the Von Neumann entropy of associated CFT. This turned out to be incorrect,
but the connections of RT surfaces with bulk reconstruction [104] suggests
there should be a way to relate the sweeping transition with the entanglement
structure of the ICFT.

Another interesting direction was alluded in chap.3, and it is the computation
of RT surfaces in non-equilibrium solutions with deformed horizons. Indeed, it
is still an open question how is the Bekenstein-Hawking formula modified
whenever we consider out-of-equilibrium horizons, which are non-Killing. Is it
the event horizon or the apparent horizon that must appear in the formula
[203]? In equilibrium situations, those two coincide, so the distinction
cannot be made. Additionally, this may also allow us to understand and confirm
the entropy production at the interface of the NESS state described in the
previous chapter.

Finally, entanglement entropy computations allow one to compute the Quantum
Null Energy Condition (QNEC) [204]. As we will explain in the main text, the
QNEC is an inequality (which should hold in any QFT) relating derivatives of
the entanglement entropy and the one-point function of the Stress-Energy
tensor. As such, it is a powerful tool to determine if simple holographic
bottom-up models are at least consistent. For instance, the QNEC has been used
to restrict quenches of CFTs [205, 206, 207], by using the RT prescription in
the dual bulk. Such a quench geometry is obtained in much the same way as the
models of chap.2, the difference being the membrane is now spacelike, instead
of timelike. Whenever the QNEC is not obeyed in such a quench, it signals that
it is probably unphysical. In the same vein, it would be interesting to
compute the QNEC in the various geometries that were obtained in the previous
chapters to check for consistency.

Since this chapter is composed of partial, incomplete and sometimes
inconclusive results, it will necessarily seem much more scattered and less
polished than the previous ones. While there isn’t a clear-cut goal or thread,
but rather several different directions, we believe that the work we present
here is valuable, and hope that it will be completed and ”packaged” in the
more standard form of a paper sometime.

Throughout this chapter, we use units in which $8\pi G=1$.

###  4.1 Geodesics in asymptotically Anti-de-Sitter spaces

The main object of study in this chapter are RT and HRT surfaces in
3-dimensional asymptotically AdS spacetimes. In fact, we can restrict to the
study of spacetimes which are locally AdS, since the only matter content of
our models is the membrane, and everywhere else the vacuum equations of motion
hold. In three dimensions, finding RT curves amounts to computing spatial
geodesics anchored at the asymptotic boundary.

In this section, we compute and outline some properties of such geodesics,
mainly in Poincaré space. Indeed, we solved the equations in one coordinate
system, by exploiting diffeomorphisms between the different geometries, we can
obtain the geodesics in any locally AdS spacetime.

Let us begin by recalling the Poincaré metric :

$\displaystyle ds^{2}=\frac{\ell^{2}}{z^{2}}(dw_{+}dw_{-}+dz^{2})\ ,$ (4.1)

where $w_{-}=x-t$, $w_{+}=x+t$ are the lightcone coordinates. Consider the
most generic spacelike geodesic, of initial point
$p_{0}=(w_{+}^{0},w_{+}^{0},z_{0})$ and tangent vector
$\dot{p}_{0}=(\dot{w}_{+}^{0},\dot{w}_{-}^{0},\dot{z}_{0})$. We will consider
WLOG affinely parametrized geodesics, namely :

$\displaystyle\dot{p}^{\mu}\dot{p}_{\mu}=\frac{\ell^{2}}{z^{2}}\left(\dot{w}_{+}\dot{w}_{-}+\dot{z}^{2}\right)=1\
,$ (4.2)

where this condition will hold not only at the initial point, but everywhere
along the geodesic, so we omit the ”0” indices.

Denoting the affine parameter $\lambda$, the action that we need to minimize
is simply :

$\displaystyle
L=\int\frac{\ell^{2}}{z^{2}}(\dot{w}_{+}\dot{w}_{-}+\dot{z}^{2})d\lambda\ .$
(4.3)

Using cyclicity of $w_{+}$, $w_{-}$ we get two integration constants :

$K_{\pm}=\frac{\dot{w}_{\pm}\ell}{z^{2}}\ .$ (4.4)

Translations of the affine parameter give us the conservation of the tangent
vector norm (4.2) :

$\displaystyle K_{-}K_{+}z^{2}+\frac{\ell^{2}\dot{z}^{2}}{z^{2}}=1\ .$ (4.5)

Note that other symmetries like scale invariance ($z\rightarrow\omega z$,
$w_{\pm}\rightarrow\omega w_{\pm}$) or boosts ($w_{+}\rightarrow\gamma
w_{+},w_{-}\rightarrow\gamma^{-1}w_{-}$) do not provide additional integration
constants.

We can solve (4.5) for $z(\lambda)$. There are three cases which we must
differentiate, $K_{-}K_{+}>0$, $K_{+}K_{-}<0$ and $K_{+}K_{-}=0$. These signs
are simply determined by the initial vectors, since ${\rm
sign}(K_{-}K_{+})={\rm sign}(\dot{w}_{-}\dot{w}_{+})$. Thus, in Poincaré
coordinates, we will have three types of qualitatively different geodesics.

####  4.1.1 Case $K_{-}K_{+}>0$

We begin by the $K_{-}K_{+}>0$ case. Upon integrating (4.5) :

$\displaystyle\pm(\frac{\lambda}{\ell}+C)={\rm
Arctanh}\left(\sqrt{1-K_{+}K_{-}z^{2}}\right)\Leftrightarrow
z(\lambda)=\sqrt{\frac{1-\tanh^{2}(\lambda/\ell+C)}{K_{+}K_{-}}}\ .$ (4.6)

The $\pm$ signs denotes two branches of the solution which join at
$z=\frac{1}{K_{+}K_{-}}$, for $\lambda=-C\ell$. The $C$ integration constant
can be absorbed into a translation of the affine parameter, we set $C=0$ so
that the ”turning point” of the geodesic is always at $\lambda=0$. Then in
(4.6) the plus sign must be chosen for $\lambda>0$ and minus sign for
$\lambda<0$. For this choice, the initial $\lambda_{0}$ is given as :

$\lambda_{0}=-\ell\;{\rm sign}(\dot{z}_{0}){\rm
Arctanh}\left(\sqrt{1-\frac{\ell^{2}}{z_{0}^{2}}\dot{w}_{+}^{0}\dot{w}_{-}^{0}}\right)\
.$ (4.7)

Plugging (4.6) into (4.4) and performing the integration yields :

$w_{\pm}=\frac{1}{K_{\mp}}\tanh(\lambda/\ell)+G_{\pm}\ ,$ (4.8)

where $G_{\pm}$ are two integration constants. They are easily fixed using the
initial conditions, and the full solution is given by :

$\displaystyle z(\lambda)$
$\displaystyle=\frac{z_{0}^{2}}{\ell}\sqrt{\frac{1-\tanh^{2}(\lambda/\ell)}{\dot{w}_{+}^{0}\dot{w}_{-}^{0}}}\Leftrightarrow\lambda=\pm\ell\sqrt{1-\frac{z(\lambda)^{2}}{z_{0}^{2}}\frac{\ell^{2}}{z_{0}^{2}}\dot{w}_{+}^{0}\dot{w}_{-}^{0}}\
,$ (4.9) $\displaystyle w_{\pm}(\lambda)$
$\displaystyle=\frac{\tanh(\lambda/\ell)-\tanh(\lambda_{0}/\ell)}{\ell\dot{w}_{\mp}^{0}}z_{0}^{2}+w_{\pm}^{0}\
,$ $\displaystyle\lambda_{0}$ $\displaystyle=-\ell\,{\rm
sign}(\dot{z}_{0}){\rm
Arctanh}\left(\sqrt{1-\frac{\ell^{2}}{z_{0}^{2}}\dot{w}_{+}^{0}\dot{w}_{-}^{0}}\right)\
.$

These are the geodesics one usually thinks of in AdS space. At
$\lambda=\pm\infty$, they are anchored on the boundary of AdS. In the case
where $\dot{w}^{0}_{+}=\dot{w}^{0}_{-}$, they are semi-circles centered on the
asymptotic boundary. By boosting them one can obtain the general shape of the
geodesic.

Note that while it would be nice to parametrize the geodesic by its initial
point on the asymptotic boundary, this can’t be done as they lie formally at
infinity. Indeed, as we send $z_{0}=\epsilon\rightarrow 0$, to satisfy the
affine condition we need $\dot{w}^{0}_{\pm}\propto\epsilon^{2}$ and
$\dot{z}_{0}\propto\epsilon$, so on the boundary the tangent vector is
degenerate.

####  4.1.2 Case $K_{-}K_{+}<0$

We treat the case $K_{-}K_{+}<0$. The resolution of the equations is
essentially the same, modulo the change ${\rm tanh}\rightarrow{\rm coth}$.
Skipping the details, we have :

$\displaystyle z(\lambda)$
$\displaystyle=\frac{z_{0}^{2}}{\ell}\sqrt{\frac{1-\coth^{2}(\lambda/\ell)}{\dot{w}^{0}_{+}\dot{w}^{0}_{-}}}\
,$ (4.10) $\displaystyle w_{\pm}(\lambda)$
$\displaystyle=\frac{\coth(\lambda/\ell)-\coth(\lambda_{0}/\ell)}{\ell\dot{w}^{0}_{\mp}}z_{0}^{2}+w^{0}_{\pm}\
,$ $\displaystyle\lambda_{0}$ $\displaystyle=-\ell\,{\rm
sign}(\dot{z}_{0}){\rm
Arccoth}\left(\sqrt{1-\frac{\ell^{2}}{z_{0}^{2}}\dot{w}^{0}_{+}\dot{w}^{0}_{-}}\right)\
.$

Note that now, the point $\lambda=0$ is not really a ”turning point” anymore
since as we approach it all the coordinates diverge as $\frac{1}{\lambda}$.
For the two signs of $\lambda$, we get two disconnected solutions, with one
end in the Poincaré horizon $z=\infty$ and the other on the boundary $z=0$.
Naturally, the change $\lambda\rightarrow-\lambda$ is equivalent to a reversal
of the initial tangent vector
$\dot{w}_{\pm}^{0}\rightarrow-\dot{w}_{\pm}^{0}$, along with a translation of
the initial point of
$-\frac{2\coth(\lambda_{0}/\ell)z_{0}^{2}}{\ell\dot{w}_{\mp}^{0}}$. See
fig.4.1 for a sketch of the geodesic projected to the $t=cst$ plane (note that
there are no geodesics of this type that stay in this plane, because
$\dot{t}_{0}\neq 0$ necessarily).

Figure 4.1: Plot of the two disconnected branches of the geodesic of equation
(4.10). Both branches escape out to the Poincaré horizon. If we maximally
extend, we would find that they end up hitting the boundary again, as all
spacelike geodesics have two anchor points on the boundary in global
coordinates.

These geodesic seem very strange, but this is due to the coordinate system we
chose. Indeed, the Poincaré horizon is a coordinate singularity, and if we
were to map these geodesics to global coordinates, we would see that they are
a portion of a geodesic that is doubly anchored at the boundary. Nonetheless,
for the purposes of the RT prescription, the coordinate system in which we
look for the geodesics will of course matter, since it will depend on the
state of the CFT.

####  4.1.3 Case $K_{-}K_{+}=0$

The last case to consider is of ”measure zero” in the set of initial
conditions, the case $K_{-}K_{+}=0$. It includes in particular geodesics that
are shot ”straight” toward the bulk interior.

Solving (4.5) in this case gives us

$z=e^{{\rm sign}{\dot{z}_{0}}\lambda/\ell}\ ,$ (4.11)

where we absorbed the integration constant into a translation of $\lambda$.
The equation (4.4) is again readily solved, regrouping all the equations we
obtain :

$\displaystyle z$ $\displaystyle={\rm exp}\left({\rm
sign}(\dot{z}_{0})\frac{\lambda}{\ell}\right)\ ,$ (4.12) $\displaystyle
w_{\pm}$ $\displaystyle=\frac{\dot{w}^{0}_{\pm}\ell}{2z_{0}^{2}}(e^{2{\rm
sign}(\dot{z}_{0})\lambda/\ell}-e^{2\lambda_{0}/\ell})+w^{0}_{\pm}\ ,$
$\displaystyle\lambda_{0}$ $\displaystyle=\ell\;{\rm
sign}(\dot{z}_{0})\ln(z_{0})\ .$

Of course, in (4.12), one of the $\dot{w}^{0}_{\pm}$ must be equal to zero if
the equations are to be satisfied.

###  4.2 Simple examples

Let us apply the formulas we have derived above to some standard examples.
Consider an interval of length $a$ on the boundary of Poincaré space. The dual
CFT state is simply the vacuum on $\mathbb{R}^{2}$, so computing the
entanglement entropy of the interval should reproduce the classic result by
Calabrese and Cardy [85].

