YITP-23-148

Entanglement Phase Transition in Holographic Pseudo Entropy

Hiroki Kandaa, Taishi Kawamotoa, Yu-ki Suzukia, Tadashi Takayanagia,b,

Kenya Tasukia and Zixia Weic,d

aCenter for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
bInamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411, Japan
c Center for the Fundamental Laws of Nature,
Harvard University, Cambridge, MA 02138, USA
d Society of Fellows, Harvard University, Cambridge, MA 02138, USA

In this paper, we present holographic descriptions of entanglement phase
transition using AdS/BCFT. First, we analytically calculate the holographic
pseudo entropy in the AdS/BCFT model with a brane localized scalar field and
show the entanglement phase transition behavior where the time evolution of
entropy changes from the linear growth to the trivial one via a critical
logarithmic evolution. In this model, the imaginary valued scalar field
localized on the brane controls the phase transition, which is analogous to
the amount of projections in the measurement induced phase transition. Next,
we study the AdS/BCFT model with a brane localized gauge field, where the
phase transition looks different in that there is no logarithmically evolving
critical point. Finally, we discuss a bulk analog of the above model by
considering a double Wick rotation of the Janus solution. We compute the
holographic pseudo entropy in this model and show that the entropy grows
logarithmically.

###### Contents

  1. 1 Introduction
  2. 2 Entanglement phase transition from AdS/BCFT
    1. 2.1 End-of-the-world brane with a localized scalar in AdS/BCFT
    2. 2.2 Complete analysis of holographic pseudo entropy in BTZ phase: $\Delta\phi<\Delta\phi_{*}$
    3. 2.3 Other profiles of EOW brane for $V(\phi)\neq 0$
    4. 2.4 Complete analysis of holographic pseudo entropy at critical point: $\Delta\phi=\Delta\phi_{*}$
    5. 2.5 Higher dimension
    6. 2.6 Null energy condition and the brane profile
    7. 2.7 Critical behavior from a simple dissipating model
    8. 2.8 Comments on the analysis from the CFT side
  3. 3 Gauge fields on the brane
    1. 3.1 EOW brane with gauge field in Poincaré AdS
      1. 3.1.1 AdS3
      2. 3.1.2 AdS4
      3. 3.1.3 AdSd+1
    2. 3.2 Localized gauge field on EOW brane in BTZ phase
      1. 3.2.1 Construction of Solution
      2. 3.2.2 Free energy in BTZ phase
    3. 3.3 Localized gauge field on EOW brane in TAdS phase
      1. 3.3.1 Construction of solution
      2. 3.3.2 Free energy in TAdS phase
    4. 3.4 Phase transitions between BTZ and TAdS phase
  4. 4 Soft wall model with bulk scalar
    1. 4.1 Pseudo entropy from Euclidean Janus solution
      1. 4.1.1 Late time behavior of the pseudo entropy
      2. 4.1.2 Early time behavior
      3. 4.1.3 Entanglement entropy on the defect; At $\tau_{0}=t_{0}=0$
    2. 4.2 Pseudo entropy from time-like Janus solution
      1. 4.2.1 The setup
      2. 4.2.2 Time evolution of pseudo entropy
  5. 5 Conclusions and discussions
  6. A Formula for the Hyperbolic Coordinate and The Elliptic Integrals
    1. A.1 Hyperbolic coordinate and geodesic in the vacuum AdS3
    2. A.2 Elliptic integrals
  7. B Another point of view for the spacelike brane
  8. C CFT analysis of a defect marginal perturbation
    1. C.1 2nd pseudo Rényi entropy
    2. C.2 3nd pseudo Rényi entropy
    3. C.3 Rényi entropy for general $n$

##  1 Introduction

Entanglement entropy is a very useful probe of quantum phases in many-body
systems and field theories[1, 2, 3, 4]. For example, it plays a role of order
parameter in quantum phase transitions [5, 6] and it can detect topological
phases [7, 8]. At the critical points, it measures the degrees of freedom of
conformal field theories (CFTs), namely central charges [9, 10]. It also
offers a nice entropic quantity of thermalization for non-equilibrium quantum
processes, such as quantum quenches [11, 12].

As a more recent development, entanglement entropy has been employed to probe
a novel dynamical quantum phase transition, so called measurement induced
phase transition or entanglement phase transitions [13, 14]. This phase
transition occurs in monitored quantum circuits, where a quantum many-body
system is time evolved by a chaotic or random Hamiltonian and subsequently by
projection measurements. Under the unitary time evolution of the Hamiltonian,
the entanglement entropy grows linearly. However, in the presence of
projection measurements, the entanglement growth is suppressed. Due to the
competition between these two effects, an intriguing phase transition occurs.
In (1+1)-dimensional quantum spin systems, the time evolution of entanglement
entropy changes as

$\displaystyle(i)\ \ p<p_{*}:\ \ \ S_{A}\propto t,$ $\displaystyle(ii)\ \
p=p_{*}:\ \ \ S_{A}\propto\log t,$ (1.1) $\displaystyle(iii)\ \ p>p_{*}:\ \ \
S_{A}=\mbox{finite},$

where $p$ is the rate of projection measurement and $p_{*}$ denotes the phase
transition point. A similar phase transition was found for quantum systems
with dissipating dynamics in [15].

It is intriguing to see if such entanglement phase transitions exist in
broader quantum systems such as interacting quantum field theories, including
those in higher dimensions. For this purpose a powerful method is the
holographic calculation of entanglement entropy [16, 17, 18] based on the
AdS/CFT [19]. Even though there have been interesting progresses on
holographic descriptions of measurement induced phase transition [20, 21, 22,
23, 24], they are either discussed in (0+1)-dimensional models [20, 21] or by
realizing a limited number of projection measurements [20, 21, 22, 23, 24],
and hence the entanglement phase transition with the behavior (1.1) has not
been realized so far.

Recently, a different approach to the entanglement phase transition in a two
dimensional holographic CFT was suggested in the paper [25]. Instead of
considering a holographic description of projection measurements, a certain
dissipative effect is taken into account as an imaginary valued scalar field
on the end-of-the-world brane (EOW brane) in the AdS/BCFT formulation [26,
27]. This can be regarded as a modification of the gravity dual of quantum
quenches [28]. The resulting gravity dual describes a time-dependent
transition matrix [29] in CFT viewpoints, instead of a regular quantum state.
Therefore, strictly speaking, the geodesic length computes the holographic
pseudo entropy [29]. The pseudo entropy is a natural generalization of
entanglement entropy so that it depends not only on the initial state (a given
quantum state) but also on the final state (a final state projection), which
has been successfully applied to detect quantum phase transitions [30, 31, 32]
in quantum many-body systems. Refer also to e.g. [33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59] for further developments. In [25], it was conjectured that the time
evolution of pseudo entropy shows the same behavior as the entanglement
transition (1.1), where the phases for $p<p_{*}$, $p=p_{*}$ and $p>p_{*}$
correspond to the three different geometries: AdS black hole, Poincaré AdS and
the thermal AdS, respectively.

The purpose of this paper is to explore this connection more quantitatively
and generalize to other setups. First we will fully work out the behavior of
holographic pseudo entropy and show the following quantitative behavior in a
holographic model constructed by introducing a massless free scalar on the
end-of-the-world brane in AdS3/BCFT2:

$\displaystyle(i)\ \ \Delta\phi<\Delta\phi_{*}:\ \ \ \Delta
S_{A}\simeq\frac{c}{6a}t,$ $\displaystyle(ii)\ \ \Delta\phi=\Delta\phi_{*}:\ \
\ \Delta S_{A}\simeq\frac{c}{3}\log t,$ (1.2) $\displaystyle(iii)\ \
\Delta\phi>\Delta\phi_{*}:\ \ \ \Delta S_{A}=\mbox{finite},$

where $\Delta\phi$ is the amount of scalar field perturbation which controls
the dissipative effect in Lorentzian time evolution and $\Delta S_{A}$ is the
pseudo entropy with the ground state entanglement entropy subtracted. The
constant $a$ increases as $\Delta\phi$ gets larger and becomes divergent at
$\Delta\phi\rightarrow\Delta\phi_{*}$, and $c$ is the central charge of the
corresponding BCFT2.

One advantage of our approach is that it is straightforward to be extended to
higher dimensional cases and we will show that the intermediate logarithmic
behavior is not present in higher dimensions. Moreover, We will discuss a
generalization of our setup with a gauge field on the EOW brane. We will also
explore a purely bulk model which mimics a general measurement by employing
the Janus solution.

A brief summary of the contents in this paper is as follows:

In section 2, we explore the model proposed in [25], where an EOW brane with a
brane-localized massless free scalar field was considered in the AdS3/BCFT2.
Especially we work out the global structures of its Lorentzian time evolution
obtained by a Wick rotation. We compute explicitly the growth of pseudo
entropy including its higher dimensional generalization and determine the
coefficient of both its linear and logarithmic growth as presented in (1.2).
We will also analyze some brane profiles when the brane-localized scalar field
has a nontrivial potential. Besides, We find a higher dimensional result for
the growth at the critical point. In this calculation the global extension
beyond the Poincaré horizon turns out to be important. We also present a
simple model of dissipating time evolution which shows the logarithmic growth
and which is universal for any two dimensional CFTs.

In section 3, we analyze another setup of AdS/BCFT with a gauge field on the
EOW brane. We give solutions of EOW branes with a localized gauge field in
Poincaré AdSd+1 setup. Interestingly, for $d>2$ we find that the solution with
the localized gauge flux in Poincaré AdSd+1 look identical to that with the
localized scalar in Poincaré AdSd. For $d=2$, we find that the EOW brane
solution with the gauge flux is not available in Poincaré AdS3. Instead, for
the three dimensional gravity dual, the EOW brane solution with the flux can
be obtained for BTZ and thermal AdS geometry. We examine the phase transition
between the two phases. Note that this section is written in an independent
manner of the above sections to be self-contained.

In section 4, which is different from the above two sections involving the EOW
brane, we consider the model with a bulk scalar field in an asymptotically AdS
spacetime. One motivation for this section is to provide the soft wall model
for the EOW brane model, similar to the argument in the gluon condensation[60,
61]. To this end, we introduce the spacelike brane in the CFT, by taking the
double Wick rotation of Janus solution, described by the deformation caused by
the line defect whose tangential direction is in the spacelike direction. We
discuss the dual Janus solution for both the Euclidean and Lorentzian time
cases. Especially in the Lorentzian model, the dual spacetime exhibits highly
exotic behavior. We compute the pseudo entropy using the HRT formula for these
spacetimes. We find that the analytic formula for the entanglement pseudo
entropy, defined with the subsystem parallel to the defect, shows
logarithmically decreasing behavior around the defect.

In section 5, we summarize our conclusions and discuss future problems.

Some math formulas used in section 4 will be summarized in appendix A. An
alternative interpretation of the setup analyzed in section 4 can be found in
appendix B. Besides, while all the analyses appearing in the main text are
performed holographically on the gravity side, we present an analogous CFT
analysis in C where the pseudo Rényi entropy is computed perturbatively.

##  2 Entanglement phase transition from AdS/BCFT

As proposed in [25], an asymptotically AdS3 background with an end-of-the-
world brane (EOW brane) where a massless free scalar field is localized,
provides an interesting holographic setup of entanglement phase transition. In
this section, we would like to analyze this setup by focusing on the behavior
of its holographic pseudo entropy and would like to quantitatively confirm
that it shows the expected behavior of entanglement phase transition (1.1)
with explicit coefficients as given in (1.2). We would like to mention that a
brane-localized scalar field was also considered in earlier works on the
holographic Kondo effect [62, 63, 64, 65]. A scalar field in AdS/BCFT was also
analyzed in a probe limit by the paper [66]. There is another interesting
approach to AdS/BCFT with multiple boundary conditions and brane localized
matter, by introducing a corner defect on the EOW brane [67, 68, 69].

###  2.1 End-of-the-world brane with a localized scalar in AdS/BCFT

Before we present our calculations of holographic entropy, we would like to
start with a brief review of the model introduced in [25]. The AdS/BCFT is an
extension of AdS/CFT to the cases where the CFT is defined on a manifold
$\Sigma$ with boundaries [26, 27, 70]. It argues that the CFT on $\Sigma$ is
dual to an asymptotically AdS region which is surrounded by the $\Sigma$ at
the asymptotic boundary and by an EOW brane $Q$. The EOW brane is a dynamical
brane beyond which no spacetime exists, and it satisfies a Neumann boundary
condition in the metric fluctuations. Refer to figure 1 for a sketch.

In the model introduced in [25], one ingredient was added to the minimal model
of AdS/BCFT explained above in three dimensions. It is a scalar field $\phi$
localized on the EOW brane. From the viewpoint of boundary conformal field
theory (BCFT), this scalar field corresponds to an external field which
couples to a boundary scalar operator. The Neumann boundary condition on $Q$
in the presence of the scalar field (with the potential energy $V(\phi)$)
reads

$K_{ab}-h_{ab}K=T_{ab}=-h_{ab}\left(\partial_{c}\phi\partial^{c}\phi+V(\phi)\right)+2\partial_{a}\phi\partial_{b}\phi,$
(2.1)

where the indices represent the coordinates on the EOW brane. Below, unless
otherwise noted, we will set $V(\phi)=0$, i.e. we will consider a massless
free scalar field.

Figure 1: A sketch of the AdS/BCFT model. The left figure shows a BCFT defined
on a strip (light grey region) with the width $\frac{\beta}{2}$. The right
figure shows the dual gravity setup which is given by the AdS3 region
surrounded by the strip and the EOW brane (blue surface). The red curve
$\Gamma_{A}$ is the geodesic whose length calculates the holographic pseudo
entanglement entropy for the subregion $A$ shaded in green.

In particular, we consider a BCFT defined on a two dimensional strip whose
coordinate is described by $(\tau,x)$ in the Euclidean signature. The width of
this strip is $-\frac{\beta}{4}\leq\tau\leq\frac{\beta}{4}$, while the
coordinate $x$ is non-compact. In the absence of the localized scalar field,
this is identical to the setup in [26, 27] and [28] and the bulk geometry is
found to be the BTZ black hole with the temperature $1/\beta$. We write the
metric of BTZ black hole as follows:

$ds^{2}=\frac{dz^{2}}{h(z)z^{2}}-\frac{h(z)dt^{2}}{z^{2}}+\frac{dx^{2}}{z^{2}},\quad
h(z)=1-\frac{z^{2}}{a^{2}},$ (2.2)

where $T_{BTZ}=\frac{1}{2\pi a}$ is the temperature of the BTZ black hole. We
have $\beta=2\pi a$ when there is no brane-localized scalar field. The EOW
brane connects the two boundaries of the strip in the bulk as in figure 1.

We then introduce the localized scalar field on $Q$ such that it takes the
values $-\frac{\Delta\phi}{2}$ and $\frac{\Delta\phi}{2}$ on each of the two
boundaries of the strip. We further analytically continue at $\tau=0$ to the
Lorentzian time as $\tau=it$. This realizes the following transition matrix in
a CFT:

$\mathcal{T}=e^{-iHt}e^{-\beta
H/4}\ket{B(\Delta\phi/2)}\bra{B(-\Delta\phi/2)}e^{-\beta H/4}e^{iHt},$ (2.3)

where $|B(\cdots)\rangle$ are boundary states (or Cardy states) [71] labeled
by different boundary values of $\phi$. The gravity dual of this transition
matrix can be found by solving (2.1) together with the bulk Einstein equation.
The non-trivial profile of the scalar field gives the backreaction, which
makes the inverse temperature $2\pi a$ deviate from $\beta$, where $\beta$ is
also twice the width of strip. As the scalar field shift $\Delta\phi$
increases, with $\beta$ fixed, $a$ grows and gets divergent at the critical
point $\Delta\phi=\Delta\phi_{*}$. Its explicit value is given by
$\Delta\phi_{*}=2K[-1]\simeq 2.6$. The fact that $a=\infty$ means that the
dual geometry now looks like the Poincaré AdS3. For larger values
$\Delta\phi>\Delta\phi_{*}$, the EOW brane is no longer connected and the dual
geometry becomes the thermal AdS3 where the EOW brane consists of two
disconnected surfaces, which disappear in the Lorentzian counterpart obtained
by performing analytic continuation. These three phases are sketched in figure
2.

Figure 2: A sketch of gravity duals in our AdS/BCFT model with a brane
localized scalar. Depending on the value of the shift of scalar field
$\Delta\phi$, there are three phases (a) BTZ phase, (b) Poincaré phase and (c)
TAdS phase. The upper three pictures describe the Euclidean geometries, while
the lower three are the corresponding Lorentzian time evolution obtained from
analytic continuation.

The quantity we focus on below is the pseudo entropy defined by

$S_{A}=-\mbox{Tr}[{\mathcal{T}}_{A}\log{\mathcal{T}}_{A}],$ (2.4)

where ${\mathcal{T}}_{A}$ is the reduced transition matrix obtained by tracing
out the complement $\bar{A}$ of the subsystem $A$ as

${\mathcal{T}}_{A}=\mbox{Tr}_{\bar{A}}[{\mathcal{T}}].$ (2.5)

Even though ${\mathcal{T}}_{A}$ is not Hermitian in general, it turns out that
$S_{A}$ is real and non-negative for the examples we study in this paper.
Since it is obvious that in the final phase $\Delta\phi>\Delta\phi_{*}$, the
time evolution of the bulk geometry and hence the time evolution of $S_{A}$ is
trivial, we will focus on the BTZ phase $\Delta\phi<\Delta\phi_{*}$ and
critical phase $\Delta\phi=\Delta\phi_{*}$ below. As a result, we will get the
behavior shown in (1.2)

Below we will calculate the holographic counterpart of $S_{A}$ from the bulk
geometry. In the AdS/CFT correspondence, the pseudo entropy $S_{A}$ for a
subsystem $A$ can be computed from the area of the extremal surface
$\Gamma_{A}$ which surrounds $A$ and takes the minimal value as [16, 17, 18]

$S_{A}=\frac{\mbox{Area}(\Gamma_{A})}{4G_{N}}.$ (2.6)

In the AdS/BCFT, the holographic pseudo entropy can be computed similarly
except that $\Gamma_{A}$ can also end on the EOW brane $Q$ [26, 27]. Since our
analysis in this paper always assumes that the subsystem $A$ covers a half of
the total space, $\Gamma_{A}$ always ends on $Q$. In this case, we need to
extremalize by changing the end points of $\Gamma_{A}$ on $Q$.

Before we proceed, we would like to point out an interesting possibility which
may give another interpretation of our results. In our example, the pseudo
entropy is always real and non-negative, though the pseudo entropy in general
can take complex values, as we have already mentioned. Moreover, it is also
natural that the Rényi extensions of pseudo entropy (pseudo Rényi entropy)
also take real and non-negative values in our setup as the replicated
geometries are expected to be described by certain real valued metrics, though
it would be difficult to construct them explicitly. If all pseudo Rényi
entropies are real and non-negative, then there is a chance that the
eigenvalues of the reduced transition matrix are also real and non-negative.
Such a matrix is called pseudo Hermitian and can be diagonalized by a
similarity transformation $\mathcal{T}_{A}=P\cdot\Lambda\cdot P^{-1}$, where
$\Lambda$ is a diagonal matrix with real and non-negative eigenvalues. This
implies, in this case, that the calculation of pseudo entropy for the
transition matrix can be regarded as that of entanglement entropy for certain
quantum states. It is an interesting future problem to examine this clearly
and relate our results to entanglement entropy of definite quantum states in
some quantum systems, though we will not pursue this in the present paper.

