Yan Liu${}^{\,a,b}$111Email<EMAIL_ADDRESS>Hong-Da
Lyu${}^{\,a,b}$222Email<EMAIL_ADDRESS>and Chuan-Yi
Wang${}^{\,a,b}$333Email<EMAIL_ADDRESS>
aCenter for Gravitational Physics, Department of Space Science
and International Research Institute of Multidisciplinary Science,
Beihang University, Beijing 100191, China

bPeng Huanwu Collaborative Center for Research and Education,
Beihang University, Beijing 100191, China

# On AdS3/ICFT2 with
a dynamical scalar field located on the brane

###### Contents

  1. 1 Introduction
  2. 2 Setups of AdS3/ICFT2
    1. 2.1 Solve the system at zero temperature
      1. 2.1.1 Simple examples of solutions
    2. 2.2 BCFT limit of the equation of motion
    3. 2.3 Null energy condition on the brane
  3. 3 Holographic entanglement entropy
    1. 3.1 Interface entropy and $g$-function
      1. 3.1.1 $g$-theorem
    2. 3.2 BCFT limits of holographic entanglement entropy
    3. 3.3 Case studies of several examples
      1. 3.3.1 Solution I with trivial scalar field
      2. 3.3.2 Solution II
      3. 3.3.3 Solution III
      4. 3.3.4 Solution IV
      5. 3.3.5 Solution V
      6. 3.3.6 Solution VI
    4. 3.4 Comment on the entanglement entropy of various subsystems
  4. 4 Finite temperature
    1. 4.1 Gluing two BTZ black holes
      1. 4.1.1 Permissible configurations at finite temperature
      2. 4.1.2 The solution with trivial scalar field: [H2,H2]
      3. 4.1.3 Solution with nontrivial scalar field: [H2,H2]
      4. 4.1.4 Solution with nontrivial scalar field: [E,E] and [E,H1]
      5. 4.1.5 Solution with nontrivial scalar field: [H1,E]
      6. 4.1.6 Solution with nontrivial scalar field: no [H1,H1]
      7. 4.1.7 BCFT limit
    2. 4.2 Gluing thermal AdS3 and BTZ black hole
  5. 5 Conclusions and open questions
  6. A Configurations of non-monotonic profile at zero temperature

##  1 Introduction

Interfaces contacting with two distinct systems are ubiquitous in nature. For
instance, an impurity or defect within a pristine system can be regarded as an
interface bridging two identical systems. Similarly, a quantum dot linked to
two interacting quantum wires or a bubble separating the interior and exterior
environments are also examples of interfaces. The investigation of such
systems not only enhances our comprehension of field theories but also
illuminates the intricate interplay between disparate domains [1].

We will investigate the system of a one-dimensional interface which can be
thought as a quantum dot, contacting two two-dimensional CFTs residing on
semi-infinite lines. When both CFTs are identical, the interface resembles a
defect or impurity. However, in general the two CFTs may differ significantly.
For instance, one of the CFTs could be trivial, leading to a boundary
conformal field theory (BCFT) system. Alternatively, folding the system when
the two CFTs are identical also yields a BCFT. Naturally, richer physical
phenomena are anticipated in the realm of interface conformal field theories
(ICFTs), where interfaces dynamically interact with CFTs. This is evidenced by
the emergence of novel observables within ICFTs, such as energy transports [2,
3], i.e. transmission or reflection coefficients quantifying the energy flux
across the interface. Additionally, ICFTs exhibit intriguing entanglement
structures [5, 4, 6], further deepening our understanding of their intricate
dynamics.

In the literature, considerable attention has been devoted to conformal
interfaces that preserves a single copy of the Virasoro algebra. Yet, it is
crucial to note that the interfaces may host internal dynamical degrees of
freedom, or coupling two CFTs nontrivially, which could break the conformal
symmetry of the interface field theory. We are focused on systems involving
strongly interacting field theories and general interfaces. In such scenarios,
the conventional tools of CFT are not applicable, while holography could
provide invaluable insights into the fundamental physics underlying such
systems.

Holographic ICFTs have undergone extensive investigation, ranging from the
top-down approaches like intersecting D-branes [7, 8, 9], which elucidate the
characteristics of supersymmetric field theories with defects, to bottom-up
approaches utilizing holography to explore the properties of interface CFTs.
One such bottom-up approach involves constructing the holographic model for
ICFTs using the Janus solution in the bulk [10, 11]. In this framework, the
interface is conceptualized as a non-local defect that smoothly dissolves
within the gravitational bulk. However, solving such systems poses significant
challenges, primarily due to the involvement of PDE’s.

A more straightforward approach to studying holographic ICFT involves
considering a localized interface brane embedded within the gravitational
system [12, 13, 14, 15]. This method has attracted significant research
attention in recent years. Various dynamics have been examined in this setups,
including the energy transports [16, 17, 18] and the entanglement structures
[15, 19, 20]. Additionally, insights into the island formula for double
holography in holographic BCFT have been provided from the perspective of
holographic ICFT [21, 15]. Moreover, the phases of interfaces in compact CFTs
have been investigated in [22].444In Euclidean CFTs, a new CFT2 state can be
prepared from a CFT1 state through a quenching operation at the interface
[14]. This procedure yields an ICFT in Euclidean spacetime, where the
properties of approximate CFT states are explored through AdS/ICFT in [14]. A
similar construction can be applied to CFTs undergoing weak measurement, as
discussed in [23]. Other progress in AdS/ICFT can be found in e.g. [24, 25,
26, 27, 28, 29, 30, 31, 32].

In all the aforementioned setups, the additional dynamics on the interface has
often been overlooked. Here, we propose to consider an AdS3/ICFT2 setup with a
dynamical interface brane involving a localised scalar field. This differs
from the studies of DCFT in [12], where the two CFTs that the defect
contacting were identical and a dynamical brane has been considered at finite
temperature. Instead, we consider a scenario where the dynamical brane is
embedded within two distinct AdS spacetime, i.e. the two CFTs that the defect
contacting could be different. On this interface brane within bulk, we
introduce a dynamical scalar field. Our primary objective is to explore the
impact of this scalar field on AdS/ICFT. We will study both zero-temperature
and finite-temperature configurations. Moreover, to gain further insights into
the interface field theory, we will study the entanglement structure and
define an interface entropy.

Our paper is organized as follows. In Sec. 2, we introduce the setups of
AdS3/ICFT2 and discuss the BCFT limit as well as the null energy condition on
the interface brane. In Sec. 3 we will study the aspect of holographic
entanglement entropy, including the interface entropy and its BCFT limit.
Moreover, we will provide several concrete examples of the zero-temperature
configuration to illustrate the behavior of the interface entropy. In Sec. 4
we turn our attention to the system at finite temperature, exploring the
influence of the scalar field on the profile of the interface brane. In Sec. 5
we conclude our study and discuss the open questions for further exploration.

##  2 Setups of AdS3/ICFT2

In this section we first show the setups of the bulk gravitational theory
which are applicable for both zero and finite temperature scenarios. Then we
will sovle it at zero temperature in Sec. 2.1. We will also discuss the BCFT
limit of the zero temperature system in Sec. 2.2 and the null energy condition
on the interface brane in Sec. 2.3.

We consider the gravitational theory

$S_{\text{bulk}}=S_{\text{I}}+S_{\text{II}}+S_{Q}$ (2.1)

with

$\displaystyle\begin{split}S_{\text{I}}&=\int_{N_{\text{I}}}d^{3}x\sqrt{-g_{\text{I}}}\,\bigg{[}\frac{1}{16\pi
G}\bigg{(}R_{\text{I}}+\frac{2}{L_{\text{I}}^{2}}\bigg{)}\bigg{]},\\\
S_{\text{II}}&=\int_{N_{\text{II}}}d^{3}x\sqrt{-g_{\text{II}}}\,\bigg{[}\frac{1}{16\pi
G}\bigg{(}R_{\text{II}}+\frac{2}{L_{\text{II}}^{2}}\bigg{)}\bigg{]},\\\
S_{Q}&=\frac{1}{8\pi
G}\int_{Q}d^{2}y\sqrt{-h}\,\bigg{[}\big{(}K_{\text{I}}-K_{\text{II}}\big{)}-(\partial\phi)^{2}-V(\phi)\bigg{]}.\end{split}$
(2.2)

Note that there is a minus sign in front of the extrinsic curvature scalar
$K_{\text{II}}$ in $S_{Q}$. This is due to that the extrinsic curvatures are
computed using the outward normal vector pointing from I to II. Without loss
of generality, we assume that $L_{\text{I}}\geq L_{\text{II}}$. Our setups
extend the discussion in [12] to the scenarios where different field theories
reside on either side of the interface. Distinguished from the previous
studies in [16, 22, 15] where there is a constant tension on the interface
brane, we here consider a localized dynamical scalar field residing on the
interface brane $Q$.

CFT${}_{\text{I}}$$x_{\text{I}}$$u_{\text{I}}$CFT${}_{\text{II}}$$x_{\text{II}}$$u_{\text{II}}$$(N_{\text{I}},g^{\text{I}}_{ab})$$(N_{\text{II}},g^{\text{II}}_{ab})$$Q$$Q$
Figure 1: Cartoon plot for the setup. The left and right bulk, denoted as
$N_{\text{I}}$ and $N_{\text{II}}$ respectively, are dual to
CFT${}_{\text{I}}$ and CFT${}_{\text{II}}$ respectively. The interface brane
$Q$ is in blue.

The configuration on a constant time slice is shown in Fig. 1. In the left
bulk $N_{\text{I}}$, we use coordinates
$x_{\text{I}}^{a}=(t_{\text{I}},x_{\text{I}},u_{\text{I}})$ (where $a=0,1,2$)
while in the right bulk $N_{\text{II}}$ we use coordinates
$x_{\text{II}}^{a}=(t_{\text{II}},x_{\text{II}},u_{\text{II}})$. On the
boundary, CFT${}_{\text{I}}$ occupies the regime $x_{\text{I}}<0$ while
CFT${}_{\text{II}}$ is in the regime $x_{\text{II}}>0$. The two conformal
field theories, CFT${}_{\text{I}}$ and CFT${}_{\text{II}}$, contact at the
interface point where $x_{\text{I}}=x_{\text{II}}=0$. On the interface brane
$Q$, we have the intrinsic coordinates $y^{\mu}=(t,z)$ where $\mu=0,1$.555In
Sec. 4, we use intrinsic coordinates $y^{\mu}=(t,w)$. The embedding equations
of $Q$ in $N_{\text{I}}$ and $N_{\text{II}}$ are $x_{\text{I}}^{a}(y^{\mu})$
and $x_{\text{II}}^{a}(y^{\mu})$, which obeys the following continuous
condition on $Q$

$\displaystyle h_{\mu\nu}=\frac{\partial x^{a}_{\text{I}}}{\partial
y^{\mu}}\frac{\partial x^{b}_{\text{I}}}{\partial
y^{\nu}}\,g^{\text{I}}_{ab}=\frac{\partial x^{a}_{\text{II}}}{\partial
y^{\mu}}\frac{\partial x^{b}_{\text{II}}}{\partial
y^{\nu}}\,g^{\text{II}}_{ab}\,.$ (2.3)

Here $h_{\mu\nu}$ is the the induced metric on $Q$, i.e.
$ds_{Q}^{2}=h_{\mu\nu}dy^{\mu}dy^{\nu}$.

The equations of motion are

$\displaystyle
R^{\text{I}}_{ab}-\frac{1}{2}g^{\text{I}}_{ab}R^{\text{I}}-\frac{1}{L_{\text{\text{I}}}^{2}}g^{\text{I}}_{ab}$
$\displaystyle=$ $\displaystyle 0\,,$ (2.4) $\displaystyle
R^{\text{II}}_{ab}-\frac{1}{2}g^{\text{II}}_{ab}R^{\text{II}}-\frac{1}{L_{\text{II}}^{2}}g^{\text{II}}_{ab}$
$\displaystyle=$ $\displaystyle 0\,,$ (2.5) $\displaystyle\Delta
K_{\mu\nu}-h_{\mu\nu}\Delta
K+\left[(\partial\phi)^{2}+V(\phi)\right]h_{\mu\nu}-2\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle=$ $\displaystyle 0\,,$ (2.6) $\displaystyle
2\partial_{\mu}(\sqrt{-h}h^{\mu\nu}\partial_{\nu}\phi)-\sqrt{-h}\frac{dV(\phi)}{d\phi}$
$\displaystyle=$ $\displaystyle 0\,,$ (2.7)

where $\Delta X\equiv X_{\text{I}}-X_{\text{II}}$ with $X$ as $K_{ab}$ or $K$.
Here (2.4) and (2.5) are the equations for the metric fields in the left bulk
$N_{\text{I}}$ and the right bulk $N_{\text{II}}$, while (2.6) and (2.7) are
the equations on the interface brane $Q$.

We can separate (2.6) into two parts, i.e. the trace part and the traceless
part. The trace part is

$\displaystyle V(\phi)=\frac{1}{2}\Delta K\,,$ (2.8)

and the traceless part is

$\displaystyle\Delta K_{\mu\nu}-\frac{1}{2}h_{\mu\nu}\Delta
K+(\partial\phi)^{2}h_{\mu\nu}-2\partial_{\mu}\phi\partial_{\nu}\phi=0\,.$
(2.9)

Now the equations on $Q$ can be summarized as follows

$\displaystyle g^{\text{I}}_{ab}dx_{\text{I}}^{a}dx_{\text{I}}^{b}\Big{|}_{Q}$
$\displaystyle=$ $\displaystyle
g^{\text{II}}_{ab}dx_{\text{II}}^{a}dx_{\text{II}}^{b}\Big{|}_{Q}\,,$ (2.10)
$\displaystyle\Delta K_{\mu\nu}-\frac{1}{2}h_{\mu\nu}\Delta
K+(\partial\phi)^{2}h_{\mu\nu}-2\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle=$ $\displaystyle 0\,,$ (2.11) $\displaystyle
V(\phi)-\frac{1}{2}\Delta K$ $\displaystyle=$ $\displaystyle 0\,,$ (2.12)
$\displaystyle
2\partial_{\mu}(\sqrt{-h}h^{\mu\nu}\partial_{\nu}\phi)-\sqrt{-h}\frac{dV(\phi)}{d\phi}$
$\displaystyle=$ $\displaystyle 0\,.$ (2.13)

We will concentrate on the static systems, anticipating the existence of a
global time within the dual field theory. The above setups are applicable for
both zero-temperature and finite-temperature scenarios. In the following we
will present the zero temperature solution and then study its properties. We
will study the finite temperature solution in Sec. 4.

###  2.1 Solve the system at zero temperature

At zero temperature, the bulk equations (2.4) and (2.5) give the planar AdS3
solution

$\displaystyle
ds^{2}_{A}=\frac{L^{2}_{A}}{u^{2}_{A}}\,\big{[}-dt^{2}_{A}+dx^{2}_{A}+du^{2}_{A}\big{]}\,,~{}~{}~{}~{}~{}~{}A=\mbox{I,\,
~{}II}\,.$ (2.14)

Here $u_{A}\in(0,\infty)$ with the boundary field theory CFTA located at the
boundary $u_{A}\to 0$. We consider the case that the spatial coordinates
$x_{A}$ are non-compact.666It is worth noting that the case of compact spatial
directions has been recently investigated in [22]. CFT${}_{\text{I}}$ lives in
the regime $x_{\text{I}}<0$ while CFT${}_{\text{II}}$ is in the regime
$x_{\text{II}}>0$. The central charges for the dual CFT${}_{\text{I}}$ and
CFT${}_{\text{II}}$ are [33]

$\displaystyle c_{\text{I}}=\frac{3L_{\text{I}}}{2G_{N}}\,,\ \
~{}~{}~{}~{}c_{\text{II}}=\frac{3L_{\text{II}}}{2G_{N}}\,.$ (2.15)

We have assumed that $L_{\text{I}}\geq L_{\text{II}}$, thus $c_{\text{I}}\geq
c_{\text{II}}$. For convenience, we define

$\nu\equiv\frac{c_{\text{II}}}{c_{\text{I}}}=\frac{L_{\text{II}}}{L_{\text{I}}}$
(2.16)

where $\nu\in(0,1]$. Then we can parameterize the system using
$L_{\text{I}},\nu$ instead of $L_{\text{I}},L_{\text{II}}$.

It will be convenient to use the rescaled coordinates $z_{A}$,

$\displaystyle
u_{\text{I}}=\sqrt{\frac{L_{\text{I}}}{L_{\text{II}}}}z_{\text{I}}=\frac{z_{\text{I}}}{\sqrt{\nu}}\,,~{}~{}~{}~{}u_{\text{II}}=\sqrt{\frac{L_{\text{II}}}{L_{\text{I}}}}z_{\text{II}}=\sqrt{\nu}z_{\text{II}}\,.$
(2.17)

Since we are interested in static configuration, the brane $Q$ is supposed to
be a timelike hypersurface. Assuming it is given by

$x_{\text{I}}=\psi_{\text{I}}(z_{\text{I}})\,~{}~{}~{}~{}\text{or
equivalently}~{}~{}~{}x_{\text{II}}=\psi_{\text{II}}(z_{\text{II}})\,,$ (2.18)

then the continuous condition of the metric on $Q$ reads

$\displaystyle\frac{\nu
L_{\text{I}}^{2}}{z^{2}_{\text{I}}}\left[-dt_{\text{I}}^{2}+\left(\frac{1}{\nu}+\psi^{\prime
2}_{\text{I}}(z_{\text{I}})\right)dz_{\text{I}}^{2}\,\right]=\frac{\nu
L_{\text{I}}^{2}}{z^{2}_{\text{II}}}\left[-dt_{\text{II}}^{2}+\left(\nu+\psi^{\prime
2}_{\text{II}}(z_{\text{II}})\right)dz_{\text{II}}^{2}\,\right]\,.$ (2.19)

We assume777In principle, one could consider the solution of
$t_{\text{I}}=\alpha\,t_{\text{II}}$, $z_{\text{I}}=\alpha\,z_{\text{II}}$ and
$\frac{1}{\nu}+\psi^{\prime 2}_{\text{I}}(z_{\text{I}})=\nu+\psi^{\prime
2}_{\text{II}}(z_{\text{II}})$. Here we set $\alpha=1$ such that time is
globally well-defined.

$t_{\text{I}}=t_{\text{II}}\equiv t\,,$ (2.20)

then the continuous condition of the induced metric (2.19) gives

$z_{\text{I}}=z_{\text{II}}\equiv z\,,~{}~{}~{}~{}\frac{1}{\nu}+\psi^{\prime
2}_{\text{I}}(z_{\text{I}})=\nu+\psi^{\prime 2}_{\text{II}}(z_{\text{II}})\,.$
(2.21)

In following calculations, we identify
$\psi_{\text{I}}(z_{\text{I}}),\psi_{\text{II}}(z_{\text{II}})$ with
$\psi_{\text{I}}(z),\psi_{\text{II}}(z)$.

Let us remark on the parameterisation of a point $p$ on the brane $Q$, which
can be expressed in three distinct coordinate systems: the intrinsic
coordinates $y^{\mu}=(t,z)$, the $N_{\text{I}}$ part of AdS
$x_{\text{I}}^{a}=(t,\psi_{\text{I}}(z),z/\sqrt{\nu})$ and the $N_{\text{II}}$
part of AdS $x_{\text{II}}^{a}=(t,\psi_{\text{II}}(z),\sqrt{\nu}z)$. We
consider the simplest static case $\phi=\phi(z)$888More generally, one could
consider $\phi=\phi(t,z)$ and then the embedding equation as well as the
induced metric of $Q$ is time dependent. It would be interesting to further
explore this case., then from the equations (2.11, 2.12, 2.13) we obtain

$\displaystyle\phi^{\prime 2}=$
$\displaystyle~{}\,\frac{L_{\text{I}}}{2z}\frac{-\psi^{\prime\prime}_{\text{I}}+\nu\psi^{\prime\prime}_{\text{II}}}{\sqrt{\nu+\psi^{\prime
2}_{\text{II}}}}\,,$ (2.22) $\displaystyle V(\phi(z))=$
$\displaystyle~{}\,\frac{\sqrt{\nu}\left(2(\psi_{\text{I}}^{\prime}-\nu\psi_{\text{II}}^{\prime})+2(\nu\psi_{\text{I}}^{\prime
3}-\psi_{\text{II}}^{\prime
3})-z(\psi_{\text{I}}^{\prime\prime}-\nu\psi_{\text{II}}^{\prime\prime})\right)}{2L_{\text{I}}(1+\nu\psi_{\text{I}}^{\prime
2})^{3/2}}\,,$ (2.23) $\displaystyle\frac{dV(\phi)}{d\phi}=$
$\displaystyle~{}\frac{2z^{2}\phi{{}^{\prime\prime}}}{L^{2}_{\text{I}}(1+\nu\psi^{\prime
2}_{\text{I}})}-\frac{2z^{2}\nu\phi^{\prime}\psi_{\text{I}}^{\prime}\psi_{\text{I}}{{}^{\prime\prime}}}{L^{2}_{\text{I}}(1+\nu\psi^{\prime
2}_{\text{I}})^{2}}\,,$ (2.24)

where $\psi_{\text{I}},\psi_{\text{II}},\phi$ are all function of $z$ and the
prime ′ is the derivative with respect to $z$. Note that the above three
equations are not independent, i.e. we can derived the third one from the
first two.