Consider the initial point to be $(t=0,x=0,z=\epsilon)$ on the boundary, where
we introduced the IR cutoff $\epsilon$. Since the geodesics exhibited in the
previous section are parametrized in terms of the initial tangent vector
$\dot{p}^{0}$ and not the final point, we must invert some relations to find
the correct values for which the geodesic ends on $(t=0,x=a,z=0)$111In
poincaré space, it is possible to invert the relation generally and to give
the geodesics as parametrized by initial and final points. We don’t find it
useful to reproduce it here, as it is more cumbersome than the parametrization
we exhibited and, while it can be convenient, we won’t really need it for our
purposes..

We already know $\dot{w}_{+}\dot{w}_{-}>0$ in order to have a geodesic
connecting two points on the boundary. The endpoint is located at
$z=\epsilon$, at $\lambda=-\lambda_{0}$. From the equation of
$w_{\pm}(-\lambda_{0})$ (4.9), we find :

$\displaystyle\dot{w}^{0}_{+}=\dot{w}^{0}_{-}\equiv\dot{w}_{0}=\frac{\epsilon}{\ell\sqrt{\frac{a^{2}}{4\epsilon^{2}}+1}}\
.$ (4.13)

What remains is to compute the length of the geodesic. Thankfully, in the
affine parametrization this is simply given by the difference in initial and
final affine parameters:

$\displaystyle L=-2\lambda_{0}=2\ell{\rm Arctanh}\left(\frac{\dot{w}_{0}\ell
a}{2\epsilon^{2}}\right)=2\ell{\rm
Arctanh}\left(\frac{a}{\sqrt{a^{2}+4\epsilon^{2}}}\right)\ .$ (4.14)

To obtain the entanglement entropy, all we have to do is divide by
$4G=\frac{1}{2\pi}$ and take the limit $\epsilon\rightarrow 0$. We obtain at
leading order :

$\displaystyle S=2\pi
L=4\pi\ell\ln\left(\frac{a}{\epsilon}+\mathcal{O}(\epsilon^{2})\right)=\frac{c}{3}\ln\left(\frac{a}{\epsilon}\right)+\mathcal{O}(\epsilon^{2})\
,$ (4.15)

which correctly reproduces the expected result[85].

Let us illustrate a slightly more complicated example, which will be crucial
for the methods of computation we will use in the ICFT setups. We consider now
the spinning string geometry, the metric (3.1). In this case, spacetime is
only stationary, but not static. The problem thus becomes immediately more
complex; instead of studying curves restricted to a 2-dimensional Cauchy
slice, we must allow for generic spatial geodesic in the full 3D geometry.
Furthermore, one can try to solve the geodesic equations for the metric (3.1),
and it is feasible but a much harder problem.

Instead, we exploit the change of coordinate to bring us back to the Poincaré
geometry. To find the diffeomorphism relating the two coordinate systems one
can for instance equate the two parametrizations of the AdS hyperboloid
described in sec.1.1, and solve the resulting equations. We find ($\sigma={\rm
sign}(J)$):

$\displaystyle w_{+}$
$\displaystyle=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}\exp\left(\frac{(r_{+}-\sigma
r_{-})(x+t)}{\ell}\right)\ ,$ (4.16) $\displaystyle w_{-}$
$\displaystyle=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}\exp\left(\frac{(r_{+}+\sigma
r_{-})(x-t)}{\ell}\right)\ ,$ $\displaystyle z$
$\displaystyle=\sqrt{\frac{r_{+}^{2}-r_{-}^{2}}{r^{2}-r_{-}^{2}}}\exp\left(\frac{xr_{+}-\sigma
tr_{-}}{\ell}\right)\ .$

Of course, (4.16) is far from unique, as we can apply any isometry of Poincaré
and still obtain a valid change of coordinates. Note in passing that this
coordinate change covers only a portion of Poincaré space, since $w_{+}>0\cap
w_{-}>0$, and only a portion of the spinning string, as the coordinate change
becomes singular at the horizon $r=r_{+}$222The outer horizon is mapped to
$w_{-}=0\cup w_{+}=0$. A more careful analysis would show that this surface is
composed in fact of two horizons, the future, and past ones. If one extends to
the rest of the Poincaré spacetime, we would obtain the two interior regions,
as well as another exterior.. There are other similar changes of coordinates
that map the region $r_{-}<r<r_{+}$ and $0<r<r_{-}$, but we won’t be needing
them here.

Consider now the boundary region in stationary CFT delimited by two points
$(t=0,r=1/\epsilon,x=0)$ and $(t=0,r=1/\epsilon,x=a)$. WLOG we take
$\sigma=1$, namely $J>0$. Under the change of coordinates (4.16), these are
mapped to

$\displaystyle p_{1}=$
$\displaystyle\left(w_{+}=1,w_{-}=1,z=\epsilon\sqrt{r_{+}^{2}-r_{-}^{2}}\right)\
,$ (4.17) $\displaystyle p_{2}=$
$\displaystyle\left(w_{+}=\exp(\frac{r_{+}-r_{-}}{\ell}a),w_{-}=\exp(\frac{r_{+}+r_{-}}{\ell}a),z=\epsilon\sqrt{r_{+}^{2}-r_{-}^{2}}\exp(\frac{xr_{+}}{\ell})\right)\
,$

where we kept only the leading order in $\epsilon$. We will write $p_{i}^{b}$
for the corresponding points on the 2D-boundary, with the $z$-coordinate
excised.

Note that we do not care about the specific shape of the boundary interval we
select between the two points. Indeed, any spacelike curve between $p_{1}$ and
$p_{2}$ will have the same entanglement entropy, because they all lie in the
same ”causal diamond”333The causal diamond of a portion of a Cauchy slice $A$
is the region that is in causal contact uniquely with $A$. For instance, if
$A$ is the interval between two points $x$, $y$, the causal diamond is
determined by shooting lightrays from $x$ and $y$, and selecting the enclosed
region. Because points in this region can only be affected by $A$, they can be
expressed as a unitary transformation applied to $A$. See fig.4.13. . Any two
of these curve houses states which are related by a unitary transformation,
and thus have the same entanglement entropy, see sec.1.8.

We can exploit this fact to simplify the problem even further. Indeed using
the points (4.17), we would still need to apply the HRT prescription, as they
do not lie on a preferred Cauchy slice. Then, we use the isometries of
Poincaré to bring these two points on a constant time slice. In this case, we
can begin by a $x$-translation to bring $p_{1}$ to the origin. Then, we
perform a boost to bring $p_{2}$ on the same $t=0$ Cauchy slice. Since these
are isometries of Minkowski on the boundary, we do not need to perform them
explicitly, as we now that the norm $|p^{b}_{1}-p^{b}_{2}|$ will be conserved.
We find :

$\displaystyle|p_{2}^{b}-p_{1}^{b}|^{2}=4\exp(\frac{r_{+}a}{\ell}){\rm
sinh}\left(\frac{r_{+}-r_{-}}{2\ell}a\right){\rm
sinh}\left(\frac{r_{+}+r_{-}}{2\ell}a\right)\equiv\tilde{a}\ .$ (4.18)

In this way, we successfully reduced the problem to the same RT computation as
previously with an interval of size $\tilde{a}$. There is one difference, and
that is in the UV cutoffs which are now different at the two endpoints.
Recycling (4.13) and computing the correct $\lambda_{f}$ given the different
cutoffs, we obtain :

$\displaystyle S=2\pi
L=2\pi(\lambda_{f}-\lambda_{0})=\frac{c}{6}\log(\frac{4}{\epsilon^{2}(r_{+}^{2}-r_{-}^{2})}{\rm
sinh}\left(\frac{r_{+}-r_{-}}{2\ell}a\right){\rm
sinh}\left(\frac{r_{+}+r_{-}}{2\ell}a\right))\ ,$ (4.19)

which correctly recovers the expected entanglement entropy, see (3.10).

With this method, inverting (4.16), it is also possible to obtain the equation
of the HRT surface in the original coordinate system. In particular, the
spacelike geodesic that we obtain does not remain in a constant time plane, as
we should expect for a non-static geometry. A sketch of the HRT surface,
projected on the $t=0$ plane, can be found on fig. 4.2. Notice that the HRT
surface cannot breach the horizon, and as $a\rightarrow\infty$, it will
approach it more and more, which gives the Bekenstein-Hawking formula for the
coarse-grained entropy (see (3.11)).

Figure 4.2: Sketch of an HRT surface between two equal time points on the
boundary, $x_{1}$ and $x_{2}$. The HRT surface is not confined to the constant
time Cauchy slice depicted here, rather it is its projection onto this plane.
We can see that the HRT surface drops very quickly close to the horizon, where
it can reduce its length. Spacelike geodesics are not forbidden from entering
the horizons, but there are no spacelike geodesics entering the horizon with
two anchor points on the same boundary, see app.D.3

###  4.3 Entanglement structure of static ICFT states

In this section, we apply the lessons we have just learned to compute the
entanglement structure of static ICFT states on $\mathbb{R}^{2}$. We will also
mention possible applications to the compact geometries of chap.2.

####  4.3.1 RT surfaces with interfaces

In this section, we wish to apply the RT prescription to spacetimes containing
a gravitating domain wall. As the wall connects two separate bulks, we need a
prescription to continue the RT surface across the wall. For instance, in the
case of end-of-the-world branes which appear in Island computations [21, 22],
the prescription states that the RT surface should end perpendicularly on the
End of the World brane444This is only in the case where the EOW brane is
described by the simplest Lagrangian, only involving its tension. If
additional fields are living on it, the prescription will change. We will show
that this can be recovered this as a special limit of the more general
interface prescription.

Consider again the action determining a spacelike geodesic, but on a spacetime
which is composed of two pieces. Each piece will contain a domain wall which
we parametrise as $x^{\mu}_{mi}(\xi^{a})$, where the $i$ denotes its embedding
on side $i$. The geodesic is parametrised by $\lambda$, and crosses the
membrane at $\lambda^{*}$ :

$\displaystyle
L=\int_{\lambda_{0}}^{\lambda^{*}}\sqrt{g^{1}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}d\lambda+\int_{\lambda^{*}}^{\lambda_{0}}\sqrt{g^{2}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}d\lambda=\int_{\lambda_{0}}^{\lambda^{*}}\mathcal{L}_{1}(x,\dot{x})d\lambda+\int_{\lambda^{*}}^{\lambda_{0}}\mathcal{L}_{2}(x,\dot{x})d\lambda\
.$ (4.20)

In what follows, we omit the second half of the action as it is treated
similarly. In the variation of (4.20), the initial and final points are fixed,
$\delta x(\lambda_{0})=\delta x(\lambda_{f})=0$. On the other hand, the
variation at $\lambda^{*}$ receives a softer constraint; it must simply remain
on the membrane. Using the parametrization of the membrane, we can write is as
:

$\displaystyle\delta x^{\rho}(\lambda^{*})=\frac{\partial
x^{\rho}_{m}}{\partial\xi^{a}}\delta\xi^{a}\ ,$ (4.21)

where the $\delta\xi^{a}$ are arbitrary variations. The variation of (4.20)
then reads :

$\displaystyle\delta
L=\int_{\lambda_{0}}^{\lambda^{*}}\frac{\delta\mathcal{L}_{1}}{\delta
x^{\sigma}(\lambda)}\delta
x^{\sigma}(\lambda)+\frac{\partial\mathcal{L}_{1}}{\partial\dot{x}^{\rho}}\delta
x^{\rho}(\lambda^{*})+1\leftrightarrow 2\ ,$ (4.22)

where we used the shorthand $\frac{\delta\mathcal{L}_{1}}{\delta
x^{\sigma}(\lambda)}=\frac{\partial\mathcal{L}_{1}}{\partial
x^{\sigma}}-\frac{d}{d\lambda}\frac{\partial\mathcal{L}_{1}}{\partial\dot{x}^{\sigma}}$,
which is the term which will give the Euler-Lagrange equations. The second
term in (4.22) appears from an integration by parts.

Thus by the arbitrariness of the variation $\delta x^{\sigma}(\lambda)$, we
obtain the geodesic equations which must be obeyed by the geodesic on both
sides. The ”crossing constraints” are obtained by requiring that the boundary
term vanishes. They are :

$\displaystyle\frac{\partial\mathcal{L}_{1}}{\partial\dot{x}^{\rho}}\frac{\partial
x^{\rho}_{m}}{\partial\xi^{a}}=\frac{\partial\mathcal{L}_{2}}{\partial\dot{x}^{\rho}}\frac{\partial
x^{\rho}_{m}}{\partial\xi^{a}}\ ,$ (4.23) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{t^{1\,\rho}_{a}\dot{x}^{\sigma}_{1}g_{1\,\rho\sigma}}{\sqrt{g_{1\mu\nu}\dot{x}_{1}^{\mu}\dot{x}_{1}^{\nu}}}=\frac{t^{2\,\rho}_{a}\dot{x}^{\sigma}_{2}g_{2\,\rho\sigma}}{\sqrt{g_{2\mu\nu}\dot{x}_{2}^{\mu}\dot{x}_{2}^{\nu}}}\
,$

where we labeled by $i$ the curves lying on side $i$. We introduced the
notation $t^{i\,\rho}_{a}=\frac{\partial x^{\rho}_{m}}{\partial\xi^{a}}$ which
are the tangent vectors to the membrane.