###  2.2 Complete analysis of holographic pseudo entropy in BTZ phase:
$\Delta\phi<\Delta\phi_{*}$

Let us consider the BTZ black hole phase and compute $S_{A}$. Since the
geodesic extends beyond the horizon in the coordinate patch (2.2), we need to
work with the Kruskal coordinates

$ds^{2}=\frac{-4dudv+(1-uv)^{2}dx^{2}/a^{2}}{(1+uv)^{2}},$ (2.7)

where $u$ and $v$ are related to $z$ and $t$ outside of the horizon as

$u=-\sqrt{\frac{a-z}{a+z}}e^{-t/a},\quad v=\sqrt{\frac{a-z}{a+z}}e^{t/a}.$
(2.8)

In this coordinate, the horizon locates at $uv=0$. The center and the
asymptotic boundary of the spacetime correspond to $uv=1$ and $uv=-1$
respectively. Note that since we do not compactify $x$, the points on $uv=1$
are not singular.

Now, the trajectory of the EOW brane and the scalar field on the EOW brane can
be described as $v=v(u)$ and $\phi=\phi(u)$, assuming the translational
invariance in $x$ direction. After some algebra, we obtain

$\displaystyle V(\phi)$
$\displaystyle=-\frac{(1-u^{2}v^{2})v^{\prime\prime}+2(3-uv)(v-uv^{\prime})v^{\prime}}{8(1-uv)v^{\prime
3/2}},$ (2.9) $\displaystyle\phi^{\prime 2}$
$\displaystyle=\frac{(1-uv)v^{\prime\prime}-2(v-uv^{\prime})v^{\prime}}{2\left(1-u^{2}v^{2}\right)\sqrt{v^{\prime}}},$
(2.10)

where primes denote the derivative with respect to $u$. As we have already
mentioned, we consider the case where $V(\phi)=0$. This implies

$v^{\prime\prime}=-\frac{2v^{\prime}(3-uv)(v-uv^{\prime})}{1-(uv)^{2}}.$
(2.11)

We impose

$v(u_{0})=-u_{0},\quad v^{\prime}(u_{0})=1,\quad
u_{0}:=-\sqrt{\frac{1-\zeta_{0}}{1+\zeta_{0}}},\quad\zeta_{0}:=\frac{z_{0}}{a}<1,$
(2.12)

as the initial conditions. Besides, $\zeta_{0}$ and $a$ are fixed by
$\Delta\phi$ and $\beta$ via

$\Delta\phi=2\left(1-\zeta_{0}^{2}\right)^{1/4}K\left(-\left(1-\zeta_{0}^{2}\right)\right),$
(2.13)
$\frac{\beta}{a}=\frac{4}{\zeta_{0}\sqrt{\left(1-\zeta_{0}^{2}\right)\left(2-\zeta_{0}^{2}\right)}}\left(\Pi\left(-\frac{\zeta_{0}^{2}}{1-\zeta_{0}^{2}}\middle|\frac{1-\zeta_{0}^{2}}{2-\zeta_{0}^{2}}\right)-\left(1-\zeta_{0}^{2}\right)K\left(\frac{1-\zeta_{0}^{2}}{2-\zeta_{0}^{2}}\right)\right).$
(2.14)

The conditions (2.12) are equivalent to $z(t=0)=z_{0}$ and $\dot{z}(t=0)=0$ in
the Poincaré coordinate. Using the Poincaré coordinate $z$ as the parameter of
the curve, we can solve this equation as follows:

$u=-\frac{1-z/a}{A(z/a)}e^{-B(z/a)},\quad v=\frac{A(z/a)}{1+z/a}e^{B(z/a)},$
(2.15)
$A(\zeta):=\frac{\sqrt{\left(1-\zeta_{0}^{2}\right)\left(\zeta_{0}^{2}+\zeta^{2}-\zeta_{0}^{2}\zeta^{2}\right)}+\zeta\sqrt{\zeta^{2}-\zeta_{0}^{2}}}{\sqrt{\zeta^{2}+\zeta_{0}^{2}-\zeta_{0}^{4}}},$
(2.16)
$B(\zeta):=\frac{\zeta_{0}}{\sqrt{\left(1-\zeta_{0}^{2}\right)\left(2-\zeta_{0}^{2}\right)}}\left[F\left(\cos^{-1}\frac{\zeta_{0}}{\zeta}\middle|\frac{1}{2-\zeta_{0}^{2}}\right)-\Pi\left(\frac{1-\zeta_{0}^{2}}{2-\zeta_{0}^{2}};\cos^{-1}\frac{\zeta_{0}}{\zeta}\middle|\frac{1}{2-\zeta_{0}^{2}}\right)\right],$
(2.17)

where $F(\phi|m)$ and $\Pi(n;\phi|m)$ are the incomplete elliptic integrals of
the first kind and of the third kind, respectively. Here, $z$ runs from
$z_{0}$ to $\infty$ as a parameter.

Consider the extended region by embedding BTZ metric into the global AdS:

$ds^{2}=\frac{1}{\cos^{2}\rho}\left(d\rho^{2}-d\tau^{2}+\sin^{2}\rho
d\theta^{2}\right),$ (2.18)

where the AdS radius is normalized to unity. Explicitly, embedding is
accomplished by

$\displaystyle u$
$\displaystyle=\frac{-\sin\rho\cos\theta+\sin\tau}{\cos\rho+\sqrt{\cos^{2}\tau-\sin^{2}\rho\sin^{2}\theta}}$
$\displaystyle v$
$\displaystyle=\frac{\sin\rho\cos\theta+\sin\tau}{\cos\rho+\sqrt{\cos^{2}\tau-\sin^{2}\rho\sin^{2}\theta}}$
$\displaystyle x$
$\displaystyle=a\sinh^{-1}\left(\frac{\sin\rho\sin\theta}{\sqrt{\cos^{2}\tau-\sin^{2}\rho\sin^{2}\theta}}\right).$
(2.19)

The original BTZ region is a subregion of the global AdS, which is

$\left\\{(\rho,\tau,\theta)|\cos^{2}\tau>\sin^{2}\rho\sin^{2}\theta,\tau\in[-\pi/2,\pi/2],\rho\in[0,\pi/2),\theta\in[0,2\pi)\right\\}.$
(2.20)

Now, we introduce the same coordinates for the extended region adjacent to the
original BTZ black hole by shifting the range of $\tau$ to $[\pi/2,3\pi/2]$.
For clarity, we use a tilde to distinguish the original coordinates and the
extended ones. Although (2.11) is the equation for the original region, it is
important to emphasize that it still governs the EOW brane even outside of the
original region, as we are using identical coordinates in the extended region.
Instead of (2.12), however, we need to impose that the EOW brane is connected
and smooth at the junction point $z=\infty$ (equivalently, $\tilde{z}=\infty$)
from the perspective of the global AdS. Thus, we have

$\tilde{u}_{3}=\frac{1}{v_{\infty}^{2}}\frac{A(\tilde{z}/a)}{1+\tilde{z}/a}e^{B(\tilde{z}/a)},\quad\tilde{v}_{3}=-v_{\infty}^{2}\frac{1-\tilde{z}/a}{A(\tilde{z}/a)}e^{-B(\tilde{z}/a)},$
(2.21)

where

$v_{\infty}:=\lim_{z\rightarrow\infty}e^{B(z/a)}=\exp\left[\frac{\zeta_{0}}{\sqrt{\left(1-\zeta_{0}^{2}\right)\left(2-\zeta_{0}^{2}\right)}}\left(K\left(\frac{1}{2-\zeta_{0}^{2}}\right)-\Pi\left(\frac{1-\zeta_{0}^{2}}{2-\zeta_{0}^{2}}\middle|\frac{1}{2-\zeta_{0}^{2}}\right)\right)\right].$
(2.22)

Figure 3 shows the result of the solutions.

Figure 3: The EOW branes in the $\Delta\phi$ phase. It is conformally
compactified using arctan. The lower square corresponds to the original BTZ
black hole, while the upper one corresponds to the extended region. In this
picture, the dual BCFT lives in the left asymptotic boundary of the lower BTZ.
The EOW brane with $z_{0}/a=1$, corresponding to $\Delta\phi=0$, forms a
vertical line, which is consistent with the configuration described in [28].

Next, we will calculate the pseudo entropy using the HRT prescription. We take
a half-line $x>0$ as the sub-region $A$. In the HRT prescription, the pseudo
entropy at time $t$ is given by

$S(t)=\max_{P_{2}\in Q_{0}}\frac{\ell(P_{1},P_{2})}{4G}.$ (2.23)

Here, $Q_{0}$ represents the set of points that are located on the EOW brane
and have an $x$ coordinate of zero. $P_{1}$ is the endpoint of $A$ at time $t$
and $\ell(P_{1},P_{2})$ represents the space-like geodesic length between
$P_{1}$ and $P_{2}$. Maximization is carried out only for points that are
spacelike separated. The explicit form of $\ell(P_{1},P_{2})$ in the global
coordinates is given by AdS-invariant function:

$\cosh\ell(P_{1},P_{2})=\sigma\left(P_{1}\middle|P_{2}\right):=\frac{\cos(\tau_{1}-\tau_{2})-\sin\rho_{1}\sin\rho_{2}\cos(\theta_{1}-\theta_{2})}{\cos\rho_{1}\cos\rho_{2}}.$
(2.24)

In our setup, $P_{1}$ is $(\epsilon,t,0)$ in the Poincaré coordinates, where
$\epsilon$ is the UV cut-off scale of the CFT. Now, using eq.(2.8) and
eq.(2.19), we can calculate the entropy. Numerical results tell us that the
point $P_{2}$, which maximizes $\sigma\left(P_{1}\middle|P_{2}\right)$, is
located in the extended region. Then, we have

$\sigma=\frac{a}{\epsilon}\frac{-1+\tilde{u}\tilde{v}+e^{t/a}\tilde{u}-e^{-t/a}\tilde{v}}{1+\tilde{u}\tilde{v}},$
(2.25)

which becomes at late times $t\gg a$

$\sigma\simeq\frac{e^{t/a}}{2v_{\infty}^{2}\epsilon}\frac{A(\tilde{z}/a)}{\tilde{z}/a}e^{B(\tilde{z}/a)}.$
(2.26)

Therefore, we need to solve

$\frac{d\sigma}{d\tilde{z}}=\frac{e^{t/a}a}{2v_{\infty}^{2}\epsilon}\frac{e^{B(\tilde{z}/a)}}{\tilde{z}^{2}/a}(A^{\prime}(\tilde{z}/a)\cdot\tilde{z}/a+A(\tilde{z}/a)B^{\prime}(\tilde{z}/a)\cdot\tilde{z}/a-A(\tilde{z}/a))=0.$
(2.27)

In fact, this equation can be solved analytically and the real solution
$\tilde{z}$ that is greater than $z_{0}$ is

$\frac{\tilde{z}}{a}=\tilde{\zeta}_{0}:=\frac{\zeta_{0}}{\left(1-\zeta_{0}^{2}\right)^{1/4}}.$
(2.28)

Finally, we obtain

$S(t)=\frac{c}{6}\cosh^{-1}\sigma\simeq\frac{c}{6}\frac{1}{a}t+\frac{c}{6}\ln\frac{a}{\epsilon}+\log
g,$ (2.29)

where

$\log
g=\frac{c}{6}\ln\left(\frac{A(\tilde{\zeta}_{0})e^{B(\tilde{\zeta}_{0})}}{v_{\infty}^{2}\tilde{\zeta}_{0}}\right).$
(2.30)

Since eq.(2.29), excluding the $\log g$ term, is identical to the entropy of
$\Delta\phi=0$ if $a$ is fixed, we can identify $\log g$ — which is determined
only by $\Delta\phi$ via $\zeta_{0}=z_{0}/a$ — as the boundary entropy.

Figure 4: Graphs of entropy behavior. Left: $t$-dependency, i.e., we fix $a=1$
and $z_{0}=0.5.$ Middle: $z_{0}$-dependency, i.e., fix $a=1$ and $t=10.$
Right: $\Delta\phi$-dependency, i.e., fix $\beta=1$ and $t=10.$ For all
graphs, we set $\epsilon=10^{-5}.$

###  2.3 Other profiles of EOW brane for $V(\phi)\neq 0$

Before we move on, we would like to pause here to explore more solutions of
EOW branes for $V(\phi)\neq 0$.

First, the solution of $\phi^{\prime 2}=0$, i.e. $\phi=$const. is

$\tan^{-1}v=\tan^{-1}u-2\tan^{-1}u_{0},\quad
V(\phi)=\frac{2u_{0}}{1+u_{0}^{2}},\quad-1<u_{0}<1.$ (2.31)

where $u_{0}$ is a negative constant. Since $\phi^{\prime}=0$, this solution
corresponds to the one without the brane scalar and the constant potential is
the tension of the brane. In fact, the value of the potential is constrained
within the range of $-1<V(\phi)<1,$ and the shape of the brane becomes a
straight line in the Kruskal coordinates like figure 5. These are the standard
solutions in the pure gravity model of AdS/BCFT studied in [26, 27].

Figure 5: Brane configurations with $\phi^{\prime}=0.$ These correspond to
constant tension branes without the brane scalar.

There is another solution with the constant potential:

$vu=-u_{0}^{2},\quad
V(\phi)=\frac{1+6u_{0}^{2}+u_{0}^{4}}{4u_{0}(1+u_{0}^{2})},\quad-1<u_{0}<0.$
(2.32)

Unlike the solution in eq.(2.31), $\phi^{\prime}$ is not zero, in fact,

$\phi^{\prime 2}=-\frac{u_{0}}{u^{2}\left(1+u_{0}^{2}\right)}>0,$ (2.33)

and $V(\phi)\geq 1.$ As we can see in figure 6, this brane does not extend
across with the horizon.

Figure 6: Brane profile of eq.(2.32). $uv=\rm{const.}$ means $z=\rm{const.}$

We can analytically calculate the entanglement entropy. Since
$u_{2}v_{2}=-u_{0}^{2}$, using eq.(2.24), we have

$\ell=\cosh^{-1}\left(1+2\frac{-\Delta u\Delta
v}{\alpha}\right),\quad\alpha=\frac{2\epsilon(1-u_{0}^{2})}{a+\epsilon}.$
(2.34)

To maximize $\ell$ with respect to $u_{2}$, we need to find the maximum value
of $-\Delta u\Delta v$. We can easily find it as follows:

$-\Delta u\Delta
v=-\sqrt{\frac{a-\epsilon}{a+\epsilon}}\left(e^{-t/a}v_{2}+e^{t/a}\frac{u_{0}^{2}}{v_{2}}\right)+\frac{a-\epsilon}{a+\epsilon}+u_{0}^{2}\leq\left(u_{0}+\sqrt{\frac{a-\epsilon}{a+\epsilon}}\right)^{2}.$
(2.35)

Namely, since $D$ is time-independent, the entanglement entropy is also a
constant. The value of the entanglement entropy is

$S(t)=\frac{c}{6}\ln\left(\frac{(1+u_{0})^{2}}{1-u_{0}^{2}}\frac{2a}{\epsilon}\right).$
(2.36)

This solution has unusual features compared to previous ones. Since the EOW
brane is located at $z=\rm{const.}$ in Lorentzian coordinates, through
analytical continuation, we find that it remains constant, which means the EOW
brane does not end on the asymptotic boundary even in Euclidean metric and the
description of the path integral from some boundary state fails111We note that
a similar situation, where the brane does not intersect the asymptotic
boundary in the Euclidean signature, was also discussed in [72].. The
gravitational force toward the horizon direction is cancelled by the effect of
time-dependent localized scalar field. As the left side and right side of the
BTZ black hole are connected through the Euclidean region, the EOW brane
exists also in the right side.

Finally let us mention possible CFT interpretations of this EOW brane. One
possibility is that it is a certain thermofield double-like state without the
IR degree of freedom. This may look similar to the prescription of the
$T\bar{T}$ deformation[73], where the finite radius boundary conceals the UV
degree of freedom, the EOW brane is hovering. Unlike the $T\bar{T}$
deformation, however, the EOW brane is subject to the Neumann boundary
condition. Another possibility is the final state projection [74, 39].

In order to think of the projection proposal, we consider a geometry which may
be corresponding to the final state projection at a finite time:

$(u+\alpha)(v-\alpha)=-u_{0}^{2}.$ (2.37)

By the analytical continuation, we have the corresponding Euclidean brane:

$\sqrt{\frac{a-z}{a+z}}=\alpha\cos(\tau/a)+\sqrt{u_{0}^{2}-\alpha^{2}\sin^{2}(\tau/a)}.$
(2.38)

This implies that $\beta=2\pi a.$

For another topic, there exists a configuration that does not have the well-
defined Euclidean brane. For instance, $z=z_{0}\exp(-t^{2}/2\sigma)$ poses no
problem at least in Lorentzian space if we choose the parameters not to make
the EOW brane space-like. On the other hand, in the Euclidean space, this EOW
brane follows $z=z_{0}\exp(\tau^{2}/\sigma),$ which obviously encounters
difficulties. Namely, this brane penetrates $z=a$ in the Euclidean space and
has no periodicity.

###  2.4 Complete analysis of holographic pseudo entropy at critical point:
$\Delta\phi=\Delta\phi_{*}$

For the critical phase $\Delta\phi=\Delta\phi_{*}$, the corresponding geometry
is the Poincaré AdS

$ds^{2}=\frac{dz^{2}-dt^{2}+dx^{2}}{z^{2}},$ (2.39)

and the EOW brane is determined by the following differential equation

$1-\dot{z}^{2}=\frac{z_{0}^{4}}{z^{4}},$ (2.40)

with the initial condition $z(t=0)=z_{0}$. The solution of this equation can
be described by using the hypergeometric function as follows:

$|t|=z_{0}f(z/z_{0}),\quad
f(x):=x\cdot{}_{2}F_{1}\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{1}{x^{4}}\right)-\alpha,\quad\alpha=\frac{\sqrt{\pi}\Gamma(3/4)}{\Gamma(1/4)}.$
(2.41)

As we can see in figure 7, this EOW brane will fall at a light speed at late
time:

$z\simeq t+\alpha z_{0}.$ (2.42)

This implies that the EOW brane can be extended to the global AdS where the
Poicaré patch is embedded.

Figure 7: The blue line represents an EOW brane with $V(\phi)=0$ in the
Poincaré patch. At late times, this EOW brane moves at a speed approaching
that of light.

The Poincaré AdS is embedded into the global AdS via $\mathbb{R}^{2,2}$ as
follows:

$\displaystyle
X_{0}=\frac{z}{2z_{0}}\left(1+\frac{1}{z^{2}}\left(x^{2}-t^{2}+z_{0}^{2}\right)\right)=\frac{\cos\tau}{\cos\rho},$
(2.43) $\displaystyle X_{1}=\frac{x}{z}=\tan\rho\sin\phi,$ (2.44)
$\displaystyle
X_{2}=\frac{z}{2z_{0}}\left(1+\frac{1}{z^{2}}\left(x^{2}-t^{2}-z_{0}^{2}\right)\right)=\tan\rho\cos\phi,$
(2.45) $\displaystyle X_{3}=\frac{t}{z}=\frac{\sin\tau}{\cos\rho}.$ (2.46)

Thus, using the global coordinates, the Poincaré coordinates are represented
as

$\frac{z}{z_{0}}=\frac{\cos\rho}{\cos\tau-\sin\rho\cos\theta},\
\frac{t}{z_{0}}=\frac{\sin\tau}{\cos\tau-\sin\rho\cos\theta},\
\frac{x}{z_{0}}=\frac{\sin\rho\sin\theta}{\cos\tau-\sin\rho\cos\theta}.$
(2.47)

The Poincaré AdS corresponds to the region where
$\cos\tau-\sin\rho\cos\theta>0.$ Now we define new coordinates for the region
out of the Poincaré patch, namely $\cos\tau-\sin\rho\cos\theta<0,$ as the
following:

$\frac{\tilde{z}}{z_{0}}=\frac{\cos\rho}{\cos\tau-\sin\rho\cos\theta},\
\frac{\tilde{t}}{z_{0}}=\frac{\sin\tau}{\cos\tau-\sin\rho\cos\theta},\
\frac{\tilde{x}}{z_{0}}=\frac{\sin\rho\sin\theta}{\cos\tau-\sin\rho\cos\theta}.$
(2.48)

Note that $\tilde{z}$ is negative. Obviously, since eq.(2.47) and eq.(2.48)
have the same form, new coordinates have the same metric with the original
Poincaré metric (eq.(2.39))and are embedded into the global AdS in the same
way (eq.(2.43)). Therefore, the differential equation of the EOW brane outside
of the Poincaré patch is also

$1-\dot{\tilde{z}}^{2}=\frac{z_{0}^{4}}{\tilde{z}^{4}}.$ (2.49)

Thus, we can find the solution

$|\tilde{t}-\tilde{t}_{0}|=z_{0}f(-\tilde{z}/z_{0}),\quad\tilde{t}_{0}=-2\alpha
z_{0},$ (2.50)

by imposing $\tilde{z}\simeq\tilde{t}+\alpha z_{0}$ for $\tilde{t}\to-\infty$.