The Ricci tensor of the induced metric on the brane is

$\displaystyle R^{Q}_{\mu\nu}=\lambda(z)\,h_{\mu\nu}\,,$ (2.25)

where

$\displaystyle\lambda(z)=-\frac{1+\nu\psi^{\prime}_{\text{I}}\,(\psi^{\prime}_{\text{I}}+z\psi^{\prime\prime}_{\text{I}})}{L_{\text{I}}^{2}\,(1+\nu\psi^{\prime
2}_{\text{I}})^{2}}\,.$ (2.26)

Obviously when $\psi^{\prime}_{\text{I}}=\text{constant}$, we have
$\lambda(z)=\text{constant}$. This is consistent with the picture that in the
case of trivial scalar field, the brane is a straight line and has the induced
metric with pure AdS2 [21, 15].999One can show that when
$\lambda(z)=\text{constant}$, the only allowed profile for the brane is a
straight line. Another unphysical solution may exist that the brane terminates
at a special point in the bulk and we do not consider such solution.

We are interested in the case that the brane is asymptotically AdS2 in UV.
Then in the UV limit $z\to 0$, $\lambda(z)$ should be a negative constant. One
class of such solution is

$\psi_{\text{I}}\to\gamma\,z^{n},~{}~{}~{}\text{with}~{}n\geq 1\,,$ (2.27)

where $\gamma$ is a constant. With the above solution, from (2.21) we have

$\psi_{\text{II}}\to\pm\sqrt{\frac{1}{\nu}-\nu+n^{2}\gamma^{2}\delta_{n1}}\,z\,,~{}~{}~{}\text{when}~{}z\to
0\,.$ (2.28)

####  2.1.1 Simple examples of solutions

With the above setups, given the potential $V(\phi)$, one can obtain
consistent solutions $\phi(z),\psi_{\text{I}}(z),\psi_{\text{II}}(z)$ of the
system. In practice, we construct the consistent solution as follows. We start
from an “arbitrary” function $\psi_{\text{I}}(z)$ which should satisfy the
null energy condition discussed in Sec. 2.3. Then $\psi_{\text{II}}(z)$ can be
solved from (2.21). The consistent scalar field $\phi(z)$ should satisfy
(2.22) which are determined by $\psi_{\text{I}}(z)$ and $\psi_{\text{II}}(z)$,
and $V(\phi)$ satisfies (2.23). In the following we will show two simple
examples. More examples will be discussed in Sec. 3.3.

The first example is the case that the brane $Q$ is a straight line. In this
case we have solutions,

$\displaystyle\begin{split}\psi_{\text{I}}&=\gamma z\,,\\\
\psi_{\text{II}}&=\pm\sqrt{\frac{1+\gamma^{2}\nu-\nu^{2}}{\nu}}z\,,\\\
\phi(z)&=\phi_{0}\,,\\\
V(\phi)&=\frac{\gamma\nu^{3/2}\mp\sqrt{1+\gamma^{2}\nu-\nu^{2}}}{L_{\text{I}}\nu\sqrt{1+\gamma^{2}\nu}}\,,\end{split}$
(2.29)

where $\gamma,\phi_{0}$ are constants. Note that the sign in front of
$\psi_{\text{II}}$ is related to if the configuration of $Q$ is an acute angle
or an obtuse angle in the bulk $N_{\text{II}}$.

This case has been studied in [14, 15] with $\phi_{0}=0$ and the tension
$T=V$. Similar to the result in [14, 15], from (2.29) the tension is
constrained by

$T_{\text{min}}<|T|<T_{\text{max}}\,,$ (2.30)

where

$T_{\text{min}}=\frac{1-\nu}{L_{\text{I}}\nu}\,,\ \
T_{\text{max}}=\frac{1+\nu}{L_{\text{I}}\nu}\,.$ (2.31)

For a given value of the tension satisfying (2.30), the profiles of the branes
are uniquely determined, as shown in (2.29).

The second example is the case when $\nu=1$, i.e. the two CFT’s have the same
central charge. In this case, the system can be viewed as the presence of a
defect located at the origin of field theory. The equations (2.21) become

$\displaystyle\psi^{\prime 2}_{\text{I}}$ $\displaystyle=$
$\displaystyle\psi^{\prime 2}_{\text{II}}\,,$ (2.32)

which have two kinds of solutions. The first one is

$\psi^{\prime}_{\text{I}}=\psi^{\prime}_{\text{II}}\,,~{}~{}~{}~{}\phi=c_{1}\,,~{}~{}~{}~{}V=0\,.$
(2.33)

This is a trivial one and the brane does not play any role, i.e. there is no
defect at all on the field theory. The second one is

$\displaystyle\begin{split}\psi_{\text{I}}(z)=&-\psi_{\text{II}}(z)=~{}\varphi(z)\,,\\\
\phi^{\prime
2}(z)=&~{}-\frac{L_{\text{I}}\,\varphi^{\prime\prime}(z)}{z\sqrt{1+\varphi^{\prime
2}(z)}}\,,\\\ V(\phi)=&~{}\frac{2\varphi^{\prime}+2\varphi^{\prime
3}-z\varphi^{\prime\prime}}{L_{\text{I}}(1+\varphi^{\prime
2})^{\frac{3}{2}}}\,.\end{split}$ (2.34)

This could be understood as unfolding a holographic BCFT system. We will
further comment on this solution in the next subsection.

###  2.2 BCFT limit of the equation of motion

ICFTs are more generic than BCFTs in the sense that BCFT can be viewed as
special limit of ICFT. There are two different ways to obtain the BCFT limit
from ICFT. The first way is to consider the limit

$\nu=\frac{L_{\text{II}}}{L_{\text{I}}}\to 0\,,$ (2.35)

which can be realized by setting $L_{\text{II}}\to 0$ while $L_{\text{I}}$ is
finite. This means that the central charge of CFT${}_{\text{II}}$ is
relatively small and we can ignore it in the whole system.

To study the limit, it is more convenient to work in the coordinates of $t,u$
on $Q$ where $u=u_{\text{I}}$. Using the relation in (2.17), we can fix
$u_{\text{II}}=\nu u$. The profile of the brane could be parameterized as
$x_{\text{I}}=\tilde{\psi}_{\text{I}}(u)$, or equivalently
$x_{\text{II}}=\tilde{\psi}_{\text{II}}(u)$ and $\phi=\tilde{\phi}(u)$. The
equations for these fields can be obtained by rewriting (2.21) and (2.22,
2.23, 2.24) with

$\psi^{\prime}_{\text{I}}(z)\to\frac{\tilde{\psi}^{\prime}_{\text{I}}(u)}{\sqrt{\nu}}\,,~{}~{}~{}\psi^{\prime\prime}_{\text{I}}(z)\to\frac{\tilde{\psi}^{\prime\prime}_{\text{I}}(u)}{\nu}\,,~{}~{}~{}\psi^{\prime}_{\text{II}}(z)\to\frac{\tilde{\psi}^{\prime}_{\text{II}}(u)}{\sqrt{\nu}}\,,~{}~{}~{}\psi^{\prime\prime}_{\text{II}}(z)\to\frac{\tilde{\psi}^{\prime\prime}_{\text{II}}(u)}{\nu}\,,~{}~{}~{}\phi^{\prime}(z)\to\frac{\tilde{\phi}^{\prime}(u)}{\sqrt{\nu}}\,.$

Here the prime in functions of $z$ (e.g. $\psi^{\prime}_{\text{I}}(z)$)
represents the derivative with respect to $z$, while the prime in functions of
$u$ represents the derivative with respect to $u$. We take the $\nu\to 0$
limit, assuming that in this limit $\tilde{\psi}_{\text{I}}(u)=\varphi(u)$
which is a $\nu$-independent and smooth function, then we can obtain

$\displaystyle\begin{split}\tilde{\psi}_{\text{I}}(u)=&~{}\varphi(u)\,,\\\
\tilde{\psi}^{\prime}_{\text{II}}(u)=&~{}\pm\sqrt{1+\varphi^{\prime
2}(u)}+\mathcal{O}(\nu)\,,\\\ \tilde{\phi}^{\prime
2}(u)=&~{}-\frac{L_{\text{I}}\varphi^{\prime\prime}(u)}{2u\sqrt{1+\varphi^{\prime
2}(u)}}+\mathcal{O}(\nu)\,,\\\
V(\tilde{\phi})=&~{}\mp\frac{1}{L_{\text{I}}\nu}+\frac{2\varphi^{\prime}(u)+2\varphi^{\prime
3}(u)-u\varphi^{\prime\prime}(u)}{2L_{\text{I}}(1+\varphi^{\prime
2}(u))^{\frac{3}{2}}}+\mathcal{O}(\nu)\,.\end{split}$ (2.36)

Note that there is a divergent term in the potential $V$. This also has been
seen in [14, 15] where the tension is divergent in the AdS/BCFT limit. For the
special case $\phi=0,V=T$, the above system reduced to the results in [14, 15]
and solution of (2.36) gives to a straight line.

We can make a comparison to the case without regime $N_{\text{II}}$, i.e. the
framework of AdS/BCFT. In the case of AdS/BCFT with a dynamical scalar field
on $Q$, we parameterize $Q$ as $x=\tilde{\psi}_{\text{I}}(u)=\varphi(u)$. Then
the equations of motion are

$\displaystyle\begin{split}\phi^{\prime
2}(u)=&~{}-\frac{L_{\text{I}}\,\varphi^{\prime\prime}(u)}{2u\sqrt{1+\varphi^{\prime
2}(u)}}\,,\\\ V(\phi)=&~{}\frac{2\varphi^{\prime}(u)+2\varphi^{\prime
3}(u)-u\varphi^{\prime\prime}(u)}{2L_{\text{I}}(1+\varphi^{\prime
2}(u))^{\frac{3}{2}}}\,.\end{split}$ (2.37)

These equations have been studied in [34]. The above equations are the same as
(2.36) except that they do not have divergent term in the potential $V(\phi)$.

The second way to obtain a BCFT is to consider the limit $\nu=1$ where we can
perform the folding trick[19]. In this case the dynamical equations in ICFT
are listed in (2.34). Note that when $\nu=1$ we have
$u_{\text{I}}=u_{\text{II}}=z=u$. Obviously, it has exactly the same form as
(2.37) after setting $\phi^{\prime 2},V(\phi)$ in the ICFT (2.34) as twice of
those in (2.37).

###  2.3 Null energy condition on the brane

The energy conditions on the brane is important to constrain the dynamics of
the interface brane. Particularly within the framework of AdS/BCFT, the null
energy condition (NEC) is widely used.101010For discussions regarding other
energy conditions in AdS/DCFT, wherein the NEC is deemed the most fundamental,
see e.g. [12]. In the following we derive the constraints on the profiles of
the interface brane from the NEC for the matter field residing on the brane.

The coordinate system on $Q$ is $y^{\mu}=(t,z)$. The null vector on $Q$ is

$\displaystyle N^{\mu}=\left(\pm\sqrt{\frac{1}{\nu}+\psi^{\prime
2}_{\text{I}}}\,,~{}1\right)\,,$ (2.38)

and therefore the NEC is

$\displaystyle(\Delta K_{\mu\nu}-h_{\mu\nu}\Delta
K)N^{\mu}N^{\nu}=\frac{L_{\text{I}}}{z}\,\frac{(-\psi^{\prime\prime}_{\text{I}}+\nu\psi^{\prime\prime}_{\text{II}})}{\sqrt{\frac{1}{\nu}+\psi^{\prime
2}_{\text{I}}}}\geq 0\,.$ (2.39)

From (2.22), the NEC is equivalent to $\phi^{\prime 2}(z)\geq 0$. This means
that whenever we have a consistent background solution, then the NEC is
satisfied.

However, as we show in the following, this condition actually constrains the
possible choices of $\psi_{\text{I}}$. From the junction condition (2.21), we
have

$\displaystyle\psi^{\prime}_{\text{II}}=\pm\sqrt{\frac{1}{\nu}-\nu+\psi^{\prime
2}_{\text{I}}}\,.$ (2.40)

The sign above is related to the tangent direction of the brane is form an
obtuse angle or an acute angle. When $0<\nu<1$, the NEC (2.39), at $z$ where
$\psi^{\prime 2}_{\text{I}}\not=\nu-\frac{1}{\nu}$, can be simplified as

$\displaystyle-\psi^{\prime\prime}_{\text{I}}+\nu\psi^{\prime\prime}_{\text{II}}=\left(-1\pm\frac{\nu\psi^{\prime}_{\text{I}}}{\sqrt{\frac{1}{\nu}-\nu+\psi^{\prime
2}_{\text{I}}}}\right)\psi^{\prime\prime}_{\text{I}}\geq 0\,,$ (2.41)

or equivalently,

$\displaystyle\psi^{\prime\prime}_{\text{I}}\leq 0$ (2.42)

by noticing that
$\Big{|}\frac{\nu\psi^{\prime}_{\text{I}}}{\sqrt{\frac{1}{\nu}-\nu+\psi^{\prime
2}_{\text{I}}}}\Big{|}<1\,.$ When $\nu=1$, i.e. the two CFTs have the same
central charge, the NEC (2.39) can be simplified as

$\displaystyle\psi^{\prime}_{\text{I}}=\psi^{\prime}_{\text{II}}\ \
~{}~{}~{}\text{or}\ \
~{}~{}~{}\psi^{\prime}_{\text{I}}=-\psi^{\prime}_{\text{II}}\,,\ \
\psi^{\prime\prime}_{\text{I}}\leq 0\,.$ (2.43)

Note that the first case in (2.43) is trivial in the sense that the brane does
not play any role, which has been discussed in (2.33). Therefore we will focus
on the case with $\psi^{\prime\prime}_{\text{I}}\leq 0$ for all the values of
$\nu$. This condition constraints the allowed configurations of $Q$.

We can use NEC to study the configuration of $\psi_{\text{II}}$. From the
junction condition (2.21), we have

$\psi^{\prime}_{\text{I}}=\pm\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}\,.$ (2.44)

Note that for $\nu\in(0,1]$, the expression in the square root needs to be
non-negative and this constraints the possible values of
$\psi^{\prime}_{\text{II}}$. Now the NEC (2.39), at $z$ where $\psi^{\prime
2}_{\text{II}}\not=\frac{1}{\nu}-\nu$, becomes

$\displaystyle-\psi^{\prime\prime}_{\text{I}}+\nu\psi^{\prime\prime}_{\text{II}}=\left(\nu\mp\frac{\psi^{\prime}_{\text{II}}}{\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}}\right)\psi^{\prime\prime}_{\text{II}}\geq 0\,,$ (2.45)

where the sign in the bracket is aligned with the sign in (2.44). When
$0<\nu<1$ we have $|\psi^{\prime}_{\text{II}}|\geq\sqrt{\frac{1}{\nu}-\nu}>0$.
In this case the possible constraints from the NEC for the monotonic profiles
are

  1. (1)

$0<\nu<1$, $\psi^{\prime}_{\text{II}}\not=\text{constant}$,
$\psi^{\prime}_{\text{II}}\geq\sqrt{\frac{1}{\nu}-\nu}$,
$\psi^{\prime}_{\text{I}}=+\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}$, $\psi^{\prime\prime}_{\text{II}}\leq 0$.

  2. (2)

$0<\nu<1$, $\psi^{\prime}_{\text{II}}\not=\text{constant}$,
$\psi^{\prime}_{\text{II}}\leq-\sqrt{\frac{1}{\nu}-\nu}$,
$\psi^{\prime}_{\text{I}}=+\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}$, $\psi^{\prime\prime}_{\text{II}}\geq 0$.

  3. (3)

$0<\nu<1$, $\psi^{\prime}_{\text{II}}\not=\text{constant}$,
$\psi^{\prime}_{\text{II}}\geq\sqrt{\frac{1}{\nu}-\nu}$,
$\psi^{\prime}_{\text{I}}=-\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}$, $\psi^{\prime\prime}_{\text{II}}\geq 0$.

  4. (4)

$0<\nu<1$, $\psi^{\prime}_{\text{II}}\not=\text{constant}$,
$\psi^{\prime}_{\text{II}}\leq-\sqrt{\frac{1}{\nu}-\nu}$,
$\psi^{\prime}_{\text{I}}=-\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}$, $\psi^{\prime\prime}_{\text{II}}\leq 0$.

  5. (5)

$0<\nu<1$, $\psi^{\prime}_{\text{II}}=\text{constant}$,
$|\psi^{\prime}_{\text{II}}|\geq\sqrt{\frac{1}{\nu}-\nu}$,
$\psi^{\prime}_{\text{I}}=\pm\sqrt{-\frac{1}{\nu}+\nu+\psi^{\prime
2}_{\text{II}}}$.

In the first four cases, we always have $\psi^{\prime\prime}_{\text{I}}\leq 0$
as shown in (2.42). In the case (5) the interface brane is a straight line,
which corresponds to the scenario involving a trivial scalar field. Obviously
with a nontrivial scalar field, the brane can bend in different ways. Note
that in the cases (1-4), for the specific point in the bulk where
$\psi^{\prime}_{\text{II}}=\pm\sqrt{\frac{1}{\nu}-\nu}$, we have
$\psi^{\prime}_{\text{I}}=0$ which is a turning point for the profile
$\psi_{\text{I}}$ of the brane. This indicates that the brane is no longer
monotonic. We will only consider the case that the brane is monotonic in the
main text.111111In appendix A, an example of a profile $\psi_{\text{I}}$ of
the brane which is a non-monotonic function is shown in Fig. 27 .

We show the cartoon picture of the bending branes in Fig. 2. We have used the
blue line to represent the profile of $Q$ with $\psi_{\text{I}}$ and the red
line for $\psi_{\text{II}}$. In the cases (1)&(2) the brane will extend to
infinity while in the cases (3)&(4), the brane can extend to infinity or a
finite value of $u$.

(2)(1)(3)(4) Figure 2: Cartoon plot for the four kinds of monotonic
configuration of brane profile when $0<\nu<1$ and
$\psi^{\prime}_{\text{II}}\not=\text{constant}$. These configurations shows
clearly about the allowed convexity of the brane.

When $\nu=1$, we have (2.33) with the relation
$\psi^{\prime}_{\text{I}}=+\psi^{\prime}_{\text{II}}\,$; or (2.34) with the
relation
$\psi^{\prime}_{\text{I}}=-\psi^{\prime}_{\text{II}}\,,\psi^{\prime\prime}_{\text{II}}\leq
0\,,$ i.e. the configurations of (2)&(3) in Fig. 2.

##  3 Holographic entanglement entropy

The presence of a nontrivial scalar field on the interface brane $Q$ leads to
more complicated brane profiles, indicating that the ICFT is away from a fixed
point. In this section, we will first study the entanglement entropy of a
specific regime within field theory using holographic techniques.
Subsequently, we will introduce a $g$-function for the ICFT to quantify the
effective degrees of freedom in Sec. 3.1 and discuss the BCFT limits of
holographic entropy in Sec. 3.2. In Sec. 3.3 we will present several
illustrative examples to demonstrate the characteristic features of the
system. A consistent observation is that whenever the null energy condition is
satisfied, the $g$-function is monotonically decreasing from UV to IR. We will
comment on the entanglement entropy for other intervals in Sec. 3.4.