The constraints (4.23) tell us that the projection of membrane tangent vectors
on the geodesic should be conserved while crossing.

One can get a bit more intuition on the meaning of this constraint by
considering the membrane to contain only spacelike directions. In this case,
the tangent vectors $t^{i\,\mu}_{a}$ are all spacelike, and using the metric
matching condition (2.4), we have that the scalar products
$t^{i\,\mu}_{a}t^{i\,\nu}_{b}g_{i\mu\nu}=h_{ab}$ are equal on both sides,
since they are the membrane’s induced metric. This allows us to divide on both
sides of (4.23) by the norm of the tangent vectors. The resulting equations
then equate the angles of the geodesic with the membrane :

$\displaystyle\sphericalangle\left(t^{1}_{a},\dot{x}_{1}\right)=\sphericalangle\left(t^{2}_{a},\dot{x}_{2}\right)\
.$ (4.24)

Note that the condition (4.23) can only be interpreted as (4.24) in the case
where the metric matching is satisfied. Indeed, there is an intuitive way of
understanding why is this the case. The metric matching condition is
essentially a continuity equation on the metric. This should imply that the
Christoffel symbols are at most step-wise discontinuous. Then by the geodesic
equation :

$\displaystyle\ddot{x}^{\mu}=\Gamma^{\mu}_{\nu\rho}\dot{x}^{\nu}\dot{x}^{\rho}\
.$ (4.25)

By integrating we find that $\dot{x}^{\mu}$ should be continuous across the
interface. The precise way this is realized is simply (4.24).

In the case where some directions of the membrane are timelike, the norm
squared of the tangent vectors will be negative. We could still divide on both
sides by the absolute value of the norm, but the expression that we obtain in
this way is difficult to visualize as an ”angle”.

####  4.3.2 Vacuum ICFT

We are now ready to compute RT surfaces in setups containing a gravitating
wall. We consider in this section the joining of two CFTs in their vacuum
state, on an infinite interval. The solution is described in sec.2.3.2, we
rewrite here the crucial formulas to set slightly different conventions. The
dual metrics can be given in Poincaré coordinates :

$\displaystyle
ds^{2}_{j}=\frac{\ell_{j}^{2}}{z_{j}^{2}}\left(-dt^{2}+dx_{j}^{2}+dz_{j}^{2})\right)\
,$ (4.26)

and the membrane equation is very simple in these coordinates :

$\displaystyle x_{j}(z_{j})=-\frac{\tan(\psi_{j})}{z_{j}}\ ,$ (4.27)

where the solution is given in the folded picture, and the conventions are to
keep the part of spacetime that lies to the ”increasing x” side of the
membrane (for convenience we inverted here the convention with respect to
previous chapters). The identification is done by setting
$\ell_{1}/z_{1}=\ell_{2}/z_{2}\Leftrightarrow\cos(\psi_{1})/z_{1}=\cos(\psi_{2})/z_{2}$
along the membrane.

Figure 4.3: Example of a glued geometry dual to a vacuum ICFT state. The
depicted angles $\psi_{i}$ have positive values in the picture. The arrows
denote that the two membranes should be identified.

Remember that in the minimal model, the only parameters available are the AdS
radii $\ell_{i}$ and the tension $\lambda$ of the membrane, as here, the state
is fixed to the vacuum. From the gluing equations (2.27), we can trade
$\ell_{2}/\ell_{1}$ and $\lambda$ for the angles $\psi_{i}$. The WLOG
assumptions $\ell_{1}\leq\ell_{2}$ as well as the Israel equations then impose
:

$\displaystyle 0<\psi_{1}\leq\pi/2\;\;;-\psi_{1}<\psi_{2}<\psi_{1}\ .$ (4.28)

So, for any two angles in the range (4.28), we have an ICFT dual for definite
$\frac{c_{2}}{c_{1}}$ and membrane tension $\lambda$.

The simplicity of the solution in this case allows us to explicitly construct
a coordinate system unifying the two spacetime pieces[50]. Then, the full
solution can be expressed in a single coordinate chart, see fig.4.4.

Figure 4.4: The same solutions depicted in fig.4.3, but presented in an
unified coordinate chart. One can verify that the way we glued the two
spacetimes, they respect the gluing condition
$cos(\psi_{1})/z_{1}=\cos(\psi_{2})/z_{2}$.

We will see that in this picture, crossing RT surfaces can be expressed by a
beautiful geometric construction, which is due to [183]. In the following
section, we reproduce the results already obtained there, and we push the
analysis of the RT surfaces in slightly more detail than in the aforementioned
paper.

####  4.3.3 Intervals containing the interface

We consider first the application of the RT prescription to boundary intervals
containing the interface. Consider first two points on the boundary at equal
times, located on each side of the interface,
$p_{1}=(\tau_{1}=0,x_{1}=\sigma_{1})$ and
$p_{2}=(\tau_{2}=0,x_{2}=\sigma_{2})$ (in the folded picture). Because the
geometry is static, we can restrict ourselves to a constant time Cauchy slice
to search for the RT surface. As mentioned in sec.4.1.1, spacelike geodesics
confined to the Cauchy slice are simply semi-circles whose center lies on the
AdS boundary.

The sought-out RT surface will be composed of circle arcs meeting on the
interface. To satisfy the crossing constraints (4.23), we need to
appropriately choose the radii of the two geodesics so that the circles are
tangent on the membrane (such that their incident angle is continuous). As
first noticed in [183], this problem can be solved by a simple Euclidean
geometry construction, outlined in fig.4.5

Figure 4.5: Example of a crossing geodesic, for $\psi_{2}<0$. The angle
$\varphi$ together with $O_{2}=(\lambda,0)$ determines $\sigma_{2}$ as well as
the membrane intersection $x$. By extending the line $xO_{2}$ to intersect the
boundary of side 1 (green), we find $O_{1}$ which is the center of the second
semi-circle geodesic. The angles $\theta_{1}=\psi_{1}+\psi_{2}$ and
$\theta_{2}=\frac{\pi}{2}+\psi_{2}$ are simply introduced for convenience for
the computations of app.D.1.

The only caveat of the construction of fig.4.5 is that the RT surface is
parametrized in terms of $O_{1}=(\lambda,0)$ and $\varphi$, instead of the
boundary points $\sigma_{1}$ and $\sigma_{2}$.

By repeated use of the law of sines, we obtain the following formulas (see
app.D.1 for the detailed derivation) :

$\displaystyle\sigma_{2}$
$\displaystyle=(-\lambda)\left(\frac{\cos(\psi_{2})}{\cos(\alpha)}-1\right)\
,$ (4.29) $\displaystyle\sigma_{1}$
$\displaystyle=\frac{(-\lambda)\sin(\alpha+\psi_{2})}{\sin(\alpha-\psi_{1})}\left(1+\frac{\cos(\psi_{1})}{\cos(\alpha)}\right)\
,$ $\displaystyle\mu$
$\displaystyle=\frac{\sigma_{2}}{\sigma_{1}}=\frac{\sin(\alpha-\psi_{1})\left(\frac{\cos(\psi_{2})}{\cos(\alpha)}-1\right)}{\sin(\alpha+\psi_{2})\left(\frac{\cos(\psi_{1})}{\cos(\alpha)}+1\right)}\
,$

where we defined $\alpha=\varphi-\psi_{2}$ simply because it simplifies the
look of the expressions. There are some constraints on $\lambda$ and $\alpha$
to generate a well-defined RT surface (for instance, to avoid values for which
the construction yields an RT surface traversing on the wrong side of the
asymptotic boundary). The conditions read :

$\displaystyle\lambda<0$
$\displaystyle\rightarrow\psi_{1}\leq\alpha\leq\pi/2\Leftrightarrow
0<\frac{\sigma_{2}}{\sigma_{1}}<1\ ,$ (4.30) $\displaystyle\lambda>0$
$\displaystyle\rightarrow\pi/2\leq\alpha\leq\pi-\psi_{1}\Leftrightarrow\frac{\sigma_{2}}{\sigma_{1}}>1\
.$

Thus, for $\psi_{1}<\alpha<\pi-\psi_{1}$ we cover all possible boundary
intervals. Notice that the absolute value of $\lambda$ does not influence the
ratio $\mu=\sigma_{2}/\sigma_{1}$, which is a consequence of scale invariance.
By a lengthy computation, one can show that $\mu(\alpha)$ is a monotonically
increasing function, which shows that there is a unique RT surface given
$\sigma_{1}$ and $\sigma_{2}$.

What remains is to compute the length of the geodesic to obtain the
entanglement entropy. Parametrizing the semicircle as ($x=R\cos(\theta)$,
$z=R\sin(\theta)$) the length of the geodesic depends only on the initial and
final opening angles :

$\displaystyle
L=\ell\ln(\frac{\sin\theta_{f}(1+\cos\theta_{0})}{\sin\theta_{0}(1+\cos\theta_{f})})\
.$ (4.31)

Using (4.31), and denoting the $z_{i}$-cutoffs $\epsilon_{i}$, we find the
length of the full geodesic (see app.D.1):

$\displaystyle L=$
$\displaystyle\ell_{1}\ln(\frac{\sigma_{1}}{\epsilon_{1}})+\ell_{2}\ln(\frac{\sigma_{2}}{\epsilon_{2}})$
(4.32)
$\displaystyle\underbrace{+\ell_{1}\log\left(\frac{2}{\tan\left(\frac{\alpha-\psi_{1}}{2}\right)\left(1+\frac{\cos(\alpha)}{\cos(\psi_{1})}\right)}\right)+\ell_{2}\log\left(\frac{2\tan\left(\frac{\alpha+\psi_{2}}{2}\right)}{1-\frac{\cos(\alpha)}{\cos(\psi_{2})}}\right)}_{g(\xi)}\
.$

The quantity on the second line is independent of $\lambda$, and thus depends
only on the ratio $\mu$. The fact that (4.32) takes the form
$L=\ell_{1}\ln(\frac{\sigma_{1}}{\epsilon_{1}})+\ell_{2}\ln(\frac{\sigma_{2}}{\epsilon})+g(\mu)$
is no coincidence. To understand why, consider the more general case in which
$p_{1}=(\tau_{1}=0,x_{1}=1)$, $p_{2}=(\tau_{2}\neq 0,x_{2}=\sigma_{2})$ (where
we used a scale transformation and $\tau$-translation to choose the $p_{1}$
coordinates). As we have shown in 1.5, there is one Virasoro that survives the
introduction of the interface. Constraining ourselves to the global conformal
transformations to remain in the vacuum state, we can use those to bring
$\tau_{2}$ to zero (more specifically, we use a special conformal
transformation centered on $p_{1}$). These CFT transformations translate to
isometries of the bulk so it follows that the geodesic length between the two
points is also unchanged. Using these isometries, it is possible to restrict
the general form of the geodesic distance between two points.

However, we can identify several quantities which are invariant under all the
isometries such as
$\xi=\frac{-(t_{1}-t_{2})^{2}+(x_{1}-x_{2})^{2}}{4x_{1}x_{2}}$,
$\frac{x_{i}}{z_{i}}$ or
$\frac{-(t_{1}-t_{2})^{2}+(x_{1}-x_{2})^{2}+(z_{1}-z_{2})^{2}}{4z_{1}z_{2}}$.
Thus, this isn’t sufficient to constrain it to the form identified above. We
must take into account the additional fact that the geodesics are anchored to
the boundary (or more precisely on the IR cutoff surface). A more elegant way
to restrict the generic form, is to exploit the conformal symmetry of the dual
theory. Indeed, as described in [183](see also [208]) another interpretation
of boundary-anchored geodesic is that they compute two-point functions of
operators of large scaling dimension. Then, we can transfer the form of the
generic two-point function into the form of the geodesic length between two
points $p_{i}=(t=\tau_{i},x=\sigma_{i},z=\epsilon_{i})$ :

$\displaystyle L$
$\displaystyle=\ell_{1}\ln(\frac{\sigma_{1}}{\epsilon_{1}})+\ell_{2}\ln(\frac{\sigma_{2}}{\epsilon_{2}})+g(\xi)\
,$ (4.33) $\displaystyle\xi$
$\displaystyle=\frac{-(\tau_{1}-\tau_{2})^{2}+(x_{1}-x_{2})^{2}}{4x_{1}x_{2}}\
.$

This formula embodies the procedure we used to obtain (4.19), by a succession
of coordinate changes. We can use (4.33) together with (4.32) to compute the
entanglement entropy of any boundary interval for which the two points lie on
either side of the interface.

