =
†
0 (t ) Bk Bk 0 (t ) = sinh2 ( k t ) ,
and the same can be done for SB . Notice also that the vacuum
at a finite t can be written in terms of Wn(t )
|0(t ) =
Wn(t ) |n, n .
6 These operators were first studied in TFD context in [
48
] and
further developed in [
49
]. The relation with dissipative dynamics was first
studied in [
22
].
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
The expressions above allow us to make an important
connection with the density operator. Considering
This can be easily seen from Eq. (25) where e− 21 SA(t) ≈ I
for t 4/γ . We shall return to this point in Sect. 5.7
ρ = |0(t ) 0(t )| .
then the reduced density operator is
ρA = T rB [|0(t ) 0(t )|]
=
n
Wn(t ) |n n|
3.1 DCFT renormalized entanglement entropy
Let us return now to our original example, where k ≡
γ ∀ k. In this case, the expressions above simplify
considerably and we can compute explicitly the entanglement entropy
as a function of time. In fact,
∂ S
∂t ≈ 0
t
4/γ .
(32)
(33)
(34)
(35)
(36)
(37)
(38)
that is, the entropy operator coincides with the entanglement
Hamiltonian defined in the literature [
29
]. In fact, by tracing
out the B-degrees of freedom, we obtain the time-dependent
reduced matrix density [
50
]
ρA(t ) = e−SA(t),
showing that the entropy operator SA(t ) is nothing but the
so-called modular Hamiltonian for this state [
29
]. Then, the
expectation value of any A-operator OA in |0(t ) can be
written as
0(t )|OA|0(t ) = T rρAOA
Using Eq. (24), we can verify that the time evolution of
the vacuum state is generated by the time derivative of the
entropy operator or entropy production, namely
∂ |0(t )
∂t
1 ∂ S
= − 2 ∂t |0(t ) ,
which has an interesting holographic interpretation, as it will
be discussed in the next section. Remarkably, let us observe
that this equation resembles thermodynamics, even though its
derivation has nothing to do with the thermodynamic laws. It
is interesting how, by virtue of this equation, the basic notion
of equilibrium ∂ |0(t ) ≈ 0 is equivalent to the maximum
∂t
entropy condition
∂ S
∂t |0(t ) ≈ 0,
which is fulfilled if the operator S achieves its maximum
value for some time t . This might be recognized precisely as
the second law of thermodynamics in a quantum-mechanical
sense. For instance, in the example given previously, we can
observe explicitly that
where
Nˆ A =
Defining the functions
F (t ) := − ln tanh2 (γ t ) ,
G(t ) :=
ln cosh2 (γ t ) ,
k
k
†
Ak Ak .
k
+
(39)
(40)
(41)
(42)
(43)
(44)
then the entanglement entropy operator reads simply as
SA(t ) = F (t )Nˆ A + G(t ).
The Eq. (43) is very useful since, in principle, we can
follow the change in the area of an extremal surface in a dual
holographic geometry. Actually, this is an operator whose
expectation value in the time dependent state defined in (19)
provides the area of an extremal surface. For instance, taking
the expectation value in the time dependent ground state, we
have
sA(t ) =
δkk [− sinh2 (γ t ) ln tanh2 (γ t ) − ln cosh2 (γ t )]
Note that when the volume of the space is finite, there is an
UV finite term in entropy densities. Differently from the UV
behavior of the usual entanglement entropy (which scales
with the area), one expects the leading UV behavior of this
entanglement entropy to scale with the volume [
52
]. In order
to obtain a finite result, we introduce an UV regulator. Using
periodic boundary conditions in d = 2, the result is
7 In [
51
] it is argued that the entanglement entropy for a very small
subsystem obeys a property which is analogous to the first law of
thermodynamics; here this kind of relation naturally arises.
Fig. 1 The state
|0 = |0 A ⊗ |0 B is dual to
two disconnected copies of AdS
S(t ) = −2
e− 2πn/L
∞
n=1
× sinh2 (γ t ) ln tanh2 (γ t ) − ln cosh2 (γ )
= − sinh2 (γ t ) ln tanh2 (γ t )
L
π
− ln cosh2 (γ t )
− 1 + O( /L) ,
(45)
where L is the circumference of the S1 space and is the UV
regulator. Following the usual procedure, we can pick up the
UV finite term by differentiating the entropy density in the
limit → 0:
∂ S(t ) 1
∂ L L = − L2 sinh2 (γ t ) ln tanh2 (γ t )
− ln cosh2 (γ t ) ,
and the renormalized entropy is
sARen(t ) = − sinh2 (γ t ) ln tanh2 (γ t ) + ln cosh2 (γ t )
= f(t ) sinh2 (γ t ) + g(t ),
where
f(t ) = − ln tanh2 (γ t ) , g(t ) = ln cosh2 (γ t ) .
For γ t
1
sARen(t ) ≈ −(γ t )2 ln (γ t )
Notice the similarity with the results found in [
41,42
].
Now, let us consider the asymptotic limit γ t 1:
sARen(t ) ≈ ln(cosh(γ t )) ≈ γ t.