$x_{\text{I}}$$u_{\text{I}}$$z_{*}$$\gamma_{1}$$-\sigma_{\text{I}}$$x_{\text{II}}$$u_{\text{II}}$$(N_{\text{I}},g^{\text{I}}_{ab})$$(N_{\text{II}},g^{\text{II}}_{ab})$$Q$$Q$$z_{*}$$\gamma_{2}$$\sigma_{\text{II}}$$(0,0)$$(0,0)$
Figure 3: Cartoon plot for the configuration of the extremal surfaces
$\gamma_{1}$ and $\gamma_{2}$ for chosen the boundary subsystem
$[-\sigma_{\text{I}},0]\cup[0,\sigma_{\text{II}}]$.

We are interested in the boundary subsystems including the interface. We
consider the subsystem $[-\sigma_{\text{I}},0]\cup[0,\sigma_{\text{II}}]$, the
corresponding extremal surface contains two parts, denoted by $\gamma_{1}$ and
$\gamma_{2}$ as shown in Fig. 3. In the coordinate system (2.14) and using
(2.17), at a constant time $t$, $\gamma_{1}$ and $\gamma_{2}$ can be written
as

$\displaystyle\gamma_{1}=\left(t,x_{\text{I}}[\xi_{1}],\frac{z_{\text{I}}[\xi_{1}]}{\sqrt{\nu}}\right)\,,~{}~{}~{}~{}~{}\gamma_{2}=(t,x_{\text{II}}[\xi_{2}],{\sqrt{\nu}}z_{\text{II}}[\xi_{2}])\,,$
(3.1)

where $\xi_{1}$ and $\xi_{2}$ are the parameters of the two curves. The length
functional is

$\displaystyle\mathcal{A}=\int
d\xi_{1}\,\frac{L_{\text{I}}\sqrt{\nu}}{z_{\text{I}}}\sqrt{\dot{x}_{\text{I}}^{2}+\frac{\dot{z}_{\text{I}}^{2}}{\nu}}\,+\int
d\xi_{2}\,\frac{L_{\text{I}}\sqrt{\nu}}{z_{\text{II}}}\sqrt{\dot{x}_{\text{II}}^{2}+\nu\dot{z}_{\text{II}}^{2}}\,,$
(3.2)

where the dots are derivatives with respect to $\xi_{1}$ or $\xi_{2}$.

For these two curves, we first perform the variation to obtain the boundary
terms and then we choose $\xi_{1}=z_{\text{I}}$ and $\xi_{2}=z_{\text{II}}$.
We get the boundary condition on the brane

$\displaystyle\frac{x^{\prime}_{\text{I}}(z)\psi^{\prime}_{\text{I}}(z)+\frac{1}{\nu}}{\sqrt{x^{\prime
2}_{\text{I}}(z)+\frac{1}{\nu}}}+\frac{x^{\prime}_{\text{II}}(z)\psi^{\prime}_{\text{II}}(z)+\nu}{\sqrt{x^{\prime
2}_{\text{II}}(z)+\nu}}\Bigg{|}_{\text{Q}}=0\,,$ (3.3)

where the prime is the derivative with respect to $z$. Note that
$x_{\text{I}}(z_{\text{I}})$ and $\psi_{\text{I}}(z)$ are curves for
$\gamma_{1}$ and $Q$ respectively, similarly for the other functions.

Let us give a geometrical interpretation of (3.3). The tangent vectors for the
curves $\gamma_{1}$ and $\gamma_{2}$ are

$\displaystyle
V_{1}=\left(0,x^{\prime}_{\text{I}}[z_{\text{I}}],\frac{1}{\sqrt{\nu}}\right)\,,\
\ \ V_{2}=(0,x^{\prime}_{\text{II}}[z_{\text{II}}],\sqrt{\nu})\,,$ (3.4)

where we have chosen the parameter
$\xi_{1}=z_{\text{I}},~{}\xi_{2}=z_{\text{II}}$. The tangent vectors of the
brane $Q$ are

$\displaystyle
W_{1}=\left(0,\psi^{\prime}_{\text{I}}[z],\frac{1}{\sqrt{\nu}}\right)\,,\ \ \
W_{2}=(0,\psi^{\prime}_{\text{II}}[z],\sqrt{\nu})\,.$ (3.5)

The boundary condition (3.3) on the brane $Q$ can be rewritten as

$\displaystyle\frac{g^{\text{I}}_{ab}V^{a}_{1}W^{a}_{1}}{\sqrt{g^{\text{I}}_{ab}V^{a}_{1}V^{a}_{1}}\sqrt{g^{\text{I}}_{ab}W^{a}_{1}W^{a}_{1}}}+\frac{g^{\text{II}}_{ab}V^{a}_{2}W^{a}_{2}}{\sqrt{g^{\text{II}}_{ab}V^{a}_{2}V^{a}_{2}}\sqrt{g^{\text{II}}_{ab}W^{a}_{2}W^{a}_{2}}}=0\,,$
(3.6)

where we have used the continuous condition (2.21). The above equation can be
formally written as

$\displaystyle\hat{V}_{1}\cdot\hat{W}_{1}+\hat{V}_{2}\cdot\hat{W}_{2}=0\,,$
(3.7)

where the hat means the normalized vector and the dot means the contraction of
two vectors by the AdS metric. (3.7) indicates that the angles between the
tangent vector of $\gamma_{1}$ and $Q$ and the one of $\gamma_{2}$ and $Q$ are
supplementary, which is consistent with the construction of geodesic in [15]
when $Q$ is a straight line.

The equations of motion from (3.2) give

$\begin{split}x^{\prime\prime}_{\text{I}}-\frac{x^{\prime}_{\text{I}}+\nu
x^{\prime
3}_{\text{I}}}{z_{\text{I}}}&=0\,,~{}~{}~{}~{}~{}~{}~{}x^{\prime\prime}_{\text{II}}-\frac{\nu
x^{\prime}_{\text{II}}+x^{\prime 3}_{\text{II}}}{\nu
z_{\text{II}}}=0\,,\end{split}$ (3.8)

where the prime are derivatives with respect to $z_{\text{I}}$ and
$z_{\text{II}}$. Obviously they take the same form as the ones in AdS/BCFT
[35, 36]. For the case that the bulk system is a portion of AdS3 (2.14), the
solutions $x_{\text{I}}(z_{\text{I}})$ and $x_{\text{II}}(z_{\text{II}})$ are
circle arcs passing through the points marked as $-\sigma_{\text{I}}$ and
$z_{*}$ as well as $\sigma_{\text{II}}$ and $z_{*}$ in Fig. 3. Plugging these
solutions into (3.3), the boundary condition can be written as

$\displaystyle\begin{split}&\frac{z^{2}-\nu(\sigma_{\text{I}}+\psi_{\text{I}})(\sigma_{\text{I}}+\psi_{\text{I}}-2z\psi^{\prime}_{\text{I}})}{z^{2}+\nu(\sigma_{\text{I}}+\psi_{\text{I}})^{2}}+\frac{\nu\left(\nu
z^{2}-(\sigma_{\text{II}}-\psi_{\text{II}})(\sigma_{\text{II}}-\psi_{\text{II}}+2z\psi^{\prime}_{\text{II}})\right)}{\nu
z^{2}+(\sigma_{\text{II}}-\psi_{\text{II}})^{2}}\Bigg{|}_{\text{Q}}=0\,.\end{split}$
(3.9)

After solving (3.9), we obtain the location
$z_{*}(\sigma_{\text{I}},\sigma_{\text{II}})$ of the extremal surface
intersecting on the interface brane. Then we can obtain the entanglement
entropy in terms of the geodesics length

$\displaystyle\begin{split}S_{\text{E}}(\sigma_{\text{I}},\sigma_{\text{II}})=&\,\frac{\mathcal{A}(\sigma_{\text{I}},\sigma_{\text{II}})}{4G}\\\
=&\,\frac{L_{\text{I}}}{4G}\,\left[\cosh^{-1}\frac{\frac{z^{2}}{\sqrt{\nu}}+\sqrt{\nu}(\sigma_{\text{I}}+\psi_{\text{I}}(z))^{2}}{2z\epsilon_{\text{I}}}+\nu\cosh^{-1}\frac{z^{2}\nu+(\sigma_{\text{II}}-\psi_{\text{II}}(z))^{2}}{2z\sqrt{\nu}\epsilon_{\text{II}}}\right]\Bigg{|}_{z=z_{*}}\\\
=&\,\frac{c_{\text{I}}}{6}\log\frac{\frac{z^{2}_{*}}{\sqrt{\nu}}+\sqrt{\nu}(\sigma_{\text{I}}+\psi_{\text{I}}(z_{*}))^{2}}{z_{*}\epsilon_{\text{I}}}+\frac{c_{\text{II}}}{6}\log\frac{z^{2}_{*}\nu+(\sigma_{\text{II}}-\psi_{\text{II}}(z_{*}))^{2}}{z_{*}\sqrt{\nu}\epsilon_{\text{II}}}\,,\end{split}$
(3.10)

where $z_{*}=z_{*}(\sigma_{\text{I}},\sigma_{\text{II}})$,
$\epsilon_{\text{I}}$ and $\epsilon_{\text{II}}$ are UV cutoffs along the
radial direction in the left and right AdS3 and we have ignored the terms of
order
$\mathcal{O}(\epsilon^{2}_{\text{I}}),\mathcal{O}(\epsilon^{2}_{\text{II}})$.121212If
we demand that the two cutoffs are the same on the brane $Q$, then we have
$\epsilon_{\text{I}}=\epsilon_{\text{II}}/\nu$. In (3.10), we have assumed
that the extremal curves are smooth and within the bulk. There exists
situations that the extremal surface passing the solution of (3.9) and the
boundary point might be out of bulk of one side, e.g. for
$\sigma_{\text{I}}\gg\sigma_{\text{II}}$ in the configuration (3) shown in
Fig. 2. In this case, we could use the squashed geodesics [37] for the
extremal curves out of the bulk, i.e. demanding the extremal curve spanning
along the interface brane instead of out of the bulk regime. We will not
consider this case in our study.

Another equivalent method to obtain the boundary condition (3.9) is to
minimise the length of the geodesic with respect to the intersecting point
between the geodesics and the interface brane. Suppose that the intersecting
point between the extremal surface and the brane takes the intrinsic
coordinate $(t,z)$, the location can be solved from

$\displaystyle\frac{d}{dz}\left[\cosh^{-1}\frac{\frac{z^{2}}{\sqrt{\nu}}+\sqrt{\nu}(\sigma_{\text{I}}+\psi_{\text{I}}(z))^{2}}{z\epsilon_{\text{I}}}+\nu\cosh^{-1}\frac{z^{2}\nu+(\sigma_{\text{II}}-\psi_{\text{II}}(z))^{2}}{z\sqrt{\nu}\epsilon_{\text{II}}}\right]=0\,,$
(3.11)

which is equivalent to (3.9).

###  3.1 Interface entropy and $g$-function

Setting $\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$, the interface
entanglement entropy can be defined for the interface field theory as

$S_{\text{iE}}(\sigma)=S_{\text{E}}-\frac{1}{2}S_{\text{E}}^{L}-\frac{1}{2}S_{\text{E}}^{R}$
(3.12)

where $S_{\text{E}}$ is the holographic entanglement entropy of the interval
$[-\sigma,\sigma]$ from AdS/ICFT (3.10), while $S_{\text{E}}^{L}$ and
$S_{\text{E}}^{R}$ are the holographic vacuum entanglement entropy of the same
interval that are obtained from the holographic CFT${}_{\text{I}}$ and
CFT${}_{\text{II}}$ respectively, i.e.

$S_{\text{E}}^{L}=\frac{c_{\text{I}}}{3}\log\frac{2\sigma}{\epsilon_{\text{I}}}\,,~{}~{}~{}~{}~{}S_{\text{E}}^{R}=\frac{c_{\text{II}}}{3}\log\frac{2\sigma}{\epsilon_{\text{II}}}\,.$
(3.13)

Note that similar proposal has also been used in [14] for AdS/ICFT without
scalar field. One nice feature of the above definition (3.12) is that when the
the brane is completely trivially connecting the two AdS spacetime, i.e. the
case of (2.33), we have $S_{\text{iE}}(\sigma)=0$.

Then we can define a $g$-function $g(\sigma)$ as follows

$\displaystyle\begin{split}\log g(\sigma)=&~{}S_{\text{iE}}(\sigma)\\\
=&~{}\frac{c_{\text{I}}}{6}\log\frac{\frac{z^{2}_{*}}{\sqrt{\nu}}+\sqrt{\nu}(\sigma+\psi_{\text{I}}(z_{*}))^{2}}{2z_{*}\sigma}+\frac{c_{\text{II}}}{6}\log\frac{z^{2}_{*}\nu+(\sigma-\psi_{\text{II}}(z_{*}))^{2}}{2\sqrt{\nu}z_{*}\sigma}\,,\end{split}$
(3.14)

with $z_{*}=z_{*}(\sigma)$ determined from (3.9) by setting
$\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$. The $g$-function is finite and
independent with the UV cutoffs $\epsilon_{\text{I}}$ and
$\epsilon_{\text{II}}$.

Note that $\log g(\sigma)\in(-\infty,\infty)$. When the brane has a constant
tension, i.e. the solution (2.29), the interface entropy $S_{\text{iE}}=\log
g(\sigma)$ is a constant so that the above $g$-function is also a constant
[14]. This is expected as the induced metric on $Q$ is an AdS2 slice. We will
show this as the first example in the section 3.3.1. Specially, when the
profiles satisfy the relations $\psi_{\text{I}}(z)\geq 0$ and
$\psi_{\text{II}}(z)\leq 0$, we have $\psi_{\text{I}}(z_{*})\geq 0$ and
$\psi_{\text{II}}(z_{*})\leq 0$, then from (3.14), $\log g(\sigma)\geq 0$.

The monotonic behavior of $\log g(\sigma)$ is expected to reflect the RG flow
the interface-associated degrees of freedom of the interface field theory. In
Sec. 3.1.1 we will prove $g$-theorem for the profile of the case (2) in Fig.
2. In Sec. 3.3 we will present specific examples demonstrating that $\log
g(\sigma)$ consistently exhibits a monotonic decrease when the NEC is
satisfied, which includes other cases in Fig. 2.131313It is interesting to
connect the $g$-function to the defect RG flow in the field theory, see e.g.
[38].

####  3.1.1 $g$-theorem

We will first prove that when the induced metric on the interface $Q$ is
asymptotic to AdS2 in both the UV and IR limits, we have $\lim_{\sigma\to
0}S_{\text{iE}}(\sigma)\geq\lim_{\sigma\to\infty}S_{\text{iE}}(\sigma)$. Then
will give a proof of $g$-theorem for the profile of case (2) in Fig. 2.

In the UV limit $z\to 0$, the brane is asymptotically AdS2 if we assume that
the brane $Q$ has the following asymptotic profile

$\displaystyle\psi_{\text{I}}(z)\simeq\gamma z^{n}+\cdots\,,$ (3.15)

and

$\displaystyle\psi_{\text{II}}(z)\simeq\pm\sqrt{\frac{1}{\nu}-\nu+\gamma^{2}n^{2}\delta_{1n}}\
z+\cdots\,,$ (3.16)

where the dots are the higher order term in $z$. From the above asymptotic
behaviors, expanding (3.9) by assuming that $z_{*}$ and $\sigma$ are of same
order, we have

$\displaystyle z_{*}=\sqrt{\frac{\nu}{1+\gamma^{2}\nu\delta_{1n}}}\
\sigma+\cdots\,.$ (3.17)

Therefore the interface entropy in the UV limit is

$\displaystyle\begin{split}S_{\text{iE}}^{\text{UV}}=&~{}\lim_{\sigma\to
0}\,S_{\text{iE}}(\sigma)\\\
=&~{}\frac{c_{\text{I}}}{6}\log\left(\gamma\sqrt{\nu}\delta_{1n}+\sqrt{1+\gamma^{2}\nu\delta_{1n}}\right)\\\
&~{}~{}~{}~{}+\frac{c_{\text{II}}}{6}\log\left(\frac{\sqrt{1+\gamma^{2}\nu\delta_{1n}}\mp\sqrt{1-\nu^{2}+\gamma^{2}\nu\delta_{1n}}}{\nu}\right)+\mathcal{O}(\sigma)\,,\end{split}$
(3.18)

which is a constant in the leading term. The sign in (3.18) corresponds to the
choice of sign in (3.16). When $n=1$, $S_{\text{iE}}^{\text{UV}}$ is the same
as the result with trivial scalar field, which will be written out explicitly
in (3.42). When $n>1$, the left brane is perpendicular to the boundary in the
near-boundary limit thus the left part contributes nothing to
$S_{\text{iE}}^{\text{UV}}$, i.e. we have
$S_{\text{iE}}^{\text{UV}}=\frac{c_{\text{II}}}{6}\log(\frac{1\mp\sqrt{1-\nu^{2}}}{\nu})$.

In the IR limit $z\to+\infty$, the brane is asymptotically AdS2 if we assume
that the brane $Q$ has the asymptotic profile

$\displaystyle\psi_{\text{I}}(z)\simeq\psi^{\prime}_{\text{I}}(+\infty)z+...\,,$
(3.19)

and

$\displaystyle\psi_{\text{II}}(z)\simeq\pm\sqrt{\psi^{\prime
2}_{\text{I}}(\infty)+\frac{1}{\nu}-\nu}\,z+...\,,$ (3.20)

with the dots the subleading terms compared to $z$. Note that the above
asymptotic behaviors are not the most generic ones while a special one which
would lead to the IR geometry asymptotically AdS2. From the above asymptotic
behavior and (3.9), we have

$\displaystyle\sigma\simeq\sqrt{\frac{1}{\nu}+\psi^{\prime
2}_{\text{I}}(+\infty)}\,z+...\,.$ (3.21)

The interface entropy in the IR limit is

$\displaystyle\begin{split}S_{\text{iE}}^{\text{IR}}=&\lim_{\sigma\to+\infty}S_{\text{iE}}(\sigma)\\\
=&~{}\frac{c_{\text{I}}}{6}\log\left(\psi^{\prime}_{\text{I}}(+\infty)\sqrt{\nu}+\sqrt{1+\psi^{\prime}_{\text{I}}(+\infty)^{2}\nu}\right)\\\
&~{}~{}~{}~{}+\frac{c_{\text{II}}}{6}\log\frac{\sqrt{1+\psi^{\prime}_{\text{I}}(+\infty)^{2}\nu}\mp\sqrt{1+\psi^{\prime}_{\text{I}}(+\infty)^{2}\nu-\nu^{2}}}{\nu}\,,\end{split}$
(3.22)

where the sign corresponds to the choice of sign in (3.20).

When the matter field on $Q$ satisfies the NEC as discussed in Sec. 2.3, i.e.
when $\nu\neq 1$, or the condition
$\nu=1,\psi^{\prime}_{\text{I}}=-\psi^{\prime}_{\text{II}}$, we have
$\psi^{\prime\prime}_{\text{I}}\leq 0$. This indicates that
$\psi^{\prime}_{\text{I}}(0)\geq\psi^{\prime}_{\text{I}}(+\infty)$. From
(3.18) and (3.22), we always have

$\displaystyle S_{\text{iE}}^{\text{UV}}\geq S_{\text{iE}}^{\text{IR}}\,.$
(3.23)

When $\nu=1,\psi^{\prime}_{\text{I}}=\psi^{\prime}_{\text{II}}$, we have
$S_{\text{iE}}=0$ indicating the interface is trivial.

Note that (3.23) is a relation between the values of $g$-function at the
specific UV and IR fixed points. When we are away from the UV and IR limit,
the $g$-function is expected to satisfy the $g$-theorem
$\frac{d}{d\sigma}S_{\text{iE}}(\sigma)\leq 0$ along the whole RG flow. In
Sec. 2.3 we have proved that the existence of the profile indicates the NEC.
With the assumption of a given profile, now let us analyze the $g$-theorem in
detail.