The only missing piece is that we must invert an equation to extract the
function $\alpha(\mu)$. This can in fact obtained analytically, although the
expression is not very elegant. The defining equation is obtained with (4.29),
and after some massaging :

$\displaystyle\mu=\frac{c_{2}\sqrt{1-c_{\alpha}^{2}}+c_{\alpha}s_{2}}{c_{1}\sqrt{1-c_{\alpha}^{2}}-c_{\alpha}s_{1}}\times\frac{c_{\alpha}+c_{1}}{c_{2}-c_{\alpha}}\
,$ (4.34)

where we use the shorthands $c_{i}=\cos(\psi_{i})$, $c_{\alpha}=\cos(\alpha)$.
Eq. (4.34) can be reduced to a 4th-order polynomial equation after squaring.
Solving yields 4 candidate solutions for $c_{\alpha}(\mu)$, but after plugging
them back in (4.34), only two remain. Imposing the further condition
$|c_{\alpha}|<1$, only one remains, as we expected from the monotonicity of
$\mu(\alpha)$ :

$\displaystyle\cos(\alpha)=\frac{(-1+\mu^{2})(c_{1}+c_{2})-\sqrt{2}(\mu-1)\sin(\frac{\psi_{1}+\psi_{2}}{2})\sqrt{(1+\mu^{2})-(\mu-1)^{2}\cos(\psi_{1}-\psi_{2})+4\mu\cos(\psi_{1}+\psi_{2})}}{2(1+\mu^{2}+2\mu\cos(\psi_{1}+\psi_{2}))}\
.$ (4.35)

The equation is quite unpalatable, but by plugging it into (4.32), together
with the RT prescription we obtain an analytic expression of the entanglement
entropy of any interval in the boundary CFT, which is remarkable.

Let us conclude by considering the ”End of the World” membrane limit, where
side 1 disappears. This is realized by taking the limit
$\frac{\ell_{1}}{\ell_{2}}\rightarrow 0$, which is $\psi_{1}\rightarrow\pi/2$
in terms of the angles. In this limit, since side 1 effectively disappears,
the geodesic in fig.4.5 becomes anchored at the membrane. Because the green
boundary and the membrane are co-linear in this limit, $O_{1}$ moves to the
origin, and the geodesic becomes perpendicular to the wall. We recover in this
way the BCFT prescription [22] for RT surfaces.

####  4.3.4 Same-side intervals

We consider now the case where the two endpoints of the boundary interval are
located on the same side. Let us begin with the case where the two points are
located on the true vacuum, i.e. the CFT with lower central charge. In this
case, it is easy to see that the RT surface will be the same as in the trivial
case with no boundary. From now on, we will call ”trivial” the geodesics that
do not cross the membrane.

From the constraints (4.28) it is clear that a trivial geodesic will always be
allowed, as the semicircle will never intersect the membrane. Let us consider
a putative non-trivial geodesic, that crosses into the false vacuum. This
curve will be necessarily longer than the trivial geodesic. Indeed, from
(4.31), the portion of the geodesic in the false vacuum will obtain a
multiplicative factor of $\ell_{2}/\ell_{1}>1$, with respect to the length it
would have had in the true vacuum. Adding this to the fact we are deviating
from the trivial geodesic path for this excursion into the other side, it is
clear that its length must be higher than the trivial one.

Although this argument alone does not exclude the existence of non-minimal
extremal curves of this type, we will confirm they do not exist through the
geometric construction that will follow. The entanglement structure of
intervals lying on the true vacuum is thus unchanged from the ”pure” case
given by (4.15).

Things get more interesting as we move the boundary segment to the false
vacuum. Both arguments given in the previous paragraphs fail here. For once,
$\psi_{2}$ can be negative, and so given big enough intervals the trivial
geodesic will intersect the membrane, such that the true geodesic must
necessarily be non-trivial. On the other hand, even when $\psi_{2}>0$, taking
some extra distance to cross onto the other side might be worth it. Indeed,
this can be seen as taking a ”shortcut” through the false vacuum, where
distances are shrunk by a factor of $\ell_{1}/\ell_{2}$.

The typical non-trivial geodesic is depicted in fig.4.6.

Figure 4.6: Example of a double crossing geodesic, for $\psi_{2}<0$. The full
geodesic is composed of three semicircles of centers $O_{1}$, $O_{2}$ and
$O_{3}$. The construction goes similarly as the single crossing case. We begin
with $O_{1}=(\lambda,0)$ and $\varphi$. By extending the line $O_{1}x_{1}$, we
intersect the green boundary at $O_{2}$. We find $x_{2}$ as the other
intersection with the semi-circle of center $O_{2}$, and repeat by extending
$x_{2}O_{2}$, constructing $O_{3}$. The semi-circle of center $O_{3}$
intersects the boundary at $\sigma_{2}$.

The Euclidean construction is again parametrized by the angle $\varphi$ and
center $O_{1}$. The full geodesic is composed of three circle arcs that are
tangent to each other on the membrane. Again, with repeated applications of
the sine law one can obtain expressions for the final and initial points (see
app. D.2 for details) :

$\displaystyle\sigma_{1}$
$\displaystyle=-\lambda\left(\frac{\cos(\psi_{2})}{\cos(\alpha)}-1\right)\ ,$
(4.36) $\displaystyle\sigma_{2}$
$\displaystyle=-\frac{\lambda\sin(\alpha+\psi_{2})}{\sin(\psi_{1}-\alpha)}\frac{\sin(\alpha+\psi_{1})}{\sin(\alpha-\psi_{2})}\left(1+\frac{\cos(\psi_{2})}{\cos(\alpha)}\right)\
,$ $\displaystyle\mu$
$\displaystyle=\frac{\sigma_{2}}{\sigma_{1}}=\frac{\sin(\alpha+\psi_{2})}{\sin(\psi_{1}-\alpha)}\frac{\sin(\alpha+\psi_{1})}{\sin(\alpha-\psi_{2})}\frac{\left(\cos(\alpha)+\cos(\psi_{2})\right)}{\left(\cos(\psi_{2})-\cos(\alpha)\right)}\
.$

While the formulas seem only slightly more complicated than (4.29), the
situation is quite a bit more involved in this case. Let us first look at the
allowed range for $\alpha=\varphi-\psi_{2}$. To find it, we enforce
$\sigma_{1}>0$ and $\sigma_{2}>0$, as well as that the intersection on the
membrane takes place at $z>0$. By symmetry we will later restrict further to
$\sigma_{2}>\sigma_{1}$. After tedious but straightforward computations, we
find :

$\displaystyle\lambda<0$
$\displaystyle\rightarrow|\psi_{2}|\leq\alpha\leq\psi_{1}\Rightarrow\frac{\sigma_{2}}{\sigma_{1}}>1\
,$ (4.37) $\displaystyle\lambda>0$
$\displaystyle\rightarrow\pi-\psi_{1}\leq\alpha\leq\pi-{\rm
Max}(\psi_{2},0)\Rightarrow\frac{\sigma_{2}}{\sigma_{1}}<1\ .$

From (4.36), $\alpha\rightarrow\pi-\alpha$ sends
$\frac{\sigma_{1}}{\sigma_{2}}\rightarrow\frac{\sigma_{2}}{\sigma_{1}}$. Thus,
it should suffice to consider only one of the two cases of (4.37). But as we
can see, the ranges of the two options for ${\rm sign}(\lambda)$ don’t seem to
coincide under this mapping. In fact, this is simply an artifact of our
Euclidean construction that introduced a dissymmetry between $\sigma_{1}$ and
$\sigma_{2}$, because we considered the intersections $x_{1}$ to lie in the
”positive” side of the half-line $O_{1}x_{1}$. By considering intersections on
the other half-line, the range ${\rm Max}(\psi_{2},0)<\alpha<0$ (with
$\lambda>0$) is indeed allowed, restoring the symmetry
$\sigma_{2}\leftrightarrow\sigma_{1}$.

For what follows we will usually consider ${\rm
Max}(\psi_{2},0)<\alpha<\psi_{1}$ and thus $\mu>1$, since it is the situation
depicted in fig.4.6. Unlike in the previous section, the function
$\mu(\alpha)$ is not monotonic. Indeed, it has a unique minimum at
$\alpha_{\rm crit}$ :

$\displaystyle\sin(\alpha_{\rm
crit})=\frac{1}{4}\left(-\frac{\sin(2\psi_{1})}{\cos(\psi_{2})}+\sqrt{\frac{\sin^{2}(2\psi_{1})}{\cos^{2}(\psi_{2})}+8\left(1-\frac{\cos(2\psi_{1}+\psi_{2})}{\cos(\psi_{2})}\right)}\right)\
,$ (4.38)

which we verify to always lie in the allowed $\alpha$ range. This implies that
at least in some range, there are two distinct non-trivial geodesics.

Looking at the boundary limits, we have to distinguish the two cases
$\psi_{2}<0$ and $\psi_{2}>0$ :

$\displaystyle\lim_{\alpha\rightarrow\psi_{1}}\mu(\alpha)$
$\displaystyle=\infty\ ,$ (4.39) $\displaystyle\lim_{\alpha\rightarrow{\rm
Max}(\psi_{2},0)}\mu(\alpha)$
$\displaystyle=\begin{cases}\infty\quad&\text{if}\,\psi_{2}\geq 0\\\
\frac{1-\cos(\psi_{2})}{1+\cos(\psi_{2})}\quad&\text{if}\,\psi_{2}<0\end{cases}\
.$

In fig. 4.7 we plot the curve $\mu(\alpha)$ for both signs of $\psi_{2}$.

(a) Example plot of $\mu(\alpha)$ in the case $\psi_{2}>0$.

(b) Example plot of $\mu(\alpha)$ in the case $\psi_{2}<0$. The green line
denotes the value of $\mu$ above which the trivial solution ceases to exist.

Figure 4.7: Plots of $\mu(\alpha)$ for both signs of $\psi_{2}$. The red line
indicates the value of $\mu$ under which no non-trivial solutions exist,
attained at $\alpha_{crit}$. For $\psi_{2}>0$, above $\mu_{min}$ two distinct
non-trivial solutions always exist, while it is not the case when
$\psi_{2}<0$. The geodesics corresponding to values of $\alpha<\alpha_{crit}$
are never the shortest.

An interesting remark arising from this analysis is yet again the appearance
of the mysterious ”critical tension” $\lambda_{0}$ (see 2.4), which is the
tension corresponding to $\psi_{2}=0$. For $\psi_{2}>0$ or overcritical
tensions, there are always two candidate non-trivial geodesics, while for
$\psi_{2}<0$ or subcritical tensions, this is only the case for small enough
$\mu$. In this case, this will have no bearing on the RT surface as we will
see that the geodesics for $\alpha<\alpha_{crit}$ are never dominant. However,
in the interpretation of [183], where the geodesics compute two-point
functions, the subdominant ones do contribute to corrections, so in this case,
the crossing of the critical tension does produce a measurable field theory
effect. In spite of all these hints, a deeper understanding of the meaning of
this critical tension is still lacking.

A last remark concerns the disappearance of trivial geodesics. It is easy to
see that when $\psi_{2}<0$ and
$\mu>\frac{1-\cos(\psi_{2})}{1+\cos(\psi_{2})}=\mu(\alpha=0)\equiv\mu_{min}$,
they cross the membrane and thus cease to exist. As we approach this limit
from below, the crossing geodesics with $\alpha<\alpha_{crit}$ approaches the
trivial one, matching at $\mu=\mu_{min}$. Perhaps there is a deeper
explanation for the simultaneous disappearance of these two geodesic paths
from the field theory POV, but we have not been able to elucidate it.

As in the previous case, we can obtain the geodesic length analytically (see
D.2 for more details) :

$\displaystyle
L=\ell_{2}\ln(\frac{\sigma_{1}}{\epsilon_{1}})+\ell_{2}\ln(\frac{\sigma_{2}}{\epsilon_{2}})+\underbrace{\ell_{2}\ln(\frac{4\tan(\frac{\alpha+\psi_{2}}{2})}{\tan(\frac{\alpha-\psi_{2}}{2})\left(1-\frac{\cos^{2}(\alpha)}{\cos^{2}(\psi_{2})}\right)})+\ell_{1}\ln(\frac{\tan(\frac{\alpha+\psi_{1}}{2})}{\tan(\frac{\psi_{1}-\alpha}{2})})}_{g(\xi)}\
.$ (4.40)

As expected, the length is invariant under $\alpha\rightarrow\pi-\alpha$ which
is the inversion $\sigma_{2}\leftrightarrow\sigma_{1}$, and we have arranged
it in the form (4.33). This length is to be compared to the trivial one which
can be written as:

$\displaystyle L_{\rm triv}$
$\displaystyle=\ell_{2}\ln(\frac{4R^{2}}{\epsilon_{1}\epsilon_{2}})\quad
R=\frac{\sigma_{2}-\sigma_{1}}{2}\ .$ (4.41)

To compare the two lengths we should use (4.36) to express $L_{\rm triv}$ in
terms of $\alpha$. Let us call $\Delta L(\alpha)=L(\alpha)-L_{\rm
triv}(\alpha)$ the difference between trivial and non-trivial geodesic.