In the asymptotic limit, the entropy grows linearly with
time according to the Cardy/Calabrese results. Actually, the
asymptotic entropy resembles the finite temperature results
of [
46
], if γ is interpreted in terms of the temperature. This
suggests that asymptotically the system thermalizes, which
is no surprise when dealing with a dissipative system. Before
(46)
(47)
(48)
(49)
(50)
(51)
(52)
carrying out the holographic interpretation of this result, let us
understand how to deal with the time dependence of entropy
in holographic computations. This is the main goal of the
next section.
4 The time dependent holographic computations
The renormalized entropy found in (47) would be 4G times
the area of the minimal surface in the dual holographic space
whose boundary is anchored by ∂ . However, one does not
know a priori how much of the variation of this entropy is
due to the dual metric change or to the extremal surface, or
both. So, let us discuss this point in detail.
According to the AdS/CFT dictionary, the state |0(t ) is
dual to a background spacetime metric, namely gμν (t ).8 In
particular, |0(t = 0) = |0 corresponds to the globally AdS
spacetime; actually, |0 = |0 A ⊗ |0 B is dual to two
disconnected copies of AdS as in Fig. 1 (see [
30,38
] for
interpretations of such geometries). On the other hand, the
entanglement entropy is given by the RT formula,
s(t ) =
1
0(t )| SA(t ) |0(t ) = 4G N a[gμν (t ), t ]
1
:= 4G N
t
det g(ind)(t ),
where g(ind)(t ) denotes the induced metric on the minimal
(co-dimension two) surface t . The surface represented by
the integral on the right hand side of (51) must be properly
regularized with a cut-off such that t has a finite area (see
details on this calculation in the context of Wilson loops in
[
53–55
]).
The limit case
lim 0(t )| SA(t ) |0(t ) = 0,
t→0
8 We are assuming implicitly that the fields are in the adjoint
representation of a SU(N) group in the large N limit.
reflects the fact that the minimal surface 0, embedded in
the exact AdS spacetime (whose conformal boundary is Sd )
and anchored by ∂ , has vanishing area as → Sd . Then,
using (51), the metric induced on t is
1
f(t ) sinh2 (γ t ) + g(t ) = 4G N
t
det g(ind)(t ).
(53)
In other words, a holographic dual of our model consists
of a metric gμν (t ) equipped with a minimal surface t that
satisfies (53).
Considering first order variations of the state |0(t ) , we
can verify that
where the variation is normalized as TrA δρA = 0(t )|δ0(t )
= 0. The Eq. (54) can be written as
δsA(t ) = 0(t ) + δ0(t ) |SA(t )| 0(t ) + δ0(t )
− 0(t ) |SA(t )| 0(t ) .
In particular, one can think of t (or γ t ) as a parameter for the
states’ metrics and compute
0 |SA(t )| 0 = g(t ) = (γ t )2 + O3(γ t ) · · ·
For instance,
while, using (52),
0(t ) |SA(t )| 0(t ) − 0 |SA(t )| 0 = f(t ) sinh2 (γ t ) ,
δsA(t ) ≡ 0(t ) |SA(t )| 0(t ) − 0 |SA(0)| 0
= f(t ) sinh2(γ t ) + g(t ),
and the difference between both expressions is precisely
O2(δρA) = (γ t )2 + O3(γ t ), which expresses (54) in terms
of the parameter t (or γ t ).
On the other hand, since the first variation of the area with
respect to t vanishes [
56
], we have that
1
δsA(t ) = T r A δρA SA(t ) = 4G N
δa
δg jk
t
δg jk (t )
t
where, since the co-dimension t is two, the area is entirely
embedded in a spacelike hypersurface N , so here g jk stands
for the spacelike Riemannian metric on N .
Therefore, one can conclude that for states |ψ ≡
|0(t ) + δ0(t ) near to |0(t ) the interpretation of the quantity
1
= 4G N a[g jk (t ) + δg jk , t ]
1
:= 4G N t
det (g + δg)(ind)(t )
ψ |SA(t )| ψ
is the area of the same surface t calculated in the deformed
metric g(ψ) ≡ g jk (t ) + δg jk . Note that t is the extremal
one for the metric g jk (t ), then in general it doesn’t need to be
the extremal surface for g(ψ). Actually, for states arbitrarily
different from |0(t ) , one does not know on which surface
should evaluate this functional, nor if the area law is even
valid or there is corrections to it. Nevertheless, one can find a
holographic (differential) formula that may be a useful recipe
to track the dual metric in certain specific cases, where the
minimal embedded surface does not change as the metric is
deformed. In principle one must consider a one-parameter
family of Riemannian metrics {g jk (θ )}θ∈R on a fixed
manifold (space) N , but in the cases we are going to study, we
can identify the real parameter θ with t , or γ t .
We can differentiate the expression (59) with respect to
the time parameter and obtain the holographic formula for
the entropy production
σ R ≡
ds(t )
dt
1
= 4G N
δa
δg jk
t
dg jk
dt
,
where the time/parameter derivative is the total derivative
since t shall be interpreted as a parameter rather than a
spacetime coordinate. However, if the whole spacetime is built up
as a foliation of spaces {(N , g jk )t , }t ≡ M , there is a frame
where the parameter t is the time coordinate and it is on the
same foot as the other coordinates of the spacetime,
dg jk
dt =
∂g jk (xi , t )
∂t
xi ∈ t
.