From (3.14), we have

$\frac{d}{d\sigma}S_{\text{iE}}(\sigma)=\frac{c_{\text{I}}}{6}\left[\frac{2\nu(\sigma+\psi_{\text{I}}(z_{*}))}{z_{*}^{2}+\nu(\sigma+\psi_{\text{I}}(z_{*}))^{2}}+\frac{2\nu(\sigma-\psi_{\text{II}}(z_{*}))}{z_{*}^{2}\nu+(\sigma-\psi_{\text{II}}(z_{*}))^{2}}-\frac{1+\nu}{\sigma}\right]\,,$
(3.24)

where we have used $\frac{\partial S_{\text{E}}}{\partial z_{*}}=0$.

When $\nu=1$, we consider (2.34), i.e.
$\psi_{\text{I}}(z)=-\psi_{\text{II}}(z)$. Eq. (3.24) can be simplified
as141414In this case, the system can be viewed as an unfolding of BCFT. It was
proved in [39, 36] that for AdS/BCFT when NEC is satisfied, the $g$-theorem
$\frac{d}{d\sigma}S_{\text{iE}}(\sigma)\leq 0$ holds.

$\frac{d}{d\sigma}S_{\text{iE}}(\sigma)=-\frac{c_{\text{I}}(z_{*}^{2}-\sigma^{2}+\psi^{2}_{\text{I}}(z_{*}))}{3\sigma(z_{*}^{2}+(\sigma+\psi_{\text{I}}(z_{*}))^{2})}\,.$
(3.25)

The profile of system is case (2) or (3) in Fig. 2, where we can always find
$z^{2}+\psi^{2}_{\text{I}}(z_{*})\geq\sigma^{2}$ using the convex property of
the interface brane and the boundary condition (3.38) introduced in the next
subsection, thus $\frac{d}{d\sigma}S_{\text{iE}}(\sigma)\leq 0$ and
$g$-theorem holds.

When $0<\nu<1$, using $\frac{\partial S_{\text{E}}}{\partial z_{*}}=0$ we can
simplify (3.24) as

$\frac{d}{d\sigma}S_{\text{iE}}(\sigma)=\frac{c_{\text{I}}}{6\sigma}\left[-\frac{2\nu(\sigma+\psi_{\text{I}}(z_{*}))(\psi_{\text{I}}(z_{*})-z\psi^{\prime}_{\text{I}}(z_{*}))}{z_{*}^{2}+\nu(\sigma+\psi_{\text{I}}(z_{*}))^{2}}+\frac{2\nu(\sigma-\psi_{\text{II}}(z_{*}))(\psi_{\text{II}}(z_{*})-z\psi^{\prime}_{\text{II}}(z_{*}))}{z_{*}^{2}\nu+\nu(\sigma-\psi_{\text{II}}(z_{*}))^{2}}\right]\,.$
(3.26)

For the case (2) in Fig. 2, we have
$\psi_{\text{I}}(0)=\psi_{\text{II}}(0)=0$, $\psi_{\text{I}}(z)\geq
0,\psi_{\text{II}}(z)\leq 0$, and the NEC constraints
$\psi^{\prime\prime}_{\text{I}}(z)\leq
0,\psi^{\prime\prime}_{\text{II}}(z)\geq 0$. Then

$(\sigma+\psi_{\text{I}}(z_{*}))(\psi_{\text{I}}(z_{*})-z\psi^{\prime}_{\text{I}}(z_{*}))\geq
0\,,\ \
~{}~{}~{}(\sigma-\psi_{\text{II}}(z_{*}))(\psi_{\text{II}}(z_{*})-z\psi^{\prime}_{\text{II}}(z_{*}))\leq
0\,,$ (3.27)

thus $\frac{d}{d\sigma}S_{\text{iE}}(\sigma)\leq 0$, i.e. the $g$-theorem
holds. For other cases in Fig. 2, it is not obvious to prove the $g$-theorem.
Moreover, the possible existence of squashed geodesics makes the proof more
complicated. Instead we will provide some examples in Sec. 3.3, to show that
it is indeed satisfied for the specific examples we considered.

###  3.2 BCFT limits of holographic entanglement entropy

The BCFT limit of AdS/ICFT was discussed in Sec. 2.2 and there are two
different BCFT limits, i.e. $\nu\to 0$ and $\nu=1$. Here we take these two
BCFT limits to the entanglement entropy and the interface entropy.

Let us first study the limit $\nu\to 0$. In this case, the BCFT limit of the
geodesic can be found by solving the boundary condition (3.9). Similar to the
discussion in Sec. 2.2, it is more convenient to work in the $t,u$ coordinate
on $Q$. We first rewrite (3.9) in $u$ coordinate

$\displaystyle\begin{split}&\frac{u^{2}-(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}})(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}}-2u\tilde{\psi}^{\prime}_{\text{I}})}{u^{2}+(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}})^{2}}+\nu\frac{\nu^{2}u^{2}-(\sigma_{\text{II}}-\tilde{\psi}_{\text{II}})(\sigma_{\text{II}}-\tilde{\psi}_{\text{II}}+2u\tilde{\psi}^{\prime}_{\text{II}})}{\nu^{2}u^{2}+(\sigma_{\text{II}}-\tilde{\psi}_{\text{II}})^{2}}\Bigg{|}_{\text{Q}}=0\,,\end{split}$
(3.28)

where the conventions in Sec. 2.2 have been used and the prime denotes the
derivative with respect to $u$. In the limit $\nu\to 0$, it becomes

$\displaystyle\frac{u^{2}-(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}})(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}}-2u\tilde{\psi}^{\prime}_{\text{I}})}{u^{2}+(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}})^{2}}\Bigg{|}_{\text{Q}}=0\,.$
(3.29)

Using the fact that the normalized tangent vector of $\gamma_{1}$ and the
brane take the following form in $u$ coordinate from (3.4) and (3.5),

$\hat{V}_{1}=\frac{2(\tilde{\psi}_{\text{I}}+\sigma_{\text{I}})}{(\tilde{\psi}_{\text{I}}+\sigma_{\text{I}})^{2}+u^{2}}\left(-u,~{}\frac{(\tilde{\psi}_{\text{I}}+\sigma_{\text{I}})^{2}-u^{2}}{2(\tilde{\psi}_{\text{I}}+\sigma_{\text{I}})}\right)\,,~{}~{}~{}~{}~{}~{}\hat{W}_{1}=\frac{1}{\sqrt{\tilde{\psi}^{\prime
2}_{\text{I}}+1}}\left(\tilde{\psi}^{\prime}_{\text{I}},~{}1\right)\,,$ (3.30)

it is straightforward to prove that (3.29) is equivalent to

$\displaystyle\hat{V}_{1}\cdot\hat{W}_{1}=0\,.$ (3.31)

Therefore (3.29) is exactly the boundary condition of minimal surface in
AdS/BCFT [35]. Given the solution $u_{*}$ of (3.29), we can calculate the
entanglement entropy in AdS/BCFT

$\displaystyle S_{\text{E}}$
$\displaystyle=\frac{c_{\text{I}}}{6}\log\frac{u^{2}_{*}+(\sigma_{\text{I}}+\tilde{\psi}_{\text{I}}(u_{*}))^{2}}{u_{*}\epsilon_{\text{I}}}\,.$
(3.32)

The interface entanglement entropy or $g$-function takes the following form in
AdS/BCFT,

$\displaystyle\begin{split}\log
g(\sigma)=&~{}S_{\text{iE}}(\sigma)=~{}\frac{c_{\text{I}}}{6}\log\frac{u^{2}_{*}+(\sigma+\tilde{\psi}_{\text{I}}(u_{*}))^{2}}{2u_{*}\sigma}\,.\end{split}$
(3.33)

In the case of trivial scalar field with $\phi=0,V=T$, the configuration for
the ICFT is (2.29). From (2.36) the BCFT limit is given by

$\displaystyle\tilde{\psi}_{\text{I}}=\frac{L_{\text{I}}T}{\sqrt{1-L^{2}_{\text{I}}T^{2}}}\,u\,,~{}~{}~{}~{}\tilde{\psi}_{\text{II}}=\pm\frac{1}{\sqrt{1-L^{2}_{\text{I}}T^{2}}}\,u+\mathcal{O}(\nu)\,.$
(3.34)

From (3.29), (3.32) and (3.33), the entanglement entropy takes the following
form

$\displaystyle\begin{split}u_{*}&=\sqrt{1-L^{2}_{\text{I}}T^{2}}\sigma\,,\\\
S_{\text{E}}&=\frac{c_{\text{I}}}{6}\log\frac{2\sigma}{\epsilon_{\text{I}}}+\log
g=\frac{c_{\text{I}}}{6}\log\left(\sqrt{\frac{1+L_{\text{I}}T}{1-L_{\text{I}}T}}\,\frac{2\sigma}{\epsilon_{\text{I}}}\right)\,,\\\
\log
g&=\frac{c_{\text{I}}}{12}\log\left(\frac{1+L_{\text{I}}T}{1-L_{\text{I}}T}\right)\,.\end{split}$
(3.35)

This is precisely the result obtained in [35]. For nontrivial scalar field
case, boundary entropy is not a constant anymore [34] and the results depend
on the detailed profile of the brane $Q$.

Another type of BCFT limit is the $\nu=1$ limit. Noticing that now we have
$z=u_{\text{I}}=u_{\text{II}}=u$ and we will use $z$-coordinate for
convenient. We set $\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$ and from
(2.34) we also set $\psi_{\text{I}}(z)=-\psi_{\text{II}}(z)=\varphi(z)$. The
boundary condition (3.3) becomes

$\displaystyle\frac{x^{\prime}_{\text{I}}(z)\varphi(z)+1}{\sqrt{x^{\prime
2}_{\text{I}}(z)+1}}+\frac{-x^{\prime}_{\text{II}}(z)\varphi(z)+1}{\sqrt{x^{\prime
2}_{\text{II}}(z)+1}}\Bigg{|}_{\text{Q}}=0\,.$ (3.36)

Note that when $\nu=1$, from (3.8) the equations for $x_{\text{I}}$ and
$x_{\text{II}}$ are the same, so they should have the same solution. From the
$Z_{2}$ folding symmetry between the left part and right part of the system,
we have $x^{\prime}_{\text{I}}=-x^{\prime}_{\text{II}}=x^{\prime}$ on the
brane. Then (3.36) becomes

$\displaystyle 2\frac{x^{\prime}(z)\varphi(z)+1}{\sqrt{x^{\prime
2}(z)+1}}\Bigg{|}_{\text{Q}}=0\,.$ (3.37)

Repeating the analysis below (3.3), (3.37) indicates that

$\displaystyle\hat{V}_{1}\cdot\hat{W}_{1}=\hat{V}_{2}\cdot\hat{W}_{2}=0\,,$
(3.38)

which is precisely the perpendicular boundary condition of the minimal surface
on the EOW brane in AdS/BCFT. We can also use (3.8) to write (3.37) in the
following form

$\displaystyle\frac{z^{2}-(\sigma+\varphi)(\sigma+\varphi-2z\varphi^{\prime})}{z^{2}+(\sigma+\varphi)^{2}}\Bigg{|}_{\text{Q}}=0\,,$
(3.39)

which is just a rewrite of (3.9) by setting $\nu=1$,
$\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$ and
$\psi_{\text{I}}(z)=-\psi_{\text{II}}(z)=\varphi(z)$. Note that this is also
the same as (3.29) in the $\nu\to 0$ limit. Similarly, after solving above
equation to obtain the intersecting point $z_{*}$, we can calculate EE and
$g$-function and they just have the form of (3.32) and (3.33) by identifying
$u_{*}\to z_{*}$ and $\tilde{\psi}_{\text{I}}(u_{*})\to\varphi(z_{*})$.

###  3.3 Case studies of several examples

In this subsection, we present six distinct examples with different monotonic
profiles of the interface brane $Q$. An example with non-monotonic profile is
shown in appendix A. With a nontrivial scalar field, the interface field
theory is expected to deviate from a fixed point. The bending of the interface
brane reflects the RG flow of interface, revealing numerous significant
physical phenomena.

Before proceeding, let us summarize our findings based on the specific
examples examined:

  1. (1)

In cases where the induced metric on the interface brane $Q$ flows from a UV
AdS to an IR AdS, the scalar potential evolves from a locally maximal in UV to
a globally minimal in IR, as shown in the solutions II, III, V. In solution I
where the scalar field is trivial, the induced metric is a pure AdS2.

  2. (2)

When the induced metric on the interface brane $Q$ flows from a UV AdS to an
IR flat spacetime, the scalar potential evolves from a locally maximal in UV
(solution VI or the solution in Appendix A) or a locally minimal in UV
(solution IV with $n>3$) to a globally minimal in IR.

  3. (3)

The relation
$\partial_{\phi}V(\phi)|_{\phi_{\text{UV}}}=\partial_{\phi}V(\phi)|_{\phi_{\text{IR}}}=0$
is always satisfied for all examples.

  4. (4)

When the scalar potential exhibits non-monotonic behavior, i.e. the solutions
IV and V, multiple extremal surfaces may exist for the interval we studied.
Consequently, a first order phase transition occurs for the interface entropy
$S_{\text{iE}}(\sigma)$ as $\sigma$ increases.

  5. (5)

When the induced metric is asymptotically flat in IR, i.e. the solutions VI
and IV and the solution in Appendix A, the interface entropy
$S_{\text{iE}}(\sigma\to\infty)$ goes to $-\infty$. Notably, in such cases,
the scalar field could be divergent (or finite) if the interface brane spans
to $z\to\infty$ (or a finite value) in the IR limit.

  6. (6)

The $g$-theorem is always consistently satisfied, i.e. $S_{\text{iE}}(\sigma)$
monotonically decreases when $\sigma$ increases. This is consistent with the
analytical proof in Sec. 3.1.1.

####  3.3.1 Solution I with trivial scalar field

The simplest example is the case with trivial scalar field, i.e. $\phi=0,V=T$.
This case was studied in [15, 14]. Now the interface $Q$ is a straight line
and the solution is given by (2.29) and the tension is bounded by (2.30). The
induced metric on the interface brane is AdS2.

In this case, the intersection point between the minimal surface and the brane
is

$\displaystyle z_{*}=\sqrt{\frac{\nu}{1+\gamma^{2}\nu}}\ \sigma\,.$ (3.40)

Then we can calculate the entanglement entropy

$\displaystyle
S_{\text{E}}=\frac{c_{\text{I}}}{6}\log\frac{2\sigma}{\epsilon_{\text{I}}}+\frac{c_{\text{II}}}{6}\log\frac{2\sigma}{\epsilon_{\text{II}}}+\log
g\,,$ (3.41)

with the $g$-function

$\displaystyle\log
g=\frac{c_{\text{I}}}{6}\log\left(\gamma\sqrt{\nu}+\sqrt{1+\gamma^{2}\nu}\right)+\frac{c_{\text{II}}}{6}\log\left(\frac{\sqrt{1+\gamma^{2}\nu}\mp\sqrt{1+\gamma^{2}\nu-\nu^{2}}}{\nu}\right)\,,$
(3.42)

where the signs $\mp$ in (3.42) corresponds to the choices of sign $\pm$ in
(2.29). Note that the $g$-function is a constant and independent of the
interval length $\sigma$. This is consistent with the fact that the induced
metric on the interface $Q$ is an AdS2 and the interface field theory on the
boundary is expected to be a CFT. Notably, when $\nu=1$ and
$\psi_{\text{I}}=\psi_{\text{II}}=\gamma z$ we have $\log g=0$ which is
precisely the case that the interface $Q$ does not play any role in the
construction.

We can also perform the coordinate transformation on Poincare AdS coordinate
(2.14) via

$\displaystyle
u_{\text{A}}=\frac{y_{\text{A}}}{\cosh(\rho_{\text{A}}/L_{\text{A}})}\,,\qquad
x_{\text{A}}=y_{\text{A}}\tanh(\rho_{\text{A}}/L_{\text{A}})\,,~{}~{}~{}\text{A}=(\text{I},\text{II})$
(3.43)

then the interface entropy can be expressed as

$\displaystyle S_{\text{iE}}=\log
g(\sigma)=\frac{1}{4G}\left(\rho^{*}_{\text{I}}+\rho^{*}_{\text{II}}\right)\,,$
(3.44)

where $\rho^{*}_{\text{I}},\rho^{*}_{\text{II}}$ are

$\displaystyle\rho^{*}_{\text{I}}=L_{\text{I}}\log(\lambda\sqrt{\nu}+\sqrt{1+\lambda^{2}\nu}),\
\ \
\rho^{*}_{\text{II}}=L_{\text{II}}\log\frac{\sqrt{1+\lambda^{2}\nu}\mp\sqrt{1+\lambda^{2}\nu-\nu^{2}}}{\nu}.$
(3.45)

which parameterize the positions of the branes. This is the same as the result
in [15, 14].

####  3.3.2 Solution II

The second example of the interface is described by

$\displaystyle\psi_{\text{I}}(z)=$ $\displaystyle\frac{az+bz^{2}}{1+z}\,.$
(3.46)

The NEC (2.42) imposes the constraint $a\geq b$. The solution for
$\psi_{\text{II}}$ can be represented using elliptic integrals, which are
quite complicated. Fig. 4 illustrates an example of the brane configuration
when $a=2$, $b=1$, $L_{\text{I}}=1$, and $\nu=1/2$, with the minus sign
selected for $\psi_{\text{II}}$. It belongs to the case (2) classified in Fig.
2.

Figure 4: The profiles of the interface brane $Q$ for the example (3.46) with
$a=2$, $b=1$, $L_{\text{I}}=1$, and $\nu=0.5$.

In the UV limit $z\to 0$ and the IR limit $z\to\infty$, $\psi_{\text{I}}$
behaves as

$\displaystyle\begin{split}\psi_{\text{I}}(z)=&~{}az+(b-a)z^{2}+\mathcal{O}(z^{3})\,,\qquad(z\to
0)\,,\\\
\psi_{\text{I}}(z)=&~{}bz-(b-a)+\mathcal{O}\left(\frac{1}{z}\right)\,,\qquad(z\to\infty)\,.\end{split}$
(3.47)

Therefore, we can approximate it with a straight line in both the UV and IR
regimes, and it exhibits asymptotic AdS2 behavior in both the UV and IR.
Therefore the profile can be though of a precise example in Sec. 3.1.1. The
nature of fixed points can also be confirmed by examining equation (2.26),
where we find

$\displaystyle\begin{split}\lambda(z)=&-\frac{1}{L^{2}_{\text{I}}(1+a^{2}\nu)}+\mathcal{O}(z)\,,~{}~{}~{}~{}~{}\qquad(z\to
0)\,,\\\
\lambda(z)=&-\frac{1}{L^{2}_{\text{I}}(1+b^{2}\nu)}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\,,\qquad(z\to\infty)\,.\end{split}$
(3.48)

The leading term is a negative constant in both the UV and IR regimes,
indicating that the interface brane asymptotically approaches AdS2 in these
regions. Interestingly, the condition derived from the NEC is equivalent to
requiring that the effective AdS2 radius on the interface $Q$ satisfy
$L_{Q}^{\text{UV}}\geq L_{Q}^{\text{IR}}$ for $b>0$.151515The relation is no
longer true for $a<0$. Nevertheless, we have checked several examples and find
that the $g$-theorem is still satisfied. The ICFT dual to the interface is
anticipated to flow from a UV fixed point to an IR fixed point.

The interface entropy at both the UV and IR fixed points can be calculated
using equation (3.42)

$\displaystyle\begin{split}S^{\text{UV}}_{\text{iE}}&=\frac{c_{\text{I}}}{6}\log\left(a\sqrt{\nu}+\sqrt{1+a^{2}\nu}\right)+\frac{c_{\text{II}}}{6}\log\left(\frac{\sqrt{1+a^{2}\nu}\mp\sqrt{1+a^{2}\nu-\nu^{2}}}{\nu}\right)\,,\\\
S^{\text{IR}}_{\text{iE}}&=\frac{c_{\text{I}}}{6}\log\left(b\sqrt{\nu}+\sqrt{1+b^{2}\nu}\right)+\frac{c_{\text{II}}}{6}\log\left(\frac{\sqrt{1+b^{2}\nu}\mp\sqrt{1+b^{2}\nu-\nu^{2}}}{\nu}\right)\,.\end{split}$
(3.49)

For the condition $a\geq b$, we have

$\displaystyle S^{\text{UV}}_{\text{iE}}\geq S^{\text{IR}}_{\text{iE}}\,,$
(3.50)

as discussed already in Sec. 3.1.1. A special case is that the $g$-function
becomes a constant thus $S^{\text{UV}}_{\text{iE}}=S^{\text{IR}}_{\text{iE}}$
when $a=b$ and the brane becomes $\psi_{\text{I}}(z)=az$ which is precisely
the example discussed in Sec. 3.3.1.