Figure 8: Left: The EOW brane embedded in the global AdS is represented as a
red surface. The original Poincaré AdS is the region surrounded by two blue
Poincaré horizons. Right: The cross-sectional view of a cylinder cut in half.
Figure 9: The graph of the entanglement entropy $S$ vs time $t$. We set the
Newtonian constant $G=1$ and UV cut-off $\epsilon=10^{-5}.$

Figure 8 shows the extended EOW brane. In a similar way in the small
$\Delta\phi$ phase, we can evaluate the entanglement entropy using HRT surface
$\Gamma$. We set subregion $A$ of a time slice of CFT, as a half line $x>0.$
Then, $\Gamma$ ends $(z,t,x)=(\epsilon,t,0)$ where $\epsilon$ is UV cut-off.
Another endpoint of $\Gamma$ is on the red line in the right side of figure 8.
Therefore, to calculate the entanglement entropy, we need to find maximum
value of the length of $\Gamma$ for a fixed $t$.

Figure 9 tells us that for any $z_{0}$, the entanglement entropy grows
logarithmically at late time.

$S\simeq\frac{1}{2G}\log\frac{t}{\epsilon}-\frac{1}{4G}\log\frac{z_{0}}{\epsilon}=\frac{c}{3}\log\frac{t}{\epsilon}-\frac{c}{6}\log\frac{z_{0}}{\epsilon}.$
(2.51)

It is worth noting that taking the part of the spacetime beyond the Poincaré
horizon into account is crucial to get the $\frac{c}{3}\log t$ term [75, 76].
Typically, both the inside-Poincaré piece and the outside-of-Poincaré piece of
the HRT surface contribute to the $\log t$ term and together give
$\frac{c}{3}\log t$. The same behavior is also observed in the study of
holographic joining quench [75, 76].

Combining the results obtained so far, we have worked out the behavior of the
entanglement phase transition shown in (1.2).

###  2.5 Higher dimension

Now we move on to the analysis of holographic entanglement phase transition in
higher dimensions. Since the black hole and thermal AdS phase are very similar
to the previous three dimensional setup, here we concentrate on the critical
point at $\Delta\phi=\phi_{*}$.

In a similar way to $d=2$, we can analyze the Poincaré case in higher
dimension, in which the metric is

$ds^{2}=\frac{dz^{2}-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{d-1}^{2}}{z^{2}},\quad(d\geq
3).$ (2.52)

The differential equation of the EOW brane in a $d$-dimensional Poincaré space
is

$1-\dot{z}^{2}=\left(\frac{z_{0}}{z}\right)^{4(d-1)},\quad z(t=0)=z_{0},$
(2.53)

and its solution is

$|t|=z_{0}f_{d}(z/z_{0}),$ (2.54)

where

$f_{d}(x):=x\cdot{}_{2}F_{1}\left(\frac{1}{2},\frac{-1}{4(d-1)};\frac{4d-5}{4(d-1)};\frac{1}{x^{4(d-1)}}\right)-\alpha_{d},\quad\alpha_{d}:=\frac{\sqrt{\pi}\Gamma\left(\frac{4d-5}{4(d-1)}\right)}{\Gamma\left(\frac{2d-3}{4(d-1)}\right)}.$
(2.55)

For higher dimensions, we can also define new coordinates for the region
outside of the Poincaré patch in a similar way. Then, outside of the horizon,
the EOW brane follows

$|\tilde{t}-\tilde{t}_{0}|=z_{0}f_{d}(-\tilde{z}/z_{0}),\quad\tilde{t}_{0}=-2\alpha_{d}z_{0}.$
(2.56)

Figure 10 shows behaviors of EOW branes.

Next, we consider a subregion $A$ of a CFT time slice at a certain time $t$,
defined as $A=\\{(\epsilon,t,x_{1},\dots,x_{d-1})|x_{1}>0\\}.$ Hence, the HRT
surface $\Gamma=\\{(z,t(z),0,x_{2},\dots,x_{d-1})\\}$ ends on $\partial A$ and
on the EOW brane. The HRT surface maximizes the area of $\Gamma$,

$\operatorname{Area}\Gamma=\int_{\Gamma}\frac{\sqrt{1-(dt/dz)^{2}}}{z^{d-1}}dzdx_{2}\cdots
dx_{d-1}=V_{d-2}\int_{i}^{f}\frac{\sqrt{dz^{2}-dt^{2}}}{z^{d-1}},$ (2.57)

where label $i$ and $f$ represent endpoints of $\Gamma$. The HRT surface holds
the Euler Lagrange equation with respect to the area function

$\dot{z}^{2}=1+\left(\frac{t_{*}}{z}\right)^{2(d-1)},$ (2.58)

where $t_{*}$ is some positive constant and a solution of this equation is

$\frac{t_{f}}{t_{*}}-\frac{t_{i}}{t_{*}}=B_{d}\left(\frac{z_{f}}{t_{*}}\right)-B_{d}\left(\frac{z_{i}}{t_{*}}\right),\quad
B_{d}(x):=x\cdot{}_{2}F_{1}\left(\frac{1}{2},\frac{-1}{2(d-1)};\frac{2d-3}{2(d-1)};\frac{-1}{x^{2(d-1)}}\right).$
(2.59)

Furthermore, by substituting eq.(2.58) into eq.(2.57), we have

$\operatorname{Area}(\Gamma)=V_{d-2}\int_{z_{i}}^{z_{f}}\frac{dz}{z^{d-1}\sqrt{1+(z/t_{*})^{2(d-1)}}}=\frac{V_{d-2}}{t_{*}^{d-2}}\left(A_{d}\left(\frac{z_{f}}{t_{*}}\right)-A_{d}\left(\frac{z_{f}}{t_{*}}\right)\right),$
(2.60)

where

$A_{d}(x):=\frac{\sqrt{1+x^{2(d-1)}}}{(d-2)x^{d-2}}\left[{}_{2}F_{1}\left(1,\frac{d-2}{2(d-1)};\frac{2d-3}{2(d-1)};\frac{-1}{x^{2(d-1)}}\right)-1\right].$
(2.61)

Since the EOW brane locates outside of the Poincaré horizon, we need to modify
the area of the EOW brane as follows:

$\displaystyle\operatorname{Area}(\Gamma)$
$\displaystyle=\frac{V_{d-2}}{t_{*}^{d-2}}\left(A_{d}(\infty)-A_{d}\left(\frac{\epsilon}{t_{*}}\right)\right)+\frac{V_{d-2}}{t_{*}^{d-2}}\left(A_{d}(\infty)-A_{d}\left(-\frac{\tilde{z}_{i}}{t_{*}}\right)\right)$
(2.62)
$\displaystyle=-\frac{V_{d-2}}{t_{*}^{d-2}}\left(A_{d}\left(\frac{\epsilon}{t_{*}}\right)+A_{d}\left(-\frac{\tilde{z}_{i}}{t_{*}}\right)\right).$
(2.63)

Here we use $A_{d}(\infty)=0$ and $\tilde{z}_{i}$ to represent the coordinates
of the endpoint of $\Gamma$ on the EOW brane. To compute the area, we need to
know $t_{*}$. This is determined via

$B_{d}\left(\frac{\epsilon}{t_{*}}\right)+B_{d}\left(-\frac{\tilde{z}_{i}}{t_{*}}\right)=\frac{t}{t_{*}}-\frac{\tilde{t}_{i}}{t_{*}}.$
(2.64)

Figure 10: Left: Solid lines and dotted lines represent EOW branes and HRT
surface respectively at time $t$. Right: The horizontal axis is time
coordinate of the region out of Poincaré patch. A dotted line corresponds to
$\tilde{t}=\tilde{t}_{0}$. The vertical axis shows the area of the HRT surface
in $d=4$ connecting $(\epsilon,t)$ and $(\tilde{z}(\tilde{t}),\tilde{t})$. We
set $t=10$, $z_{0}=1$ and $\epsilon=10^{-5}$ in both graphs.

We can determine the position of the endpoint of $\Gamma$ on the EOW brane
numerically. The right side of figure 10 tells us that the HRT surface ends at
$(\tilde{z}_{i},\tilde{t}_{i})=(-z_{0},\tilde{t}_{0}).$

Furthermore, at late time, $t_{*}$ behaves as

$t_{*}\simeq\frac{\sqrt{\pi}}{2\Gamma\left(\frac{d}{2d-2}\right)\Gamma\left(\frac{2d-3}{2d-2}\right)}t.$
(2.65)

Hence, both $\frac{\epsilon}{t_{*}}$ and $-\frac{\tilde{z}_{i}}{t_{*}}$
decrease in the time evolution. Now, for small $x\ll 1$, $A_{d}(x)$ behaves as

$A_{d}(x)\simeq-\frac{1}{d-2}x^{-(d-2)}.$ (2.66)

Therefore, using eq.(2.63), we have

$\operatorname{Area}(\Gamma)\simeq
A_{\rm{max}}^{d}:=\frac{V_{d-2}}{d-2}\left(\frac{1}{z_{0}^{d-2}}+\frac{1}{\epsilon^{d-2}}\right),$
(2.67)

namely, at late time, the leading term of the entanglement entropy is
constant. Numerical calculation supports this result. See figure 11. In this
way, in the higher dimensional case, there is no logarithmic evolution of
$S_{A}$ at the critical point, as opposed to the AdS3 case.

Figure 11: The graph of the entanglement entropy $S_{\rm{EE}}$ vs time $t$ in
$d=4$ dimension. We set $G=1$, $z_{0}=1$ and $\epsilon=10^{-5}$.

###  2.6 Null energy condition and the brane profile

It is useful to reconsider our EOW profile in terms of null energy condition.
Consider the Poincaré AdSd+1:
$ds^{2}=z^{-2}(dz^{2}-dt^{2}+\sum_{i=1}^{d-1}dx_{i}^{2})$ and place a EOW
brane with the profile $z=z(t)$. The physical region is taken to the boundary
side of the EOW brane $z\leq z(t)$. In this case, the extrinsic curvature
reads

$\displaystyle K_{tt}-h_{tt}K=-\frac{(d-1)\sqrt{1-\dot{z}^{2}}}{z^{2}},\ \ \ \
K_{x_{i}x_{i}}-h_{xx}K=\frac{(d-1)(1-\dot{z}^{2})-z\ddot{z}}{z^{2}(1-\dot{z}^{2})^{3/2}}.$
(2.68)

The null vector on this EOW brane is taken to be
$(N^{t},N^{x_{1}},N^{x_{2}},\cdot\cdot\cdot)=(1,\pm\sqrt{1-\dot{z}^{2}},0,\cdot\cdot\cdot)$.
Then the null energy condition reads

$\displaystyle
N^{a}N^{b}(K_{ab}-h_{ab}K)=-\frac{\ddot{z}}{z\sqrt{1-\dot{z}^{2}}}\geq 0.$
(2.69)

This constrains the profile of the EOW brane.

In our model of EOW brane with a localized scalar, the boundary energy stress
tensor $T^{(Q)}_{ab}$ on the EOW brane, which is related to the extrinsic
curvature via the Neumann boundary condition $K_{ab}-h_{ab}K=T^{(Q)}_{ab}$,
reads

$\displaystyle T^{(Q)}_{tt}=\dot{\phi}^{2}+\frac{V}{z^{2}}(1-\dot{z}^{2}),$
$\displaystyle
T^{(Q)}_{x_{i}x_{i}}=\frac{\dot{\phi}^{2}}{1-\dot{z}^{2}}-\frac{V}{z^{2}}.$
(2.70)

The null energy condition reads

$\displaystyle N^{a}N^{b}T^{(Q)}_{ab}=2\dot{\phi}^{2}\geq 0.$ (2.71)

The profile of our previous EOW brane solutions with the scalar (2.50) and
(2.54) violates this condition (2.69). One quick way to see this is to note
that the scalar field takes imaginary values in the Lorentzian solution and
thus $\dot{\phi}^{2}<0$. Since it has an imaginary value of scalar field,
(2.71) is also violated as expected. Thus the EOW brane solution which shows
the entanglement phase transition violates the null energy condition. This is
not surprising because, in order to realize this entanglement phase
transition, we need a dissipative effect, which may lead to a non-unitary
evolution, to suppress the entanglement growth.

It might also be intriguing to consider the energy conservation in the gravity
dual. By taking a derivative of $K_{ab}-h_{ab}K=T^{(Q)}_{ab}$, we obtain

$\displaystyle\nabla^{a}(K_{ab}-h_{ab}K)=\nabla^{a}T^{(Q)}_{ab}.$ (2.72)

It is straightforward from the differential geometry to see
$\nabla^{a}(K_{ab}-h_{ab}K)=R_{nb}$, where $n$ is in the direction normal to
the EOW brane. If the bulk dynamics is described by a pure Einstein equation
(in any dimensions), it is clear that $R_{nb}\propto g_{nb}=0$. Thus the
boundary energy stress tensor is conserved $\nabla^{a}T^{(Q)}_{ab}=0$, which
means that the energy on EOW brane is preserved and there is no energy flux
from the bulk to the brane. This is true in all setups we consider in this
paper.

If there are matter fields in the bulk, they may lead to the bulk energy flux
$T^{\rm(bulk)}_{nb}$, which is proportional to $R_{nb}$ via the Einstein
equation. In this case we have $\nabla^{a}T^{(Q)}_{ab}=R_{nb}\propto
T^{\rm(bulk)}_{nb}$ and thus there is energy flow between the EOW brane and
the bulk.

###  2.7 Critical behavior from a simple dissipating model

Before we move on, we would like to point out that the critical logarithmic
time evolution of entanglement entropy can be found in a simple dissipating
model in a two dimensional CFT.

A simple modeling of quantum quenches is to use the BCFT [11]. The time
evolution under quantum quenches in a two dimensional CFT can be described by

$\displaystyle|\Psi(t)\rangle\propto e^{-itH}e^{-\frac{\beta}{4}H}|B\rangle,$
(2.73)

where $|B\rangle$ is the boundary state. At late time, this state looks like a
thermal state at the temperature $1/\beta$ as long as we examine coarse-
grained observables. The evolution of entanglement entropy $S_{A}$, where the
subsystem $A$ is a half space, can be found from a standard replica method
calculation in the CFT, as follows [28].

$\displaystyle
S_{A}=\frac{c}{6}\log\left[\frac{\beta}{2\pi\epsilon}\cosh\left(\frac{2\pi
t}{\beta}\right)\right],$ (2.74)

up to a constant term. At late time, this grows linearly as
$S_{A}\simeq\frac{\pi c}{3\beta}t$.

We can take into account the dissipating effect by introducing a damping
factor $e^{-\gamma tH}$ under the time evolution, where $\gamma$ is the
parameter which controls the amount of the dissipation. This leads to the
quantum state

$\displaystyle|\Psi_{d}(t)\rangle\propto e^{-itH}e^{-\gamma
tH}e^{-\frac{\beta}{4}H}|B\rangle.$ (2.75)

To calculate the evolution of entanglement entropy, we can replace $\beta$ in
(2.74) with $\beta+4\gamma t$, which leads to

$\displaystyle S_{A}=\frac{c}{6}\log\left[\frac{\beta+4\gamma
t}{2\pi\epsilon}\cosh\left(\frac{2\pi t}{4\gamma t+\beta}\right)\right].$
(2.76)

At late time, this behaves like

$\displaystyle S_{A}\simeq\frac{c}{6}\log\left[\frac{2\gamma
t}{\pi\epsilon}\cosh\left(\frac{\pi}{2\gamma}\right)\right].$ (2.77)

This shows the critical logarithmic behavior. Note that the coefficient of the
logarithmic evolution gets halved when compared with the result in our
holographic model (2.51). Also note that, when such a damping factor is
involved, the logarithmic time evolution is generic in two dimensional CFTs
even if the starting point is not the boundary state [77].

###  2.8 Comments on the analysis from the CFT side

We have so far realized an entanglement phase transition in AdS/BCFT by
introducing a brane localized scalar field on the gravity side. It is a
natural question to consider the corresponding setup on the CFT side, i.e. a
BCFT defined on a strip where the boundary conditions on the two boundaries
are related by a marginal transformation.

In appendix C, we consider a slightly different but parallel setup and compute
the pseudo Rényi entropy at the leading order of the marginal perturbation
purely in the CFT language. Denoting the perturbation parameter as $\lambda$,
our holographic computation implies that the leading order correction of the
pseudo entanglement entropy is at order $\lambda^{2}$ and gives a negative
linear $t$ contribution. In the CFT computation summarized in appendix C, we
find that, for even $n$, the $\lambda^{2}$ order correction of the $n$-th
pseudo Rényi entropy is zero, while for odd $n$, it turns out to give a
positive linear $t$ contribution.

Although the quantities computed on the two sides are different and we can not
compare them with each other, it would be an intriguing future question to
understand how the analyses on the two sides are related with each other.

##  3 Gauge fields on the brane

As a natural generalization of matter fields localized on the EOW brane, we
can also consider the gauge field instead of scalar field. Again, we aim to
construct an AdS/BCFT solution where the BCFT is defined on a strip. The
action we consider is the Einstein-Hilbert action with gauge fields on the
brane (we will work in Euclidean signature and set $V=0$ for simplicity)

$\displaystyle S$ $\displaystyle=S_{\rm EH}+S_{\rm brane},$ $\displaystyle
S_{\rm EH}$ $\displaystyle=-\frac{1}{16\pi G}\int d^{d+1}x\sqrt{g}(R+d(d-1)),$
$\displaystyle S_{\rm brane}$ $\displaystyle=-\frac{1}{8\pi
G}\int\sqrt{h}\left(K-\frac{1}{4}h^{cd}h^{ef}F_{ce}F_{df}\right).$ (3.1)

Taking a variation with respect to the induced metric, we obtain the equations
of motion (or Neumann boundary condition) for the EOW brane

$K_{ab}-Kh_{ab}=-\frac{1}{4}h_{ab}h^{ce}h^{df}F_{cd}F_{ef}+F_{ac}F_{bd}h^{cd}.$
(3.2)

###  3.1 EOW brane with gauge field in Poincaré AdS

We start with the analysis of EOW brane placed in the $d+1$ dimensional
Poincaré vacuum AdS, given by

$ds^{2}=\frac{dz^{2}+d\tau^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}}{z^{2}}.$ (3.3)

We specify the profile of the EOW brane by $z=z(\tau)$, assuming the
translational invariance along $x^{i}$ direction. The induced metric reads

$ds^{2}=\frac{(1+\dot{z}^{2})d\tau^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}}{z^{2}}.$
(3.4)

and the extrinsic curvature on this brane is computed as follows:

$\displaystyle K_{\tau\tau}-Kh_{\tau\tau}$
$\displaystyle=\frac{(d-1)\sqrt{1+\dot{z}^{2}}}{z^{2}},$ $\displaystyle
K_{xx}-Kh_{xx}$
$\displaystyle=\frac{(d-1)(1+\dot{z}^{2})+z\ddot{z}}{z^{2}(1+\dot{z}^{2})^{\frac{3}{2}}}.$
(3.5)

Here, we choose the physical region of AdS/BCFT setup to be $0<z<z(\tau)$ and
the normal vector is pointed in the outer direction. Below, we will study the
solution to (3.2) in each dimension.