The formula (61) relates the entropy production σ R in the
field theory and the rate of change of the spacetime metric
g jk (xi , t ). In (62), xi (i = 1, . . . d) denotes the spatial
coordinates, and xi ∈ t means that these coordinates shall be
evaluated on the minimal surface after taking partial time
derivative. This equation should be supplemented with the
RT equation in order to determine such a surface:
(55)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
δa
δ t g jk
= 0.
This prescription is useful for the specific example we are
investigating. Equation (61) can be integrated out in the time
parameter to obtain the (induced) metric, during an interval
(t−, t+) provided that
d t
dt
= 0, ∀ t ∈ (t−, t+).
In fact, regarding the limit in which covers the whole
sphere Sd , one can do an ansatz for the metric with
spherical symmetry and AdS asymptotics. In d+1 dimensions, the
spacetime metric is given by
Fig. 2 Spatial section t =
constant of the wormhole
solution
Φ
1
ds2 = −h(r )dt 2 + h(r ) dr 2 + g
(r, t )d 2d−1,
and the spatial metric g jk on Nt is nothing but
g jk d x j d x k ≡ h(r )−1dr 2 + g
(r, t )d 2d−1,
where h(r ) → R−2r 2 as r → ∞ is the asymptotic (AdS)
condition. In (65), t is the time coordinate, r is the radial
coordinate, d−1 stands for the polar coordinates, and R is
interpreted as the AdS curvature. The exact AdS spacetime
is given by h = 1 + R−2r 2 and the induced metric on the
spheres r = constant is g (r, t )d 2d−1 = r 2d 2d−1. Now
assume that h(r ) = 0 ∀ r ≥ 0.9 Therefore, the extremal
surface is a sphere r0(t ) whose area is
a =
r0(t)
gd−1(r, t )d
=
gd−1(r0(t ), t )
d d−1 ≡
gd−1(r0(t ), t ) k,
with k ≡ Vol( d−1). All the computations made above hold
for any spacetime dimension.
If d = 2, then k = 2π and Eq. (61) reduces to
s˙ =
r0(t) 2√g
k
= 4G N 2√g
1
∂g
(r, t )
1
(r0(t ), t )
(r, t )
∂t
∂g
r=r0(t)
(r, t )
∂t
r=r0(t)
,
(68)
where r0 would be the position of the minimal surface. Let
us propose the following ansatz for the solution
(r, t ) = r 2 + α(t ),
α ≥ 0,
d
g
Then, Eq. (63) gives
r0 = 0.
9 Here we are using radial coordinates such that gr−r1 = gtt ≡ h(r ) for
simplicity, but this is not crucial and depends on changes of the radial
coordinates r → r (r ).
(65)
(66)
Sd−1
(67)
(69)
(70)
Σo
So, condition (64) is satisfied since t is given by the
constant embedding → (r ( ) = 0, ) ∈ Nt . Then Eq. (68)
becomes
d k 1
dt f(t ) sinh2(γ t ) + g(t ) = 4G N 2√α(t ) α˙ (t )
k d
= 4G N dt
α(t ),
whose solution is
α(t ) =
4G N s(t ) 2
k
Thus, in principle, the function h(r ) could be determined by
solving the Einstein equations (EE) with this input, and the
method would allow to determine the space metric from the
behavior of the extremal surface.
As it will be clear in the next section, the later times
behavior of the entropy indicates that the system
thermalizes (see Eq. (50)) [
46,57
]. So, the later times regime might
be holographically modeled by a two sided geometry
consisting of slight (time dependent) deformations from the
maximally extended AdS-Schwarschild solution, whose
boundaries (where and live) are causally disconnected by an
event horizon (see reference [40]), at least classically. The
following model is going to be built up more precisely along
these lines of thought.
The example above is a one-sided geometry that can be
interpreted as the dual to the state described by the reduced
density matrix ρ ≡ T r |0(t ) 0(t )| in the field theory.
This can also be expressed as a two-sided geometry dual
to the full state |0(t ) ∈ H ⊗ H . For example, for the
AdS-Schwarzschild solution with h(r ) = 1 − 2M/r d−2 +
R−2r 2, this state is the thermal matrix density ρ(β), while
its TFD representation |0(β) corresponds to the Kruskal
extension of the solution (see the construction of [
40
]). This
type of spacetimes describes a sort of wormhole such that
the minimal area surface is clearly placed at the throat of the
geometry (Fig. 2).
Moreover, if the metric is globally a General Relativity
(GR) solution (for some physical energy-momentum tensor),
the topological censorship theorems [
58
] state that there is
an event horizon separating causally these two conformal
boundaries placed at r ∼ ±∞. Nevertheless, if the boundary
field theories are assumed to interact as in the example above
(Eq. (72)), then the boundaries should be causally connected
(see [
59
]). Classically, this conflict can be bypassed only
if we assume that the solution above, with h(r ) maximally
extended to all the real values of r with a throat of radius
√α at r = 0, does not contain event horizons anywhere;
that is, it can be only a solution of some deformation from
GR dynamics (e.g., Lovelock theory [
60
], Lorentz violating
gravities [
61
], etc.). However, a mechanism that makes the
black hole traversable, based on a teleportation protocol, was
presented in [
28
]. This mechanism is suitable to understand
the holographic picture for the entropy in the asymptotic limit
(γ t 1).