In the UV limit $z\to 0$, the scalar field and its potential exhibit the
following asymptotic behavior

$\displaystyle\begin{split}\phi^{2}(z)=&~{}\frac{4(a-b)L_{\text{I}}\sqrt{\nu}(\sqrt{1+a^{2}\nu-\nu^{2}}\mp
a\nu^{3/2})}{\sqrt{1+a^{2}\nu}\sqrt{1+a^{2}\nu-\nu^{2}}}z+\cdots\,,\\\
V(\phi)=&~{}\frac{a\nu^{3/2}\mp\sqrt{1+a^{2}\nu-\nu^{2}}}{L_{\text{I}}\nu\sqrt{1+a^{2}\nu}}-\frac{1}{4L_{\text{I}}^{2}(1+a^{2}\nu^{2})}\phi^{2}+\cdots\,.\end{split}$
(3.51)

In the IR limit as $z\to\infty$, the scalar field and its potential
demonstrate the following asymptotic behavior

$\displaystyle\begin{split}(\phi(z)-\phi_{0})^{2}=&~{}\frac{(a-b)L_{\text{I}}\sqrt{\nu}(\sqrt{1+b^{2}\nu-\nu^{2}}\mp
b\nu^{3/2})}{\sqrt{1+b^{2}\nu}\sqrt{1+b^{2}\nu-\nu^{2}}}\frac{1}{z^{2}}+\cdots\,,\\\
V(\phi)=&~{}\frac{b\nu^{3/2}\mp\sqrt{1+b^{2}\nu-\nu^{2}}}{L_{\text{I}}\nu\sqrt{1+b^{2}\nu}}+\frac{2}{L_{\text{I}}^{2}(1+b^{2}\nu)}(\phi-\phi_{0})^{2}+\cdots\,.\end{split}$
(3.52)

where $\phi_{0}$ is a constant of integration. From the above expressions, the
potential $V$ exhibits a maximum at $\phi=0$ and a minimum at $\phi=\phi_{0}$.
These points respectively correspond to the UV and IR fixed points.

We show an example of this solution in Fig. 5, where the parameters are set to
be $a=2$, $b=1$, $L_{\text{I}}=1$, and $\nu=1/2$. The upper two plots are for
$\phi(z)$ and $V(\phi)$. The integration constant $\phi_{0}$ in (3.52) is
$1.62$. The lower two plots are for the entanglement entropy and the interface
entropy. Obviously the $g$-theorem is satisfied.

Figure 5: The $\phi(z)$, $V(\phi)$ (upper plots) and the entanglement entropy
and interface entropy (lower plots) for the profile (3.46). Here we set
$a=2\,,b=1\,,L_{\text{I}}=1\,,\nu=0.5$.

####  3.3.3 Solution III

The third example of the interface brane is described by

$\displaystyle\psi_{\text{I}}(z)=a\arctan(bz)+cz\,,$ (3.53)

where $a,b,c$ are constants. We choose $b>0$ since its sign can be moved out.
The NEC (2.42) demands $ab\geq 0.$ The solution for $\psi_{\text{II}}$ can
only be obtained numerically. In Fig. 6, we show an example of the brane
configuration. Similar to the discussion of the previous example, the minus
sign for $\psi_{\text{II}}$ is chosen and it belongs to the case (2)
classified in Fig. 2.

Figure 6: The profile of the interface brane $Q$ for the example (3.53). Here
we choose $a=b=c=1\,,\nu=0.5\,,L_{\text{I}}=1$.

$\psi_{\text{I}}(z)$ has the following asymptotic behaviour,

$\displaystyle\begin{split}\psi_{\text{I}}(z)=&~{}(ab+c)z-\frac{ab^{3}}{3}z^{3}+\mathcal{O}(z^{5})\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(z\to
0)\,,\\\ \psi_{\text{I}}(z)=&~{}cz+\frac{\pi
a}{2}-\frac{a}{bz}+\frac{a}{3b^{3}z^{3}}+\mathcal{O}\left(\frac{1}{z^{5}}\right)\,,~{}~{}~{}~{}(z\to\infty)\,.\end{split}$
(3.54)

The induced metric on the brane and the Ricci tensor satisfy the relation
$R_{Q\mu\nu}=\lambda(z)h_{\mu\nu}$, where

$\displaystyle\begin{split}\lambda(z)=&~{}-\frac{1}{(1+(ab+c)^{2}\nu)L^{2}_{\text{I}}}+\mathcal{O}(z^{4})\,,~{}~{}~{}(z\to
0)\,,\\\
\lambda(z)=&~{}-\frac{1}{(1+c^{2}\nu)L^{2}_{\text{I}}}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\,,~{}~{}~{}~{}~{}~{}~{}~{}(z\to\infty)\,.\end{split}$
(3.55)

This indicates that the brane is also asymptotically AdS in the UV and IR
limits.

The entanglement entropy can also computed from (3.42). Following the
discussion in Sec. 3.1.1, once the NEC is satisfied, we have

$S^{\text{UV}}_{\text{iE}}\geq S^{\text{IR}}_{\text{iE}}\,.$ (3.56)

A special case is that the $g$-function becomes a constant thus
$S^{\text{UV}}_{\text{iE}}=S^{\text{IR}}_{\text{iE}}$ when $ab=0$ and the
brane becomes $\psi_{\text{I}}(z)=cz$, i.e. the solution in Sec. 3.3.1.

In the UV limit $z\to 0$, the scalar field and its potential behave as
follows,

$\displaystyle\begin{split}\phi^{2}(z)=&~{}\frac{ab^{3}L_{\text{I}}\sqrt{\nu}(\sqrt{1+((ab+c)^{2}-\nu)\nu}\mp(ab+c)\nu^{3/2})}{\sqrt{1+(ab+c)^{2}\nu}\sqrt{1+((ab+c)^{2}-\nu)\nu}}z^{2}+\dots\,,\\\
V(\phi)=&~{}\frac{(ab+c)\nu^{3/2}\mp\sqrt{1+((ab+c)^{2}-\nu)\nu}}{L_{\text{I}}\nu\sqrt{1+(ab+c)^{2}\nu}}+C_{1}\cdot\phi^{4}+\dots\,,\end{split}$
(3.57)

where $C_{1}$ are complicated functions depending on $a,b,c$ and $\nu$. It is
interesting to see that the effective mass of the scalar field vanishes in the
UV limit which means that the scalar field is sourceless on the brane. In the
IR limit $z\to\infty$, the scalar field and its potential have following
asymptotic behaviour,

$\displaystyle\begin{split}(\phi(z)-\phi_{0})^{2}=&~{}\frac{aL_{\text{I}}\sqrt{\nu}(\sqrt{1+c^{2}\nu-\nu^{2}}\mp
c\nu^{3/2})}{b\sqrt{1+c^{2}\nu}\sqrt{1+c^{2}\nu-\nu^{2}}}\frac{1}{z^{2}}+\dots\,,\\\
V(\phi)=&~{}\frac{c\nu^{3/2}\mp\sqrt{1+c^{2}\nu-\nu^{2}}}{L_{\text{I}}\nu\sqrt{1+c^{2}\nu}}+\frac{2}{L_{\text{I}}^{2}(1+c^{2}\nu)}(\phi-\phi_{0})^{2}+\dots\,.\end{split}$
(3.58)

In Fig. 7, we show an example of the scalar field, the potential, as well as
the entanglement entropy and the interface entropy $S_{\text{iE}}$. Obviously
the $g-$theorem holds and $\phi_{0}\simeq 1.28$. Different from the behavior
of $S_{\text{iE}}$ in Fig. 5, here when $\sigma\to 0$, we have
$\partial_{\sigma}S_{\text{iE}}\to 0$.

Figure 7: The $\phi(z)$, $V(\phi)$ (upper plots) and the entanglement entropy
and interface entropy (lower plots) for the profile (3.53). Here we chose
$a=b=c=1\,,\nu=0.5\,,L_{\text{I}}=1$ and we have $\phi_{0}\simeq 1.28$.

####  3.3.4 Solution IV

We have considered the curved interface branes that can be approximated as
straight lines in both the UV and IR regimes in Sec. 3.3.2 and Sec. 3.3.3.
Here, we will study a more general solution that is asymptotically AdS2 in the
UV regime, as shown in (2.27).

The profile $\psi_{\text{I}}$ of the interface brane has the form

$\displaystyle\psi_{\text{I}}=\gamma\ z^{n}$ (3.59)

where $n\geq 1$ and $\gamma$ are constants. When $n=1$, this exactly matches
the case we studied in Sec. 3.3.1 and there is no constraint on $\gamma$ from
NEC. In this subsection we focus on the regime $n>1$, where the NEC further
constraints $\gamma<0$. Evaluating (2.26) we obtain

$\displaystyle\lambda(z)=$
$\displaystyle-\frac{1}{L^{2}_{\text{I}}(1+\gamma^{2}\nu\delta_{n1})}+\cdots\,,\qquad~{}~{}(z\to
0)\,,$ (3.60) $\displaystyle\lambda(z)=$
$\displaystyle-\frac{1}{n(L^{2}_{\text{I}}\gamma^{2}\nu)z^{2(n-1)}}+\cdots\,,\qquad(z\to\infty)\,.$
(3.61)

This implies that the brane asymptotically approaches AdS2 in the UV regime.
However, the condition $\lambda(z)\to 0$ as $z\to\infty$ indicates that it
becomes flat in the deep IR.

The solution of $\psi_{\text{II}}$ can be obtained as

$\displaystyle\psi_{\text{II}}=\pm\sqrt{\frac{1-\nu^{2}}{\nu}}\ z\
_{2}F_{1}\left(-\frac{1}{2},\ \frac{1}{2n-2},\ \frac{2n-1}{2n-2},\
-\frac{n^{2}\gamma^{2}\nu}{1-\nu^{2}}\ z^{2n-2}\right)\,,$ (3.62)

where ${}_{2}F_{1}$ denotes the hypergeometric function. Fig. 8 illustrates an
example of the brane configuration when $\gamma=-2$, $n=10$, $L_{\text{I}}=1$,
and $\nu=1/2$, with the minus sign selected for $\psi_{\text{II}}$. It belongs
to the case (4) classified in Fig. 2.

Figure 8: Plots of configurations of $\psi_{\text{I}}$ of (3.59) and
$\psi_{\text{II}}$ of (3.62). Here we have set $\gamma=-2$, $n=10$,
$L_{\text{I}}=1$ and $\nu=0.5$.

The behaviors of the scalar field and its potential at the UV limit ($z\to 0$)
are

$\displaystyle\begin{split}\phi^{2}&=-\frac{2n}{n-1}L_{\text{I}}\sqrt{\nu}\gamma\
z^{n-1}+\cdots\,,\\\ V&=\mp\frac{\sqrt{1-\nu^{2}}}{\nu
L_{\text{I}}}+\frac{(n-1)(n-3)}{4L^{2}_{\text{I}}}\
\phi^{2}+\cdots\,,\end{split}$ (3.63)

where we have assumed $\phi(0)=0$, i.e. the source of the scalar field on $Q$
in UV is set to be zero. From (3.63), we observe that $V^{\prime}(0)=0$ and
$V^{\prime\prime}(0)>0$ and $<0$ when $n<3$ and $n>3$ respectively. Here, the
prime ′ denotes the derivative with respect to $\phi$. This suggests that
$\phi=0$ (i.e. UV) corresponds to a local maximum, and a local minimum for
$n<3$, and $n>3$, respectively.

The behaviors of the scalar field and its potential at the IR limit
($z\to\infty$) are

$\displaystyle\begin{split}|\phi|&=\sqrt{\frac{(n-1)L_{\text{I}}(1\pm\nu)}{2}}\log(z)+\cdots\,,\\\
V&=\frac{\mp 1-\nu}{\nu L_{\text{I}}}+\cdots\,.\end{split}$ (3.64)

In contrast to the previous examples, the scalar field $\phi$ undergoes a
monotonic increase until infinity while the potential $V$ converges towards a
constant. Fig. 9 shows a typical example of the scalar field $\phi$ and the
potential $V$ with the same parameters in Fig. 8. The scalar field $\phi$ is
divergent when $z\to\infty$. The potential is non-monotonic: $V$ has a local
minimum at $\phi=0$ and a global minimum for large $\phi$, while it has a
maximum at a finite value of $\phi$.

Figure 9: The typical configuration of the scalar field $\phi$ and the
potential $V$. Here we have set $\gamma=-2$, $n=10$, $L_{\text{I}}=1$ and
$\nu=0.5$.

Numerically, we find that the non-monotonic nature of the scalar potential can
lead to the existence of multiple extremal surfaces for the specific spatial
regime we have considered. More precisely, for the profile
$\psi_{\text{II}}<0$ that we analyzed in Fig. 8, there are three possible
intersection points $z_{*}$ within the interval $(0.708,0.774)$ of
$\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$, as illustrated explicitly in
the upper two plots of Fig. 10. Furthermore, the lower plots in Fig. 10 show
the entanglement entropy $S$ and the interface entropy $S_{\text{iE}}=\log g$
as functions of $\sigma$. A notable observation is that there is a first-order
phase transition for both the entanglement entropy and the boundary entropy
when we increase $\sigma$. This transition is a result of the competition
among three extremal curves. We anticipate that this result stems from the
non-monotonic behavior of the scalar field’s potential. Moreover, the
interface entropy is monotonically decreasing and satisfies the $g$-theorem.

Figure 10: The upper two figures are the typical configurations of the
extremal surface for $\sigma=0.75$. The lower two figures are the entanglement
entropy and interface entropy as functions of $\sigma$ for the profile (3.59)
and (3.62) with the minus sign selected. Here we have set $\gamma=-2$, $n=10$,
$L_{\text{I}}=1$ and $\nu=0.5$.

####  3.3.5 Solution V

The studies in Sec. 3.3.4 demonstrate the occurrence of multiple extremal
curves in scenarios where the scalar potential exhibits a non-monotonic
behavior. To further support this observation, we will now delve into another
intriguing example. We consider the following profile for the interface brane

$\displaystyle\psi_{\text{I}}=\gamma\ a^{-z}-\gamma\,,$ (3.65)

where $\gamma,a$ are constants. The NEC requires $\gamma<0$. We set $a>1$ to
ensure $\psi_{\text{I}}\geq 0$. After evaluating (2.26), we find

$\displaystyle\begin{split}\lambda(z)=&-\frac{1}{L^{2}_{\text{I}}(1+\gamma^{2}\nu(\log
a)^{2})}+\cdots\,,~{}~{}~{}(z\to 0)\,,\\\
\lambda(z)=&-\frac{1}{L^{2}_{\text{I}}}+\cdots\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad(z\to\infty)\,.\end{split}$
(3.66)

This implies that the brane asymptotically approaches AdS2 in both the UV and
deep IR regions.

In this case, the solution of $\psi_{\text{II}}$ is quite complicated and its
expression will be shown here. In Fig. 11 we show an example of the profiles
where $\gamma=-5,a=10,L_{\text{I}}=1$ and $\nu=0.5$. Note for the left portion
of bulk spacetime, the brane only approaches a finite value of
$x_{\text{I}}=-\gamma$.161616This feature reminds us the profiles of the end-
of-world (EOW) brane in AdS/BCFT for gapped phases [40]. It belongs to the
case (2) classified in Fig. 2.

Figure 11: Plots of interface brane for the example (3.65). Here we have set
$\gamma=-5,a=10,L_{\text{I}}=1$ and $\nu=0.5$.

Obtaining the analytical behavior for scalar field and the potential up to
subleading order is challenge. Here we instead present the numerical plots of
$\phi$ and $V$ in Fig. 12. We observe that the scalar field $\phi$ tends
towards a constant value $\phi_{0}$ as $z$ increases, indicating that the
brane asymptotically approaches AdS2 in the deep IR region. Regarding the
potential $V$, its behavior is non-monotonic, similar to the potential in Sec.
3.3.4. It exhibits a local maximum at $\phi=0$, corresponding to the UV fixed
point, and a global minimum at $\phi=\phi_{0}$, corresponding to the IR fixed
point. Additionally, between these two points, the potential has a local
minimum and a local maximum.

Figure 12: Typical configuration of the scalar field $\phi(z)$ and potential
$V(\phi)$. We choose $\gamma=-5,a=10,L_{\text{I}}=1$ and $\nu=0.5$ and we have
$\phi_{0}=2.09$.

Similar to the observation in Sec. 3.3.4, with a non-monotonic potential, we
might have multiple extremal curves. More precisely, as illustrated in the
upper two plots, Fig. 13, for the $\psi_{\text{II}}<0$ case we analyzed, there
are three possible intersecting points $z=z_{*}$ within the regime $(0,1.099)$
of the spatial interval $\sigma_{\text{I}}=\sigma_{\text{II}}=\sigma$. In the
lower two figures in Fig. 13, we show the entanglement entropy $S$ and the
interface entropy $S_{\text{iE}}$, as functions of $\sigma$. We observe again
there is a first-order phase transition in both the entanglement entropy and
the boundary entropy when we increase $\sigma$. This is due to the competition
among the multiple extremal surfaces. This observation further confirms that
the potential $V$’s non-monotonic nature leads to the phase transitions of the
$g$-function of ICFT. Here the $g$-theorem is also satisfied. Note that the
boundary entropy has a lower bound and is always greater than $0$ in this case
and this is different from the case in Sec. 3.3.4.

Figure 13: The upper two plots are the multiple extremal surfaces and the
lower two plots are the entanglement entropy and the interface entropy
$S_{\text{iE}}=\log{g}$ as functions of $\sigma$ for the profile (3.65). Here
we have set $\gamma=-5,a=10,L_{\text{I}}=1$ and $\nu=0.5$.

####  3.3.6 Solution VI

In the previous discussions, we have provided several examples where, in the
IR regime, the interface brane extend to $z\to\infty$. Here we present a
specific example where the interface brane only spans upto a finite regime of
$z$. We consider the following profile of the interface brane

$\displaystyle\psi_{\text{I}}=\frac{a}{z-b}+\frac{a}{b}\,,$ (3.67)

where $a$ and $b$ are constants with $b>0$. The NEC constraints $a>0$. Note
that when $x$ goes to $-\infty$, $z$ only approaches a finite constant value
$b$, therefore the brane exists solely within the finite range $0<z<b$. From
(2.26) we obtain

$\displaystyle\begin{split}\lambda(z)&=-\frac{b^{4}}{L^{2}_{\text{I}}(b^{4}+a^{2}\nu)}+\mathcal{O}(z)\,,\quad(z\to
0)\,,\\\
\lambda(z)&=\mathcal{O}((z-b)^{3})\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad(z\to
b)\,.\end{split}$ (3.68)

These results suggest that the brane is asymptotic AdS2 near the UV boundary,
yet becomes flat in the vicinity of $z=b$, similar to the studies in Sec.
3.3.4.

The solution for $\psi_{\text{II}}$ in this scenario is

$\displaystyle\begin{split}\psi_{\text{II}}=&~{}\pm\Bigg{(}\frac{a}{b-z}\,_{2}F_{1}\left(-\frac{1}{2}\,,-\frac{1}{4}\,,\frac{3}{4}\,,-\frac{(b-z)^{4}(1-\nu^{2})}{a^{2}\nu}\right)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{a}{b}\,_{2}F_{1}\left(-\frac{1}{2}\,,-\frac{1}{4}\,,\frac{3}{4}\,,-\frac{b^{4}(1-\nu^{2})}{a^{2}\nu}\right)\Bigg{)}\,,\end{split}$
(3.69)

where ${}_{2}F_{1}$ is a hypergeometric function. Note that (3.69) has a pole
at $z=b$ where the brane is asymptotically to and the brane is defined within
the interval $0<z<b$. In Fig. 14 we show an example configurations of
$\psi_{\text{I}}$ and $\psi_{\text{II}}$ by setting $a=b=1$. This is a special
example of case (3) classified in Fig. 2. When we choose the plus sector in
(3.69), the configuration becomes to case (4) classified in Fig. 2.