####  3.1.1 AdS3

As the simplest example, consider the AdS3 case. The equations of motion for
the gauge field

$\partial_{a}(\sqrt{h}F^{ab})=0,$ (3.6)

can be solved as

$F_{\tau x}=E_{x}\cdot\frac{\sqrt{1+\dot{z}^{2}}}{z^{2}},$ (3.7)

where $E_{x}$ is a constant. The boundary condition (3.2) leads to

$\displaystyle 2\sqrt{1+\dot{z}^{2}}$ $\displaystyle=F_{\tau x}^{2}z^{4},$
$\displaystyle 2(1+\dot{z}^{2}+z\ddot{z})$ $\displaystyle=F_{\tau
x}^{2}z^{4}\sqrt{1+\dot{z}^{2}},$ (3.8)

which ends up with

$\ddot{z}=0.$ (3.9)

Therefore, the brane profile is solved as

$z=\sqrt{\frac{4}{E_{x}^{4}}-1}\cdot\tau+\mbox{const}.$ (3.10)

Indeed, we can see that the above equation of motion is identical to the known
vacuum solution [26] with the nonzero potential (or constant tension) given by
$V=-\frac{E_{x}^{2}}{2}$. Since the brane profile is now a straight line, the
solution does not include the gravity dual of BCFT on a strip, as opposed to
the localized scalar field case.

####  3.1.2 AdS4

In the AdS4 case, there are three components of gauge flux and by solving
(3.6) we can set them in the following form:

$F_{\tau x_{i}}=E_{i}\cdot\frac{\sqrt{1+\dot{z}^{2}}}{z},\ \ \
F_{x_{1}x_{2}}=B_{12},$ (3.11)

where $E_{1},E_{2}$ and $B_{12}$ are all constants. On the other hand, (3.2)
lead to

$\displaystyle\frac{2\sqrt{1+\dot{z}^{2}}}{z^{2}}$
$\displaystyle=\frac{z^{2}}{2}\left(F_{\tau x_{1}}^{2}+F_{\tau
x_{2}}^{2}\right)-\frac{z^{2}(1+\dot{z}^{2})}{2}\,F_{x_{1}x_{2}}^{2},$
$\displaystyle\frac{2(1+\dot{z}^{2})+z\ddot{z}}{z^{2}(1+\dot{z}^{2})^{\frac{3}{2}}}$
$\displaystyle=-\frac{z^{2}}{2(1+\dot{z}^{2})}\left(F_{\tau x_{2}}^{2}-F_{\tau
x_{1}}^{2}\right)+\frac{z^{2}}{2}\,F_{x_{1}x_{2}}^{2},$ (3.12)
$\displaystyle\frac{2(1+\dot{z}^{2})+z\ddot{z}}{z^{2}(1+\dot{z}^{2})^{\frac{3}{2}}}$
$\displaystyle=-\frac{z^{2}}{2(1+\dot{z}^{2})}\left(F_{\tau x_{1}}^{2}-F_{\tau
x_{2}}^{2}\right)+\frac{z^{2}}{2}\,F_{x_{1}x_{2}}^{2}.$ (3.13)

In the presence of both electric and magnetic fields, these equations reduce
to

$\displaystyle\begin{cases}\displaystyle\frac{2}{z^{2}\sqrt{1+\dot{z}^{2}}}&=\displaystyle
E^{2}-\frac{B^{2}z^{2}}{2}\\\
4(1+\dot{z}^{2})+z\ddot{z}&=E^{2}z^{2}(1+\dot{z}^{2})^{3/2}\end{cases},$
(3.14)

where $E=E_{1}=E_{2}$ and $B=B_{12}$. These can be solved as

$\displaystyle\pm\tau$
$\displaystyle=\int_{0}^{z}\cfrac{\zeta^{2}|2E^{2}-B^{2}\zeta^{2}|}{\sqrt{16-\zeta^{4}(2E^{2}-B^{2}\zeta^{2})^{2}}}\,d\zeta+\text{const.}.$
(3.15)

The value of the integral on the right-hand side behaves as depicted in figure
12, depending on the values of $E$ and $B$. For $E^{2}/|B|>2$, i.e.
$\lambda>2/3$ in figure 12, the smallest positive root of the denominator is
$\zeta=\sqrt{E^{2}-\sqrt{E^{4}-4B^{2}}}/|B|$, and the shape of the EOW brane
is convex. On the other hand, for $E^{2}/|B|<2$, i.e. $\lambda<2/3$, the
smallest positive root of the denominator is
$\zeta=\sqrt{E^{2}+\sqrt{E^{4}+4B^{2}}}/|B|$, and the shape has a plateau at
$\zeta=\sqrt{2}|E|/|B|$.

Figure 12: The profiles of the EOW brane with the gauge fields
$E=\sqrt{\lambda},B=1-\lambda$. For comparison, the results using expressions
involving hypergeometric functions for $\lambda=0$ and $\lambda=1$ are shown
in the bottom.

Especially, in the absence of magnetic/electric fields, we obtain connected
solutions:

$\displaystyle\pm\tau$
$\displaystyle=\frac{B^{2}z^{5}}{20}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{5}{8};\frac{13}{8};\frac{B^{4}z^{8}}{16}\right)-\frac{\sqrt{2\pi}}{5\sqrt{|B|}}\cdot\frac{\Gamma(13/8)}{\Gamma(9/8)}\qquad(E=0),$
(3.16) $\displaystyle\pm\tau$
$\displaystyle=\frac{E^{2}z^{3}}{6}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{E^{4}z^{4}}{4}\right)-\frac{\sqrt{2\pi}}{3E}\cdot\cfrac{\Gamma(7/4)}{\Gamma(5/4)}\qquad(B=0).$
(3.17)

The profiles of the EOW branes in these cases are shown in figure 13.

Figure 13: Left: the profile of the EOW brane with $E=0,B=1$. Right: the
profile of the EOW brane with $E=1,B=0$.

####  3.1.3 AdSd+1

Finally, we analyze the higher dimensional solution in Poincaré AdSd+1. For
simplicity, we turn on only electric fields and they take the form so that
they satisfy (3.6):

$F_{\tau x_{i}}=E_{i}\cdot\frac{\sqrt{1+\dot{z}^{2}}}{z^{4-d}}.$ (3.18)

On the other hand, (3.2) leads to

$\displaystyle\frac{2(d-1)\sqrt{1+\dot{z}^{2}}}{z^{4}}$
$\displaystyle=\sum_{i}F_{\tau x_{i}}^{2},$
$\displaystyle\frac{(d-1)(1+\dot{z}^{2})+z\ddot{z}}{z^{4}\sqrt{1+\dot{z}^{2}}}$
$\displaystyle=-\sum_{j}\frac{F_{\tau x_{j}}^{2}}{2}+F_{\tau x_{i}}^{2}.$
(3.19)

These can be solved by setting

$\displaystyle E_{i}=E,\ \ \ \ \ (i=1,2,\cdot\cdot\cdot,d-1),$
$\displaystyle\dot{z}^{2}=\frac{4}{E^{4}}z^{8-4d}-1.$ (3.20)

Note that the differential equation for the brane profile (3.20) is the same
as the previous scalar case with replacing $d-2$ with $d-1$ as can be seen
from the Lorentzian version (2.53). This suggests the following observation:
the gauge fixing kills degrees of freedom for one dimension. We can state that
the scalar field on the EOW brane in the vacuum AdSd describes the same
configuration of the gauge field on the EOW brane in the vacuum AdSd+1.

###  3.2 Localized gauge field on EOW brane in BTZ phase

Here, we consider the EOW brane solution with the localized gauge field in the
BTZ phase. Again, we work with the BTZ metric in the Euclidean signature,
obtained by setting $\tau=it$ in (2.2)

$ds^{2}=\frac{h(z)d\tau^{2}}{z^{2}}+\frac{dz^{2}}{h(z)z^{2}}+\frac{dx^{2}}{z^{2}},$
(3.21)

where $h(z)=1-\frac{z^{2}}{a^{2}}$. We choose the brane profile in the form
$z=z(\tau)$ as before. The induced metric on the brane reads

$ds^{2}=\frac{h(z)^{2}+\dot{z}^{2}}{h(z)z^{2}}d\tau^{2}+\frac{dx^{2}}{z^{2}},$
(3.22)

where $\dot{f}=\frac{df}{d\tau}$ and $f^{\prime}=\frac{df}{dz}$. The extrinsic
curvature reads

$\displaystyle K_{xx}-Kh_{xx}$
$\displaystyle=\frac{2h^{3}-zh^{2}h^{\prime}-3zh^{\prime}\dot{z}^{2}+2h\dot{z}^{2}+2hz\ddot{z}}{2z^{2}z(h+\frac{\dot{z}^{2}}{h})^{\frac{3}{2}}},$
$\displaystyle K_{\tau\tau}-Kh_{\tau\tau}$
$\displaystyle=\frac{h\sqrt{h+\frac{\dot{z}^{2}}{h}}}{z^{2}},$ (3.23)

where $f^{\prime}=\frac{df}{dz}$.

####  3.2.1 Construction of Solution

The Neumann boundary condition (3.2) gives

$\displaystyle K_{xx}-Kh_{xx}$ $\displaystyle=\frac{hz^{2}F_{\tau
x}^{2}}{2(h^{2}+\dot{z}^{2})},$ $\displaystyle K_{\tau\tau}-Kh_{\tau\tau}$
$\displaystyle=\frac{F_{\tau x}^{2}z^{2}}{2}.$ (3.24)

The equations of motion of gauge fields (3.6) are solved as

$\displaystyle F_{\tau
x}=\frac{f}{z^{2}}\sqrt{\frac{h(z)^{2}+\dot{z}^{2}}{h(z)}},$ (3.25)

where $f$ is a constant described as the strength of gauge flux.

Combining the above form, we obtain the differential equation for the brane
profile

$\dot{z}^{2}=\frac{4h(z)^{3}}{f^{4}}-h(z)^{2}.$ (3.26)

This is solved as follows

$\frac{z}{a}=\frac{\sqrt{1-\frac{f^{4}}{4}}\cos\frac{\tau}{a}}{\sqrt{1-\left(1-\frac{f^{4}}{4}\right)\sin\frac{\tau}{a}^{2}}},\
\ \ \ \ \ F_{\tau x}=\frac{2h(z)}{fz^{2}}.$ (3.27)

for the range $-\frac{\pi}{2}a\leq\tau\leq\frac{\pi}{2}a$. Refer to figure 14
for the plot.

Figure 14: The profiles $z=z(\tau)$ of the EOW branes with the gauge field in
BTZ geometry. The horizontal and vertical axis describes $\tau$ and $z$,
respectively and we set $a=1$. The blue, orange and green curves correspond to
the values $f=0.1,1$ and $1.4$.

Note that compared with the EOW brane solution with the localized scalar [25],
our solution with the gauge field takes a much simpler form. In particular,
the width of the strip is given by $\pi a$, which coincides with a half of the
black hole inverse temperature.

####  3.2.2 Free energy in BTZ phase

Now let us evaluate the free energy i.e. the on-shell action for this BTZ
solution with the EOW brane $Q$. This is dual to a BCFT on $\Sigma$, which is
a cylinder. The action we consider is

$\displaystyle I_{\rm BTZ}$ $\displaystyle=I_{\rm EH}+I_{\rm
Q}+I_{\rm\Sigma}-I_{\rm c.t.},$ $\displaystyle I_{\rm EH}$
$\displaystyle=-\frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{g}(R+2),$
$\displaystyle I_{\rm brane}$ $\displaystyle=-\frac{1}{8\pi
G}\int_{Q}\sqrt{h}(K-\frac{1}{4}F_{ab}F^{ab}),$ $\displaystyle I_{\rm\Sigma}$
$\displaystyle=-\frac{1}{8\pi G}\int_{\Sigma}\sqrt{h}K,$ $\displaystyle I_{\rm
c.t.}$ $\displaystyle=-\frac{1}{16\pi
G}\int_{\mathcal{M}}\sqrt{g^{\rm(vac)}}(R+2)-\frac{1}{8\pi
G}\int_{Q}\sqrt{h^{\rm(vac)}}K-\frac{1}{8\pi
G}\int_{\Sigma}\sqrt{h^{\rm(vac)}}K,$ (3.28)

where $I_{\rm c.t}$ is the same action evaluated on the vacuum AdS (i.e.
Poicate AdS3) with the same asymptotic AdS boundary $\Sigma$, which provides
the standard holographic counter term.

Now, we would like to evaluate $I_{\rm BTZ}$ for our AdS/BCFT solution
surrounded by the EOW brane (3.27) with the gauge flux. We choose the UV cut
off to be $z\geq\epsilon$ and the length in $x$ direction to be $L$ via the
compactification $x\sim x+L$, so that $\Sigma$ looks like a cylinder. Each
term in the on-shell action is evaluated as follows:

$\displaystyle I_{\rm EH}$ $\displaystyle=\frac{1}{4\pi
G_{N}}\int{g}=\frac{L}{2\pi
G_{N}}\int^{\frac{\pi}{2}a-\delta}_{0}d\tau\left[\frac{1}{2\epsilon^{2}}-\frac{1}{2z^{2}}\right],$
$\displaystyle I_{\rm Q}$ $\displaystyle=\frac{3f^{2}}{16\pi
G_{N}}\int_{Q}\sqrt{h}=\frac{3L}{4\pi
G_{N}}\int^{\frac{\pi}{2}a-\delta}_{0}d\tau\frac{h(z)}{z^{2}},$ $\displaystyle
I_{\rm\Sigma}-I_{\rm c.t}$ $\displaystyle=-\frac{(\pi\tilde{a}-2\delta)L}{8\pi
G_{N}\epsilon^{2}},$ (3.29)

where $\tilde{a}$ is the rescaled value of $a$ to adjust the BTZ metric and
the Poincaré AdS3 at $z=\epsilon$ defined by
$\tilde{a}=a\sqrt{1-\epsilon^{2}/a^{2}}$. The infinitesimal parameter $\delta$
is the cut off $\tau$ which corresponds to the asymptotic boundary
$z=\epsilon$ on the EOW brane, which leads to the relation
$\delta=\epsilon/\sqrt{4/f^{4}-1}$. The evaluation of $I_{\rm\Sigma}-I_{\rm
c.t}$ goes very similarly as in our earlier work [25] of EOW brane with the
localized scalar field.

By combining these estimations, we eventually find

$\displaystyle I_{\rm BTZ}$ $\displaystyle=$ $\displaystyle\frac{L}{2\pi
G_{N}}\int^{\frac{\pi}{2}a-\delta}_{0}\frac{d\tau}{z(\tau)^{2}}-\frac{5L}{16G_{N}a}$
(3.30) $\displaystyle\simeq$ $\displaystyle\frac{L}{2\pi
G_{N}\epsilon\sqrt{\frac{4}{f^{4}}-1}}-\frac{L}{16G_{N}a}.$

In the zero flux limit $f\to 0$, we find $I_{\rm BTZ}\to-\frac{L}{16G_{N}a}$
and this reproduces the known result of the AdS/BCFT without the gauge flux in
[26, 27].

###  3.3 Localized gauge field on EOW brane in TAdS phase

Now we look at the EOW brane solution with the localized gauge field in
thermal AdS3 (TAdS), which is the other phase for BCFT on a cylinder. The
metric is given by

$ds^{2}=\frac{dz^{2}}{k(z)z^{2}}+\frac{d\tau^{2}}{z^{2}}+\frac{k(z)dx^{2}}{z^{2}},$
(3.31)

where $k(z)=1-\frac{z^{2}}{b^{2}}$. We compactify $x$ as $x\sim x+L$ as
before. The requirement of smooth geometry fixes $L$ to be

$\displaystyle L=2\pi b.$ (3.32)

####  3.3.1 Construction of solution

For the EOW brane defined by $z=z(\tau)$, the induced metric reads

$ds^{2}=\frac{k(z)+\dot{z}^{2}}{k(z)z^{2}}d\tau^{2}+\frac{k(z)dx^{2}}{z^{2}}.$
(3.33)

The extrinsic curvature is found to be

$\displaystyle K_{\tau\tau}-Kh_{\tau\tau}$
$\displaystyle=\frac{z(2k^{2}-zk^{\prime}\dot{z}^{2}+2k\dot{z}^{2}+2kz\ddot{z})}{2z^{2}(k+\dot{z}^{2})^{\frac{3}{2}}},$
$\displaystyle K_{xx}-Kh_{xx}$
$\displaystyle=\frac{(2k-zk^{\prime})\sqrt{k+\dot{z}^{2}}}{2z^{2}k}.$ (3.34)

The equations of motion for brane (3.2) reads

$\displaystyle K_{xx}-Kh_{xx}$ $\displaystyle=\frac{kz^{2}F_{\tau
x}^{2}}{2(k+\dot{z}^{2})},$ $\displaystyle K_{\tau\tau}-Kh_{\tau\tau}$
$\displaystyle=\frac{F_{\tau x}^{2}z^{2}}{2k}.$ (3.35)

The gauge field equation of motion (3.6) is solved to be

$\displaystyle F_{\tau x}=f\frac{\sqrt{k(z)+\dot{z}^{2}}}{z^{2}}.$ (3.36)

After short algebra, we obtain the differential equation for the brane profile

$\dot{z}^{2}=\frac{4}{f^{4}}-k(z),$ (3.37)

which is solved as

$z=C_{1}e^{\frac{\tau}{b}}+C_{2}e^{-\frac{\tau}{b}},$ (3.38)

where $C_{1}$ and $C_{2}$ are arbitrary constants with the constraint
$4C_{1}C_{2}=-b^{2}(\frac{4}{f^{4}}-1)$. However, this is not an expected
connected solution dual to a BCFT on a strip. Therefore in the thermal AdS3
case, the only available dual solution is the two disconnected EOW branes
localized at $z=\pm\frac{\pi}{2}a$.

As before, we are considering a BCFT on a cylinder whose width is $\pi a$ and
the circumference is $L=2\pi b$. Since the EOW brane can end on the tip $z=b$
in the TAdS geometry, we choose the two disconnected EOW branes from the
general solution (3.38):

$\displaystyle z=b\sqrt{\frac{4}{f^{4}}-1}\sinh\frac{\tau}{b},$ (3.39)

and

$\displaystyle z=b\sqrt{\frac{4}{f^{4}}-1}\sinh\frac{\pi a-\tau}{b},$ (3.40)

which is plotted in figure 15. We introduce the cut off $\delta$ again such
that $z=\epsilon$ at $\tau=\delta$, which leads to
$\delta=\frac{\epsilon}{\sqrt{4/f^{4}-1}}$.

Figure 15: The profiles $z=z(\tau)$ of the EOW branes with the gauge field in
TAdS geometry. The horizontal and vertical axis describes $\tau$ and $z$,
respectively. We choose $b=1$, $\pi a=5$ and $f=1$.

We introduce a positive constant $\tau_{0}$ such that
$\sqrt{\frac{4}{f^{4}}-1}\sinh\frac{\tau_{0}}{b}=1$. Then the existence of
such a disconnected solution is possible only when

$\displaystyle\tau_{0}<\frac{\pi}{2a},$ (3.41)

to avoid an intersection of EOW branes.