5 Constructing a holographic dual model
Different types of dual geometries corresponding to the
behavior described in the previous section could be built up.
As emphasized in [
52
], it is not clear whether a system of two
interacting CFTs can be realized holographically. In the case
studied here, we have two peculiarities: a time-dependent
entropy and the breaking of Lorentz symmetry by
dissipative processes. In fact, we will present a different scenario
of those studied in [
41,42
] and [52]. Let us shed some light
on the kind of model we intend to present. From Eq. (49),
it can be seen that the early times behavior of the entropy
is similar to the results of [
41,42
], where it was calculated
the entropy for two interacting theories. On the other hand,
the asymptotic behavior of the entropy, defined in Eq. (50),
suggests that there is thermalization. So, in the later times,
there is a formation of a horizon.
5.1 Early times
In early times, the geometry of a spatial slice should be
schematically similar to that shown in Fig. 2, with two locally
asymptotically AdS regions such that the two fields and
live on the respective asymptotic boundaries. Notice that
the time dependent state is |0(γ t ) , which, according to
the gauge/gravity correspondence, is dual to a metric that
depends on time in the same way, that is, gμν = gμν (γ t );
also, for γ = 0, the field theory is conformal. Since for the
time dependent state γ = 0 is equivalent to t = 0, the system
remains in its fundamental state |0 ⊗ |0 , which is dual to
the geometry described precisely by two disconnected AdS
spaces as shown in Fig. 1.10 We can encompass these two
disconnected copies in the same expression for the metric by
the introduction of a “radial” global coordinate , defined in
10 The holographic interpretation of these disconnected geometries
have been discussed in [
30,38
].
the intervals < 0 and > 0, respectively.11 Both
manifolds are analytically completed by adding the limit points
→ 0±. Expressed in terms of these coordinates, the metric
in d + 1 dimensions reads
ds2 = R2
where d2−1 denotes the metric on a compact Euclidean
manifold and we are taking the limit → 0. The two asymptotic
boundaries correspond to = ±1. In the limit → 0, this
is the dual geometry for t = 0 describing two AdS spaces in
global coordinates, and there is no causal curves connecting
both manifolds. We argue that when the interaction is turned
on (or t = 0), there is formation of a throat that entangles
the vacuum dynamically. In this scenario we have a topology
change ( = 0 regime), and the two separate spaces are now
connected. This mechanism is hard to understand using a GR
solution; we need some GR deformation or some quantum
gravity effect that are beyond the scope of this work. We are
not going to draw a holographic picture of the early times
entropy and we will concentrate just in the later times.
5.2 Later times
The later times’ entropy is ruled by Eq. (50). Note that, if γ is
proportional to the temperature, we get the Cardi–Calabrese’s
result, suggesting thermalization. This implies that the
holographic dual of this theory is BTZ black hole and the fields
live, respectively, on the two asymptotically AdS boundaries.
As shown in [
27
] and [
28
], it is possible to make the
wormhole traversable by attaching a coupling between the fields.
Effectively, somehow there is an attractive force between
the boundaries that makes the radius of the event horizon to
decrease, exposing the interior of the black hole. The
process can be viewed as a teleportation protocol that sets up a
quantum coupling of the form eigOA OB , where OA and OB
represent operators of the two boundary theories. Let us show
that we have a similar structure. The (leading first-order in
gamma) interaction term between the boundaries is
γ
Hi1nt = i 2
k
( A−k Bk − A†−k Bk ).
†
In fact, the theory studied here can be seen as a
deformation in two decoupled conformal field theories living on the
two boundaries, namely
e g ∂ M OA OB β = ZC F T (g, β),
11 The coordinate is defined in terms of the usual global radial
coordinate χ as cosh2 χ ≡ 1 −1 2 .
(74)
(75)
and the mechanism of information exchange proposed in
[
27
], and studied in [
28
], works for g > 0. The indice β
in this equation means that the expectation value is taken
on the TFD vacuum. By using the BDHM (Banks, Douglas,
Horowitz, Martinec) recipe [
62,63
], one can quantize a field
near the boundary of the AdS black hole, and OA, OB can
be some linear combination of A†−k , Ak and B†, B−k
respeck
tively. In particular, if we define hermitian operators as
OA =
OB =
A† − A
,
2i
B† − B ,
2i
OA =
OB =
A + A†
2
B + B†
2
,
.
the hamiltonian (74) has the form H = γ2 (OA OB + OA OB ),
according to prescription of [
28
]. Finally, we have
0(β)| exp(−i Hi1nt t ) |0(β)
γ t
= 0(β)| exp 2
OA OB + OA OB
|0(β) ,
and then g ≡ γ2 is positive such as we have thought.