Figure 14: Plots of configurations of $\psi_{\text{I}}$ in (3.67) and
$\psi_{\text{II}}$ with minus sector in (3.69) when $a=b=1$. We set
$L_{\text{I}}=1$ and $\nu=0.5$.

In the UV limit $z\to 0$, the scalar and its potential exhibit the following
asymptotic behavior

$\displaystyle\begin{split}\phi^{2}&=\frac{4aL_{\text{I}}\sqrt{\nu}(\sqrt{b^{4}+a^{2}\nu-b^{4}\nu^{2}}\pm
a\nu^{3/2})}{b\sqrt{(b^{4}+a^{2}\nu)(b^{4}+a^{2}\nu-b^{4}\nu^{2})}}\
z+\cdots\,,\\\
V&=\frac{-a\nu^{3/2}\mp\sqrt{b^{4}+a^{2}\nu-b^{4}\nu^{2}}}{L_{\text{I}}\nu\sqrt{b^{4}+a^{2}\nu}}-\frac{b^{4}}{4L^{2}_{\text{I}}(b^{4}+a^{2}\nu)}\phi^{2}+\cdots\,,\end{split}$
(3.70)

where we have assumed that $\phi(z=0)=0$. Near the $z=b$, the behaviors become

$\displaystyle\begin{split}(\phi-\phi_{0})^{2}&=\frac{4L_{\text{I}}(1\pm\nu)}{b}(b-z)+\cdots\,,\\\
V&=\frac{\mp
1-\nu}{L_{\text{I}}\nu}+\frac{b^{4}}{64a^{2}L^{4}_{\text{I}}\nu(\pm
1+\nu)^{2}}(\phi-\phi_{0})^{6}+\cdots\,,\end{split}$ (3.71)

where $\phi_{0}$ is the constant of integration and denotes the position at
which the potential $V$ attains its minimum value.

Figure 15: The upper two figures are plots of scalar field $\phi(z)$ and its
potential $V(\phi)$. The lower two are plots of entanglement entropy and
interface entropy $S_{\text{iE}}$ as functions of $\sigma$. Here we set
$a=b=1$, $L_{\text{I}}=1$, $\nu=1/2$ and consider the profile with negative
sign in (3.69).

In Fig. 15, we choose the minus sign for $\psi_{\text{II}}$ in (3.69) and set
$a=b=L_{\text{I}}=1$ and $\nu=1/2$. The two upper figures illustrate the
configurations of the scalar field $\phi(z)$ and its potential $V(\phi)$. We
observe that the potential $V$ monotonically decreases as a function of the
scalar field $\phi$ yet remains always positive. Additionally, the scalar
field approaches a maximum value of $\phi_{0}=2.15$ as $z\to b$, corresponding
to a local minimum of the potential $V$. The evolution from $\phi=0$ to
$\phi=\phi_{0}$, which moves from a local maximum to a local minimum of the
potential $V$, can as thought of a RG flow from UV to IR for ICFT. The two
lower figures of Fig. 15 illustrate the EE and interface entropy as functions
of the subsystem size $\sigma$. We observe that the boundary entropy
$S_{\text{iE}}$ decreases monotonically, and this decrease does not have a
lower bound. Together with the results of solution IV in Fig. 10, we suspect
that this behavior is an intrinsic characteristic of a brane that is
asymptotically flat in the IR.

In Fig. 16 we consider the case that $\psi_{\text{II}}$ in (3.69) is positive.
We also set $a=b=L_{\text{I}}=1$ and $\nu=1/2$. The two upper figures in Fig.
16 illustrate the configurations of the scalar field $\phi(z)$ and its
potential $V(\phi)$. Similar to the case with a negative $\psi_{\text{II}}$,
the potential $V$ monotonically decreases as $\phi$ increases, and the maximum
value of the scalar field $\phi_{0}=3.48$ corresponds to the point where the
potential reaches its minimum value. However, in this scenario, the potential
$V$ is always negative. The two lower figures illustrate the entanglement
entropy and the interface entropy as functions of the subsystem size $\sigma$.
The interface entropy $S_{\text{iE}}$ again monotonically decreases from a
negative value towards $-\infty$.

Figure 16: The upper two figures are plots of scalar field $\phi(z)$ and its
potential $V(\phi)$. The lower two are plots of entanglement entropy and
interface entropy $S_{\text{iE}}$ as functions of $\sigma$. Here set $a=b=1$,
$L_{\text{I}}=1$, $\nu=0.5$ and consider the profile with positive sign in
(3.69).

###  3.4 Comment on the entanglement entropy of various subsystems

In comparison to CFT or BCFT, ICFT presents a more intricate entanglement
structure, where the entanglement entropy of diverse subsystems offers a
detailed exploration of the interface dynamics [6, 41]. For example, the
entanglement entropy serves as a tool to characterize the impact of
interaction strength on the defect [6]. In [19, 20], the entanglement
structure has been studied within a class of holographic ICFT model.
Interestingly, a universal relation between the coefficients of divergent
terms in the entanglement entropy of various subsystems has been uncovered, in
both Janus solutions and RS braneworld models. Compared to the holographic
models in[19, 20], due to the presence of a dynamical scalar field on the
interface brane the induced metric might no longer be AdS2. However, we can
extract information regarding the divergent term in the entanglement entropy
from the position of the extremal surface endpoint on the boundary for the
specific case (2) classified in Fig. 2.: i.e. we assume
$\psi^{\prime}_{\text{I}}\geq 0,\ \psi^{\prime}_{\text{II}}\leq 0$. As in
previous discussions, we consider the case $\nu\leq 1$.

$X$(a)$X$(b)$X$(c) Figure 17: Cartoon plot for various boundary subsystems.
CFT${}_{\text{I,II}}$ are two halves of the straight line and the red dot is
the interface. $X$ is the subsystem. (a) the interface is at the interior of
the subsystem and the subsystem is finite; (b) the interface is at one end of
the subsystem and the subsystem is finite; (c) the interface is at one end of
the subsystem and the subsystem is infinite.

Let us first consider the subsystem (a) in Fig. 17, which is the union of
$-\sigma_{\text{I}}\leq x_{\text{I}}\leq 0$ and $0\leq
x_{\text{II}}\leq\sigma_{\text{II}}$. Suppose the intersecting point $S$
between the brane and the extremal surface is $z_{*}$. Denote the left, right
end points of the extremal surface by $A,B$. Then the left Poincare
coordinates of $A,S$ are $(0,-\sigma_{\text{I}},\epsilon_{\text{I}})\,,$ and
$(0,\psi_{\text{I}}(z_{*}),\frac{z_{*}}{\sqrt{\nu}})$ and the right Poincare
coordinates of $S,B$ are $(0,\sigma_{\text{II}},\epsilon_{\text{II}})$ and
$(0,\psi_{\text{II}}(z_{*}),\sqrt{\nu}z_{*})$. Then from (3.10) the
entanglement entropy is

$\displaystyle\begin{split}S_{\text{E}}=&~{}\frac{L_{\text{I}}}{4G}\cosh^{-1}\frac{(\sigma_{\text{I}}+\psi_{\text{I}}(z_{*}))^{2}+\frac{z_{*}^{2}}{\nu}}{2\frac{1}{\sqrt{\nu}}z_{*}\epsilon_{\text{I}}}+\frac{L_{\text{I}}\nu}{4G}\cosh^{-1}\frac{(\sigma_{\text{II}}-\psi_{\text{II}}(z_{*}))^{2}+\nu
z_{*}^{2}}{2\sqrt{\nu}z_{*}\epsilon_{\text{II}}}\\\
=&~{}\frac{c_{\text{I}}+c_{\text{II}}}{6}\log\frac{(\sigma_{\text{I}}+\sigma_{\text{II}})}{\epsilon}+\text{(regular
terms)}\,.\end{split}$ (3.72)

In the last line, we have chosen
$\epsilon_{\text{I}}=\frac{\epsilon_{\text{II}}}{\nu}=\epsilon$.

Next we consider the subsystem (b) in Fig. 17, which is $0\leq
x_{\text{II}}\leq\sigma_{\text{II}}$. For the case we considered, the extremal
surface does not cross the brane. Suppose the two end points of the extremal
surface $\gamma_{b}$ are
$A:\,(t_{\text{II}},x_{\text{II}},u_{\text{II}})=(0,\sigma_{\text{II}},\epsilon)$
and $B:\,(t_{\text{II}},x_{\text{II}},u_{\text{II}})=(0,0,\epsilon).$ The
entanglement entropy is

$\displaystyle
S_{\text{E}}=\frac{L_{\text{I}}\nu}{4G}\cosh^{-1}\frac{\sigma_{\text{II}}^{2}+2\epsilon^{2}}{2\epsilon^{2}}=\frac{c_{\text{II}}}{3}\log\frac{\sigma_{\text{II}}}{\epsilon}+\text{(regular
terms)}\,.$ (3.73)

Finally, let us consider the subsystem (c) in Fig. 17, which is $+\infty\geq
x_{\text{II}}\geq 0$. Noticing that $c_{\text{I}}\geq c_{\text{II}}$, then the
extremal surface $\gamma_{c}$ is $x_{\text{II}}=0$ and the induced metric on
$\gamma_{c}$ is

$\displaystyle
ds^{2}_{\gamma}=L^{2}_{\text{I}}\nu^{2}\frac{dy^{2}_{\text{II}}}{y^{2}_{\text{II}}}.$
(3.74)

The entanglement entropy is

$\displaystyle
S_{\text{E}}=\frac{L_{\text{I}}\nu}{4G}\int^{\ell_{IR}}_{\epsilon}\frac{dy_{\text{II}}}{y_{\text{II}}}=\frac{c_{2}}{6}\log\frac{\ell_{\text{IR}}}{\epsilon}\,,$
(3.75)

where $\epsilon$ is the UV cutoff and $\ell_{\text{IR}}$ is the IR cutoff.

From the divergence term in (3.72), (3.73) and (3.75), we can obtain an
effective central charge that satisfies the universal relation proposed in
[19, 20]. In [19] it was suggested that the endpoint of a finite interval
could characterize the underlying field theory, thus $c_{\text{II}}$ can be
viewed as an effective central charge of the defect, satisfying the upper
bound proposed in [25]. However, as it only appears within divergent terms, it
appears to lack information regarding the dynamics of the defect along RG
flow, merely reflecting the coupling properties between the defect and CFT. A
deeper understanding on the effective central charge from the perspective of
field theory, particularly when the scaling symmetry of interface is broken,
is necessary.

##  4 Finite temperature

We have explored the zero temperature solution and its associated interface
entropy. In this section, we extend our previous study to the finite
temperature case, focusing on the configurations of brane solutions. We will
first glue two BTZ black holes along a brane in Sec. 4.1, then glue a thermal
AdS solution with a BTZ black hole in Sec. 4.2. We mostly concern with the
impact of the interface-located scalar field on the configuration.

###  4.1 Gluing two BTZ black holes

The planar BTZ black hole metric is given by

$\displaystyle
ds^{2}=\frac{L^{2}_{\text{A}}}{u^{2}_{\text{A}}}\left[-f_{\text{A}}(u_{\text{A}})dt^{2}_{\text{A}}+\frac{du^{2}_{\text{A}}}{f_{\text{A}}(u_{\text{A}})}+dx^{2}_{\text{A}}\right]\,,\
\ ~{}~{}~{}\text{A=\,I\,,~{}II}\,,$ (4.1)

where

$\displaystyle
f_{\text{A}}(u_{\text{A}})=1-\frac{u^{2}_{\text{A}}}{(u^{H}_{\text{A}})^{2}}\,,\
\ ~{}~{}~{}\text{A=\,I\,,~{}II}\,.$ (4.2)

Similar to the zero temperature case, $L_{A}$ is the AdS radius. Note that
$u=u^{H}_{\text{A}}$ is the location of the horizon. We consider the case that
$x_{\text{A}}$ is non-compact. The Hawking temperatures of the black holes
are171717We use $\Theta$ for temperature to avoid possible confusion with the
tension $T$.

$\Theta_{\text{I}}=\frac{1}{2\pi
u^{H}_{\text{I}}}\,,~{}~{}~{}~{}\Theta_{\text{II}}=\frac{1}{2\pi
u^{H}_{\text{II}}}\,.~{}~{}~{}~{}$ (4.3)

The dual field theory lives at the conformal boundary $u\to 0$.

The boundary field consists of two semi-infinite CFTs intersecting at the
interface. Same as the zero temperature case, CFT${}_{\text{I}}$ lives in the
regime $x_{\text{I}}<0$ while CFT${}_{\text{II}}$ is in the regime
$x_{\text{II}}>0$. The left (or right) portion of the bulk has three possible
different configurations for the brane $Q$ related to one side of the system
$N_{\text{I}}$ (or $N_{\text{II}}$), as shown in Fig. 18. The gray regime
denotes the black hole interior, whereas the orange regime is the bulk system
$N_{\text{I}}$ (or $N_{\text{II}}$). In the left figure “E”, there is no
horizon for the bulk geometry. In the middle figure “H1”, there is a complete
horizon for the bulk geometry. In the right figure “H2”, there is only a
portion of horizon for the bulk geometry. These notations follow those in
[22].

EH1H2 Figure 18: Cartoon plot for the configuration of different types of
spacetime slice. The gray region is the interior of the black hole. The orange
region is one part of the bulk. “E” means there is no horizon in the bulk and
“H” means there is a complete or a part of horizon in the bulk.

Suppose the intrinsic coordinate system on the interface brane $Q$ is $(t,w)$
and the brane is parameterized as

$t_{\text{I}}(t,w)=t_{\text{II}}(t,w)=t\,,~{}~{}$ (4.4)

and $u_{\text{I}}(w),u_{\text{II}}(w),x_{\text{I}}(w),x_{\text{II}}(w)$.
Assuming the induced metric on $Q$ takes the form

$ds^{2}_{Q}=-F(w)dt^{2}+G(w)dw^{2}\,,$ (4.5)

satisfying

$\displaystyle\begin{split}&u_{\text{I}}(w)=\frac{u_{\text{I}}^{H}L_{\text{I}}}{\sqrt{L_{\text{I}}^{2}+(u_{\text{I}}^{H})^{2}F(w)}}\,,~{}~{}~{}~{}u_{\text{II}}(w)=\frac{u_{\text{II}}^{H}L_{\text{II}}}{\sqrt{L_{\text{II}}^{2}+(u_{\text{II}}^{H})^{2}F(w)}}\,,~{}~{}~{}~{}\\\
&\left(\frac{dx_{\text{I}}}{dw}\right)^{2}=\frac{(u_{\text{I}}^{H})^{2}\left(4(L_{\text{I}}^{2}-(u_{\text{I}}^{H})^{2}F(w))F(w)G(w)+L_{\text{I}}^{2}(u_{\text{I}}^{H})^{2}(\frac{dF}{dw})^{2}\right)}{4L_{\text{I}}^{2}F(w)\left(L_{\text{I}}^{2}+(u_{\text{I}}^{H})^{2}F(w)\right)^{2}}\,,\\\
&\left(\frac{dx_{\text{II}}}{dw}\right)^{2}=\frac{(u_{\text{II}}^{H})^{2}\left(4(L_{\text{II}}^{2}-(u_{\text{II}}^{H})^{2}F(w))F(w)G(w)+L_{\text{II}}^{2}(u_{\text{II}}^{H})^{2}(\frac{dF}{dw})^{2}\right)}{4L_{\text{II}}^{2}F(w)\left(L_{\text{II}}^{2}+(u_{\text{II}}^{H})^{2}F(w)\right)^{2}}\,.\end{split}$
(4.6)

One convenient parameterization is that

$\displaystyle\begin{split}F(w)&=w\,,\\\
G(w)&=\frac{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}+\eta^{2}_{\text{I}}=\frac{L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}}{4L^{2}_{\text{II}}w+4(u^{H}_{\text{II}})^{2}w^{2}}+\eta^{2}_{\text{II}}\,.\end{split}$
(4.7)

We will solve the system under this assumption. The expression for $u_{A}$ and
$x_{A}$ becomes

$\displaystyle\begin{split}&u_{\text{I}}(w)=\frac{u_{\text{I}}^{H}L_{\text{I}}}{\sqrt{L_{\text{I}}^{2}+(u_{\text{I}}^{H})^{2}w}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{\text{II}}(w)=\frac{u_{\text{II}}^{H}L_{\text{II}}}{\sqrt{L_{\text{II}}^{2}+(u_{\text{II}}^{H})^{2}w}}\,,~{}~{}~{}~{}\\\
&x_{\text{I}}(w)=-\int^{+\infty}_{w}\frac{u^{H}_{\text{I}}\eta_{\text{I}}(\sigma)}{\sqrt{L_{\text{I}}^{2}+(u^{H}_{\text{I}})^{2}\sigma}}d\sigma\,,~{}~{}~{}~{}x_{\text{II}}(w)=-\int^{+\infty}_{w}\frac{u^{H}_{\text{II}}\eta_{\text{II}}(\sigma)}{\sqrt{L_{\text{II}}^{2}+(u^{H}_{\text{II}})^{2}\sigma}}d\sigma\,.\end{split}$
(4.8)

Note that the parameter $w$ ranges from $w_{0}$ to $\infty$, where $w_{0}$ is
a non-negative integration constant determined by (LABEL:eq:xIxII). When
$w\to\infty$, both $u_{\text{I}}$ and $u_{\text{II}}$ tend to zero, marking
the boundary of bulk geometry, which corresponds to the UV limit.

On the hand hand, when $\omega_{0}\to 0$, $u_{\text{I}}$ and $u_{\text{II}}$
converge to finite values $u_{\text{I}}^{H}$ and $u_{\text{II}}^{H}$ at
specific, finite values of $x_{\text{I}}$ and $x_{\text{II}}$. In this case,
the brane should touch the horizon, i.e. the configuration “H2”. Consequently,
as $w$ decreases towards zero, $w\to 0,x_{\text{I}},x_{\text{II}}$ are
expected to approach finite values. Lastly, when $w_{0}$ is a positive nonzero
number, the brane should only approach a finite value $u_{A}$, correponding to
either the “E” or “H1” configuration. In this case, as $w\to w_{0}$, it is
expected that $x_{\text{I}}\to\pm\infty,x_{\text{II}}\to\pm\infty$. Obviously,
from the possible choice of $w_{0}$, the configurations [E, H2], [H1, H2],
[H2, E] and [H2, H1] are not allowed.181818We use [E, H2] to refer that the
left bulk $N_{\text{I}}$ is empty while the right bulk $N_{\text{II}}$ belongs
to type H2 shown in Fig. 18. The other pairs follow the same convention.

With the above setup (LABEL:eq:xIxII), the profile of the interface brane $Q$
in $N_{\text{I}}$ or $N_{\text{II}}$ is parameterized by

$\frac{dx_{\text{A}}}{du_{\text{A}}}=-2\,\frac{L_{\text{A}}}{u^{2}_{\text{A}}}\eta_{A}(\xi)\Bigg{|}_{\xi=\frac{L^{2}_{\text{A}}}{u^{2}_{\text{A}}}-\frac{L^{2}_{\text{A}}}{(u^{H}_{\text{A}})^{2}}}\,,~{}~{}~{}~{}~{}~{}~{}\
\ A=\text{I,~{}II}\,,$ (4.9)

with the boundary condition $x_{\text{A}}(u_{\text{A}}=0)=0.$

For the simplest static case $\phi=\phi(w)$, there are only three independent
equations in (2.10, 2.11, 2.12, 2.13). Here we write out the explicit
expressions as follows

$\displaystyle\begin{split}&0=\frac{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}+\eta^{2}_{\text{I}}-\frac{L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}}{4L^{2}_{\text{II}}w+4(u^{H}_{\text{II}})^{2}w^{2}}-\eta^{2}_{\text{II}}\,,\\\
&\frac{V(\phi)}{2}=-\frac{(L^{2}_{\text{I}}+2w(u^{H}_{\text{I}})^{2})\eta_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+2w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}})+L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{\prime}_{\text{I}}}{L_{\text{I}}u^{H}_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}})^{\frac{3}{2}}}\\\
&~{}~{}+\frac{(L^{2}_{\text{II}}+2w(u^{H}_{\text{II}})^{2})\eta_{\text{II}}(L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+2w(L^{2}_{\text{II}}+w(u^{H}_{\text{II}})^{2})\eta^{2}_{\text{II}})+L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}w(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)\eta^{\prime}_{\text{II}}}{L_{\text{II}}u^{H}_{\text{II}}(L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+4w(L^{2}_{\text{II}}+w(u^{H}_{\text{II}})^{2})\eta^{2}_{\text{II}})^{\frac{3}{2}}}\,,\\\
&\phi^{\prime
2}(w)=-\frac{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)w\eta^{2}_{\text{I}}}{4(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)w^{2}}\,\bigg{[}-wV(\phi)\\\
&~{}~{}-\frac{2w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta_{\text{I}}}{L_{\text{I}}u^{H}_{\text{I}}\sqrt{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)w\eta^{2}_{\text{I}}}}+\frac{2w(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)\eta_{\text{II}}}{L_{\text{II}}u^{H}_{\text{II}}\sqrt{L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+4(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)w\eta^{2}_{\text{II}}}}\bigg{]}\,.\end{split}$
(4.10)

Note that in the above equations only up to first order derivative of $\eta$
is involved, different from the zero temperature case (2.22). This is due to
the fact that $\eta$ plays the role of the slop of the brane which can be seen
in (4.9). The first equation in (LABEL:eq:q-finiteT) is from the continuity of
the metric field at the interface brane.