Unfortunately, the expression is much too complicated to hope to have a closed
expression for $\alpha_{\rm trans}$, the value where dominating geodesic
switches. However, we can again inspect the different limits to distill some
properties. For instance, we can show that no matter what, when
$\ell_{1}<\ell_{2}$, there will always be non-trivial geodesics dominating for
large enough intervals. Indeed, in the limit $\alpha\rightarrow\psi_{1}$,
where $\sigma_{2}/\sigma_{1}\rightarrow\infty$ :

$\displaystyle L-L_{\rm triv}$
$\displaystyle=(\ell_{2}-\ell_{1})\ln(\psi_{1}-\alpha)+\mathcal{O}(1)\ .$
(4.42)

Thus in this limit, the non-trivial geodesic with $\alpha>\alpha_{\rm crit}$
always dominates. In doing this analysis, we expected to find a ”critical
angle” for the brane, over which the non-trivial geodesic stopped existing, as
found in [209], in the context of Boundary CFT. However, as they mention in
the paper, this critical angle appears only in larger than three dimensions
for the bulk, so the absence of a critical angle in our model is consistent
with their BCFT findings (the BCFT can be obtained in the limit
$\ell_{1}\rightarrow 0$, where CFT1 disappears and the interface becomes a
boundary). It would be interesting to extend the results of [209] in higher
dimensions to the more general case of Interface CFTs.

Similarly, we can inspect the lower limit $\alpha\rightarrow{\rm
Max}(\psi_{2},0)$. The interesting case is when $\psi_{2}>0$, otherwise the
non-trivial and trivial geodesics coincide as $\alpha\rightarrow 0$. For
$\psi_{2}>0$ :

$\displaystyle L-L_{\rm triv}(\alpha=\psi_{2})$
$\displaystyle=\ell_{2}\ln(\frac{\sin(\psi_{1}-\psi_{2})}{\cos^{2}(\psi_{2})\sin(\psi_{1}+\psi_{2})})+\ell_{1}\ln(\frac{\tan(\frac{\psi_{1}+\psi_{2}}{2})}{\tan(\frac{\psi_{1}-\psi_{2}}{2})})\
,$ (4.43)

which is always positive, and therefore in this limit the trivial geodesic is
always preferred.

By plotting the curve and playing with the parameters, we can make further
statements. The typical curve $L-L_{\rm triv}$ is depicted in fig.4.8.

Figure 4.8: Plot of $\Delta L=L-L_{\rm triv}$ as a function of $\alpha$, in
the case $\psi_{2}<0$. The local maximum is located at $\alpha_{crit}$. The
orange line denotes the value of $\alpha$ above which the trivial geodesics
cease existing (in particular, values of $\Delta L$ after this point are
meaningless). As we can see, this happens after the non-trivial geodesic
becomes dominant. We verified numerically that this is always the case; namely
the geodesic switch always happens because the non-trivial geodesic becomes
shorter, not because the trivial one ceases existing.

As we can see, and as we verified numerically by swiping over the parameter
range for $\psi_{i}$, it has a single maximum. Incidentally, we verified that
this maximum is attained precisely for $\alpha_{\rm crit}$ (4.38), which is
when $\mu(\alpha)$ is minimal. There is probably a way to prove this
analytically, but we haven’t been able to do so yet. It can however be
understood intuitively, because, at $\alpha=\alpha_{crit}$, the non-trivial
geodesic is tangent at the membrane which is the worst-case scenario since we
don’t benefit of the shortcut through side 1. As we grow $\mu$, either by
increasing or decreasing $\alpha$, the geodesic can pass deeper into the true
vacuum, thus shortening its path. Combining this with the limit (4.43), this
shows that the non-trivial geodesics which lie at $\alpha<\alpha_{\rm crit}$
are always subleading when they exist.

To complete our analysis we would like to invert (4.36) to find $\alpha(\mu)$.
This appears to be a straightforward exercise much like in the previous
section, and it is in theory, but in practice, we have encountered hurdles
that have prevented us from obtaining a closed analytical expression in this
case. Indeed, consider the equation that is to be solved :

$\displaystyle\mu=\frac{c_{\alpha}+c_{2}}{c_{2}-c_{\alpha}}\times\frac{\left(\sqrt{1-c_{\alpha}^{2}}c_{2}+s_{2}c_{\alpha}\right)\left(\sqrt{1-c_{\alpha}^{2}}c_{1}+s_{1}c_{\alpha}\right)}{\left(s_{1}c_{\alpha}-\sqrt{1-c_{\alpha}^{2}}c_{1}\right)\left(\sqrt{1-c_{\alpha}^{2}}c_{2}-s_{2}c_{\alpha}\right)}\
.$ (4.44)

This can be solved analytically by squaring away the square roots which yields
a 4th order polynomial in $c_{\alpha}$555Strictly speaking we obtain a 6th
order polynomial, but there are 2 roots that are evident, and aren’t valid
solutions to (4.44).. We do not include the full expressions for the roots as
they are two cumbersome, but they take the form :

$\displaystyle c_{\alpha}$ $\displaystyle=A+\sqrt{q}\pm\sqrt{p_{2}-p_{1}}\ ,$
(4.45) $\displaystyle c_{\alpha}$
$\displaystyle=A-\sqrt{q}\pm\sqrt{p_{2}+p_{1}}\ ,$

where $A,\,q$ and $p_{i}$ depend on $\psi_{i}$ and $\mu$. The complicated step
lies in choosing which of the four candidate solutions are actually solutions
of (4.44). This is done by replacing (4.45) into (4.44) and verifying it is a
solution, as well as checking that the $c_{\alpha}$ lies in the range (4.37).
We have attempted this with the aid of Mathematica, but the resulting
expressions were humongous, and their simplification relied on several
assumptions on the range of parameters, which made it computationally
impossible to simplify analytically.

Thus, we turn to numerical verifications. This allows us to find the correct
solutions among (4.44), but only for a given value of $\mu$, $\psi_{i}$. One
could hope that the correct solutions do not change as we move the parameters,
but this is not the case. See fig.4.9 for an example

Figure 4.9: The four candidate solutions of (4.45) are plotted in different
colors. One can verify that the true solutions of (4.44) are the ones
constituting the paraboloïd shape in the picture. We see on the plot both a
discontinuous jump in the solutions, as well as a kink. By mixing and matching
the different candidates, we obtain a smooth curve.

Looking at fig. 4.9, we need to do some ”cut and patch” of different solutions
to actually obtain the inverse of the curve displayed in fig.4.7. This can be
understood by looking at the form (4.45). Say that $q$ has a double root for
some $\mu=\mu^{*}$, at fixed $\psi_{i}$. Then locally around that point,
$\sqrt{q}\approx\sqrt{q^{\prime\prime}(\mu^{*})(\mu-\mu^{*})^{2}}=\sqrt{q^{\prime\prime}(\mu^{*})}|\mu-\mu^{*}|$.
Because of the absolute value, the solutions (4.45) will have a kink at
$\mu^{*}$. Then, to obtain smooth solutions we must combine the $\pm\sqrt{q}$
solutions for $\mu<\mu^{*}$, with the $\mp\sqrt{q}$ solutions for
$\mu>\mu^{*}$. Another situation is of the type $f(\mu)/\sqrt{q(\mu)}$, where
$f(\mu)$ has a simple root at $\mu^{*}$, while $q(\mu)$ has a double root. In
that case, this will induce a discontinuity of
$2f^{\prime}(\mu^{*})/\sqrt{q^{\prime\prime}(\mu*)}$ in the function. Both
cases are displayed in fig.4.9.

As it turns out, we found that those ”inversions” of sign can happen for
$\sqrt{q}$, $\sqrt{p_{2}-p_{1}}$ and $p_{1}$ as well as it contains a term
$\propto 1/\sqrt{q}$. To be able to obtain an inverse of (4.36) for any
$\psi_{i}$, one would need to analytically determine the roots of the
aforementioned quantities. Unfortunately, this turns out to be impossible as
the equations involved are much too complicated, involving high powers and
various trigonometric functions.

Thus, in the double-crossing case a closed form like (4.35) is lacking. Of
course, (4.45) combined with a quick numerical check to select the correct
solutions still allow us to compute entanglement entropies relatively quickly,
as locally we have even an analytic expression.

###  4.4 Application to more general geometries

The extended analysis of the previous sections was in the hopes of being able
to bootstrap these computations to compute the entanglement structure of the
solution chap.3 and possibly also for the solutions of chap.2. Unfortunately,
for many cases we will see that this method is not applicable.

####  4.4.1 The thermal ICFT state

Let us begin analysing the non-compact thermal static solution with an
interface, which is the limit $J\rightarrow 0$ of the NESS state in chap.3. We
remind the bulk metric :

$\displaystyle
ds^{2}=-(r_{j}^{2}-M\ell_{j}^{2})dt^{2}+\frac{\ell_{j}^{2}dr_{j}^{2}}{r_{j}^{2}-M\ell_{j}^{2}}+r_{j}^{2}dx^{2}\
.$ (4.46)

The solution is given by (3.57) for $J=0$, $M_{1}=M_{2}$. One can verify that
in this case, the wall’s induced metric is AdS2, and we can integrate (3.57)
to obtain the explicit form for the wall :

$\displaystyle x_{j}(r)=-\frac{{\rm
sign}(\lambda^{2}\pm\lambda_{0}^{2})}{\sqrt{M}}{\rm
arcoth}\left(\frac{\sqrt{b_{\pm}^{2}\ell_{j}^{2}+r^{2}}}{r}\right)\ ,$ (4.47)

where the $\pm$ is $+$($-$) for $j=1$($j=2$), while
$b_{\pm}^{2}=\frac{(\lambda^{2}-\lambda_{\rm min}^{2})(\lambda_{\rm
max}^{2}-\lambda^{2})}{M(\lambda^{2}\pm\lambda_{0}^{2})^{2}}$. The gluing is
made by identifying $r_{j}^{2}-M\ell_{j}^{2}$ along the membrane.

The goal is now to find a coordinate transformation that brings this solution
to the vacuum one above. The additional difficulty to what was done in sec.4.2
is that now we have the membrane shape to keep track of. Since the membrane
breaks some of the isometries of Poincaré, it will not be sufficient to find
the coordinate change mapping (4.47) to (4.1), but the shape of the membrane
will also be important. The procedure we employ to identify the correct
coordinate change is as follows.

First, we find an arbitrary coordinate change which brings us to the correct
bulk metric, namely Poincaré in this case. After that, we restrict ourselves
only to isometries of Poincaré space. Generally, some of these isometries will
act non-trivially on the membrane, but they will leave the bulk metric
unchanged. Using those, we attempt to bring the membrane to the form (4.27).
For this step, one can use the dual field theory to greatly facilitate the
problem. Indeed, according to sec.1.7.2, given the boundary state and the
interface shape, the bulk dual is completely fixed. Thus, instead of trying to
match the 2-dimensional membranes in the bulk, it is sufficient to match the
1-dimensional interfaces in the dual theory, by using only global conformal
transformations.

Following this procedure, we begin by with the following change of coordinates
:

$\displaystyle x_{p}^{\prime}\pm t_{p}^{\prime}$
$\displaystyle=\frac{\sqrt{r^{2}-M\ell^{2}}}{r}e^{\sqrt{M}(x\pm t)}\ ,$ (4.48)
$\displaystyle z_{p}^{\prime}$
$\displaystyle=\frac{\sqrt{M\ell^{2}}}{r}e^{x\sqrt{M}}\ ,$

which brings (4.46) to the Poincaré metric. By taking the limit
$r\rightarrow\infty$, we see that on the boundary the interface is mapped from
$x=0$ to :

$\displaystyle-t_{p}^{\prime}{}^{2}+x_{p}^{\prime}{}^{2}=1\ .$ (4.49)

This initially seems strange as (4.49) describes two disconnected interfaces.
However, from the coordinate change (4.48) we reach only the upper right
quadrant of the plane
$(x_{p}^{\prime}-t_{p}^{\prime},x_{p}^{\prime}+t_{p}^{\prime})$, which ensures
that we reach only one of the two disconnected pieces. The other 3 quadrants
would be reached by considering the maximally extended black hole string
(which, as it contains another boundary, it also contains another membrane),
whereas our change of coordinates covers only the exterior of the solution
(4.46). In any case, that is all we need to compute RT surfaces, since for the
static situation they will not enter the horizon.