Based on these discussions, we will present in this section
a gravitational system that fits to our dissipative theory. In
particular, it has similar field’s equation of motion and it is
described in a coordinate system covering a region inside the
event horizon.
5.3 The BTZ black hole with a time-dependent boundary
Although the results presented so far do not depend on the
spacetime dimension, we will now show that in two
dimensions it is possible to obtain a clearer geometric interpretation
of the entanglement entropy we have calculated. It is well
known that, at finite temperature, two dimensional CFT is
dual to a BTZ black hole. In general, starting from a bulk
metric with a flat boundary, it is possible to perform a coordinate
transformation that generates a conformally flat boundary.
In order to have a simple way to use the holographic
renormalization and to calculate the stress-energy tensor [
64,65
],
the bulk metric is written in Fefferman–Graham coordinates
[66]. In reference [
36
], this procedure was carried out using
a particular coordinate transformation, in order to produce
boundary metrics with conformal factors that have explicit
time dependence, such that the boundary metric is of the
Friedmann–Robertson–Walker (FRW) type
ds2 = (−dt 2 + a2(t )d 2D).
For Ad S5 black holes, D = 3, it is possible to perform
a time transformation such that this system perfectly fits the
dissipative system presented here. Using the conformal time
1
η = a(t ) , the D-dimensional boundary metric has the
(79)
(81)
(82)
(83)
(84)
(85)
(87)
conformal form ds2 = a(η)2(−dη2 + d 2D). The equation
of motion becomes
(∂η2 − ∇2)
+ (D − 2) a˙ (η) ˙ = 0,
a(η)
where the dot is the derivative relative to η. Now, this equation
of motion is exactly the dissipative equation of motion, where
γ
for C F T4 we have, for a(η) = eHη, H = 2 . For Ad S3 an
approximation must be done. We are going to focus on the
Ad S3 case, where the coordinate transformation is produced
in an analytical form and it is easier to get a minimal area in
order to use the RT formula.
We will discuss in this section the main results of [
36
].
We start with the BTZ metric
ds2 = − f (r )dt 2 + fd(rr2) + r 2dφ2, f (r ) = r 2 − μ, (80)
where the temperature and the entropy are given by
T = 21π √μ, S = 4VG3 √μ.
As mentioned earlier, for our holographic applications it is
easier to define Fefferman–Graham type coordinates.
Defining the variable z
z = μ2 (r − √r − μ),
the metric takes the form
In reference [
36
], it is shown that it is possible to write this
metric as follows
1
ds2 = z2 [d z2 − N 2(τ, z)dτ 2 + A2(τ, z)dφ2],
with
A(τ, z) = a(τ ) 1 +
μ − a˙ 2(τ ) z2
4a(τ )2
(78)
N (τ, z) = 1 − μ − a˙42a+2 2aa¨ z2 = A˙(τa˙, z) , (86)
for some function a(t ) constrained by Einstein equations (see
[
36
] for details). Comparing (84) with (86), we have
Now, taking the limit z → ∞, we have a time-dependent
boundary metric of the FRW type
ds2 = dτ 2 − a(t )dφ2,
where φ is a periodic variable. By coupling a scalar field to
this metric and neglecting the self-interaction of the field, we
have the equation of motion
2 1
∂t − a2 ∇
2
φ + H ∂t φ = 0,
a
where H = a˙ .
Note that, in general, the term H φ˙ is absorbed in the
frequency term by using the conformal time variable and
rescaling the fields. However, if we keep the structure of Eq. (89),
the approach of doubling the degrees of freedom to make
canonical quantization used in [
22
] allows to explore the
subtleties involving the definition of the vacuum of the
dissipative theory. In our case, the auxiliary system is in fact a
physical system, defined as the degrees of freedom behind
the horizon.
Before discussing the kind of approximation we intend
to do in order to use this system to give a geometric
interpretation of the previous one, let us see another important
feature of the coordinate system defined in [
36
]. As in the
static case, the coordinates (τ, z) do not span the full BTZ
geometry. The horizon is defined by re = √μ. If we find the
two roots of the equation r (z, τ ) = re, we discover that the
coordinate system covers the two regions outside the event
horizons, located at
6 Gravity computations
In this part, we explain an approximated holographic method
to analyze the DCFT. It assumes a holographic model for our
dissipative system by a little time interval around (before)
certain specific instant of its evolution t0, belonging to certain
(stable) regime of interest.
Let us use a BTZ black hole in the coordinates discussed
in the previous section, simply referred as a-coordinates. For
this calculation, let us assume that there exists a t0 ∈
such that the holographic model simply stops to be a suitable
description of our system, so the physical time parameter is
taken to be t ≤ t0. We are going to assume that μ in (80)
varies slowly over time, such that the BTZ is in fact a sort of
μ
Vaidya solution in the adiabatic approximation, where ˙
μ ≈ 0
[
6,67
] (see Appendix). The Vaidya solution is the simplest
example of a time dependent gravitational background which
is explored in the context of the RT conjecture [
6,68,69
].
It represents the black hole collapse and has a time
dependent horizon defining a time dependent temperature [
70–73
].