For a given potential $V(\phi)$, by solving (LABEL:eq:q-finiteT) one can
derive $\eta_{\text{I}}(\omega),\eta_{\text{II}}(\omega)$ and $\phi(\omega)$.
Subsequently, one can obtain the induced metric (4.5) and all the information
about the system. In practice, we solve the system in a reverse manner,
similar to the zero temperature case. More precisely, for a specified $G(w)$,
we know the form of $\eta_{\text{I}}$ and $\eta_{\text{II}}$ from (4.7), i.e.
the profile of the system. Then, by solving $\eqref{eq:q-finiteT}$, we obtain
the scalar field $\phi$ and its potential $V(\phi)$, ensuring a consistent
system.

Note that our study differs from [22], where the dual field theory was
analyzed in a compact spatial direction. In contrast, the system we
investigate lacks any additional scale, affording us the freedom to choose the
unit for temperature in the dual field theory. Similar to the discussion in
Section 2.1, our focus is on a globally static configuration, implying the
existence of a global time in the dual field theory. Furthermore, we assume
that the periods of the imaginary time are identical. It is crucial to
emphasize that the temperature associated with the H1 and H2 configurations is
determined by (4.3), whereas the temperature of the type E configuration is
independent of the parameter $u_{H}$. Additionally, it is worth noting that
for the type E configuration, any potential conifold singularity that arises
due to an arbitrary choice of the periodicity of the imaginary time is hidden
behind the interface brane.

In all these different configurations, the brane should intersect with the
boundary. We can find a class of solutions by imposing the following
asymptotical behaviours

UV $\displaystyle\,,~{}~{}~{}w\to+\infty\,,~{}~{}~{}\ \
\eta_{\text{I}}(w)\to\frac{C}{w^{a}}\,,~{}~{}~{}a\geq 1\,.$ (4.11)

Note that the intersecting condition only constraints $a>1/2$. However, when
we demand that the brane is asymptotically AdS in the UV limit, we should have
$a\geq 1$. On the other hand, in the deep IR, depending on the possible
configures we have different conditions on the functions $\eta_{\text{I}}$ and
$\eta_{\text{II}}$. If the interface brane intersects with the horizon of the
BTZ black holes, then
$\eta_{\text{I}}(w)/\sqrt{L_{\text{I}}^{2}+(u^{H}_{\text{I}})^{2}w},~{}\eta_{\text{II}}(w)/\sqrt{L_{\text{II}}^{2}+(u^{H}_{\text{II}})^{2}w}$
must be integrable at $w=0$ and there should be no singularity for
$\eta_{\text{A}}$ when $0<w<\infty$. We expect that

IR $\displaystyle\,,~{}~{}~{}~{}w\to
0\,,~{}~{}~{}~{}~{}\eta_{\text{A}}(w)\to\frac{B}{w^{b}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(b<1)\,,~{}~{}~{}~{}~{}~{}~{}\text{for
H2}$ (4.12) IR $\displaystyle\,,~{}~{}~{}~{}w\to
w_{0}\,,~{}~{}~{}~{}\eta_{\text{A}}(w)\to\frac{B}{(w-w_{0})^{b}}\,,~{}~{}~{}~{}~{}~{}(b\geq
1)\,,~{}~{}~{}~{}~{}~{}\text{for E or H1}$ (4.13)

where $w_{0}>0$.

In the finite temperature case, the NEC, expressed as $(\Delta
K_{\mu\nu}-h_{\mu\nu}\Delta K)N^{\mu}N^{\nu}\geq 0\,$ with $N^{\mu}$ bing an
arbitrary null vector on the brane, is again equivalent to $\phi^{\prime
2}(w)\geq 0$. This is similar to the behavior observed in the zero temperature
case. The NEC can be simplified as

$\displaystyle\begin{split}&-\frac{(u^{H}_{\text{I}})^{4}\,\eta_{\text{I}}+(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)(-2\eta^{3}_{\text{I}}+(u^{H}_{\text{I}})^{2}\eta^{\prime}_{\text{I}})}{\nu
u^{H}_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{2}_{\text{I}})^{\frac{3}{2}}}\,+\\\
&~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(u^{H}_{\text{II}})^{4}\,\eta_{\text{II}}+(L^{2}_{\text{I}}\nu^{2}+(u^{H}_{\text{II}})^{2}w)(-2\eta^{3}_{\text{II}}+(u^{H}_{\text{II}})^{2}\eta^{\prime}_{\text{II}})}{u^{H}_{\text{II}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}\nu^{2}+4w(L^{2}_{\text{I}}\nu^{2}+(u^{H}_{\text{II}})^{2}w)\eta^{2}_{\text{II}})^{\frac{3}{2}}}\geq
0\,.\end{split}$ (4.14)

In the special case $u^{H}_{\text{I}}=u^{H}_{\text{II}}$, i.e. the
configuration [H1,H1]191919We will show that the NEC will forbid this type of
configuration in Sec. 4.1.6. or [H2,H2], the NEC (4.14) can be further
simplified. When $\nu=1$, from the first equation in (LABEL:eq:q-finiteT), we
have $\eta_{\text{I}}=\pm\eta_{\text{II}}$. When $\nu=1$ and
$\eta_{\text{I}}=\eta_{\text{II}}$, the NEC holds automatically. If $\nu=1$,
$\eta_{\text{I}}=-\eta_{\text{II}}$, or if $0<\nu<1$ the NEC can be further
simplified as

$\displaystyle\left(2\eta_{\text{I}}^{2}-\frac{(u^{H}_{\text{I}})^{4}}{(u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}}}\right)\eta_{\text{I}}-(u^{H}_{\text{I}})^{2}\,\eta^{\prime}_{\text{I}}\geq
0\,,$ (4.15)

or equivalently,

$\displaystyle\left(2\eta_{\text{II}}^{2}-\frac{(u^{H}_{\text{I}})^{4}}{(u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}}\nu^{2}}\right)\eta_{\text{II}}-(u^{H}_{\text{I}})^{2}\,\eta^{\prime}_{\text{II}}\geq
0\,.$ (4.16)

Note that here it is quite similar to the zero temperature case discussed in
Sec. 2.3, i.e. we have nontrivial constraint on the ansatz of the interface
brane $Q$.

The Ricci tensor of the induced metric on the brane is

$\displaystyle R^{Q}_{\mu\nu}=\lambda(w)h^{Q}_{\mu\nu}\,,$ (4.17)

where

$\displaystyle\lambda(w)=\frac{-L^{2}_{\text{I}}(u^{H}_{\text{I}})^{4}+4(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)^{2}(\eta^{2}_{\text{I}}+2w\eta_{\text{I}}\eta^{\prime}_{\text{I}})}{\left[L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{2}_{\text{I}}\right]^{2}}\,.$
(4.18)

From (4.11), the condition that the brane is asymptotically AdS in the UV
limit (i.e. $w\to\infty$) is $a\geq 1$ in (4.11). When $a=1$, we have the
following asymptotic behaviour

$\displaystyle\begin{split}\eta_{\text{II}}(w)&\to\pm\frac{\sqrt{4C^{2}+L^{2}_{\text{I}}(1-\nu^{2})}}{2w}\,,\\\
\phi^{\prime}(w)&\to 0\,,\\\
V(\phi)&\to\frac{-2C\nu\pm\sqrt{4C^{2}+L^{2}_{\text{I}}(1-\nu^{2})}}{L_{\text{I}}\nu\sqrt{4C^{2}+L^{2}_{\text{I}}}}\,,\\\
\lambda^{\text{UV}}&\to-\frac{1}{4C^{2}+L^{2}_{\text{I}}}\,.\end{split}$
(4.19)

When $a>1$, we have the following asymptotic behaviour

$\displaystyle\begin{split}\eta_{\text{II}}(w)&\to\pm\frac{L_{\text{I}}\sqrt{1-\nu^{2}}}{2w}\,,\\\
\phi^{\prime}(w)&\to 0\,,\\\
V(\phi)&\to\pm\frac{\sqrt{1-\nu^{2}}}{L_{\text{I}}\nu}\,,\\\
\lambda^{\text{UV}}&\to-\frac{1}{L^{2}_{\text{I}}}\,.\end{split}$ (4.20)

The absolute value of the UV limit of the potential always satisfies the bound
for the tension with trivial scalar field

$\displaystyle T_{\text{min}}<|V^{\text{UV}}|<T_{\text{max}}\,,$ (4.21)

where $T_{\text{min}}$ and $T_{\text{max}}$ are defined in (2.30).

####  4.1.1 Permissible configurations at finite temperature

We have shown above the equations of the systems at finite temperature. We
will solve them and illustrates sample profiles. It turns out when the scalar
field is trivial, the only allowed profile of the interface brane is [H2,H2].
However, with the presence of a nontrivial scalar field, we find four
permissible configurations: [E, E], [E, H1], [H1, E] and [H2, H2]. These
configurations are summarized in table 1. One common feature among all the
profiles we have found is that the scalar potential flows from a global
maximal in the UV region to a global minimum in the IR region. Furthermore, in
the IR region, the induced metric on the brane exhibits distinct behaviors: it
could be asymptotically AdS (in the case of [H2,H2] with a trivial scalar
field), dS (for [H2,H2] with a nontrivial scalar field) or flat (for other
permissible configurations). Moreover, when the induced metric is
asymptotically flat in the IR region the scalar field diverges.

| E | H1 | H2
---|---|---|---
E | $\surd$ | $\surd$ | $\times$
H1 | $\surd$ | $\times$ | $\times$
H2 | $\times$ | $\times$ | $\surd$
Table 1: The permissible configurations with a dynamical scalar filed located
on the interface brane. The column denotes the profile of the interface brane
in $N_{\text{I}}$ while the row signifies the profile in $N_{\text{II}}$.

####  4.1.2 The solution with trivial scalar field: [H2,H2]

Let us first study the case with $\phi=0$ and $V(\phi)=T$. The zero
temperature solution has been discussed in Sec. 2.1.1 and 3.3.1. Here we
consider the finite temperature generalizations.

The equations of motion on $Q$ (LABEL:eq:q-finiteT) become

$\displaystyle\begin{split}&\frac{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}+\eta^{2}_{\text{I}}=\frac{L^{2}_{\text{I}}(u^{H}_{\text{II}})^{2}\nu^{2}}{4L^{2}_{\text{I}}\nu^{2}w+4(u^{H}_{\text{II}})^{2}w^{2}}+\eta^{2}_{\text{II}}\,,\\\
&\frac{L_{\text{I}}T}{2}=\frac{-((u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}})\eta_{\text{I}}}{u^{H}_{\text{I}}\sqrt{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w((u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}})\eta^{2}_{\text{I}}}}+\frac{((u^{H}_{\text{II}})^{2}w+L^{2}_{\text{I}}\nu^{2})\eta_{\text{II}}}{\nu
u^{H}_{\text{II}}\sqrt{L^{2}_{\text{I}}(u^{H}_{\text{II}})^{2}\nu^{2}+4w((u^{H}_{\text{II}})^{2}w+L^{2}_{\text{I}}\nu^{2})\eta^{2}_{\text{II}}}}\,.\end{split}$
(4.22)

We eliminate $\eta_{\text{II}}$ in above equations and obtain the solution of
$\eta_{\text{I}}$ which takes the following form

$\displaystyle\eta_{\text{I}}=\pm\frac{L_{\text{I}}u^{H}_{\text{I}}\left(L^{2}_{\text{I}}\nu^{2}((u^{H}_{\text{II}})^{2}-(u^{H}_{\text{I}})^{2})+w(u^{H}_{\text{I}})^{2}(u^{H}_{\text{II}})^{2}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2})\right)}{2\sqrt{w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})}\,\sqrt{A(w-w_{+})(w-w_{-})}}\,,$
(4.23)

where

$\displaystyle w_{\pm}=\frac{-B\pm\sqrt{B^{2}-AC}}{A}$ (4.24)

and

$\displaystyle\begin{split}A&=(u^{H}_{\text{I}})^{4}\left(u^{H}_{\text{II}}\right)^{4}L_{\text{I}}^{4}\nu^{4}\left(\left(\frac{1+\nu}{\nu
L_{\text{I}}}\right)^{2}-T^{2}\right)\,\left(T^{2}-\left(\frac{1-\nu}{\nu
L_{\text{I}}}\right)^{2}\right)\,,\\\
B&=L^{2}_{\text{I}}\nu^{2}(u^{H}_{\text{I}})^{2}(u^{H}_{\text{II}})^{2}\left((u^{H}_{\text{I}})^{2}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2})+(u^{H}_{\text{II}})^{2}\left(1+(-1+L^{2}_{\text{I}}T^{2})\nu^{2}\right)\right)\,,\\\
C&=-L^{4}_{\text{I}}\left((u^{H}_{\text{I}})^{2}-(u^{H}_{\text{II}})^{2}\right)^{2}\nu^{4}\,.\end{split}$
(4.25)

In the above equations (4.25), we always have $C\leq 0$ where $C=0$ only
occurs when $u^{H}_{\text{I}}=u^{H}_{\text{II}}$. Meanwhile, $A$ has to be
positive otherwise the square root in (4.23) is negative and therefore no
consistent solution for $\eta_{\text{I}}$. We should impose the constraint

$\displaystyle\frac{1-\nu}{\nu L_{\text{I}}}<|T|<\frac{1+\nu}{\nu
L_{\text{I}}}\,.$ (4.26)

Note that this constraint is exactly the same as the zero temperature result
discussed in (2.30).

Similarly, the solution of $\eta_{\text{II}}$ is obtained to be

$\displaystyle\eta_{\text{II}}=\mp\frac{u^{H}_{\text{II}}L_{\text{I}}\nu}{2}\,\frac{L^{2}_{\text{I}}\nu^{2}((u^{H}_{\text{I}})^{2}-(u^{H}_{\text{II}})^{2})+w(u^{H}_{\text{I}})^{2}(u^{H}_{\text{II}})^{2}(1+(-1+L^{2}_{\text{I}}T^{2})\nu^{2})}{\sqrt{w(L^{2}_{\text{I}}\nu^{2}+w(u^{H}_{\text{II}})^{2})}\,\sqrt{A(w-w_{+})(w-w_{-})}}\,,$
(4.27)

with the $w_{\pm},A$ defined in (4.24) and (4.25).

Therefore, with a consistent solution, i.e. (4.26), we always have $AC\leq 0$
and $\omega_{+}\geq 0$. In this case where $u_{\text{I}}^{H}\neq
u_{\text{II}}^{H}$, we have $AC<0$, leading to $\omega_{+}>0$. At first
glance, one might infer that $\omega_{+}>0$ implies the brane does not
intersect with the horizon. However, (LABEL:eq:xIxII) reveals finite value for
$x_{\text{I}}$ and $x_{\text{II}}$ when $w_{0}=w_{+}$, indicating that the
brane terminates at a specific point. This behavior suggests that the
interface brane does not effectively divide the bulk into two distinct
regions, making it seem unphysical. Therefore, a static solution does not
exist when the temperatures $\Theta_{\text{I}}$ and $\Theta_{\text{II}}$
differ. To address the issue, one has to consider a dynamical brane solution,
e.g. the holographic NESS state [42, 43], or possibly a stationary state as
described in [44]. It is noteworthy that this situation differs from the
compact case, where the brane can curve at this ending point and intersects
with the boundary again [22].

Therefore, the only allowed static solution exists when
$u_{\text{I}}^{H}=u_{\text{II}}^{H}$, which implies
$\Theta_{\text{I}}=\Theta_{\text{II}}$. In this scenario, the only allowed
configuration is [H2, H2]. The solutions (4.23) and (4.27) can be further
simplified. Using (LABEL:eq:xIxII) we have

$\displaystyle\begin{split}x_{\text{I}}&=u^{H}_{\text{I}}\,\text{arctanh}\left(\frac{u_{\text{I}}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2})}{\sqrt{u^{2}_{\text{I}}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2})^{2}+(u^{H}_{\text{I}})^{2}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2}-(-1+L^{2}_{\text{I}}T^{2})^{2}\nu^{4})}}\right)\,,\\\
x_{\text{II}}&=u^{H}_{\text{I}}\,\text{arctanh}\left(\frac{u_{\text{II}}(-1+(1-L^{2}_{\text{I}}T^{2})\nu^{2})}{\sqrt{u^{2}_{\text{II}}(1+(-1+L^{2}_{\text{I}}T^{2})\nu^{2})^{2}+(u^{H}_{\text{I}})^{2}(-1+(1+L^{2}_{\text{I}}T^{2})\nu^{2}-(-1+L^{2}_{\text{I}}T^{2})^{2}\nu^{4})}}\right)\,.\end{split}$
(4.28)

In Fig. 19, we show an example of the profiles when
$L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=u_{\text{II}}^{H}=2$ and $T=2.99$. We
see that the black hole horizon attracts the interface brane to a curved
shape.

Figure 19: An example of the configurations [H2,H2] for trivial scalar field.
Here we set $L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=u_{\text{II}}^{H}=2$ and
$T=2.99$.

Note that from (4.18), we have

$\displaystyle\lambda(w)=-\frac{1}{4T^{2}}\left[\left(\frac{1}{L_{\text{I}}\nu}+\frac{1}{L_{\text{I}}}\right)^{2}-T^{2}\right]\left[T^{2}-\left(\frac{1}{L_{\text{I}}\nu}-\frac{1}{L_{\text{I}}}\right)^{2}\right]\,.$
(4.29)

This indicates that the induced metric on the interface brane is AdS2.

####  4.1.3 Solution with nontrivial scalar field: [H2,H2]

We have shown that the [H2,H2] configuration is the only possibility in the
trivial scalar field case. Here we will explore if such configuration exists
when there is a non-trivial scalar field.

To obtain an [H2,H2] configuration, we set

$\displaystyle\begin{split}\eta_{\text{I}}=\frac{1}{2L_{\text{I}}}\frac{1}{\sqrt{a+bw^{2}}}\,,\end{split}$
(4.30)

with $a>0\,,b>0$. Obviously it is a profile of type H2. The NEC in (4.15)
constraints $2a(u^{H}_{\text{I}})^{4}-1\leq 0$.

We can solve the first equation in (LABEL:eq:q-finiteT) to derive

$\displaystyle\eta_{\text{II}}=\pm\frac{1}{2L_{\text{I}}}\sqrt{\frac{1}{a+bw^{2}}+\frac{L^{4}_{\text{I}}(u^{H}_{\text{I}})^{4}(1-\nu^{2})}{((u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}})((u^{H}_{\text{I}})^{2}w+L^{2}_{\text{I}}\nu^{2})}}\,,$
(4.31)

where we have set $u^{H}_{\text{II}}=u^{H}_{\text{I}}$. With the condition
$a>0,b>0$, the NEC is satisfied. The brane can be found by solving the
equation

$\displaystyle\frac{dx_{\text{I}}}{du_{\text{I}}}=-\frac{(u^{H}_{\text{I}})^{2}}{\sqrt{au^{4}_{\text{I}}(u^{H}_{\text{I}})^{4}+bL^{4}_{\text{I}}((u^{H}_{\text{I}})^{2}-u^{2}_{\text{I}})^{2}}}\,,$
(4.32)

with the boundary condition $x_{\text{I}}(u_{\text{I}}=0)=0\,.$ Similarly we
have an differential equation for $x_{\text{II}}(u_{\text{II}})$ which we do
not write out here. We find that the functions
$x_{\text{I}}(w),x_{\text{II}}(w)$ are finite in the limit $w\to 0$ and vanish
in the limit $w\to+\infty$ thus this solution is a [H2,H2] solution. A typical
example for the profile is shown in Fig. 20.