####  3.3.2 Free energy in TAdS phase

The evaluation of free energy in the TAdS case can be conducted similarly by
computing each term in (3.28) as follows:

$\displaystyle I_{\rm EH}$ $\displaystyle=\frac{1}{4\pi
G_{N}}\int{g}=\frac{L}{2\pi
G_{N}}\int^{\frac{\pi}{2}a}_{\delta}d\tau\left[\frac{1}{2\epsilon^{2}}-\frac{1}{2z^{2}}\right],$
$\displaystyle I_{\rm Q}$ $\displaystyle=\frac{3f^{2}}{16\pi
G_{N}}\int_{Q}\sqrt{h}=\frac{3L}{4\pi
G_{N}}\int^{\tau_{0}}_{\delta}d\tau\frac{1}{z^{2}},$ $\displaystyle
I_{\rm\Sigma}-I_{\rm c.t}$ $\displaystyle=-\frac{(\pi
a-2\delta)\tilde{L}}{8\pi G_{N}\epsilon^{2}},$ (3.42)

where we defined $\tilde{L}=L\sqrt{1-\epsilon^{2}/\beta^{2}}$ and $I_{\rm
c.t}$ is the action for the vacuum Poincaré configuration with two
disconnected straight branes. Notice also that $L$ is determined by $b$ as in
(3.32).

By combining all terms, we eventually find

$\displaystyle I_{\rm TAdS}$ $\displaystyle=$ $\displaystyle\frac{L}{2\pi
G_{N}}\int^{\tau_{0}}_{\delta}\frac{d\tau}{z(\tau)^{2}}-\frac{L}{4\pi
G_{N}b^{2}}\left(\frac{\pi}{2}a-\tau_{0}\right)+\frac{La}{16\pi G_{N}b^{2}}$
(3.43) $\displaystyle\simeq$ $\displaystyle\frac{L}{2\pi
G_{N}\epsilon\sqrt{\frac{4}{f^{4}}-1}}-\frac{L}{\pi
G_{N}b}\cdot\frac{f^{2}}{4-f^{4}}+\frac{L}{4\pi
G_{N}b^{2}}\left(-\frac{\pi}{4}a+\tau_{0}\right).$

In the zero flux limit $f\to 0$, we obtain $I_{\rm
TAdS}\to-\frac{La}{16G_{N}b^{2}}$ which reproduces the known result of
AdS/BCFT without the gauge field [26, 27].

###  3.4 Phase transitions between BTZ and TAdS phase

In summary, the finite part of free energies of EOW brane background with the
localized gauge field in the BTZ and TAdS phase read

$\displaystyle I^{\rm Finite}_{\rm BTZ}=-\frac{\pi b}{8G_{N}a},$
$\displaystyle I^{\rm Finite}_{\rm
TAdS}=-\frac{2f^{2}}{(4-f^{4})G_{N}}-\frac{\pi
a}{8G_{N}b}+\frac{\tau_{0}}{2G_{N}b},$ (3.44)

where we set $L=2\pi b$ using (3.32).

These two solutions are possible ones for the gravity dual of a BCFT on a
cylinder with the width $\pi a$ and the circumference $2\pi b$. In general we
expect that when $a\ll b$ the BTZ phase is favored, while the TAdS phase does
when $b\ll a$.

The phase structure is plotted in figure 16 for various values of $f$. The
TAdS phase becomes dominant when the blue and orange colored regions overlap.
In the zero flux limit $f\to 0$, the TAdS is realized when $b<a$, which
reproduces the known result in [26, 27]. As the flux gets larger, the TAdS
phase gets squeezed as can be seen from figure 16. It is also intriguing that
there is not Poincaré AdS3 phase for an EOW brane with the localized gauge
flux, as opposed to the one with the localized scalar field.

Figure 16: Comparison of free energy between BTZ and TAdS phases of the EOW
brane with gauge field. The blue regions satisfy the condition (3.41) where
the EOW brane solution in TAdS exists. The orange region describes the region
where the TAdS solution is favored over the BTZ one i.e.
$I_{\mathrm{TAdS}}<I_{\mathrm{BTZ}}$. The left, middle and right panels
correspond to $f=0.1,f=1$ and $f=1.4$, respectively. Note that at $f=0$ we
recover the known result $b<a$.

##  4 Soft wall model with bulk scalar

So far we have discussed the pseudo entropy defined by the transition matrix
constructed from two different boundary states. In the bulk gravity side, we
consider the brane-localized field which interpolates these boundary states
with thin branes. It is also natural to consider the bulk field analogue of
this interpolating brane-localized field. In other words, we have so far
discussed the hard wall model and there should also be the corresponding soft
wall model. To this end, we consider a bulk dilaton field which interpolates
two different field values at the conformal boundaries. In the conformal field
theory side, this is nothing but interface CFTs [78, 79, 80]. See figure 18
and figure 18 for the boundary path integral dual to the previous AdS/BCFT
model and that for our current bulk field model, respectively.

In our present model, the path integral prepares a transition matrix defined
by two vacuum states for two CFTs with different marginal deformation. More
concretely, we consider the two CFT with Hamiltonian
$H_{\pm}=H_{0}\pm\phi_{\pm}\int O$ and each of the Hamiltonian has their own
vacuum, say $\ket{0_{\pm}}$. Here $\phi_{\pm}$ are boundary values of the
dilaton field which is dual to the brane deformation. Then we consider a
constant Euclidean time slice $\tau=\tau_{0}$, where the transition matrix is
defined. In this setup, the transition matrix $\mathcal{T}$ for $\tau_{0}>0$
can be expressed as

$\begin{split}\mathcal{T}(\tau_{0}>0)&=\mathcal{N}\cdot\lim_{T\to\infty}e^{-\tau_{0}H_{+}}e^{-TH_{-}}\ket{0_{-}}\bra{0_{+}}e^{-H_{+}(T-t_{0})}\\\
&=\mathcal{N}\cdot e^{-\tau_{0}H_{+}}\ket{0_{-}}\bra{0_{+}},\end{split}$ (4.1)

where $\mathcal{N}$ is a normalization factor. Similarly for $\tau_{0}<0$, we
have

$\mathcal{T}(\tau_{0}<0)=\mathcal{N}\cdot\ket{0_{-}}\bra{0_{+}}e^{\tau_{0}H_{-}}.$
(4.2)

One difference from the usual holographic model of the interface CFTs is that
we put the defect on the time slice and compute the pseudo entropy of the sub-
region which is parallel to this defect.

$\ket{B(\phi_{1})}$$\bra{B(\phi_{2})}$$\tau$$A$$\tau_{0}$$\tau=0$$\frac{\beta}{2}$
Figure 17: The model with different boundary states discussed in the above
sections.

$\tau$$A$$\tau_{0}$$\tau=0$CFT${}_{\Phi_{2}}$CFT${}_{\Phi_{1}}$defect Figure
18: The model with bulk fields. The bulk fields interpolate two CFTs,
CFT${}_{\Phi_{1}}$ and CFT${}_{\Phi_{2}}$. In the CFT side, there is a defect
lies on $\tau=0$.

The solution of the Einstein equation with back-reacting dilaton field is
known as the Janus solution [78, 79, 80]. Here we focus on the AdS3 case [80].
Explicitly, the Janus solution with the defect on Euclidean time slice
$\tau=0$ can be written as follows,

$\begin{split}ds^{2}&=dr^{2}+f(r)\frac{dx^{2}+d\xi^{2}}{\xi^{2}},\quad
f(r)=\frac{1}{2}(1+\sqrt{1-2\gamma^{2}}\cosh{2r}),\\\
\Phi(r)&=\frac{1}{\sqrt{2}}\log\quantity(\frac{1+\sqrt{1-2\gamma^{2}}+\sqrt{2}\gamma\tanh{r}}{1+\sqrt{1-2\gamma^{2}}-\sqrt{2}\gamma\tanh{r}}),\end{split}$
(4.3)

where we adopt the hyperbolic slice coordinate $(r,\xi,x)$. Here $\gamma$ is a
constant related to the strength of deformation by spacelike defect as usual
defect. Especially, when $\gamma=0$, there is no defect on the CFT side. In
this case, the gravity dual is the vacuum AdS3. We consider the range of gamma
$\absolutevalue{\gamma}<\gamma_{c}=\frac{1}{\sqrt{2}}$. Also the $\gamma$ is
related to the asymptotic value of the dilaton field $\Phi$ as

$\Phi(r\to\pm\infty)=\frac{1}{\sqrt{2}}\log\quantity(\frac{1+\sqrt{1-2\gamma^{2}}\pm\sqrt{2}\gamma}{1+\sqrt{1-2\gamma^{2}}\mp\sqrt{2}\gamma}).$
(4.4)

$\tau$$z$$r$ Figure 19: Figure for the Janus solution (4.3). The dashed line
denotes the constant $r$ surfaces, $r=\text{const.}$

###  4.1 Pseudo entropy from Euclidean Janus solution

Let us move on to the computation of the pseudo entropy in our setup. As in
figure 18, we take the subsystem $A$ to be parallel to the defect. Note that
this is different from situations explored in [81, 82] where the subsystem is
taken to be perpendicular to the defect. We compute both the Euclidean and the
Lorentzian time evolution of the pseudo entropy with the same trick as above.
That is, we take the subsystem $A$ as the interval $[-l,l]$ on the constant
Euclidean time slice $\tau=\text{const.}$ and perform the Wick rotation after
all the computations. Thus our computation of the pseudo entropy is basically
entropy of the transition matrix obtained from the two distinct vacuum states
and its Lorentzian time evolution. Note that in our computation, the bulk
geometry is not AdS spacetime itself, the choice of the time coordinate is not
the same as the usual one. Here we relate the bulk time coordinate and the
boundary time coordinate by using the analogue of the coordinate
transformation in the vacuum AdS3. We see this point in detail later.

Finally we move to the computation of the holographic pseudo entropy. What we
need to do is to compute the geodesic length connecting the entangling surface
$\partial A$ in the Janus geometry (4.3).

##### The geodesic equation

To compute the geodesic $x=x(r),\xi=\xi(r)$ and its length, we consider the
Lagrangian

$\mathcal{L}=\sqrt{1+f(r)\frac{\dot{x}^{2}+\dot{\xi}^{2}}{\xi^{2}}},$ (4.5)

where the dots denote the $r$ derivative. To compute the geodesic, it is
useful to utilize the symmetries in this setup. The Lagrangian (4.5) has two
symmetries,

  1. 1.

AdS2 isometry

$x\to x+\lambda x,\quad\xi\to\xi+\lambda\xi.$ (4.6)

The corresponding Noether charge is

$Q_{\mathrm{AdS}_{2}}=\frac{f(r)}{\mathcal{L}}\frac{\xi\dot{\xi}+x\dot{x}}{\xi^{2}}.$
(4.7)

  2. 2.

$x$-translation

$x\to x+c.$ (4.8)

The corresponding Noether charge is

$Q_{T}=\frac{f(r)}{\mathcal{L}}\frac{\dot{x}}{\xi^{2}}.$ (4.9)

By dividing (4.7) by (4.9), we have

$\begin{split}\frac{Q_{\mathrm{AdS}_{2}}}{Q_{T}}&=\frac{1}{2}\frac{d}{dx}\quantity(\xi^{2}(x)+x^{2}),\\\
\xi^{2}(x)+x^{2}&=2\frac{Q_{\mathrm{AdS}_{2}}}{Q_{T}}x+A.\end{split}$ (4.10)

where $A$ is an integration constant. Thanks to the $\mathbb{Z}_{2}$ symmetry
$x\to-x$, we find $Q_{\mathrm{AdS}_{2}}=0$. Thus, we have

$\xi^{2}+x^{2}=A,$ (4.11)

with some constant $A>0$. Note that to derive (4.11), we do not use the
concrete form of $f(r)$. By combining (4.5), (4.11) and (4.9), we obtain
geodesic equations,

$\begin{split}\left(\frac{\sqrt{A}\dot{x}}{A-x^{2}}\right)^{2}&=\frac{\alpha}{f(r)(f(r)-\alpha)},\\\
\mathcal{L}^{2}&=\frac{f(r)}{f(r)-\alpha},\quad\alpha=AQ_{T}^{2}.\end{split}$
(4.12)

From the first equation of (4.12) we see

$\begin{split}\frac{x(r)}{\sqrt{A}}&=\tanh{\quantity(-\sqrt{\frac{-k}{\chi}}F\left(\left.G(r)\right|k\right)+B)},\\\
G(r)&:=\mathrm{Arcsin}{\quantity(\frac{1}{2}\sqrt{\frac{(2\alpha-\chi-1)\left(\chi\cosh{2r}+1\right)\text{csch}^{2}{r}}{\alpha\chi}})},\\\
k&=-\frac{4\alpha\chi}{(1-\chi)(2\alpha-\chi-1)},\end{split}$ (4.13)

where $\chi=\sqrt{1-2\gamma^{2}}$, $B$ is an integration constant and
$-\frac{\pi}{2}<\mathrm{Arcsin}{\quantity(\cdot)}<\frac{\pi}{2}$.
$F\left(\left.z\right|m\right)$ denotes the incomplete elliptic integral of
the first kind. See for details of the elliptic integrals in appendix A.2.

##### Cutoff surface

In AdS/CFT correspondence, we need to regularize the physical quantities of
the boundary theory by introducing a cutoff surface in the bulk gravity. For
asymptotically AdS spacetime, the regularization procedure is based on the
Fefferman-Graham expansion of the asymptotically AdS spacetime[83, 84].
However, it is known that for the Janus geometry dual to the interface CFT we
cannot construct the FG coordinates which cover all spacetime[85, 86]. In
other words, there may be several ways for defining the coordinate
transformation from the H2 slice coordinate $(r,\xi,x)$ to the Poincaré-like
coordinate $(z,t,x)$. There are several considerations about ways of putting
the cutoff surface for these geometries[87]. Here, we define the AdS
(regularized) boundary by

$\displaystyle\xi=\epsilon\sqrt{f(r)}.$ (4.14)

This definition is motivated by two reasons. One is that, in the pure AdS
case, $\gamma=0$ and $\chi=1$, this surface completely mathcs $z=\epsilon$ in
the Poincaré coordinates. We can easily see this from (A.1). The second reason
is that, by properly defining time $\tau$, we can set the induced metric of
the cutoff surface which will be the CFT metric as the flat metric. Indeed,
from (4.14), the induced metric reads

$\displaystyle ds^{2}|_{\mathrm{ind}}$ $\displaystyle=$
$\displaystyle\frac{dx^{2}}{\epsilon^{2}}+\frac{1}{\epsilon^{2}}\left(1+\frac{4f}{f^{\prime
2}}\right)d\xi^{2}.$ (4.15)

We now introduce the boundary time $\tau$ by setting

$\displaystyle d\tau^{2}=\left(1+\frac{\epsilon^{2}}{\xi^{\prime
2}}\right)d\xi^{2}=\quantity(1+\frac{\quantity(\frac{\xi}{\epsilon})^{2}}{-\chi^{2}+\quantity(2\quantity(\frac{\xi}{\epsilon})^{2}-1)^{2}})d\xi^{2},$
(4.16)

so that the boundary metric looks like

$\displaystyle ds^{2}=\frac{dx^{2}+d\tau^{2}}{\epsilon^{2}}.$ (4.17)

We can find $\tau=\tau(\xi)$ by solving (4.16). Note that for $\tau\gg
O(\epsilon)$, we find $\tau\sim\xi$, but for $\tau\sim O(\epsilon)$ there is a
bit difference from $\xi$.

Figure 20: Plots for the $\tau(\xi)$ with $\chi=0.7,\epsilon=0.01$. For
$\tau_{0}\gg\epsilon$, we confirm that $\tau(\xi)\sim\xi$. The right figure
shows the plots for the time $\tau_{0}=O(\epsilon)$.

##### Determination of the constant $A,B,\alpha$

Firstly let us determine $A$. As mentioned above, (4.11) is derived just by
the AdS2 structure of the metric and it does not depend on the specific form
of $f(r)$. Thus we suppose the value of $A$ is universal and is the same as
the vacuum case, i.e.

$A=l^{2}+\tau_{0}^{2}.$ (4.18)

This will be valid at least when $\tau_{0}\gg\epsilon$. The constant $B$ can
be determined from the boundary condition at the turning point $x=0,r=r_{t}$.
At the turning point, $\dot{x}(r)$ should be divergent. From the first
equation of (4.12), we find that $f(r_{t})=\alpha$, i.e.

$1+\chi\cosh{2r_{t}}=2\alpha,$ (4.19)

holds. By using this identity, we also find

$G(r_{t})=\frac{\pi}{2}.$ (4.20)

Finally, by evaluating (4.13) at the turning point $r=r_{t},x=0$, we obtain

$B=\sqrt{\frac{-k}{\chi}}K(k),$ (4.21)

where $K\left(k\right)=F\left(\left.\frac{\pi}{2}\right|k\right)$ denotes the
complete elliptic integral of the first kind.

To determine the constant $\alpha$ we introduce the value of $r_{\infty}$
where geodesic pass through the cutoff surface (4.14). By

$\frac{\xi(r_{\infty})}{\sqrt{f(r_{\infty})}}=\epsilon$ (4.22)

The constant $\alpha$ is determined with (4.13) at the cutoff surface $r\to
r_{\infty}$ and $x\to l$

$\frac{l}{\sqrt{A}}=-\tanh{\quantity(\sqrt{\frac{-k}{\chi}}\quantity(F\left(\left.G(r_{\infty})\right|k\right)-K\left(k\right)))}.$
(4.23)

##### The geodesic length

Next we evaluate the geodesic length. This is done with an analogy of the
coordinate change between the hyperbolic coordinate and the usual Poincaré
coordinate in the vacuum AdS3[85]. See the appendix A.1.

As we can solve the geodesic equation in terms of the elliptic integrals, we
can also evaluate the geodesic length in an explicit manner. Indeed, the
length of the HRT surface $\Gamma_{A}$ is given by

$\begin{split}\mathrm{Area}[\Gamma_{A}]&=2\int_{r_{t}}^{r_{\infty}}dr\sqrt{\frac{f(r)}{f(r)-\alpha}}\\\
&=\quantity[\sqrt{\frac{m}{\alpha\chi}}\quantity((\chi+1)F\left(\left.H(r)\right|m\right)+(2\alpha-\chi-1)\Pi\left(\left.n;H(r)\right|m\right))]^{r_{\infty}}_{r_{t}},\end{split}$
(4.24)

where $\Pi\left(\left.n;z\right|m\right)$ is the incomplete elliptic integral
of the third kind and we introduce

$\begin{split}H(r)&:=\mathrm{Arcsin}{\left(\frac{1}{2}\sqrt{\frac{(\chi+1)(-2\alpha+\chi\cosh(2r)+1)\text{csch}^{2}(r)}{\alpha\chi}}\right)},\\\
m&:=\frac{4\alpha\chi}{(\chi+1)(2\alpha+\chi-1)},\quad
n:=\frac{2\alpha}{\chi+1}.\end{split}$ (4.25)

By using some identity involving the elliptic integrals and the formula for
the cutoff (4.22) and the turning point (4.19), we can simplify the expression
of the geodesic length. At the turning point (4.19), we can easily see
$H(r_{t})=0$ and

$F\left(\left.H(r_{t})\right|m\right)=0,\quad\Pi\left(\left.n;H(r_{t})\right|m\right)=0.$
(4.26)

Next, we try to simplify the $r=r_{\infty}$ part of (4.24) in the specific
range. To this end, let us recall the following formula [88].