We assume also that for the Ad S3, the holographic model
describes only the system approximately, i.e. in a
neighborhood (immediately before) of t0, so to first order
μ = μ0 + (t − t0)μ1 + o2,
a = a0 + (t − t0)a1 + o2.
(88)
(89)
(90)
(91)
(92)
(93)
(94)
(95)
(96)
(97)
(98)
(99)
In order to describe our dissipative system as a scalar field
propagating on anti-de Sitter boundary, we must demand also
that
a˙ /a ≈ γ , a ≈ 1, ∀t ≈ t0
so the equation of motion (89) agrees with that of our original
system.
In this gravity model, entanglement entropy can be
computed using the Ryu–Takayanagi formula. As the fields are
defined in all space spanned by the board coordinate system,
the minimal area to use in the Ryu–Takayanagi formula is
the area computed with the minimal value of r (τ, z), written
in (92). The entropy is
1
s = 4G3
μ − a˙ 2.
For a t0 γ −1, this shall be demanded to agree with s ∼ γ t
(therefore, the first order expansion of (97)), and using (94)
and (95) we obtain
This is a very important result, because this implies that, for
constant τ , the minimal value of r (τ, z) is
Therefore, the lowest value of r is less than re and the
boundary system “sees” inside the horizon. According to RT
prescription, in this coordinate system we have the following
equation for the entanglement entropy [
36
]:
2a
ze1 = √μ + a˙ ,
2a
ze2 = √μ − a˙ .
rm =
μ − a˙ 2.
1
S = 4G3 rm .
Note that this result is in agreement with the reference [
46
]
where, for non static situations, the minimum area is behind
the event horizon.
μ1 = 8G3γ
1
t0 = 4G3
μ0 − γ 2,
γ −1
μ0 − γ 2,
which implies the physical bound
μ0 ≥ γ 2.
Our description would be inconsistent otherwise. Now, the
limit γ t0 >> 1 corresponds to small G3, implying the
large N limit, since, according to the AdS/CFT dictionary,
G3 ∝ 1/N 2. These results mean that the spacetime
parameters are determined for the parameters of our model up
to first order (in a Taylor expansion on time). Note that
the adiabatic approximation corresponds to the constraint
μ1/μ0 ∼ γ G3 1, again according to large N limit. All
the approximations are consistent with small G3 limit.
6.1 On the holographic energy density
Using the typical AdS/CFT dictionary equation
4G3 (μ0 − γ 2) − (μ0 − γ 2) , (106)
which is negative (decreasing) as (μ0 − γ 2) > 0 for smal
G3. Therefore, let us assume that it can decay up to some
Tμ(Cν F T )
where
,
dρ 1
dt = 16π G3
dρ
ρ˙0 = dt (t = 0)
≈ 2γ
1
16π G3
near the final time t0. Then, using (98),
where ρ is the energy density for the boundary theory. Using
the linear approximation, it follows
1
ρ = 16π G3
μ − a˙ 2
a2
≈ 16π1G3 (μ0 − γ 2 + o(t − t0)),
so that our bound (100) is consistent with the non-negativity
of this quantity. Compatible with dissipative behavior, the
energy of the actual system decreases as
(μ1 − 2γ (μ0 − γ 2)) + o(t − t0),
(105)
(100)
(101)
(102)
(103)
(104)
minimal value, say ρ0,
ρ0 = 16π1G3 (μ0 − γ 2) ≥ 0.
6.2 Relating γ with temperature
Now we are going to show how the coupling γ can be
holographically interpretated as the temperature. In the adiabatic
approximation, as the BTZ black hole radius slowly varies
over time, the black hole stays in equilibrium during the
whole evolution and it is possible to define a time dependent
temperature. For the Vaidya/BTZ black hole, the temperature
is
1 √μ ≈ 2π
1
T = 2π
√μ0 + 2√μμ10 (t − t0) + o2 . (108)
Although it grows with time, its approximated value as t →
t0 is
which exactly coincides with the Cardy–Calabrese result.
The last term denotes the subleading contribution, which
is proportional to ργ0 . This calculation can be viewed as a
holographic check on the consistency that the entanglement
in DCFT behaves as the entanglement of an ordinary CFT at
finite temperature [
57
].
Our approach discussed in the last section can be described
as an approximation on the entangled state.
We know that the BTZ spacetime corresponds to the TFD
thermal state 0β in the boundary theory [
40
]. Thus, the state
corresponding to the gravitational system should be
represented as
0a(t) = Ga(t) 0β = Ga(t)Gβ |0 ,
(111)
where Ga(t) represents a (unknown) unitary transformation
associated to certain specific conformal transformation in
CFT, such that in the limit a(t ) → 1, Ga(t) → 1. Gβ is
the usual thermal Bogoliubov transformation. Actually, we
(107)
(109)
(110)
propose that our DCFT system, described by Eq. (19), is
approximately given by the state (111),
| (t ) ≈ 0a(t) = Ga(t) 0β = Ga(t)Gβ |0 ,
(112)
as t approaches t0. This (approximated) equation allows us
to interpret correctly our previous calculations, and to
understand correctly the ranks of validity. Notice that the state on
the right-hand side 0a(t) is the conformal ground state dual
to the BTZ, in a(t )-coordinates.