What remains to be done is to find a global conformal transformation that maps
(4.49) to the $x=0$ interface. This case is fairly simple, as (4.49) is a
hyperbola, which becomes a circle if we wick rotate to Euclidean coordinates.
As is well known, any circle can be conformally mapped to a line by a special
conformal transformation combined with a translation. In this case, the
combination of the two transformations read :

$\displaystyle x_{p}$ $\displaystyle=\frac{2(x^{\prime 2}-t^{\prime
2}-1)}{1+x^{\prime 2}-t^{\prime 2}+2x^{\prime}}\ ,$ (4.50) $\displaystyle
t_{p}$ $\displaystyle=\frac{4t^{\prime}}{1+x^{\prime 2}-t^{\prime
2}+2x^{\prime}}\ ,$

which indeed maps (4.49) to the interface $x=0$. Notice that from the allowed
quadrant mentioned before, we reach only the portion $-2<t_{p}<2$ of the
interface. At the risk of repeating ourselves, the rest of the geometry
corresponds to portions of the maximally extended black string solutions,
which are of no interest to us when computing entanglement entropies of
intervals lying on a single asymptotic boundary.

From (4.50) we can compute the associated coordinate transformation in the
bulk (see sec.1.2.1), which is simply a special conformal transformation of
the three coordinates $t,x,z$ and it is an isometry of Poincaré since the
$\frac{1}{z^{2}}$ prefactor will cancel the overall scale factor :

$\displaystyle x_{p}$
$\displaystyle=\frac{2(x^{\prime}_{p}{}^{2}-t^{\prime}_{p}{}^{2}+z^{\prime}_{p}{}^{2}-1)}{1+x^{\prime}_{p}{}^{2}-t^{\prime}_{p}{}^{2}+z^{\prime}_{p}{}^{2}+2x^{\prime}_{p})}\
,$ (4.51) $\displaystyle t_{p}$
$\displaystyle=\frac{4t^{\prime}_{p}}{1+x^{\prime}_{p}{}^{2}-t^{\prime}_{p}{}^{2}+z^{\prime}_{p}{}^{2}+2x^{\prime}_{p}}\
,$ $\displaystyle z_{p}$
$\displaystyle=\frac{4z^{\prime}_{p}}{1+x^{\prime}_{p}{}^{2}-t^{\prime}_{p}{}^{2}+z^{\prime}_{p}{}^{2}+2x^{\prime}_{p}}\
.$

Combining everything, we have the coordinate change that brings the static
ICFT black string solution to the static ICFT vacuum solution :

$\displaystyle z_{p}$
$\displaystyle=\frac{2\sqrt{M\ell^{2}}}{\cosh(\sqrt{M}t)\sqrt{r^{2}-M\ell^{2}}+r\cosh(x\sqrt{M})}\
,$ (4.52) $\displaystyle t_{p}$
$\displaystyle=\frac{2\sqrt{r^{2}-M\ell^{2}}\sinh(\sqrt{M}t)}{\cosh(\sqrt{M}t)\sqrt{r^{2}-M\ell^{2}}+r\cosh(x\sqrt{M})}\
,$ $\displaystyle x_{p}$
$\displaystyle=\frac{2r\sinh(\sqrt{M}x)}{\cosh(\sqrt{M}t)\sqrt{r^{2}-M\ell^{2}}+r\cosh(x\sqrt{M})}\
.$

We can then leverage the equations of the previous section to compute, in
principle analytically, the entanglement structure of the theory in the
thermal equilibrium state. Analyzing the resulting formulas for the
entanglement entropy is unfeasible because of their complexity, so we must
content ourselves with plots.

Consider boundary intervals at constant time, $t=0$, and $x_{i}>0$ in the
folded picture. Notice that under (4.52), these are mapped to constant time
intervals with $0<x_{p}^{i}=\sigma_{i}<2$. However, since the difference
$L-L_{triv}$ depends only on the ratio $\sigma_{2}/\sigma_{1}$, we do not
expect a qualitative change in behavior for the transitions between trivial
and non-trivial geodesics, in the case where the two points lie on the same
side. Since this transition generally takes place for
$\sigma_{2}/\sigma_{1}\gg 1$ (or $\sigma_{2}/\sigma_{1}\ll 1$), and
$\sigma_{2}\approx 2$ if $x_{2}\gg 1$, we estimate that the switching to the
double-crossing geodesic happens for $x_{1}<<\frac{2}{\sqrt{M}}$.

Indeed, this can be understood intuitively because the horizon of the BH puts
a cap to how deep can the geodesic go. Thus, crossing to the true vacuum side
is not as enticing as in the vacuum solution, since the geodesic will have to
curve back to the false vacuum side as soon as it reaches the horizon.
Therefore for this excursion to be worthwhile, the geodesic should start ever
closer to the interface as $M$ increases (and the horizon approaches the
boundary), which is confirmed by our estimation $x_{1}\ll\frac{2}{\sqrt{M}}$.

Note that one difference is that in the BTZ case, the inversion symmetry
$x_{i}\rightarrow 1/x_{i}$ no longer holds, as the cross-ratio is expressed
differently in this coordinate system. In particular, non-trivial geodesics
will exist only if one of the points satisfies $x_{i}\ll\frac{1}{\sqrt{M}}$.
This is in contrast to the vacuum case, where we have non-trivial geodesics as
soon as $\sigma_{1}/\sigma_{2}\gg 1$ (or $\ll 1$), regardless of the value of
the individual $\sigma_{i}$.

Beyond that, there is not much to say about the entanglement structure in the
black string case, save for linear growth of the entropy as the interval grows
larger, which is the tell-tale of a thermal state. We plot two curves
depicting the entanglement entropy for intervals containing the interface, and
intervals contained in the false vacuum in fig.4.10.

(a) Entanglement entropy plot, zoomed in for small values of $x_{1}$, to see
the transition between the trivial and non-trivial RT surface. At
$1/\mu_{min}$, the non-trivial geodesic ceases existing

(b) Entanglement entropy plot, zoomed out to see the linear increase in
entropy which is characteristic of thermal states. Notice that the slope is
different for the positive and negative values; this is expected as the
horizons are located at ${\scriptscriptstyle r^{i}=\sqrt{M\ell_{i}^{2}}}$,
thus yielding different proportionality constants. We obtain a singularity at
$x_{1}/x_{2}=1$, where the points coincide. The geodesic length tends to 0,
but as we subtracted the cutoff value, it becomes singular.

Figure 4.10: Two plots of the entanglement entropy in the BTZ solution. We fix
$x_{2}=1$ in side 2, and vary $x_{1}$ (negative values mean $x_{1}$ is on side
1). To be precise, we plot the entanglement entropy regularised by subtracting
the divergent parts $\ell_{i}\ln(\frac{1}{\epsilon})$, where $\epsilon$ is the
IR cutoff. This explains why we obtain negative values. The plots are made for
$\psi_{2}<0$, but they are qualitatively the same for $\psi_{2}>0$.

The study of RT surfaces in the thermal state of the ICFT has very interesting
applications in itself, especially in Page-curve computations in eternal black
hole setups [183, 123]. However from the point of view of the entanglement
structure, we did not expect any surprises, and the difference between vacuum
and thermal ICFT is similar to the case without interface.

####  4.4.2 The NESS ICFT state

The computation of the previous section was done as a preparation for the real
goal, which is computing HRT surfaces in the NESS state of chap.3. Indeed, in
this state we hope to find exotic behavior for the HRT surfaces, since the
apparent horizons do not match at the membrane. While spacelike geodesics with
two anchor points on the boundary cannot cross these, it would be conceivable
for an HRT surface to cross the membrane outside one horizon, and emerge
inside the other, see fig.4.11. Indeed, while in the spinning string, a
geodesic with two anchor points on the boundary cannot cross the apparent
horizon, there are geodesics with a single anchor point on the boundary that
do (see app.D.3).

Figure 4.11: A depiction of an HRT surface (in blue) dipping inside the
apparent horizon of the hotter side. The blackened region is the inside of the
event horizon, while the grey one sits inside the apparent horizon. While
there do not exist spacelike geodesics which cross the apparent horizon and
have two anchors at the boundary, single-anchored geodesic can cross the
apparent horizon. By clipping such a geodesic at the membrane, and crossing
outside the apparent horizon on side 2, we could obtain the depicted HRT
surface. Here, we depicted the geodesic as avoiding the event horizon, but we
do not know if this is always true.

In fact, such an occurence seems almost inevitable as we consider a growing
boundary interval containing the interface. As shown in fig. 4.2, spacelike
geodesics tend to approach the (apparent) horizon, and thus as they hit the
membrane from the colder side, we would expect them to emerge behind the
apparent horizon of the hotter side.

The other compelling reason to look at the entanglement structure of this
state is to definitively prove that the interface indeed acts as a perfect
scrambler, and to understand the entropy production from the gravity
perspective. It is not immediately clear if such a thing can be computed
within the stationary NESS state. Indeed, to compute the entropy production at
the interface, one would naturally consider the entanglement entropy for an
interval containing it. In the state preparation depicted in fig.3.3, the
shockwave due to the quench will alter the dual geometry, and the RT surface
will be modified, resulting in a time-varying entanglement entropy for the
interval. Then, taking the derivative of this quantity would yield the entropy
production at the interface.

However, from the geometries computed in chap.3, we have access only to the
gray area of fig.3.3, i.e. only the stationary state. Then, the entropy of an
interval containing the interface will naturally be constant simply because
the HRT surface computing its value will not have time-dependence. This is
unavoidable by the very fact we are considering a stationary state. Still, in
the microscopic model, even in the stationary state, the knowledge of the
”S-matrix” of the interface should allow us to compute the entropy production.
The question of whether this information is accessible from the minimal model
under consideration is still open.

One possible direction would be to consider the entropy (left and right-
moving) currents which will naturally be constant in time. While the incoming
currents are certainly thermal(they are prepared to be such) the outgoing
currents from the interface can have a non-trivial _spatial_ profile, which
tells us how different wave modes get scrambled after scattering[183, 184,
181]. Therefore, looking at the behavior of the intervals entropy under a
spatial variation of the endpoints might be a way to probe the entropy
production at the interface. Note that at infinity these currents become also
thermal, as we have verified by applying the HRT prescription for intervals
far removed from the interface, and confirming the trivial surface does not
intersect the membrane (thus recovering 4.19, which shows that the dual state
in the considered region is indeed thermal).

At any rate, all of that rests on our ability to compute said HRT surfaces. We
will make a first attempt by mirroring the methods of the previous subsection.
We will be considering the metric in Eddington-Finkelstein coordinates, as we
expect the interiors of the black strings to play an important role here,
unlike the static case. We remind the form of the relevant metric :

$\displaystyle
ds^{2}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}dv^{2}+r^{2}(dy-\frac{{\rm
sign}(J)r_{-}r_{+}}{r^{2}}dv)^{2}+2\ell dvdr\ ,$ (4.53)

where we expressed it in terms of the location of the horizons $r_{\pm}$, see
(3.2) for their definitions in terms of $M$, $J$.

Composing (4.16) with (3.4) we can find a change of coordinates mapping to
Poincaré :

$\displaystyle x_{p}+t_{p}=w_{+}$
$\displaystyle=\frac{r+r_{+}}{r+r_{-}}\exp(\frac{(r_{+}-r_{-})(sy+v)}{\ell})\
,$ (4.54) $\displaystyle x_{p}-t_{p}=w_{-}$
$\displaystyle=\frac{r-r_{+}}{r+r_{-}}\exp(\frac{(r_{+}+r_{-})(sy-v)}{\ell})\
,$ $\displaystyle z$
$\displaystyle=\frac{\sqrt{r_{+}{}^{2}-r_{-}{}^{2}}}{r+r_{-}}\exp(\frac{syr_{+}-vr_{-}}{\ell})\
,$

where $s={\rm sign}(J)$, which we will take to be positive from now on. In
(4.54), we chose the change of coordinates such that the geometry is mapped to
the quadrants $w_{+}>0$, where the exterior is $w_{-}>0$, and the interior
$w_{-}<0$.

Like before, let us look what is the shape of the interface in these
coordinates. By plugging $y=0$, $r=\infty$ we find that the interface in the
new coordinates satisfies the equation :

$\displaystyle w_{+}^{r_{+}+r_{-}}w_{-}^{r_{+}-r_{-}}=1\ .$ (4.55)

Of course we recover (4.49) when $J=0$ and $r_{-}=0$.

The next step in the procedure would be to find a global conformal
transformation that maps (4.55) to the interface $x=0$. Unfortunately, we are
out of luck here; it can be shown that no such transformation exists. In a
way, this outcome should have been expected from the get-go. Unlike the ”pure”
case with no membrane, here the non-equilibrium NESS state is not an
equilibrium state in disguise. In hindsight, it would have been stranger to be
able to map it to the static vacuum case, as that would imply that the NESS
state we had found was merely an equilibrium situation expressed in peculiar
coordinates.

That is however a serious setback in our hopes of computing RT surfaces in
this setup. Indeed, without access to the beautiful Euclidean geometry
description of sec.4.3.2, the problem at hand seems much harder to tackle even
numerically, let alone analytically. In the next section, we outline some
attempts at semi-numerical algorithms, which are still a work in progress.