$\begin{split}\Pi\left(\left.n;z\right|m\right)&=F\left(\left.z\right|m\right)-\Pi\left(\left.\frac{m}{n};z\right|m\right)\\\
&+\frac{1}{2\sqrt{\frac{(n-m)(n-1)}{n}}}\log\left(\frac{\sqrt{\frac{(n-m)(n-1)}{n}}\tan{z}+\sqrt{1-m\sin^{2}{z}}}{-\sqrt{\frac{(n-m)(n-1)}{n}}\tan{z}+\sqrt{1-m\sin^{2}{z}}}\right).\end{split}$
(4.27)

Finally we have an expression for the geodesic length,

$\begin{split}\mathrm{Area}[\Gamma_{A}]&=\sqrt{\frac{m}{\alpha\chi}}\quantity(2\alpha
F\left(\left.H(r_{\infty})\right|m\right)-(2\alpha-\chi-1)\Pi\left(\left.\frac{m}{n};H(r_{\infty})\right|m\right))\\\
&+\log\left(\frac{\sqrt{\frac{(n-m)(n-1)}{n}}\tan{H(r_{\infty})}+\sqrt{1-m\sin^{2}{H(r_{\infty})}}}{-\sqrt{\frac{(n-m)(n-1)}{n}}\tan{H(r_{\infty})}+\sqrt{1-m\sin^{2}{H(r_{\infty})}}}\right).\end{split}$
(4.28)

The pseudo entropy $S_{A}$ is given by the usual HRT formula (2.6). As similar
to the discussion in the previous subsection, we also consider the analytic
continuation to Lorentzian path integral at $\tau_{0}=0$. Formally, this is
done by replacing $\tau_{0}=it_{0}+\epsilon$. We show that the Euclidean and
Lorentzian time evolution of the pseudo entropy $S_{A}$ in figure 21 and
figure 22 respectively222We should take care about the branch of
$\sqrt{\frac{(n-m)(n-1)}{n}}$.. We see that the pseudo entropy grows mildly as
$S_{A}\sim\log{\frac{\tau_{0}}{l}}$ in the Euclidean time, and in Lorentzian
time at time up to the half of the subsystem size $l$. At late time, the
pseudo entropy reaches the value of vacuum entanglement entropy
$S_{0}=\frac{c}{3}\log\frac{2l}{\epsilon}$. For the Euclidean case, this is
because the large Euclidean time evolution makes the states to the vacuum of
the Hamiltonian. For the Lorentzian time case, we may interpret that the
reduced transition matrix forgets about the information, that is, one of the
defects. However, of course, the global transition matrix contains the
information. The pseudo etropies take small value around the defect. This is
because the two vacuum states are different states for $\gamma\neq 0$. For the
Lorentzian time evolution, the pseudo entropy moves up and down around
$t_{0}\sim l$. This behavior should be checked from the boundary side. We
leave this as a future problem.

Let us then introduce a parameter $\delta$ as

$\delta=\frac{2(2\alpha-\chi-1)}{\chi}e^{-2r_{\infty}}=2(2\alpha-\chi-1)\quantity(\frac{\epsilon}{2\tau_{0}})^{2}.$
(4.29)

This quantity is motivated by the formula for the definition of $r_{\infty}$.
At least for the time scale $\epsilon\ll\absolutevalue{\tau_{0}}$, we can
safely suppose that the parameter $\delta$ is small. Since
$H(r_{\infty})\approx\mathrm{Arcsin}{\sqrt{\frac{1-\delta}{n}}}$ for small
$\delta$, we obtain

$\begin{split}\Pi\left(\left.n;H(r_{\infty})\right|m\right)&=F\left(\left.H_{\infty}\right|m\right)-\Pi\left(\left.\frac{m}{n};H_{\infty}\right|m\right)+\frac{\log\left(\frac{4}{\delta}\left(\frac{n}{n-1}+\frac{m}{n-m}\right)^{-1}\right)}{2\sqrt{\frac{(n-1)(n-m)}{n}}}+O\left(\delta^{1}\right),\end{split}$
(4.30)

where we introduce $H_{\infty}=\mathrm{Arcsin}{\sqrt{\frac{1}{n}}}$. Also note
that $F\left(\left.H_{\infty}\right|m\right)$ and
$\Pi\left(\left.\frac{m}{n};H_{\infty}\right|m\right)$ is $O(\delta^{0})$.

Finally we obtain the formula for the geodesic length as follows

$\begin{split}\mathrm{Area}[\Gamma_{A}]&=\sqrt{\frac{m}{\alpha\chi}}\quantity(2\alpha
F\left(\left.H_{\infty}\right|m\right)-(2\alpha-\chi-1)\Pi\left(\left.\frac{m}{n};H_{\infty}\right|m\right))\\\
&+\log\left(\frac{4}{\delta}\left(\frac{n}{n-1}+\frac{m}{n-m}\right)^{-1}\right)+O(\delta^{1}).\end{split}$
(4.31)

Note that the pseudo entropy has the logarithmic cut-off dependence similar to
the vacuum entanglement entropy in CFT2. Note that this formula only works
when $\absolutevalue{\tau_{0}}\gg\epsilon$,i.e. $\delta\ll 1$.

Figure 21: The plots of the pseudo entropy in Euclidean time evolution
$S_{A}(\tau_{0})$. We set $\gamma=0.35,\epsilon=0.01,l=1$. The black line
expresses the entanglement entropy for undeformed CFT
$S_{0}=\frac{c}{3}\log\frac{2l}{\epsilon}$.

Figure 22: The plots of the pseudo entropy in Lorentzian time evolution after
the Wick rotation. the first picture is plot with $S_{A}(t_{0})$.
$\gamma=0.4,\epsilon=\epsilon^{\prime}=0.01,l=1$ in $t_{0}\in[0,5]$. The first
figure shows the real part of the PE and the second one is its imaginary part.

####  4.1.1 Late time behavior of the pseudo entropy

##### Late Euclidean time $\tau_{0}\gg l$

Since $\tau_{0}\to\infty$ is a limit away from the defect, we naturally expect
that the situation is close to the vacuum. When $\gamma=0$,
$\alpha=1+\frac{\tau_{0}^{2}}{l^{2}}$, we expect $\alpha\to\infty$ as
$\tau\to\infty$. To confirm this fact, we take $\tau_{0}\gg l$ and $\alpha\gg
1$ limit of (4.23). After a short algebra we find that

$\begin{split}(\text{l.h.s})&=\frac{l}{\tau_{0}}+O\quantity(\quantity(\frac{l}{\tau_{0}})^{3}),\\\
(\text{r.h.s})&=\sqrt{\frac{1}{\alpha}}+O\quantity(\frac{1}{\alpha^{\frac{3}{2}}}).\end{split}$
(4.32)

Thus we can conclude $\alpha=\quantity(\frac{\tau_{0}}{l})^{2}+\cdots$ for
general $\gamma$. Also we can estimate the geodesic length (4.31) when
$\tau_{0}\gg l\gg\epsilon$. Then we see the pseudo entropy has the late
Euclidean time behavior,

$S_{A}=\frac{c}{3}\log\quantity(\frac{2l}{\epsilon})+O\quantity(\quantity(\frac{l}{\tau_{0}})^{2}).$
(4.33)

As a result, the pseudo entropy approaches the vacuum entanglement, which
matches our expectation.

##### Late Lorentzian time $t_{0}\gg l$

With Wick rotation $\tau_{0}=it+\epsilon^{\prime},t>0$, we find the real time
dynamics with the quantum states prepared from the Janus path integral.
Following the same procedure as above, we can analyze the late time behavior
of the pseudo entropy. However, we should take care of the branch of the
argument a bit. Similar to the Euclidean analysis, we expect
$\alpha\sim\frac{\tau_{0}^{2}}{l^{2}}\sim-\frac{t^{2}}{l^{2}}\quantity(1-\frac{2i\epsilon^{\prime}}{t})$.
Then, choosing $y=\frac{1+\chi}{2\alpha}$, we find

$\begin{split}-\tanh{\quantity(\sqrt{\frac{-k}{\chi}}\quantity(F\left(\left.\mathrm{Arcsin}{\sqrt{1-y}}\right|k\right)-K\left(k\right)))}&=-\sqrt{\frac{2y}{1+\chi}}+O(y^{\frac{5}{2}})\\\
\frac{1}{\sqrt{1+\frac{\tau_{0}^{2}}{l^{2}}}}&=\frac{1}{\sqrt{-\frac{t_{0}^{2}}{l^{2}}}}+O\quantity(\frac{l^{3}}{t_{0}^{3}}),\end{split}$
(4.34)

in $\alpha\to-\infty,t_{0}\to\infty$. Thus we confirm
$\alpha=-\frac{t_{0}^{2}}{l^{2}}+\cdots$. Next, we evaluate the pseudo entropy
at late time. First, let us evaluate the inside of the square root in the
(4.25) at $r=r_{\infty}$,

$q:=\frac{(\chi+1)(-2\alpha+\chi\cosh{(r_{\infty})}+1)\text{csch}^{2}(r_{\infty})}{2\alpha\chi}=-\frac{1+\chi}{2t^{2}}(l^{2}-\epsilon^{2})\quantity(1+i\frac{4l^{2}}{l^{2}-\epsilon^{2}}\frac{\epsilon^{\prime}}{t})+\cdots.$
(4.35)

Since $q$ is a complex number, we should take care of the branch of the square
root333$\sqrt{q}=e^{\frac{1}{2}\log{\absolutevalue{q}}+\frac{i}{2}\mathrm{Arg}\quantity(q)+in\pi}=\pm\sqrt{\absolutevalue{q}}e^{\frac{i}{2}\mathrm{Arg}\quantity(q)}$,
where $\mathrm{Arg}\quantity(\cdot)$ runs from $-\pi$ to $\pi$.. We take the
positive sign here. The negative sign branch just gives the pseudo entropy
with the opposite sign of the positive one. Note that

$\begin{split}\sqrt{\absolutevalue{q}}=\frac{l}{t}\sqrt{\frac{1+\chi}{2}}\sqrt{\quantity(1-\frac{\epsilon^{2}}{l^{2}})\quantity(1+\quantity(\frac{4}{1-\frac{\epsilon^{2}}{l^{2}}}))^{2}\quantity(\frac{\epsilon^{\prime}}{t})^{2}}+\cdots\end{split}$
(4.36)

by recalling that $\mathrm{Arg}\quantity(q)=-\pi+\frac{2\epsilon^{\prime}}{t}$
for $t>0$. In this case,

$\begin{split}\sqrt{q}=-i\frac{l}{t}\sqrt{\frac{1+\chi}{2}}\sqrt{\quantity(1-\frac{\epsilon^{2}}{l^{2}})\quantity(1+\quantity(\frac{4}{1-\frac{\epsilon^{2}}{l^{2}}}))^{2}\quantity(\frac{\epsilon^{\prime}}{t})^{2}}e^{i\frac{2\epsilon^{\prime}}{t}}+\cdots,\end{split}$
(4.37)
$\begin{split}H(r_{\infty})=\mathrm{Arcsin}{\sqrt{q}}=-i\frac{\sqrt{l^{2}-\epsilon^{2}}}{\sqrt{2}t}\sqrt{1+\chi}\sqrt{1+\quantity(\frac{4l^{2}}{l^{2}-\epsilon^{2}})^{2}\quantity(\frac{\epsilon^{\prime}}{t})^{2}}e^{i\frac{2\epsilon^{\prime}}{t}}+\cdots.\end{split}$
(4.38)

Similarly, we take the same branch for $\sqrt{\alpha}$,

$\begin{split}\sqrt{\alpha}=\sqrt{\absolutevalue{\alpha}}e^{\frac{i}{2}\mathrm{Arg}\quantity(\alpha)}=i\frac{t}{l}\sqrt{1+\frac{4\epsilon^{\prime
2}}{t^{2}}}e^{-i\frac{2\epsilon^{\prime}}{t}}.\end{split}$ (4.39)

For small $z$, it holds

$2\alpha
F\left(\left.z\right|m\right)-(2\alpha-\chi-1)\Pi\left(\left.\frac{m}{n};z\right|m\right)=(1+2\alpha+2\alpha\chi)z+O(z^{3}).$
(4.40)

By using this expansion, the first term of (4.28) has the series expansion

$\sqrt{\frac{m}{\alpha\chi}}\quantity(2\alpha
F\left(\left.H(r_{\infty})\right|m\right)-(2\alpha-\chi-1)\Pi\left(\left.\frac{m}{n};H(r_{\infty})\right|m\right))=2(1+\chi)+\cdots.$
(4.41)

Similarly we see

$\log{\quantity(\frac{\sqrt{\frac{(n-m)(n-1)}{n}}\tan{H(r_{\infty})}+\sqrt{1-m\sin^{2}{H(r_{\infty})}}}{-\sqrt{\frac{(n-m)(n-1)}{n}}\tan{H(r_{\infty})}+\sqrt{1-m\sin^{2}{H(r_{\infty})}}})}=\log{\quantity(\frac{4l^{2}}{\epsilon^{2}})}+O\quantity(\frac{l^{2}}{t_{0}^{2}},\frac{\epsilon^{2}}{t^{2}}).$
(4.42)

From this, we can confirm that the pseudo entropy approaches the vacuum
entanglement entropy in the Lorentzian late time in this branch.

####  4.1.2 Early time behavior

Here we consider the behavior of the pseudo entropy in the period of
$\epsilon\ll\absolutevalue{\tau_{0}}\ll l$. Again, we consider the subsystem
$A$ with the length $2l$ (i.e. $A=[-l,l]$) at the specific time
$\tau_{0}\simeq\xi(\infty)$. We find the relation between the value of $r$ at
the turning point $r_{t}$ and $L$:

$\displaystyle\frac{l}{\xi(\infty)}=\sinh\left[\int^{\infty}_{r_{t}}\frac{dr}{\sqrt{f(r)\left(\frac{f(r)}{f(r_{t})}-1\right)}}\right].$
(4.43)

Let us consider the limit $r_{t}\to 0$ or equally $\frac{L}{\xi(\infty)}\gg 1$
. By using the fact that the dominant contribution for the integration is
around $r\sim r_{t}$ and for rough calculation we can obtain $f(r)$ to
$f(r_{t})$ and since $r_{t}$ is small $f(r_{t})=\alpha=\frac{1+\chi}{2}$, we
obtain

$\displaystyle\int^{\infty}_{r_{t}}\frac{dr}{\sqrt{f(r)\left(\frac{f(r)}{f(r_{t})}-1\right)}}\simeq-\frac{1}{\chi}\log\left[\sqrt{\frac{1-\chi}{1+\chi}}\frac{r_{t}}{4}\right]+O(r_{t}^{2}).$
(4.44)

Thus, also by using $\tau_{0}\simeq\xi(\infty)$, we find

$\displaystyle\frac{L}{\tau_{0}}\simeq\frac{1}{2}\left(\sqrt{\frac{1-\chi}{1+\chi}}\frac{r_{t}}{4}\right)^{-\frac{1}{\sqrt{\chi}}}.$
(4.45)

The UV cutoff can be estimated for $\tau_{0}\gg\epsilon$ by
$\epsilon\simeq\frac{\tau_{0}}{\sqrt{f(r_{\infty})}}$ i.e.

$r_{\infty}=\log\left(\frac{2}{\sqrt{\chi}}\frac{\tau_{0}}{\epsilon}\right).$
(4.46)

Now we can estimate the area of $\mathrm{\Gamma_{A}}$ in a similar way. We can
evaluate its behavior in the early time limit $r_{t}\to 0$ as follows

$\displaystyle\mathrm{Area}\quantity[\Gamma_{A}]$ $\displaystyle=$
$\displaystyle 2\int_{r_{t}}^{r_{\infty}}dr\sqrt{\frac{f(r)}{f(r)-\alpha}}$
(4.47) $\displaystyle\simeq$ $\displaystyle
2r_{\infty}-\frac{\sqrt{2(1+\chi)}}{\chi}\log\left(\sqrt{\frac{1-\chi}{1+\chi}}\frac{r_{t}}{4}\right)+O(1),$
$\displaystyle\simeq$
$\displaystyle-2\log\epsilon+\left(2-\frac{\sqrt{2(1+\chi)}}{\chi}\right)\log\tau_{0}+\frac{\sqrt{2(1+\chi)}}{\chi}\log
l.$

The first term comes from the integral around $r_{\infty}$. Here
$f(r_{\infty})$ is large and integral for $\mathrm{Area}\quantity[\Gamma_{A}]$
is evaluated in this large $r_{\infty}$ limit. From this result, we find that
the pseudo entropy grows logarithmically in the early time region for both
Euclidean and Lorentzian time evolution.

####  4.1.3 Entanglement entropy on the defect; At $\tau_{0}=t_{0}=0$

From the time-reversal symmetry at $\tau_{0}=t_{0}=0$, we can easily obtain
the geodesic formula for the subsystem on the defect. Especially we see that
the geodesic is on the $r=0$ surface. Fortunately, the metric on the $r=0$
surface takes the same form as that of the constant time slice of the Poincaré
AdS3, i.e.

$ds^{2}|_{r=0}=f(0)\frac{d\xi^{2}+dx^{2}}{\xi^{2}}.$ (4.48)

The cutoff surface is given by $\xi=\epsilon f(0)$. Thus the entanglement
entropy is given by the one of the vacuum entanglement entropy with the
central charge $c_{\mathrm{eff}}=\frac{2l_{\mathrm{AdS}}}{3G_{N}}\sqrt{f(0)}$
and cutoff $\epsilon_{\mathrm{eff}}=\epsilon f(0)$. As a result, we have

$S_{A}(\tau_{0}=t_{0}=0)=\frac{c_{\mathrm{eff}}}{3}\log{\frac{2l}{\epsilon_{\mathrm{eff}}}}=\sqrt{\frac{1+\chi}{2}}\frac{c}{3}\log\quantity(\frac{2}{1+\chi}\frac{2l}{\epsilon}).$
(4.49)

Note that the coefficient $c_{\mathrm{eff}}$ does not depend on the choice of
the cutoff. The effective central charge is a monotonically decreasing
function for the deformation strength $\gamma$. From this fact we may say that
the defect plays a role of the measurement.

###  4.2 Pseudo entropy from time-like Janus solution

####  4.2.1 The setup

Here we treat the Lorentzian path integral directly in the holographic CFT
dual to a spacelike brane on $t=0$, instead of performing analytical
continuation from the Euclidean setup. The path integral again gives the
transition matrix of two distinct vacuum states of the two CFTs. More
specifically, on a constant time slice $t=t_{0}$, we prepare the transition
matrix defined through two states

$\begin{split}\mathcal{T}(t_{0})&=\lim_{T\to\infty}e^{-i(T-\absolutevalue{t_{0}})H_{-}}\ket{0_{-}}\bra{0_{+}}e^{+iTH_{+}}e^{i\absolutevalue{t_{0}}H_{-}}\\\
&=\ket{0_{-}}\bra{0_{+}}e^{+i\absolutevalue{t_{0}}H_{-}}.\end{split}$ (4.50)

The second expression is the same as the analytic continuation of the
Euclidean model and as we see later we have the same plots of the pseudo
entropy. Note that if we prepare quantum states for initial and final states
which are not the eigenstates of the Hamiltonians, then we will have a
different transition matrix from the one obtained from the analytic
continuation. See figure 24 for a sketch. As we will discuss from now on, this
path integral is holographically dual to the Lorentzian Janus solution which
gives a thick spacelike brane in the gravity side.

$t$$A$$t=0$$t=t_{0}$spacelike defectLorentzian Time Figure 23: Sketch for the
path integral of the spacelike brane model for the measurement. We expect that
the spacelike defect decreases entanglement like measurement. The left figure
shows the vacuum process and the right figure shows the multiple measurement
and quantum quench. In this process, there are entanglement creation from
quench and decreasing from defect. So we expect an entanglement transition
occurs.

$t_{0}$$\ket{0_{-}}$$\bra{0_{+}}$

$T-\absolutevalue{t_{0}}$$\absolutevalue{t_{0}}$$T$$\ket{0_{-}}$$\bra{0_{+}}$

Figure 24: The path integrals in the complex time plane for two models in this
section. The vertical direction is Lorenztian and the parallel direction shows
the Euclidean. The left shows the path integral for the analytic continuation
of the Euclidean model in the last subsection. The right shows the path
integral for the current Lorenzian model. If the initial states and final
states are the vacuum states then these two are equivalent.