As said before, Ga(t) represents a (unknown) unitary
transformation associated to a specific conformal transformation
in CFT, such that in the limit a(t ) → 1, Ga(t) → 1; Gβ is
the usual thermal Bogoliubov transformation. Therefore,
| (t ) = U (t − t0) | (t0) .
Then, using (24), the time evolution can be expressed as
generated by the entropy operator as
1
| (t ) = exp 2 [S(t0) − S(t )] | (t0) .
This equation can be checked in the geometric dual through
the RT formula, i.e,
1
Tr[ρ(t ) − ρ(t0)]S(t0) = 4G [amin(t ) − amin(t0)],
(115)
where ρ is the reduced density operator computed by tracing
out the B’s degrees of freedom. The idea is that the
lefthand side is computed explicitly for the state , while the
right-hand side is computed for 0a(t) , whose entanglement
entropy can be computed directly from the formula (97).
The remark here is that, for parameters given explicitly by
Eqs. (98) and (99), both sides match perfectly. This supports
that the approximation (112) is, in a sense, consistent with the
time evolution of the states. However, the dual gravitational
description must break down for very large time. From the
point of view of the gravitational solution, if one considers a
BTZ solution with very high temperature T √μ0, then
μ0
γ 2,
and the formula for the entanglement entropy gives s =
2π √μ0, which coincides with the thermodynamic entropy,
4G3
and therefore s ∝ T . So, the state corresponding to this
solution is nothing but the TFD thermal vacuum [
40
]:
k
|0(β) = Z −1(β)
exp[e−Ek β/2 A†−k Bk†] |0 ,
while the state of our DCFT system (19) is manifestly
different, showing that for very large black hole temperature, our
original approximation (112) breaks down.
(113)
(114)
(116)
(117)
Nevertheless, the form of the states (19) and (117), and
the configuration of the systems are suggestively similar in
many aspects. So a natural question that arises here is if there
exists some way of slightly modifying the action/states of the
double CFT proposed, in order to capture more of the dual
gravitational description. The simplest possibility is to
introduce a parameter, interpreted as a temperature from the
Asubsystem, such that: (a) in the limit β → 0, the state agrees
with 117, and (b) for low (temperature) scales β−1 → ∼ γ ,
one would recover the (main) dissipative/damping
behavior (19). In the limit γ t → 0, the (sub)systems A and B
decouple, and they can be seen as the standard TFD
duplication in the state (117). This will be studied in a forthcoming
work.
7 Conclusion
In this work, we have studied a close relation between
dissipation and entanglement in the context of AdS/CFT
correspondence. This kind of relation was experimentally verified
in [
20
], where it was shown that dissipation generates
continuously entanglement between two macroscopic objects. In
order to study quantum dissipation in conformal field
theories, we have followed the strategy of [
22–24
], which consists
of doubling the degrees of freedom of the original system by
defining an auxiliary one. In particular, using the
canonical formalism presented in [
22
], we have defined an entropy
operator, responsible of controlling the dissipative dynamics.
The entropy operator, which naturally appears in the scenario
studied here, turns up to be in fact the modular Hamiltonian.
A geometrical and physical interpretation of the auxiliary
system and entropy operator is given in the context of the
AdS/CFT correspondence using the RT formula. One has
two asymptotically AdS regions such that the two theories
live on the respective asymptotic boundaries. The dissipation
at one boundary is interpreted as an exchange of energy with
the other boundary, which is controlled by a kind of constant
coupling. The scenario is similar to the one presented in [
30
],
where a teleportation protocol is explored in order to make
the wormholes traversable, introducing interaction between
the two asymptotic boundaries.
We have showed that the vacuum state evolves in time
as an entangled state and the entanglement entropy was
calculated. The entropy depends on time and has two distinct
behaviors: in the early times, it is the typical entanglement
entropy of two interacting theories; in the later times, the
entropy’s behavior suggests thermalization. In order to give
a geometric interpretation, we have looked for a gravitational
system that reproduces a similar equation of motion in the
boundary. Since this kind of dissipative behavior was
studied long time ago in the context of scalar fields coupled to
Friedmann–Robertson–Walker (FRW) metric with a
dampΦ
Σ
t
(Ψ)
Fig. 3 The figure represents schematically the geometric dual used
to (approximately) describe the system. The upper part represents the
black hole state (see [
40
]) described by (a half of) the Euclidean BTZ
solution in the Hartle–Hawking wave functional. The horizontal line
corresponds to t0. Our system is approximately described by the solution
on an evolving surface (t ≤ t0)
ing term coming from the Hubble constant [
31–35
], then the
BTZ black hole in the coordinate system developed in [
36
]
is the natural system to study here, as it has a FRW metric in
the boundary. However, in two dimensions a linear
approximation must be done. In addiction, owing to the linear time
dependence of the entropy, we have showed that the dual
geometry must be not the BTZ black hole in the coordinates
defined in [
36
], but an adiabatic approximation for the Vaidya
black hole defined in the same coordinate system.