####  4.4.3 Other static geometries

Let us finally briefly mention the possibility to map the vacuum ICFT case to
the static geometries of chap.2. Here again, the question is disappointingly
quickly answered by looking at the induced metric on the membrane. Indeed, as
remarked before, the membrane metric in the vacuum case of sec.4.3.2 is AdS2;
in particular, its Ricci scalar is a constant. For the solutions of chap.2
(eq.(2.37)), this is not the case except when $M_{1}=M_{2}$. So the analytical
results of the vacuum ICFT case are applicable only to this very narrow subset
of solutions.

For the $M<0$ case, the relevant mapping linking Poincaré (1.11) to global
coordinates (2.1) :

$\displaystyle
z_{p}=\frac{\ell^{2}\sqrt{-M}}{\sqrt{r^{2}-M\ell^{2}}\cos(t\sqrt{-M})+r\cos(x\sqrt{-M})}\
,$ (4.56) $\displaystyle
t_{p}=\frac{\ell\sqrt{r^{2}-M\ell^{2}}\sin(\sqrt{-M}t)}{\sqrt{r^{2}-M\ell^{2}}\cos(\sqrt{-M}t)+r\cos(\sqrt{-M}x)}\
,$ $\displaystyle x_{p}=\frac{\ell
r\sin(\sqrt{-M}x)}{\sqrt{r^{2}-M\ell^{2}}\cos(\sqrt{-M}t)+r\cos(\sqrt{-M}x)}\
.$

Note that we must constrain $z_{p}>0$, so that only a portion of the global
space is mapped by (LABEL:globaltopoincmap). This change of coordinates is
specifically adapted for the interface located at $x=0$ (the other interface
is located at $x=\frac{\pi}{2\sqrt{-M}}$, as can be seen by integrating the
solutions (2.37) in the case $M_{1}=M_{2}=M<0$). This also explains the
”disappearance” of one of the two interfaces under (LABEL:globaltopoincmap).
Consider the Cauchy-Slice $t=0$; under (LABEL:globaltopoincmap), the full
slice is mapped to Poincaré coordinates except for a single point, which is
$x=\frac{\pi}{2\sqrt{-M}}$, the position of the second interface. The same
transformation (LABEL:globaltopoincmap), but with translated $x$, would have
mapped the two interfaces although they would not have the simple shape
$x=cst$, which is needed if we want to apply the results of sec.4.3.2.

Since we can map the full Cauchy slice $t=0$ (except one point), the findings
of sec.4.3.2 are almost directly applicable to the global solutions. Of
particular interest are remarks made about the connections of various RT
surfaces with the critical tension $\lambda_{0}$ (see the paragraph after
fig.4.7). Indeed, in the case where $M_{1}=M_{2}<0$, the sweeping transition
happens precisely at $\lambda=\lambda_{0}$ (which does not hold when
$M_{1}\neq M_{2}$). With speculation, this can lead us to two possible
conjectures concerning the sweeping transition. In the following, ”candidate
RT surface” is an allowable spacelike geodesic in the RT prescription, which
is not necessarily the minimal one.

  * •

Centerless slices admit no trivial candidate RT surfaces for sufficiently big
boundary intervals (while centerful always do)

  * •

Centerless slices admit only one crossing candidate RT surface for
sufficiently big boundary intervals (while centerful admits two)

In the simpler case $M_{1}=M_{2}$, both conditions are satisfied.
Unfortunately, the first conjecture can be immediately falsified. Indeed,
according to the conditions (2.64), we can have centerful solutions with
tension $\lambda<\lambda_{0}$, given $\ell_{1}<\sqrt{2}\ell_{2}$. For these
solutions, the near-boundary membrane has $\psi_{2}<0$. So for big enough
boundary intervals, the trivial geodesic will intersect the membrane and cease
existing, invalidating the first conjecture. One could amend the conjecture to
link the disappearance of the trivial geodesic to the critical tension
$\lambda_{0}$, instead of the sweeping transition, but it is also unclear if
this statement holds for $M_{1}\neq M_{2}$.

The second conjecture cannot be dismissed as easily, and one must again turn
to numerical methods. This is a work in progress, as we are still in search of
a satisfying all-encompassing algorithm as explained in the following section.

###  4.5 Numerical attempts

We describe briefly in this section some attempts at obtaining HRT surfaces in
these more complicated geometries with the use of numerical methods. The
algorithms we will present were developed mainly for the use in the stationary
NESS geometry, but with the scope of applying them also to the static
geometries of chap.2. Ironically, the simplicity of the thin-brane model is
actually the main difficulty in implementing numerical methods. Indeed, for a
smooth metric standard shooting algorithms would presumably work to find the
sought-out HRT surfaces.

Let us restate the problem to set notation. The spacetime in consideration is
composed of two halves of locally AdS3 spaces (labeled $1$ and $2$), which are
joined through a codimension two membrane, whose embedding equation in each
side $x^{\mu}_{m,i}(\xi^{a})$ is known. Given any two spacelike separated
points $p_{1}$ and $p_{2}$, located on an IR cutoff surface that is
arbitrarily close to the boundary, we seek to find all possible spacelike
geodesics connecting them. We have identified two different possible
directions for the algorithms, which have both their pro and cons.

####  4.5.1 Semi-Numerical algorithms

In the case of a homogeneous spacetime, we have shown in sec.4.1 that we are
able to find analytically the geodesics connecting two given points in the
bulk. We can use that to greatly reduce the amount of numerics needed, hence
the name ”semi-numerical”. When we need to connect two points in the same
slice, we can consider that we have the geodesic path analytically (with some
caveats we will mention), as long as we have an analytic change of coordinates
mapping our metric to Poincaré space.

In what follows, we consider only possible non-trivial geodesics, i.e.
geodesics that cross the interface.

##### The minimization method

An allowable non-trivial geodesic will cross the membrane in such a way that
the crossing constraints (4.23) are satisfied. Consider first the case where
the $p_{i}$ lie on different sides of the interface. In this algorithm, we
begin by picking an arbitrary point on the interface parametrised by
$\xi^{a}_{*}$, such that the corresponding points on both sides
($x^{\mu}_{m,i}(\xi^{a}_{*})$) are spacelike separated from the boundary
points $p_{i}$. Given that, we obtain a curve composed of two geodesic pieces
joined at the interface. This path will in general not satisfy the crossing
constraints, and that is where the numerics kick in.

The crossing equations (4.23) will determine the correct $\xi^{a}_{*}$.
Although those equations can be expressed completely analytically in our
setups, it is in general not possible to solve them so. Thus, we use standard
numerical methods, such as Newton-Raphson algorithm which needs the equation
to be differentiable, or secant methods which do not, both of them being
natively implemented in Mathematica through the ”FindRoot” function.

In the case where $p_{1}$ and $p_{2}$ are on the same side, the algorithm is
similar. The difference is that now our problem is parametrised by two
arbitrary points on the membrane, and we obtain two sets of crossing
equations. In this case, we need to solve twice as many algebraic equations,
with twice as many free parameters. Note that we have additional conditions
that force all chosen points to be spacelike separated from one another.

On paper, this method looks great; after all, it requires numerics only in the
resolution of a set of algebraic equations. However, it has several important
complications in practice. The first big problem is the ”spacelike separated”
condition. In poincaré coordinates, this condition is trivially verified by
checking $-\Delta t^{2}+\Delta x^{2}+\Delta z^{2}>0$. In other coordinates,
the conditions are usually non-trivial, and one has to employ the coordinate
change to Poincaré to verify them. This is an important issue because it is
difficult to implement these constraints into the numerical algorithms that
solve the crossing equations.

Generically what tends to happen is that the algorithm attempts to evaluate
the equations at a point $\xi^{a}_{*}$ which does not satisfy the constraints,
crashing the search as we transition to timelike geodesics. One can mend this
problem somewhat by ”sprinkling” the interface with several initial points
$\xi^{a}_{*}$, in the hopes of one falling close enough to the solution such
that the numerical solver converges. This is of course not ideal, first of all
because it increases significantly the computational cost, but mainly because
one can never be certain if a solution was not found due to insufficient
sprinkling, or due to it not existing at all. Additionally, selecting an
efficient sprinkling is also non-trivial as the allowable regions are non-
compact.

This problem disappears completely in the case where we consider static
geometries. In this case we can restrict ourselves from the get-go to a
specific Cauchy slice, which reduces the dimensionality of the problem, but
more importantly removes the timelike direction. In this case, any two points
chosen in this slice will be spacelike separated. Thus, this method seems more
adapted to computation of RT surfaces in the geometries of chap.2.

Another, less prevalent but still important problem will be unwanted
intersections. Because the spacetime is excised at the interface, there will
be situations in which the geodesic crosses the interface in unintended
places, see fig.4.12. To avoid such unwanted solutions, we should check for
spurious intersections with the membrane. This is actually a difficult
problem, as the resulting equations are again not analytically solvable,
requiring another numerical resolution. These constraints are difficult to
reconcile with the numerical solving of the crossing equation, for the same
reasons explained for the spacelike condition.

Figure 4.12: Sketch of a problem that can occur when implementing numerical
algorithms. In the figure, we aim to find a double crossing geodesics anchored
at $x_{1}$ and $x_{2}$. Say in the minimization method we are looking for a
geodesic traversing at the green point. The geodesic path crosses the membrane
at the red cross, before reaching the green point, and thus there are no
geodesics crossing at the green dot. Verifying that such edge cases do not
happen generically is not an easy task.

We have attempted to use this method in the static geometries of chap.2, with
reasonable success in the case of symmetric intervals. It remains a work in
progress however to iron out the ”unwanted crossings” issue, as well as to
analyse the resulting numerical curves. For this reason, we don’t find it
useful to present the very preliminary results that we have at hand.

##### The shooting method

In this second approach, we parametrise the candidate HRT surface by its
initial conditions. We start on $p_{1}$, where we pick arbitrarily a spacelike
unit vector, $\dot{p}_{1}$. This is again best done in Poincaré coordinates,
where the unit condition (4.2) is explicitly solvable. This yields an
expression for the geodesic in terms of $(p_{1},\dot{p}_{1})$. The step here
which (generally) requires a numerical resolution is the determination of the
intersection with the membrane. One has two options to express this condition.
This equation can be written down either in Poincaré coordinates (where
presumably, the membrane equation is more complicated, but the geodesics are
simpler), or in the coordinate system naturally adapted to the membrane
solution. If the geodesic is parametrised with $\lambda$, this equation yields
$\lambda^{*}$, the intersection point with the membrane, if it exists. If no
solution is found, one draws another initial vector $\dot{p}_{1}$ and repeat
the procedure.

Then, the crossing conditions (4.23) determine new initial conditions
$(p_{m},\dot{p}_{m})$ for the geodesic on the other side, which we can follow
until we either reach the boundary, escape to infinity, or hit the membrane
again and repeat the process. In principle, we obtain in this way a
parametrisation of the final point $p_{f}$, in terms of the initial conditions
$p_{f}(p_{1},\dot{p}_{1})$. What remains to be done is solve the equation
$p_{f}(p_{1},\dot{p}_{1})=p_{2}$ for $\dot{p}_{1}$, which again can in general
only be done numerically given the complexity of the function $p_{f}$.

The pros of this approach is that we do not have to worry about the ”spacelike
separated condition”, as it is automatically satisfied by the choice of a
spacelike tangent vector $\dot{p}_{1}$. Otherwise, it is a trade-off; instead
of having to solve the crossing equations, we need to solve for the final
point of geodesic. Whether this is advantageous or not depends on the
specifics of the geometry at hand.

Generally, we run in much of the same problems which are difficult to avoid
with numerical solution-finding. First of all, we may miss the intersection of
the geodesic with the membrane with an inappropriate initialisation value for
$\lambda$, which again would force us repeat the numerical resolution by
sprinkling possible initial $\lambda_{0}$. The other problem lies in the
inversion of $p_{f}(p_{1},\dot{p}_{1})$. While it should be a continuous
function, in attempts we made we found it to be extremely sensitive on the
initial condition. Coupled to the fact that there are geodesics that escape to
infinity (fig.4.1), this makes the numerical inversion of $p_{f}$ unreliable
at best…

####  4.5.2 Application to the interface NESS state

In this small section we presents the attempts to apply the shooting method to
the stationary states obtained in chap.3. The reason we chose the shooting
method over the minimization method is because in the first place, we wanted
to search for geodesics like those depicted in fig. 4.11. As in this case the
specific final point $p_{f}$ was not the first priority, the shooting method
seemed the more appropriate.