We know that the vacuum AdS3 with the planar conformal boundary can be written
in the Poincaré coordinate.

$ds^{2}=\frac{-dt^{2}+dz^{2}+dx^{2}}{z^{2}}$ (4.51)

We can introduce new coordinate $(r,\xi,x)$ by the following coordinate
transformation from the Poincaré coordinate,

$\begin{split}z&=\frac{\xi}{\cos{r}},\quad t=\xi\tan{r},\\\
\sin{r}&=\frac{t}{z},\quad z^{2}-t^{2}=\xi^{2}.\end{split}$ (4.52)

In this coordinate, the metric takes the form of

$ds^{2}=-dr^{2}+\cos^{2}{r}\frac{d\xi^{2}+dx^{2}}{\xi^{2}}.$ (4.53)

If we suppose that $(r,\xi)$ is defined on the region such that

$-\frac{\pi}{2}\leq r\leq\frac{\pi}{2},\quad\xi\geq 0,$ (4.54)

then this region is the restricted region in the Poincaré patch

$-z\leq t\leq z.$ (4.55)

We call this region as region $\mathrm{I}\mathrm{I}$. To cover the other
regions, we can analytically continue $(r,\xi)$ to the complex value. We have
two other regions. The $t\geq z$ region, called region $\mathrm{I}$, is
defined by

$\begin{split}r&=-i\rho+\frac{\pi}{2},\quad\rho\geq
0,\quad\xi=i\eta,\quad\eta\geq 0\\\ z&=\frac{\eta}{\sinh{\rho}},\quad
t=\eta\coth{\rho}.\end{split}$ (4.56)

Similarly, the $t\leq-z$ region, called region
$\mathrm{I}\mathrm{I}\mathrm{I}$, is defined by

$\begin{split}r&=i\rho-\frac{\pi}{2},\quad\rho\geq
0,\quad\xi=i\eta,\quad\eta\geq 0\\\ z&=\frac{\eta}{\sinh{\rho}},\quad
t=-\eta\coth{\rho}.\end{split}$ (4.57)

Figure 25: Three regions in the vacuum AdS solution with dS2/H2 slice
coordinate. The blue region represents the region $\mathrm{I}\mathrm{I}$, the
yellow represents the region $\mathrm{I}$ and the red
$\mathrm{I}\mathrm{I}\mathrm{I}$.

Figure 26: The contour maps of the constant $\xi^{2}$ surface (left) and the
constant $r$ surface (right).

Now, the gravity dual of the Lorentzian time evolution of the transition
matrix (4.50) is obtained by solving the Einstein equation with the dilaton
field. The solution can also be obtained from the double Wick rotation of the
original timelike Janus solution [78]. The final solution looks like

$\text{Region $\mathrm{I}\mathrm{I}$}:\quad\begin{aligned}
ds^{2}&=-dr^{2}+f(r)\frac{d\xi^{2}+dx^{2}}{\xi^{2}},\\\
f(r)&=\frac{1}{2}\quantity(1+\sqrt{1-2\gamma^{2}}\cos{2r}),\\\
\Phi(r)&=\frac{1}{\sqrt{2}}\log\quantity(\frac{1+\sqrt{1-2\gamma^{2}}+i\sqrt{2}\gamma\tan{r}}{1+\sqrt{1-2\gamma^{2}}-i\sqrt{2}\gamma\tan{r}}).\end{aligned}$
(4.58)

If we focus on the region $\mathrm{I}\mathrm{I}$, where $r$-coordinate is
real, and if $\gamma$ is real and $\absolutevalue{\gamma}<\frac{1}{\sqrt{2}}$,
then the dilaton field is purely imaginary,

$\begin{split}\Phi(r)&=i\frac{2\gamma\tan{r}}{1+\sqrt{1-2\gamma^{2}}}.\end{split}$
(4.59)

Note that this solution matches the wick rotated geometry of our Euclidean
model (4.3) with peculiar Wick rotation $r\to ir$. By doing the appropriate
analytic continuation, we can obtain the solution for the other two regions,

$\text{Region $\mathrm{I}$}:\quad\begin{aligned}
ds^{2}&=d\rho^{2}-f(\rho)\frac{-d\eta^{2}+dx^{2}}{\eta^{2}},\\\
f(\rho)&=\frac{1}{2}\quantity(1-\sqrt{1-2\gamma^{2}}\cosh{2\rho}),\\\
\Phi(\rho)&=\frac{1}{\sqrt{2}}\log\quantity(\frac{1+\sqrt{1-2\gamma^{2}}+\sqrt{2}\gamma\coth{\rho}}{1+\sqrt{1-2\gamma^{2}}-\sqrt{2}\gamma\coth{\rho}}),\end{aligned}$
(4.60) $\text{Region $\mathrm{I}\mathrm{I}\mathrm{I}$}:\quad\begin{aligned}
ds^{2}&=d\rho^{2}-f(\rho)\frac{-d\eta^{2}+dx^{2}}{\eta^{2}},\\\
f(\rho)&=\frac{1}{2}\quantity(1-\sqrt{1-2\gamma^{2}}\cosh{2\rho}),\\\
\Phi(\rho)&=\frac{1}{\sqrt{2}}\log\quantity(\frac{1+\sqrt{1-2\gamma^{2}}-\sqrt{2}\gamma\coth{\rho}}{1+\sqrt{1-2\gamma^{2}}+\sqrt{2}\gamma\coth{\rho}}).\end{aligned}$
(4.61)

We have two comments on this metric. One is that there exist lines
$\rho=\rho_{c}$ such that

$f(\rho_{c})=0,$ (4.62)

where the dS2 slices shrink to a point. Note that on this critical line,

$\displaystyle 1+\sqrt{1-2\gamma^{2}}-\sqrt{2}\gamma\coth{\rho_{c}}=0,$ (4.63)

holds and the dilaton field diverges. These show that the solution becomes
singular at $\rho=\rho_{c}$. However, we proceed further by assuming that,
since we consider a non-Hermitian evolution, it may have such an exotic
feature in its gravity dual. Moreover, for $0\leq\rho<\rho_{c}$, $f(\rho)>0$
and $\Phi(\rho)$ takes a complex value, which looks like a feature also common
to our brane localized scalar model. The second comment is on the boundary
where CFTs live. Because of the Wick rotation, in region
$\mathrm{I}\mathrm{I}$, the warped factor for the $H_{2}$ slices does not blow
up for every $r$. On the other hand, for the region $\mathrm{I}$ and
$\mathrm{I}\mathrm{I}\mathrm{I}$, at the $\rho\to\infty$ limit, the factor
$-f(\rho)$ in (4.59, 4.61) for the dS2 blows up exponentially. Thus each
region of the spacetime (4.59) (4.61) is asymptotically AdS. Thus we consider
that the holographic theory lives on these asymptotic boundaries.

$\rho=0,r=\frac{\pi}{2}$$\rho=0,r=-\frac{\pi}{2}$$\rho=\rho_{c}$$\rho=\rho_{c}$$\rho,r$$f(\rho)>0$$f(\rho)<0$$f(\rho)>0$$f(\rho)<0$
Figure 27: Sketch for the Janus spacetime duals to the CFT with spacelike
interface with nonzero $\gamma$. The red and pink region denotes the region
$\mathrm{I}\mathrm{I}\mathrm{I}$, the yellow region denotes the region
$\mathrm{I}\mathrm{I}$ and the blue and the light blue region denotes
$\mathrm{I}$. The green line denotes the sketch of the cutoff surface we are
introducing.

####  4.2.2 Time evolution of pseudo entropy

Let us compute the entanglement pseudo entropy holographically in this
solution. Take subsystem $A$ as the interval $[x=-l,x=l]$ on the time slice at
$t=t_{0}$. Here we will focus on the $t_{0}<0$. The $t_{0}>0$ case can be done
in the same manner. In this case, the HRT surface starts from region
$\mathrm{I}\mathrm{I}\mathrm{I}$. Let us denote the co-dimension two HRT
surface as $\eta=\eta(\rho),x=x(\rho)$ and the corresponding Lagrangian as

$\mathcal{L}_{\mathrm{I}\mathrm{I}\mathrm{I}}=\sqrt{1-f(\rho)\frac{-\dot{\eta}^{2}+\dot{x}^{2}}{\eta^{2}}}.$
(4.64)

Similar to the model in the last subsection, this Lagrangian has two types of
the symmetry:

  1. 1.

AdS2 scaling;

$\eta\to(1+\lambda)\eta,\quad x\to(1+\lambda)x.$ (4.65)

The corresponding Noether charge is

$Q_{\mathrm{AdS}_{2}}=-\frac{f(\rho)}{\mathcal{L}_{\mathrm{I}\mathrm{I}\mathrm{I}}}\frac{-\dot{\eta}\eta+\dot{x}x}{\eta^{2}}.$
(4.66)

Because of the $\mathbb{Z}_{2}$ symmetry $x\to-x$, the HRT surface has a
turning point at $x=0$, where $\frac{d\eta}{dx}=0$. Thus
$Q_{\mathrm{AdS}_{2}}=0$. This leads to

$\eta^{2}=x^{2}-A,$ (4.67)

where $A$ is some constant. Similar to the Euclidean model considered above,
so far we have not used the concrete form of $f(r)$ and thus it is natural to
assume that $\eta(x)$ is independent of the value of $\gamma$. We assume that
$A$ is also independent of $\gamma$,

$A=l^{2}-t_{0}^{2}.$ (4.68)

In other words, we set a boundary condition $\eta(l)=t_{0}$.

  2. 2.

$x$ translation;

$\eta\to\eta,\quad x\to x+c.$ (4.69)

This leads us to the corresponding Noether charge

$Q_{T}=-\frac{f(\rho)}{\mathcal{L}_{\mathrm{I}\mathrm{I}\mathrm{I}}}\frac{\dot{x}}{\eta^{2}}.$
(4.70)

By combining (4.64,4.67,4.70), we obtain

$\displaystyle\sqrt{A}\left(\frac{\dot{x}}{x^{2}-A}\right)^{2}$
$\displaystyle=\frac{\alpha}{f(\rho)(f(\rho)-\alpha)},$ (4.71)
$\displaystyle\mathcal{L}_{\mathrm{I}\mathrm{I}\mathrm{I}}^{2}$
$\displaystyle=\frac{f(\rho)}{f(\rho)-\alpha},$ (4.72)

where we introduce $\alpha=AQ_{T}^{2}$. $x(\rho)$ can be determined from the
first equation. Then we have

$x(\rho)=\sqrt{A}\tanh{\quantity(-\sqrt{\frac{-k}{\chi}}F\left(\left.G(\rho)\right|k\right)+B)},$
(4.73)

where

$\begin{split}G(\rho)&=\pi-\tilde{G}(\rho),\\\
\tilde{G}(\rho)&=\mathrm{Arcsin}{\left(\sqrt{\frac{(1-\chi)(2\alpha+\chi\cosh{2\rho}-1)\text{csch}^{2}\rho}{4\alpha\chi}}\right)},\\\
k&=\frac{4\alpha\chi}{(1-\chi)(-2\alpha+\chi+1)}.\end{split}$ (4.74)

We can determine the constant $\alpha,B$ from the boundary condition for
$x(\rho)$.

  * •

Because of the $\mathbb{Z}_{2}$ symmetry, $\rho=\rho(x)$ is an even function
of $x$. In this case, there is a turning point at $x=0$, where
$\frac{d\rho}{dx}$=0. From (4.71), we find that the turning point is at
$f(\rho)=0$ or $f(\rho)=AQ_{T}^{2}=\alpha$. Suppose the correct one is
$f(\rho_{t})=\alpha$, that is,

$2\alpha+\chi\cosh{2\rho_{t}}-1=0$ (4.75)

holds. In fact, this choice is the correct one in the vacuum AdS3 solution and
$f(\rho)=0$ corresponds to $x=\pm\sqrt{A}$. It seems that we have two choices
for arcsin in (4.74). When we take $\sin^{-1}(0)=0$ then we have $B=0$. When
we take $\sin^{-1}(0)=\pi$, then

$\begin{split}B=2\sqrt{\frac{-k}{\chi}}K(k).\end{split}$ (4.76)

Especially in the vacuum AdS3 limit $\chi\to 1$, $k\to+\infty$, this branch
retrieves the correct solution. Indeed, from the asymptotic expansion of
$K(k)$ [89], we can see

$B\to\frac{1}{\sqrt{\chi}}\log\quantity(-k)\to\pm\infty.$ (4.77)

  * •

At the boundary $\rho\to\infty$, $x\to l$. From this condition we can get
$\alpha$;

$\begin{split}G(\rho\to\infty)=&\pi-\mathrm{Arcsin}{\sqrt{\frac{1-\chi}{2\alpha}}}\\\
l=&\sqrt{A}\tanh{\quantity(\sqrt{\frac{-k}{\chi}}F\left(\left.\tilde{G}(\rho_{\infty})\right|k\right))}.\end{split}$
(4.78)

We put the plots for the $\alpha(t_{0})$ in figure 28. We see $\alpha$ can
take complex values in the time period $t_{0}\gtrapprox-l$.
$-i\epsilon^{\prime}$ is prescribed so that the quantum state can be vacuum
and we can obtain clean plots.

Figure 28: Plots for the $\alpha(t_{0}-i\epsilon^{\prime})$. The parameters
are chosen such that $l=1,\gamma=0.4,\epsilon=\epsilon^{\prime}=0.01$.
$1-\frac{t_{0}^{2}}{l^{2}}$ is the $\alpha$ for the vacuum AdS3 case,i.e.,
$\gamma=0,\chi=1$.

Eventually, we get the geodesic

$x(\rho;t_{0})=\sqrt{A}\tanh{\quantity(\sqrt{\frac{-k}{\chi}}F\left(\left.\tilde{G}(\rho)\right|k\right))}.$
(4.79)

We have some comments on the turning points and geodesic. Firstly the turning
point $\rho_{t}$ is a function of $t_{0}$ for general $\gamma<\gamma_{c}$. We
naively expect that $\rho_{t}(t_{0})$ is real positive until it reaches
$\rho_{c}$. Roughly, this happens around $t_{0}\sim-l$. In this time period,
$\rho_{t}$ and also the whole geodesic lie in the parts of the region
$\mathrm{I}\mathrm{I}\mathrm{I}$ where $f(\rho)>0$. See figure 27 for a
sketch. After this time, the turning point is situated in the region
$f(\rho)<0$. Later, it reaches region $\mathrm{I}\mathrm{I}$, where $\rho_{t}$
is pure imaginary. This story is almost true for a small value of $\gamma$.
However, if we tune $\gamma$ to be larger where the correlation between the
system and probe gets strong, $\rho_{t}$ takes general complex values after
some time around $t_{0}\sim O(l)$. This comes from the fact that $\alpha$
takes complex values. Thus the geodesic lies in another region which
analytically continues from region $\mathrm{I}\mathrm{I}\mathrm{I}$ rather
than region $\mathrm{I}\mathrm{I}$. The second comment is that, for a small
enough time, the geodesic can be time-like since $f(\rho)<0$ happens in some
parts of the geodesic, which leads to a complex valued holographic pseudo
entropy. We put the plots of the profile $x(\rho)$ in figure 30.

---
Figure 29: The plots of $\rho_{t}$. The Left figure shows when $\gamma=0.1$
and the right shows $\gamma=0.4$. The other parameters are set
$\lambda=1,\epsilon=\epsilon^{\prime}=0.01$. For the small $\gamma$, we see
the transition of $\rho_{t}$ from real to pure imaginary around $t_{0}\sim-l$,
however for large $\gamma$ it looks that $\rho_{t}$ takes a general complex
number.

Figure 30: Plots of $x(\rho)$. The left figure shows the $x(\rho)$ on
$t_{0}=5$ and the right figure shows the one with on $t_{0}=0.5$. The
parameters are chosen so that
$\gamma=0.4,l=1,\epsilon=\epsilon^{\prime}=0.01$. The $x(\rho)$ approaches to
zero at $\rho=\rho_{t}$.

The geodesic length is computed as follows:

$\begin{split}\mathrm{Area}[\Gamma_{A}]&=-2\int_{C}d\rho\sqrt{\frac{f(\rho)}{f(\rho)-f(\rho_{t})}}\\\
&=\quantity[\sqrt{\frac{-k}{\alpha\chi}}\quantity((1-\chi)F\left(\left.G(\rho)\right|k\right)+(2\alpha+\chi-1)\Pi\left(\left.j;G(\rho)\right|k\right))]^{\rho_{t}}_{\rho_{\infty}},\end{split}$
(4.80)

where

$j=\frac{2\alpha}{1-\chi}.$ (4.81)

Here we introduce the cutoff surface $\rho=\rho_{\infty}$ by

$\frac{\epsilon}{-t_{0}}=\frac{1}{\sqrt{1-f(\rho_{\infty})}}.$ (4.82)

This will be a nice cutoff surface since the inside of the square root is
always positive and it matches the one of the vacuum AdS case when $\gamma=0$.
Note that this cutoff works well when $\absolutevalue{t_{0}}\gg\epsilon$.
Since $G(\rho_{t})=\pi$ and

$\begin{split}F\left(\left.\pi-\theta\right|k\right)&=2K\left(k\right)-F\left(\left.\theta\right|k\right),\\\
\Pi\left(\left.j;\pi-\theta\right|k\right)&=2\Pi\left(j|k\right)-\Pi\left(\left.j;\theta\right|k\right),\end{split}$
(4.83)

we obtain

$\mathrm{Area}\quantity[\Gamma_{A}]=-\sqrt{\frac{-k}{\alpha\chi}}\quantity((1-\chi)F\left(\left.\tilde{G}(\rho_{\infty})\right|k\right)+(2\alpha+\chi-1)\Pi\left(\left.j;\tilde{G}(\rho_{\infty})\right|k\right)).$
(4.84)

Similar to the Euclidean model in the last subsection, by using the formula
(4.27), we have the last expression

$\begin{split}\mathrm{Area}\quantity[\Gamma_{A}]&=\sqrt{\frac{-k}{\alpha\chi}}\quantity(2\alpha
F\left(\left.\tilde{G}(\rho_{\infty})\right|k\right)-(2\alpha+\chi-1)\Pi\left(\left.\frac{k}{j};\tilde{G}(\rho_{\infty})\right|k\right))\\\
&+\log\quantity(\frac{\sqrt{\frac{(j-k)(j-1)}{j}}\tan{\tilde{G}(\rho_{\infty})}+\sqrt{1-k\sin^{2}\tilde{G}(\rho_{\infty})}}{-\sqrt{\frac{(j-k)(j-1)}{j}}\tan{\tilde{G}(\rho_{\infty})}+\sqrt{1-k\sin^{2}\tilde{G}(\rho_{\infty})}}).\end{split}$
(4.85)

By using the condition (4.78), we can also write

$\begin{split}\mathrm{Area}\quantity[\Gamma_{A}]&=\sqrt{\frac{-k}{\alpha\chi}}\quantity(2\alpha\sqrt{\frac{\chi}{-k}}\operatorname{arctanh}{\quantity(\frac{l}{\sqrt{A}})}-(2\alpha+\chi-1)\Pi\left(\left.\frac{k}{j};\tilde{G}(\rho_{\infty})\right|k\right))\\\
&+\log\quantity(\frac{\sqrt{\frac{(j-k)(j-1)}{j}}\tan{\tilde{G}(\rho_{\infty})}+\sqrt{1-k\sin^{2}\tilde{G}(\rho_{\infty})}}{-\sqrt{\frac{(j-k)(j-1)}{j}}\tan{\tilde{G}(\rho_{\infty})}+\sqrt{1-k\sin^{2}\tilde{G}(\rho_{\infty})}}).\end{split}$
(4.86)

As shown in figure 31, we see that the pseudo entropy behaves in completely
the same manner as the analytic continuation of the pseudo entropy from the
Euclidean case. This is because the transition matrix is equivalent in our
setups.