We have used the RT formula to relate the BTZ/Vaidya
geometry’s throat area to the entropy. As the time evolution
of the entangled state is controlled by the entropy operator,
the RT conjecture allows us to relate the time evolution of the
throat with the time evolution of the vacuum state. It should
be emphasized that this kind of relation involving state/area
is possible because, in the canonical formalism used here,
there is a well defined entanglement entropy operator. A
possible close relation of this operator with a kind of area
operator, defined in the context of quantum gravity, deserves
further development (for some suggestions in this sense, see
for instance [
50
]).
It was shown that the approximation is consistent with the
time evolution of the states. However, the dual gravitational
description must break down for very large time, when the
entanglement entropy calculated here is in fact a
thermodynamical entropy of BTZ solution with very high temperature
T √μ0. In this limit, the asymptotically entangled state
(19) must be the TFD thermal vacuum (see Fig. 3). This
structure suggests strongly that one can adopt reciprocal point of
view and take seriously the dual gravitational description to
reconstruct the dissipative conformal field theory (DCFT); in
other words, to find some slightly deformation of the DCFT
theory in according to the dual gravitational description.
In addition to the previously discussed deformation of the
DCFT, there are some applications of this work that we intend
to do. Another important sequence to this work is to use the
approach developed here in the context of Janus solutions of
supergravity (studied in [
74
]), where two CFTs defined on
1 + 1 dimensional half spaces are glued together over a 0 + 1
dimensional interface. If we can generalize the Janus
deformation in order to include the dissipative deformation studied
here, it will be possible to have a holographic interpretation of
the dissipative system in terms of the time dependent Janus’
black hole. Also, it will be important to compute the retarded
Green’s function in order to compare with the results of
reference [
75
]. Finally, in the spirit of reference [
28
], it will
be very interesting to study the specific teleportation
protocol that reproduces the scenario presented here, as well as
to apply the technology developed here in the Ad S2 case, in
order to find a possible connection between the DCFT and
the Sachdev–Ye–Kitaev quantum mechanical model.
Acknowledgements M. Botta Cantcheff would like to thank
CONICET for financial support. The authors would like to thank Pedro
Martinez for the help with the figures.
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.
Appendix A: Maximally extended geometries and minimal surfaces
Let us verify that this type of spacetimes can be described
as two-sided geometries through a maximal (Kruskal-like)
extension; then, the resulting geometry is a sort of wormhole
such that the minimal area surface is clearly seen as the throat
of the geometry; see Fig. 4. This wormhole solution is the
analogous to the Einstein–Rosen bridge in four dimensions
and it is not traversable classically.
So, let us consider for instance the BTZ solution
1
ds2 = −h(r )dt 2 + h(r ) dr 2 + r 2dφ2,
where h(r ) = r 2 − μ. Doing the following coordinate’s
change
ρ2 ≡ r 2 − μ,
for r 2 ≥ μ; thus ρdρ = r dr and (A1) becomes
(A1)
(A2)
H
Fig. 4 The red line represents the spatial slice shown in Fig. 2. The
point H (in blue) represents the event horizon
ds2 = −ρ2dt 2 + ρ2 + μ
1
dρ2 + (ρ2 + μ)dφ2.
(A3)
So, the solution (A1), valid for 0 ≤ ρ ≤ ∞, can be extended
here to all the real line −∞ ≤ ρ ≤ ∞, which describes two
causally disconnected asymptotically AdS spacelike regions
joined by the surface (throat) ρ = 0, as shown schematically
in Fig. 1. In addition, notice that, in these coordinates, there
is a horizon precisely in ρ = 0.
By symmetry, the minimal surface is a sphere of
codimension 2, whose radius is given by (ρ2 + μ), as it can be seen
clearly from expression (A3). Therefore, the result is that the
minimal sphere is at the horizon ρ = 0 with radius √μ and
area a = 2π √μ.
The time dependent case
A time dependence of the spacetime, with similar causal
structure, can be introduced by doing μ ≡ μ(t ) in this
solution, which can be obtained by a coordinate’s change from
the Vaidya’s solution, as long as the adiabatic approximation
is valid, i.e., whether μ˙ (t )/μ(t ) is negligible. In particular,
the Vaidya’s solutions considered in the present paper can be
included in this analysis and one obtains the time dependent
wormhole metric
ds2 = −ρ2dt 2 + ρ2 + μ(t )
1
dρ2 + (ρ2 + μ(t ))dφ2, (A4)
where, again, the extremal surface corresponds manifestly to
ρ = 0, with radius √μ(t ). Therefore, by taking α ≡ μ, this
metric fits into the examples studied in Sect. 4.
Now the extremal curve shall enclose the horizon such
that the surface of extremal area becomes the event horizon,
whose area is aeh = 2π √μ. Using the RT formula, we finally
have the relation
π
S(t ) = 2G
(A5)
and once again one gets consistency of the RT formula with
the Bekenstein–Hawking law.
Therefore, as shown above, this solution (for slowly
varying μ(t )) can be extended to a similar form of the Kruskal
one, and the Penrose’s diagram corresponds to the maximally
extended AdS black hole (Fig. 4).
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