Figure 20: An example of the configurations [H2,H2] at finite temperature with
dynamical scalar field. Here we set
$a=1/5,b=1,L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=1$.

The scalar field $\phi(w)$ and the potential $V(\phi)$ are very complicated
and we just show a numerical result in Fig. 21. In the left plot, $1/w\to 0$
(or $\infty$) is the UV (or IR) limit. We find that the potential also has a
local maximal in the UV limit while a local minimal in the IR limit.

Figure 21: The scalar field and its potential for the configuration [H2,H2]
when $a=1/5,b=1,L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=1$.

Evaluating (4.18), we obtain the asymptotic behaviours

$\displaystyle\lambda(w)\simeq-\frac{bL^{2}_{\text{I}}}{1+bL^{4}_{\text{I}}}+\mathcal{O}\left(\frac{1}{w^{2}}\right)\,,\quad
w\to\infty\,,$ (4.33)

i.e. the brane is asymptotically AdS2 in the UV limit, while

$\displaystyle\lambda(w)\simeq\frac{1-a(u^{H}_{\text{I}})^{4}}{aL^{2}_{\text{I}}(u^{H}_{\text{I}})^{4}}+\mathcal{O}\left(w\right)\,,\quad
w\to 0\,,$ (4.34)

i.e. the brane is asymptotically dS2 in the IR limit for the example shown in
Fig. 19. Similar structure of spacetime has been constructed in e.g. [45]. It
would be interesting to study our setup from the perspective of double
holography [46].

####  4.1.4 Solution with nontrivial scalar field: [E,E] and [E,H1]

The presence of a nontrivial scalar field enables various types of
configurations. This subsection will present an example of the [E,E] and
[E,H1] configurations and in the next subsection we will show an example of
the [H1,E] configuration.

As we have shown in (4.11) and (4.13), a type E or type H1 brane should have a
special behavior in the UV and IR limits, respectively. A simple example can
be constructed as follows

$\displaystyle\eta_{\text{I}}=\frac{a}{w-bL^{2}_{\text{I}}}\,,$ (4.35)

where $a$ and $b$ are constants. Solving the first equation in
(LABEL:eq:q-finiteT) yields the solution for $\eta_{\text{II}}$

$\displaystyle\eta_{\text{II}}=\pm\sqrt{\frac{a^{2}}{(w-bL^{2}_{\text{I}})^{2}}+L^{2}_{\text{I}}\left(\frac{(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}-\frac{(u^{H}_{\text{II}})^{2}\nu^{2}}{4L^{2}_{\text{I}}\nu^{2}w+4(u^{H}_{\text{II}})^{2}w^{2}}\right)}\,.$
(4.36)

The NEC (4.14) further constraints the parameters in a complicated way, e.g.
when $u_{\text{I}}^{H}=u_{\text{II}}^{H}$ we have $a>0$. We further impose
$b>0$, then the brane is of type E. After substituting (4.35) into (4.9) and
performing the integration, we obtain the profile of the brane $Q$

$\displaystyle
x_{\text{I}}=-\frac{2au^{H}_{\text{I}}}{L_{\text{I}}\sqrt{1+bu^{H}_{\text{I}}}}~{}\text{arctanh}\frac{u_{\text{I}}\sqrt{1+bu^{H}_{\text{I}}}}{u^{H}_{\text{I}}}\,,$
(4.37)

where we have used $x_{\text{I}}(u_{\text{I}}=0)=0$. When
$u_{\text{I}}\to\frac{u^{H}_{\text{I}}}{\sqrt{1+bu^{H}_{\text{I}}}}$, we have
$x_{\text{I}}\to-\infty$. This indicates that the left spacetime is of type E,
i.e. there is no horizon in left spacetime.

An analytical solution for $x_{\text{II}}$ does not exist. Nevertheless, its
derivative with respect to $u_{\text{II}}$ can be derived from (4.9), from
which we know that $x_{\text{II}}$ is a monotonic increasing or decreasing
function of $u_{\text{II}}$. And there is a pole $u_{\text{II}}^{p}$ where
$x_{\text{II}}\to\pm\infty$, as can be seen from

$\displaystyle\frac{dx_{\text{II}}}{du_{\text{II}}}\propto\pm\frac{1}{u_{\text{II}}-u_{\text{II}}^{p}}\,,\quad\text{with}\quad
u_{\text{II}}^{p}=\frac{u^{H}_{\text{II}}\nu}{\sqrt{b(u^{H}_{\text{II}})^{2}+\nu^{2}}}\,.$
(4.38)

When we choose $\eta_{\text{II}}>0$ in (4.36), the right part of the bulk
contains a black hole. This is the configuration [E,H1]. The inertial observer
in the left bulk will hit the brane and the inertial observer in the right
bulk will hit the horizon. When we choose $\eta_{\text{II}}<0$ in (4.36), the
right part of the bulk does not contain a horizon of the black hole. Now the
configuration is [E,E]. In Fig. 22, we show the typical profiles of [E,E] or
[E,H1], depending on the sign we chose in (4.36). These profiles are allowed
only when we have a dynamical scalar field on the interface brane.

Figure 22: An example of the configurations [E,E] or [E, H1] at finite
temperature with dynamical scalar field. Here we set
$a=b=L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=u_{\text{II}}^{H}=1$.

Similar to the previous discussions, we can show the profile of the scalar
field and its potential for the configuration, e.g. [E,H1], as shown in Fig.
23. On the brane, the potential also has a local maximal in the UV limit and a
local minimum in the IR limit. Moreover, the scalar field diverges in the deep
IR. The profiles of the configuration [E, E] have similar behavior.

Figure 23: The scalar field and its potential for the configuration [E,H1]
when $a=b=L_{\text{I}}=1,\nu=0.5,u_{\text{I}}^{H}=u_{\text{II}}^{H}=1$. A
similar profile for the configuration [E,E] can be obtained and we do not show
it here.

The induced metric on the interface brane is

$\displaystyle
ds^{2}_{Q}=-wdt^{2}+\left(\frac{a^{2}}{(w-bL^{2}_{\text{I}})}+\frac{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}\right)dw^{2}\,.$
(4.39)

In the IR limit $w\to bL^{2}_{\text{I}}$, the induced metric becomes

$\displaystyle ds^{2}_{Q}\simeq-
bL^{2}_{\text{I}}dt^{2}+\frac{a^{2}}{(w-bL^{2}_{\text{I}})}dw^{2}\,,$ (4.40)

which implies the brane is not a black hole.

Evaluating (4.18), we obtain the asymptotic behaviours

$\displaystyle\lambda(w)\simeq-\frac{1}{4a^{2}+L^{2}_{\text{I}}}+\mathcal{O}\left(\frac{1}{w^{2}}\right)\,,~{}~{}~{}~{}\quad
w\to\infty\,,$ (4.41)

i.e. the interface brane is asymptotically AdS in the UV limit, while

$\displaystyle\lambda(w)\simeq-\frac{w-bL^{2}_{\text{I}}}{2a^{2}bL^{2}_{\text{I}}}+\mathcal{O}\left((w-bL^{2}_{\text{I}})^{2}\right)\,,~{}~{}~{}w\to
bL^{2}_{\text{I}}\,,$ (4.42)

i.e. the interface brane is asymptotically flat in IR limit.

####  4.1.5 Solution with nontrivial scalar field: [H1,E]

In this subsection we show an example of the configuration [H1,E]. Motivated
by the previous subsection, we consider the following configuration

$\displaystyle\begin{split}\eta_{\text{I}}=&-\frac{a}{w-bL^{2}_{\text{I}}}\,,\\\
\eta_{\text{II}}=&-\sqrt{\frac{a^{2}}{(w-bL^{2}_{\text{I}})^{2}}+L^{2}_{\text{I}}\left(\frac{(u^{H}_{\text{I}})^{2}}{4L^{2}_{\text{I}}w+4(u^{H}_{\text{I}})^{2}w^{2}}-\frac{(u^{H}_{\text{II}})^{2}\nu^{2}}{4L^{2}_{\text{I}}\nu^{2}w+4(u^{H}_{\text{II}})^{2}w^{2}}\right)}\,.\end{split}$
(4.43)

The NEC (4.14) constraints the choices of the parameters. We find that under
the choice of parameters $a=3,\,b=1,\ L_{\text{I}}=1,\ u^{H}_{\text{I}}=3,\
u^{H}_{\text{II}}=1,\ \nu=0.5,$ the NEC is satisfied. This case provides an
example of configuration [H1,E]. The profiles of the configuration is shown in
Fig. 24.

Figure 24: Plot of the configuration [H1, E] for
$a=3,\,b=1,\,L_{\text{I}}=1,\,\nu=0.5,\,u_{\text{I}}^{H}=3$ and
$u_{\text{II}}^{H}=1$.

In Fig. 25 we show the profiles of the scalar field and its potential. Similar
to the previous examples, the potential evolves from a local maximum in UV to
a global minimum in IR. In the deep IR, the induced metric on $Q$ is
asymptotically flat, which is again associated with a divergent scalar field
in IR.

Figure 25: The scalar field and its potential for the configuration [H1, E]
when $a=3,\,b=1,\,L_{\text{I}}=1,\,\nu=0.5,\,u_{\text{I}}^{H}=3$ and
$u_{\text{II}}^{H}=1$.

####  4.1.6 Solution with nontrivial scalar field: no [H1,H1]

We have demonstrated examples of the [E,E], [E,H1] and [H2,H2] configurations.
The nonexistence of the [E,H2], [H1,H2], [H2,E] and [H2,H1] configurations has
been mentioned below (LABEL:eq:xIxII). It can be seen as follows: Suppose
there exists a solution for [E,H2] or [H1,H2]. For E or H1 configuration,
there exists a lower bound at $w=w_{0}$ where the value of $x_{\text{I}}$
approach $\infty$ or $-\infty$. However, for the other side, from
(LABEL:eq:xIxII), $u_{\text{II}}$ can not approach the horizon when $w\to
w_{0}$. Therefore the configurations [E,H2] and [H1,H2] are not allowed.
Similar arguments apply for the [H2,E] and [H2,H1] configurations. In the
following, we will prove that the [H1,H1] configuration cannot exist neither
from the NEC.

If there exists an [H1,H1] solution, i.e. we can find a function
$\eta_{\text{I}}(w)$. Supposing $w_{0}>0$ is the lower bound of $w$ where the
brane goes to $x_{\text{I}}\to\infty$. From (4.9), we have
$x^{\prime}_{\text{I}}(u_{\text{I}})\sim-\eta_{\text{I}}(w)\to\infty\,$ when
$w\to w_{0}$. We assume this solution satisfies the NEC (4.15) which is only
valid when we impose $u^{H}_{\text{I}}=u^{H}_{\text{II}}$.202020For the
configurations [H1,E] and [E, H1] we do not have such constraint and therefore
the following argument does not apply for these two cases.  Then when $w$
decreases to $w_{0}$, we should have $\eta^{\prime}_{\text{I}}\to-\infty$ thus
$x^{\prime}_{\text{I}}(u_{\text{I}})\to\infty$. This implies that when
$u_{\text{I}}$ increase to $u_{\text{I}}(w_{0})$, we have
$x_{\text{I}}(u_{\text{I}})$ decrease. This means it is impossible to approach
the critical value $w\to w_{0}$ from below. One possibility is that we have a
turning point for the non-monotonic profile of the interface brane. However,
in this case one could repeat the aforementioned reasoning near the turning
point, where $u_{\text{I}}^{\prime}(x_{\text{I}})=0$. By doing so, we arrive
at the conclusion that it is indeed impossible to obtain the profile [H1, H1].

####  4.1.7 BCFT limit

Here we briefly discuss the the BCFT limits of the system, following the same
method for the zero temperature case. The first BCFT limit is $\nu\to 0$, the
equation of motion in (LABEL:eq:q-finiteT) becomes

$\displaystyle\begin{split}&\eta_{\text{II}}=\pm\frac{1}{2}\sqrt{\frac{L_{\text{I}}(u^{H}_{\text{I}})^{2}}{w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)}+4\eta^{2}_{\text{I}}}+\mathcal{O}(\nu^{2})\,,\\\
&\phi^{\prime
2}(w)=-\frac{L_{\text{I}}\left((u^{H}_{\text{I}})^{4}\eta_{\text{I}}+(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)((u^{H}_{\text{I}})^{2}\eta^{\prime}_{\text{I}}-2\eta^{3}_{\text{I}})\right)}{2u^{H}_{\text{I}}(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\sqrt{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}}}}+\mathcal{O}(\nu)\,,\\\
&V(\phi)=\pm\frac{1}{L_{\text{I}}\nu}\\\
&-\frac{2(L^{2}_{\text{I}}+2w(u^{H}_{\text{I}})^{2})\eta_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+2w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}})+2L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{\prime}_{\text{I}}}{L_{\text{I}}u^{H}_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{2}_{\text{I}})^{\frac{3}{2}}}\\\
&~{}~{}~{}~{}~{}+\mathcal{O}(\nu)\,.\end{split}$ (4.44)

Similar to the zero temperature case (2.36), there is a divergent term
$\frac{1}{L_{\text{I}}\nu}$ in the expression of $V(\phi)$. In above
equations, the parameter $u^{H}_{\text{II}}$ appears in higher order terms.

The second way to obtain a BCFT is to consder the limit $\nu=1$,
$u^{H}_{\text{I}}=u^{H}_{\text{II}}$ and $\eta_{\text{II}}=-\eta_{\text{I}}$ ,
where we can perform the folding trick. In this case, the nontrivial equations
of motion are

$\displaystyle\begin{split}&\phi^{\prime
2}(w)=-\frac{L_{\text{I}}\left((u^{H}_{\text{I}})^{4}\eta_{\text{I}}+(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)((u^{H}_{\text{I}})^{2}\eta^{\prime}_{\text{I}}-2\eta^{3}_{\text{I}})\right)}{u^{H}_{\text{I}}(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\sqrt{L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}}}}\,,\\\
&V(\phi)=\\\
&-\frac{4(L^{2}_{\text{I}}+2w(u^{H}_{\text{I}})^{2})\eta_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+2w(L^{2}_{\text{I}}+w(u^{H}_{\text{I}})^{2})\eta^{2}_{\text{I}})+4L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{\prime}_{\text{I}}}{L_{\text{I}}u^{H}_{\text{I}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}+4w(L^{2}_{\text{I}}+(u^{H}_{\text{I}})^{2}w)\eta^{2}_{\text{I}})^{\frac{3}{2}}}\,,\end{split}$
(4.45)

where the quantities $\phi^{\prime 2}(w),V(\phi)$ are twice of those in
(4.44).

###  4.2 Gluing thermal AdS3 and BTZ black hole

In this subsection we will glue a thermal AdS3 with a BTZ black hole along the
interface brane. The permissible configuration in thermal AdS3 spacetime is
expected to be of type empty, denoted as E${}_{\text{tAdS}}$, while the
allowed configurations for a BTZ black hole can be type E, H1 or H2 as
illustrated in Fig. 18. In the absence of a scalar field, we will show that
finding a solution is impossible. However, with the inclusion of a brane-
located scalar field, we will demonstrate that the permissible configurations
only include [E${}_{\text{tAdS}}$, E], and [E, E${}_{\text{tAdS}}$].

The metric for thermal AdS3 is (2.14), and the metric for BTZ is (4.1). Note
that one can set $u_{A}^{H}\to\infty$ with $A=\text{I}$ or II in (4.1) to make
it a thermal AdS3. As an example, we take the left portion of the bulk as
thermal AdS3 while the right portion of spacetime as BTZ black hole. The
parameterization is

$\displaystyle\begin{split}&u_{\text{I}}(w)=\frac{L_{\text{I}}}{\sqrt{w}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{\text{II}}(w)=\frac{u_{\text{II}}^{H}L_{\text{II}}}{\sqrt{L_{\text{II}}^{2}+(u_{\text{II}}^{H})^{2}w}}\,,~{}~{}~{}~{}\\\
&x_{\text{I}}(w)=-\int^{+\infty}_{w}\frac{\eta_{\text{I}}(\sigma)}{\sqrt{\sigma}}d\sigma\,,~{}~{}~{}~{}x_{\text{II}}(w)=-\int^{+\infty}_{w}\frac{u^{H}_{\text{II}}\eta_{\text{II}}(\sigma)}{\sqrt{L_{\text{II}}^{2}+(u^{H}_{\text{II}})^{2}\sigma}}d\sigma\,.\end{split}$
(4.46)

It is straightforward to generalize following analysis to the case that the
left portion is BTZ black hole while the right portion is a thermal AdS
spacetime.

Plugging the above parameterization into the equations in Sec. 2 we can obtain
the equations for this case. Another consistent way is to take the limit
$u_{\text{I}}^{H}\to\infty$ to the equations in Sec. 4.1. The independent
equations of motion are

$\displaystyle\begin{split}&0=\frac{L^{2}_{\text{I}}}{4w^{2}}+\eta^{2}_{\text{I}}-\frac{L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}}{4L^{2}_{\text{II}}w+4(u^{H}_{\text{II}})^{2}w^{2}}-\eta^{2}_{\text{II}}\,,\\\
&\frac{V(\phi)}{2}=-\frac{2w\eta_{\text{I}}(L^{2}_{\text{I}}+2w^{2}\eta^{2}_{\text{I}})+L^{2}_{\text{I}}w^{2}\eta^{\prime}_{\text{I}}}{L_{\text{I}}(L^{2}_{\text{I}}+4w^{2}\eta^{2}_{\text{I}})^{\frac{3}{2}}}\\\
&~{}~{}+\frac{(L^{2}_{\text{II}}+2w(u^{H}_{\text{II}})^{2})\eta_{\text{II}}(L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+2w(L^{2}_{\text{II}}+w(u^{H}_{\text{II}})^{2})\eta^{2}_{\text{II}})+L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}w(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)\eta^{\prime}_{\text{II}}}{L_{\text{II}}u^{H}_{\text{II}}(L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+4w(L^{2}_{\text{II}}+w(u^{H}_{\text{II}})^{2})\eta^{2}_{\text{II}})^{\frac{3}{2}}}\,,\\\
&\phi^{\prime
2}(w)=-\frac{L^{2}_{\text{I}}+4w^{2}\eta^{2}_{\text{I}}}{4w^{3}}\,\bigg{[}-wV(\phi)\\\
&~{}~{}-\frac{2w^{2}\eta_{\text{I}}}{L_{\text{I}}\sqrt{L^{2}_{\text{I}}+4w^{2}\eta^{2}_{\text{I}}}}+\frac{2w(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)\eta_{\text{II}}}{L_{\text{II}}u^{H}_{\text{II}}\sqrt{L^{2}_{\text{II}}(u^{H}_{\text{II}})^{2}+4(L^{2}_{\text{II}}+(u^{H}_{\text{II}})^{2}w)w\eta^{2}_{\text{II}}}}\bigg{]}\,.\end{split}$
(4.47)

The NEC becomes

$\displaystyle\begin{split}-\frac{\,\eta_{\text{I}}+w\eta^{\prime}_{\text{I}}}{\nu(L^{2}_{\text{I}}+4w^{2}\eta^{2}_{\text{I}})^{\frac{3}{2}}}+\frac{(u^{H}_{\text{II}})^{4}\,\eta_{\text{II}}+(L^{2}_{\text{I}}\nu^{2}+(u^{H}_{\text{II}})^{2}w)(-2\eta^{3}_{\text{II}}+(u^{H}_{\text{II}})^{2}\eta^{\prime}_{\text{II}})}{u^{H}_{\text{II}}(L^{2}_{\text{I}}(u^{H}_{\text{I}})^{2}\nu^{2}+4w(L^{2}_{\text{I}}\nu^{2}+(u^{H}_{\text{II}})^{2}w)\eta^{2}_{\text{II}})^{\frac{3}{2}}}\geq
0\,,\end{split}$ (4.48)

which is again consistent with the condition $\phi^{\prime 2}\geq 0$.