Figure 31: The plot of the time evolution of the entanglement entropy. We put
$\gamma=0.4,\epsilon=0.01,l=1$. $S_{0}$ denote the vacuum entanglement entropy
without the defect $S_{0}=\frac{c}{3}\log{\frac{2l}{\epsilon}}$.

##  5 Conclusions and discussions

In this paper, we explored entanglement phase transitions in time evolution of
pseudo entropy using holography. First we considered a model of AdS3/BCFT2
which describes the global quantum quench and added a scalar field localized
on the end-of-the-world brane. In the Euclidean signature this is dual to a
boundary conformal field theory (BCFT) on a strip where the boundary
conditions on the two boundaries are different. In the gravity dual, the
different boundary conditions are realized by requiring that the brane
localized scalar field approaches different values at the two boundaries.
Under the Lorentzian time evolution via its Wick rotation, the scalar field,
which was originally real valued in the Euclidean solution, takes imaginary
values. This effect suppresses the time evolution of area of the extremal
surface, which is properly interpreted as the pseudo entropy instead of
entanglement entropy. At a certain value of the imaginary scalar field, the
growth of the entropy changes from the linear behavior to the logarithmic
behavior. This exhibits the entanglement phase transition via the AdS/BCFT
holography. We analytically determined the coefficient of the time evolution
in each phase as shown in (1.2). In this analysis, we worked out the global
extension of our AdS/BCFT solution using the Kruskal coordinate and found the
profile of extremal surfaces in the extended geometry.

On the CFT side of this phase transition, we looked at the behavior of pseudo
entropy of a transition matrix instead of entanglement entropy for a quantum
state. However, since the gravity dual is given by a Lorentzian time evolving
geometry, it might look dual to a certain quantum state in a CFT. The only
difference from the standard gravity dual is that we have an imaginary valued
scalar field localized on the brane, which is expected to give some
dissipating effect. We noted that since the Rényi pseudo entropies computed
from holography look real and non-negative, it is possible that it is pseudo
Hermitian. If this is correct, it can be mapped to a regular Hermitian density
matrix via a similarity transformation and our gravity solution can also be
interpreted as a certain quantum state under a non-unitary time evolution
instead of a transition matrix. It will be an intriguing future problem to
explore this possibility. Besides, we also commented on the direct computation
of the pseudo Rényi entropy from the CFT side. It would be an important future
problem to understand how to compute the pseudo entanglement entropy, possibly
nonperturbatively, from the CFT side.

We also studied a class of AdS/BCFT solution with a gauge field localized on
the end-of-the-world brane instead of a scalar field. In the AdS3 case, we
showed that the profile of the end-of-the-world brane is identical to the
standard AdS/BCFT setup for pure gravity where the gauge flux gives a non-zero
tension. This means that there is no critical point where the pseudo entropy
grows logarithmically as opposed to the localized scalar case. By computing
its free energy, we worked out its phase structure. We also analyzed solutions
of the end-of-the-world brane with the gauge field in a higher dimensional
Poincaré AdS. We found that the brane profile with the localized gauge flux in
AdSd+1 is the same as the one with the localized scalar field in AdSd. One
interesting future direction is to apply this new AdS/BCFT setup to model
strongly correlated condensed matter systems with boundaries and study
transport phenomena in them.

Finally, we turned to a bulk version of such a solution where the scalar field
is not localized on the brane but spreads throughout the entire bulk
spacetime. We have discussed the transition matrices of two distinct vacuum
states. We find that for both the Euclidean and Lorentzian models, the pseudo
entropies decrease around the spacelike defect and take smaller values than
the vacuum AdS case. It is intriguing to note that the Lorentzian solution
describes a highly exotic spacetime. Especially, it has singularities and the
dilation scalar field becomes complex valued in a certain region. As a future
work, it would be interesting to demonstrate a measurement-induced phase
transition with the bulk fields. More specifically, by considering the soft
wall, the phase transition is expected to happen more mildly or in a crossover
style. An important consideration for achieving entanglement phase transition
in our bulk field model is the competition between the growth of entanglement
due to a quantum quench and the reduction of entanglement through measurement.
To realize entanglement phase transition by modifying our spacelike brane
model, we should start with a quenched state which might be obtained from a
black hole geometry.

## Acknowledgements

We are grateful to Rathindra Nath Das, Juan Maldacena, Shinsei Ryu and Mark
Van Raamsdonk for useful discussions. This work is supported by MEXT KAKENHI
Grant-in-Aid for Transformative Research Areas (A) through the “Extreme
Universe” collaboration: Grant Number 21H05187. This work is also supported by
Inamori Research Institute for Science and by JSPS Grant-in-Aid for Scientific
Research (A) No. 21H04469. TK is supported by Grant-in-Aid for JSPS Fellows
No. 23KJ1315. YS is supported by Grant-in-Aid for JSPS Fellows No.23KJ1337. ZW
is supported by the Society of Fellows at Harvard University.

##  Appendix A Formula for the Hyperbolic Coordinate and The Elliptic
Integrals

###  A.1 Hyperbolic coordinate and geodesic in the vacuum AdS3

The relation between the Poincaré coordinate $(z,t_{E})$ and the Janus
coordinate $(r,\xi)$ is

$\displaystyle\xi^{2}=\tau^{2}+z^{2},\quad\frac{z}{\tau}=\frac{1}{\sinh{r}}.$
(A.1)

We know the RT surface in the vacuum AdS3 in the Poincaré coordinate

$z(x)=\sqrt{l^{2}-x^{2}},\quad\tau=\tau_{0}$ (A.2)

Then the RT surface we see is

$x(r)=\sqrt{l^{2}-\frac{\tau_{0}^{2}}{\sinh{r}^{2}}},\quad\xi(r)=\tau_{0}\coth{r}$
(A.3)

The Noether charges can be read off as

$Q_{T}=\frac{1}{l},\quad
A=l^{2}+\tau_{0}^{2},\quad\alpha=\frac{A}{l}=1+\frac{\tau_{0}^{2}}{l^{2}}$
(A.4)

To check consistency we should see the integral of the E.L. equation in a
direct manner.

$\begin{split}\int
dr\sqrt{\frac{\alpha}{f(r)(f(r)-\alpha)}}&=\operatorname{arctanh}{g(r)}+B\\\
g(r)&={\frac{\sqrt{\alpha}\sinh{r}}{\sqrt{1-\alpha+\sinh^{2}{r}}}}\end{split}$
(A.5)

Thus the solution of the equation is

$\begin{split}\operatorname{arctanh}{\frac{x(r)}{\sqrt{A}}}=\operatorname{arctanh}{g(r)}+B\\\
\frac{x(r)}{\sqrt{A}}=\frac{g(r)+B^{\prime}}{1+g(r)B^{\prime}},\quad
B^{\prime}:=\tanh{B}\end{split}$ (A.6)

Demand there is the turning point $x=0,r=r_{t}$ where $f(r_{t})=\alpha$ we
have $B^{\prime}=+\infty$ and thus

$\frac{x(r)}{\sqrt{A}}=\frac{1}{g(r)}$ (A.7)

The geodesic length is evaluated by integrating the lagrangian. We should care
about the sign of the Lagrangian. Correct one is

$L=2\mathrm{sgn}{(\tau_{0})}\int_{r_{\infty}}^{r_{t}}dr\sqrt{\frac{f(r)}{f(r)-\alpha}}=2\mathrm{sgn}{(\tau_{0})}\left[\operatorname{arctanh}{\left(\frac{\sinh{r}}{\sqrt{-\alpha+1+\sinh^{2}{r}}}\right)}\right]_{r_{\infty}}^{r_{t}}$
(A.8)

By putting the cut-off as

$\frac{\epsilon}{\tau_{0}}=\frac{1}{\sinh{r_{\infty}}},\quad\cosh^{2}{r_{t}}=\alpha$
(A.9)

Then,

$\begin{split}\left[\operatorname{arctanh}{\left(\frac{\sinh{r}}{\sqrt{-\alpha+1+\sinh^{2}{r}}}\right)}\right]_{r_{\infty}}^{r_{t}}&=\frac{1}{2}\log\left(-1\right)-\frac{1}{2}\log\left(\frac{1+\mathrm{sgn}(\tau_{0})\left(1+\frac{\epsilon^{2}}{2\tau_{0}^{2}}\right)}{1+\mathrm{sgn}(\tau_{0})\left(1-\frac{\epsilon^{2}}{2\tau_{0}^{2}}\right)}\right)\\\
&=\mathrm{sgn}(\tau_{0})\log\frac{2l}{\epsilon}\end{split}$ (A.10)

Finally we see

$S_{A}=\frac{c}{3}\log{\frac{2l}{\epsilon}}$ (A.11)

###  A.2 Elliptic integrals

In the computation in the Janus solution we see the expression with the
elliptic integrals. In this section we summarize the definition and some
formulas of them. See [89, 88] for details.

#### Definition

There are three kinds of the incomplete elliptic integrals. Incomplete
elliptic integral of the first kind $F(z|m)$ is defined as

$F\left(\left.z\right|m\right)=\int_{0}^{z}\frac{1}{\sqrt{1-m\sin^{2}{t}}}dt.$
(A.12)

The complete elliptic integral of the first kind $K(m)$ is the specific
version of the incomplete one

$K\left(m\right)=F\left(\left.\frac{\pi}{2}\right|m\right)$ (A.13)

The definition of the incomplete elliptic function of the third kind is as
follows

$\begin{split}\Pi\left(n;z|m\right)&=\int_{0}^{z}\frac{dt}{(1-n\sin^{2}{t})\sqrt{1-m\sin^{2}{t}}}\\\
\Pi\left(n|m\right)&=\Pi\left(n;\frac{\pi}{2}|m\right)\end{split}$ (A.14)

##  Appendix B Another point of view for the spacelike brane

In section 4.2, we see a spacelike brane in the Lorentzian time evolution.
Here we want to give another viewpoint on the spacelike brane in the quantum
field theories. To this end, we review the indirect measurement shortly based
on [90, 91]. In general, it may be difficult to touch our interest system in
direct ways. For such a case, it is useful to introduce a probe system and
correlate it to our system to obtain some information. This indirect
measurement can be done in the following schemes:

  1. 1.

We prepare the quantum state which is a direct product of the system and probe
Hilbert space.

$\rho_{\mathrm{tot}}=\rho_{S}\otimes\rho_{P}$ (B.1)

  2. 2.

Act a unitary operator $U$ to create correlations between the system and the
probe.

$\rho_{\mathrm{tot}}\to U\rho_{S}\otimes\rho_{P}U^{\dagger}$ (B.2)

  3. 3.

Doing the projection measurement $\\{P_{k}\\}_{k}$ for the probe system. A
probability that the outcome of the measurement $k$ is

$p_{k}=\Tr_{SP}\quantity[P_{k}U\rho_{S}\otimes\rho_{P}U^{\dagger}P_{k}].$
(B.3)

Also, the quantum state after the measurement with the outcome of the
measurement $k$ is

$\rho_{\mathrm{tot}}^{(k)}=\frac{1}{p_{k}}P_{k}U\rho_{S}\otimes\rho_{P}U^{\dagger}P_{k}$
(B.4)

  4. 4.

Let us introduce the normalized orthogonal basis of the probe system
$\\{\ket{\Psi_{m}}\\}$ and also denote the spectral decomposition of the
initial state $\rho_{P}$ as $\rho_{P}=q_{l}\ket{\Phi_{l}}\bra{{\Phi_{l}}}$.
Then by introducing the Kraus operator and the POVM as

$\begin{split}M_{ki}&=\sqrt{q_{l}}\bra{\Psi_{m}}P_{k}U\ket{\Phi_{l}},\quad
i=(l,m)\\\ E_{k}&=\sum_{i}M^{\dagger}_{ki}M_{ki}\end{split}$ (B.5)

Then the quantum state of the system with the measurement outcome $k$ is

$\begin{split}\rho_{S}^{(k)}&=\frac{1}{p_{k}}\sum_{i}M_{ki}\rho_{S}M_{ki}^{\dagger}\\\
p_{k}&=\Tr_{S}{\quantity[E_{k}\rho_{S}]}\end{split}$ (B.6)

  5. 5.

If we do not know the measurement results, the resulting state is

$\rho_{S}^{\prime}=\sum_{k}p_{k}\rho_{k}=\sum_{ik}M_{ki}\rho_{S}M_{ki}^{\dagger}$
(B.7)

Spacelike brane from tracing out the probe

In total system point of view of the total system, we have the Schwinger
Keldysh path integral formulation.

Suppose we are only interested in the system $S$ and trace out the probe
system. Then we naturally expect that the system includes the spacelike brane
if the total unitary $U$ is instantaneous. That is the effective action of the
system will take the form of

$L_{S,\mathrm{eff}}=L_{S}+\delta(t_{0})\gamma L_{\mathrm{sp.brane}}.$ (B.8)

This will be done in the same manner as the Caldeira-Leggett model or more
generally Feynman-Vernon influence action formulation[92, 93]. Here we can
interpret $\gamma$ as a strength of the measurement. This can be explained as
follows: the spacelike brane term is emergent from the trace out of the probe
system and if the unitary $U$ is simply tensor product, then $\gamma=0$. In
terms of measurement, if unitary $U$ is decoupled, we do not have any
information about system $S$ from the probe and essentially we do not do any
measurement in this case.

Here we introduce the instant spacelike brane, i.e., the thin brane localized
on the $t=0$, we also consider the thick brane. It is known that there are
classes of soliton solutions in quantum field theories and string theory,
which are thick spacelike brane[94]. In these solutions, system field shows
dissipative radiation as a tachyon condensation. It may be possible that we
interpret this radiation as energy transfer between system and probe in the
indirect measurement scheme.

Note that it is clear that not all of the spacelike branes give an effective
picture for the indirect measurement. For example, in our models in sec. 4.2,
the pseudo entropy clearly satisfies $S_{A}=S_{A^{c}}$, which holds since the
geodesic is just determined by the entanglement surface. This means there is
no decoherence and we can think of the path integral gives the two state
amplitude.

$U_{S}$$U_{S}$$U_{P}$$P_{k}$$U$Trace out
$P$Probe$U_{S}$$U_{S}$$U_{{\mathrm{eff}},k}$ Figure 32: Sketch for the
indirect measurement.If we trace out the probe system the system action
modified to the effective action with a spacelike brane.

##  Appendix C CFT analysis of a defect marginal perturbation

In this section, we present a CFT analysis parallel to the holographic pseudo
entropy studied in this paper. The transition matrix considered in this paper
can be written as

$\displaystyle\mathcal{T}=\frac{e^{-itH}e^{-\frac{\beta}{4}H}\ket{B}\bra{B^{\prime}}e^{-\frac{\beta}{4}H}e^{itH}}{\bra{B^{\prime}}e^{-\frac{\beta}{2}H}\ket{B}},$
(C.1)

where $\ket{B^{\prime}}$ and $\ket{B}$ are two different conformal boundary
states connected by a marginal deformation. The subsystem considered in the
current setup is $A=\\{x|x>0\\}$. The corresponding path integral setup is
shown in the left of figure 33, which turns out to be a path integral over a
strip with width $\beta/2$ along the imaginary time direction, where a
marginal perturbation is performed on the one boundary located at
$\tau=\beta/4$.

It would be great to study this transition matrix directly in BCFT. However,
since it is a little bit subtle to perform the marginal perturbation exactly
on the boundary in the path integral formalism, we consider an alternative
version of the dynamics

$\displaystyle\mathcal{T}=\frac{e^{-it(H_{L}+H_{R})}\ket{\rm
TFD}\langle{\widetilde{\rm TFD}}|e^{it(H_{L}+H_{R})}}{\langle{\widetilde{\rm
TFD}}|{\rm TFD}\rangle}.$ (C.2)

Here, $|{\rm TFD}\rangle$ is the standard thermo-field double state

$\displaystyle|{\rm TFD}\rangle=\sum_{i}e^{-\beta
E_{i}/2}|n_{i}\rangle_{L}|n_{i}\rangle_{R},$ (C.3)

where $|n_{i}\rangle_{L(R)}$ is the energy eigenstate of $H_{L(R)}$ associated
to energy $E_{i}$, and $L$ and $R$ label two copies of the original system. By
contrast, $|\widetilde{\rm TFD}\rangle$ is realized by performing a marginal
perturbation to the initial condition of the thermo field double state. More
specifically, the corresponding setup is realized by considering a path
integral on a Euclidean cylinder, where $S^{1}$ is parameterized by
$\tau\in(-\beta/2,\beta/2]$, and $\mathbb{R}$ is parameterized by $x$.
Additionally, the marginal perturbation is realized by adding $\lambda\int
dx~{}O(x+i\beta/4,x-i\beta/4)$ to the action of the path integral, where
$\lambda$ is a dimensionless small parameter, and $O$ is a scalar primary with
scaling dimension $\Delta=1$. In the following, we denote
$O(x+i\beta/4,x-i\beta/4)$ as $O(x)$ for simplicity. See the right of figure
33 for a sketch. A specific way to construct a BCFT on a strip is to take a
$\mathbb{Z}_{2}$ orbifold of a standard CFT defined on a strip, by identifying
$z=x+i\tau$ and $z=x+i(\beta/2-\tau)$. This justifies what we are doing here,
i.e. considering the setup shown in the right of figure 33 instead of that
shown in the left.

Figure 33: The left figure shows a path integral realization of the transition
matrix (C.1), where the two boundaries states are taken to be $\ket{B}$ and
$\ket{B^{\prime}}$ respectively. $\ket{B^{\prime}}$ is expected to be realized
by performing a marginal perturbation (shown in orange) along $\tau=\beta/4$.
However, there are some subtleties to perform marginal perturbations exactly
along the boundary. The right figure shows an alternative path integral
realizing the density matrix (C.2). A marginal perturbation is performed along
a one-dimensional defect at $\tau=\beta/4$. The lower half $-\beta/2<\tau\leq
0$ prepares $\ket{\rm TFD}$ and the upper half $0<\tau\leq\beta/2$ prepares
$\langle{\widetilde{\rm TFD}}|$. The two setups can be regarded as connected
by a $\mathbb{Z}_{2}$ orbifold.

Now we consider the pseudo Rényi entropy for the subsystem $A_{L}\cup A_{R}$,
where $A_{L(R)}$ is the $x>0$ half of the left (right) system. Accordingly, we
should put the two edges of the subsystem at $z=i\tau$ and
$z=i(\beta/2-\tau)$, perform the replica method, and then perform the analytic
continuation $\tau\rightarrow it$. We use $\Sigma_{n}$ to denote the
$n$-replicated manifold. A sketch of $\Sigma_{2}$ is shown in figure 34, where
boundaries labeled by the same number $1\leq k\leq 4$ are identified with each
other. It is also convenient to think about the $x={\rm const.}$ slice at
$x>0$, which is shown in figure 35.

Figure 34: Replica manifold $\Sigma_{2}$ appearing in the computation of the
2nd Rényi entropy. Two copies of the original cylinder are cut along the $x>0$
half of imaginary time slices $\tau$ and $\beta/2-\tau$ and glued together by
identifying the boundaries labeled by the same number. A marginal perturbation
is performed along the orange lines.

Applying the replica method, the $n$-th pseudo Rényi entropy can be computed
as

$\displaystyle S^{(n)}(\mathcal{T}_{A})$
$\displaystyle=\frac{1}{1-n}\log\frac{\int_{\Sigma_{n}}D\phi\exp\left[\int
dx\int d\tau\mathcal{L}_{CFT}+\lambda\sum_{i=1}^{n}\int
dx_{i}~{}O(x_{i})\right]}{\left(\int_{\Sigma_{1}}D\phi\exp\left[\int dx\int
d\tau\mathcal{L}_{CFT}+\lambda\int dx~{}O(x)\right]\right)^{n}},$ (C.4)