Recall the membrane embedding (3.22):

$x_{j}=x_{m,j}(\sigma),\ \ r_{j}=r_{j}(\sigma),\ \ t_{j}=\tau+f_{j}(\sigma)\
.$ (3.22)

The derivative of the horizon entering membrane was exhibited in sec.3.5:

$\displaystyle x_{m,j}^{\prime}(\sigma)$
$\displaystyle=-\ell_{j}\frac{(\lambda^{2}\pm\lambda_{0}^{2})\sigma+\lambda
J_{i}}{2(\sigma-\sigma_{+}^{\rm Hj})(\sigma-\sigma_{-}^{\rm
Hj})\sqrt{A(\sigma-\sigma_{-})}}\ ,$ (4.57) $\displaystyle
f_{1}^{\prime}=f_{2}^{\prime}$
$\displaystyle=\frac{1}{4\sigma}\left(J_{1}\ell_{1}x_{m,1}^{\prime}-J_{2}\ell_{2}x_{m,2}^{\prime}\right)\
,$

the various constants are defined in sec.3.5. Most importantly, recall that
$\sigma=r_{j}^{2}-M_{j}\ell_{j}^{2}$.

We will be working in Eddington-Finkelstein coordinates (3.4) :

$dv=dt+\frac{\ell dr}{h(r)}\qquad{\rm and}\qquad
dy=dx+\frac{J\ell^{2}dr}{2r^{2}h(r)}\ .$ (3.4)

In the shooting method the main difficulty is the determination of the
intersection with the membrane, so it is important to express it as simply as
possible. Thus we will explicitely perform the integration of (3.4) and
(4.57). In the specific case (4.57) where the membrane enters the horizon, its
world-volume becomes AdS2666It is as of now unclear what is the condition that
determines whether the world-volume of the membrane is AdS2. It curiously
appears to be the case whenever the membrane is dual to a _single_ static
interface(i.e. of equation $x=cst$), as is the case for horizon-entering
membranes or the ones appearing in the vacuum ICFT case. It would be
interesting to explore whether this could be explained by the application of a
Fefferman-Graham-like prescription on the worldvolume of the membrane., and
$x_{j}^{\prime}$ can be integrated in terms of elementary functions, instead
of incomplete elliptic ones, which would add an additional numerical step to
the computation.

Integrating the coordinate change (with the convention that at the boundary it
is vanishing), we find :

$\displaystyle y$
$\displaystyle=x+\frac{J\ell^{2}}{4(r_{+}^{2}-r_{-}^{2})}\left(\frac{\ln(\frac{|r-r_{+}|}{r+r_{+}})}{r_{+}}-\frac{\ln(\frac{|r-r_{-}|}{r+r_{-}})}{r_{-}}\right)\
,$ (4.58) $\displaystyle v$
$\displaystyle=t+\frac{\ell}{2(r_{+}^{2}-r_{-}^{2})}\left(r_{+}\ln(\frac{|r-r_{+}|}{r+r_{+}})-r_{-}\ln(\frac{|r-r_{-}|}{r+r_{-}})\right)\
,$

where $r_{\pm}$ are the horizons.

For the membrane, the equation is a little bit more involved :

$\displaystyle x_{m}(r)$
$\displaystyle=-\frac{|J|\ell^{2}}{2(r_{+}^{2}-r_{-}^{2})}\left[{\rm
Sign}(J)\frac{g(r,\sigma^{H}_{+})}{r_{+}}-{\rm
Sign}((\lambda^{2}\pm\lambda_{0}^{2})\sigma^{H}_{-}+J\lambda)\frac{g(r,\sigma^{H}_{-})}{r_{-}}\right]\
,$ (4.59) $\displaystyle f_{1}(r)$
$\displaystyle=-\frac{J^{2}}{4}\left(\frac{\ell_{1}^{3}}{2(r_{1+}^{2}-r_{1-}^{2})}\left[{\rm
Sign}(J_{1})\frac{g(r,\sigma^{H1}_{+})}{r_{1+}\sigma_{+}^{H1}}-{\rm
Sign}((\lambda^{2}+\lambda_{0}^{2})\sigma^{H1}_{-}+J_{1}\lambda)\frac{g(r,\sigma^{H1}_{-})}{r_{1-}\sigma^{H1}_{-}}\right]+1\leftrightarrow
2\right)\ ,$ $\displaystyle g(r,a)$
$\displaystyle=\sqrt{a}\int_{\sqrt{-\sigma_{-}}}^{\sqrt{\sigma(r)-\sigma_{-}}}\frac{dt}{t^{2}-a}$
$\displaystyle=-\frac{1}{2}\left(\ln\left[\frac{\sqrt{\sigma(r)-\sigma_{-}}+\sqrt{a-\sigma_{-}}}{\left|\sqrt{\sigma(r)-\sigma_{-}}-\sqrt{a-\sigma_{-}}\right|}\right]-\ln\left[\frac{\sqrt{-\sigma_{-}}+\sqrt{a-\sigma_{-}}}{\left|\sqrt{-\sigma_{-}}-\sqrt{a-\sigma_{-}}\right|}\right]\right)\
,$

where we use the shortcut $\sigma(r)=\sqrt{r^{2}-M\ell^{2}}$ when useful, and
we omitted the indices denoting the side when possible. The convention here is
that the membrane crosses the ergosphere at $x=0$ in the spinning string
coordinates.

The membrane equation $(\tau+v_{m}(\sigma),r(\sigma),y_{m}(\sigma))$ in
Eddington-Finkelstein coordinates is obtained by plugging (4.59) into (4.58).
It becomes obvious from (4.59) that there is no hope of solving the membrane
intersection equation analytically. Note that because the membrane embedding
is invariant under $v$-translations we only need to solve one equation to find
the intersection with the geodesic. Namely :

$\displaystyle y_{m}(r_{g}(\lambda^{*}))=y_{g}(\lambda^{*})\ ,$ (4.60)

where $y_{m}$ denotes the interface embedding, and $r_{g}$, $y_{g}$ are the
coordinates of the spacelike geodesic whose equation can be obtained by
composing (4.9-4.12) with (4.54).

For the crossing equation, we define the normal and tangent vectors (omitting
the index $j$ denoting the side):

$\displaystyle t_{\tau}^{\mu}$ $\displaystyle=(1,0,0)\ ,$ (4.61)
$\displaystyle t_{\sigma}^{\mu}$
$\displaystyle=(v^{\prime}_{m,j}(\sigma),\frac{1}{2\sqrt{\sigma+M\ell^{2}}},y_{m,j}^{\prime}(\sigma))\
,$ $\displaystyle n_{\mu}$
$\displaystyle=\frac{1}{\mathcal{N}}(0,-y_{m,j}^{\prime}(\sigma),\frac{1}{2\sqrt{\sigma+M\ell^{2}}})\
,$

where $\mathcal{N}^{2}=n_{\mu}n^{\mu}$, and $n_{\mu}t^{\mu}_{a}=0$.

We can trivialize the crossing equations by expressing the geodesic tangent
vectors in the basis (4.61) :

$\displaystyle\dot{y}^{\nu}_{gi}(\lambda^{*})=a^{\tau}_{i}t_{i\tau}^{\mu}+a^{\sigma}_{i}t_{i\sigma}^{\mu}+b_{i}n^{\mu}_{i}\
.$ (4.62)

With this parametrization, and for an affinely parametrized geodesic, the
crossing equations reduce to :

$\displaystyle a_{1}^{a}a_{1}^{b}h^{1}_{ab}=a_{2}^{a}a_{2}^{b}h^{2}_{ab}\ ,$
(4.63)

where $h^{i}_{ab}=t^{\nu}_{ia}t^{\mu}_{ib}g^{i}_{\mu\nu}$ is the induced
metric on the interface. By the matching equations, $h^{1}_{ab}=h^{2}_{ab}$,
and since the metric is non-degenerate we deduce that the crossing conditions
simply force $a_{1}^{a}=a_{2}^{a}$, $b_{1}=b_{2}$ ! In other words, expressed
in the appropriate basis (4.61), the geodesic tangent vector is unchanged upon
crossing.

With these equation in mind we applied the shooting algorithm described in the
previous section which yields the numerical function
$p_{f}(p_{1},\dot{p}_{1})$. The last remaining step, which is the (numerical)
inversion of this function is until now unsuccessful, as most of the time the
numerical search diverges, for the reasons mentioned in the previous section.
By randomizing the initial condition, we were not able to find geodesics of
the type depicted in fig. 4.11, although we found crossing geodesics that were
outside both apparent horizons. Nonetheless, this does not yet show that these
peculiar geodesics do not exist. In fact, we noticed that as we move $y_{2}$
far from the interface boundary, only very precise initial conditions yield
geodesics that cross the membrane close to $r^{+}_{2}$ (refer to fig. 4.11).
Thus a random search would have a hard time finding those curves, which could
explain why we haven’t stumbled upon one.

The best way to search for such geodesics is actually to implement a hybrid of
the shooting and minimization algorithm. One picks a point on the boundary, as
well as a point on the membrane (which lies below one of the apparent
horizons), and computes the geodesic connecting those two points analytically.
Then, with the crossing condition, we obtain a shooting problem on the other
side, and we repeat until that yields a geodesic with two boundary endpoints.
This is currently being worked out.

###  4.6 QNEC in ICFT setups

The Quantum Null Energy Condition (QNEC) is an attempt to generalize the Null
Energy Condition (NEC) which states that the stress-energy tensor of classical
matter should obey $T_{\mu\nu}(x)v^{\mu}v^{\nu}\geq 0$, for any null vector
$v^{\mu}$, and at any point $x$. Such conditions are introduced ad-hoc in
General Relativity in an attempt to restrict the allowable stress-energy
tensors to ”realistic” ones, to prevent the appearance of unphysical spacetime
geometries (but they also are an essential ingredient in many general theorems
[204]), and thus such conditions should hold for types of matter that we
consider to be physical. As long as we consider only (reasonable) classical
fields, this condition is satisfied.

Unfortunately, it is violated quantum mechanically when we apply it to
$\langle T_{\mu\nu}\rangle v^{\mu}v^{\nu}$[210], as quantum effects such as
the Casimir energy can produce negative energy densities locally. To restore
the validity of these inequalities, what one usually does is to consider an
averaged version, obtaining, in this case, the Averaged NEC (ANEC) [211, 212]
:

$\displaystyle\int_{C}\langle T_{\mu\nu}\rangle v^{\mu}v^{\nu}d\lambda\geq 0\
,$ (4.64)

where the integration is done over an integral curve of the null vector field
$v^{\mu}$. The statement is that the inequality should hold for any null-
vector field and any of its associated integral curves $C$. The ANEC then has
a chance of surviving quantum effects, as they can produce local negative
energy densities, but we expect the energy to remain positive when averaging
them out for stable systems. The big disadvantage is that the condition is no
longer local, and thus becomes much less powerful.

For this reason, there have been efforts to formulate a QNEC, namely a quantum
version of the NEC which retains the locality property. The bound should of
course accommodate for the fact that locally the energy could be negative.
Such an inequality was conjectured by Bousso et. al[204], and proven using
holographic techniques [213, 214], and later also directly from field theory
techniques [215, 216]. The QNEC is particularly interesting as it relates the
energy density to variations of the entanglement entropy on an interval, which
are not usually thought to be correlated quantities in non-gravitational
systems.

We will be interested in the case of a $2$-dimensional CFT on flat space, in
which the QNEC takes a more restrictive form[217]. Consider a generic point
$x$ in the geometry, for which we want to write the QNEC. We begin by picking
a spatial slice ending at $x$, and with possibly another boundary at $y$ (see
fig. 4.13). By embedding this slice inside a Cauchy slice, we can associate
with it an entanglement entropy, which we denote $S(\rho,x,y)$, where $\rho$
explicits the dependence on the state of the CFT. We can then compute the
change in $S(\rho,x,y)$ under variations of $x$ along the lightlike
directions. The QNEC is the inequality :

$\displaystyle 2\pi\langle
T_{\pm\pm}(x)\rangle\geq\partial_{\pm}^{2}S(\rho,x,y)+\frac{6}{c}(\partial_{\pm}S(\rho,x,y))^{2}\
.$ (4.65)

In particular, notice that the term involving the central charge $c$ is
exclusive to CFTs, and makes the inequality stricter. For a QFT, this term is
missing. The derivative $\partial_{\pm}$ refer variations of $x$, keeping $y$
fixed.

Figure 4.13: The points $x$ and $y$ delimit a portion of a Cauchy slice, for
which we can compute the entanglement entropy. The shaded region is the causal
diamond, and any spacelike curve lying inside it and anchored at $x$ and $y$
will have the same entanglement entropy, as the state on these slices are
related by unitary transformations. In yellow, we denote the lightlike
direction under which we move $x$ do compute the derivatives used in the QNEC.

A consistency check is the verification that under a conformal transformation
(which will change the energy density (1.105)), the inequality conserves the
same expression. Indeed, one can verify that the quantity on the RHS of (4.65)
has the same transformation properties as the stress-energy tensor[217]. Note
also that (4.65) is really a family of inequalities for $\langle
T_{\pm\pm}(x)\rangle$, and the strictest one is obtained by maximizing the RHS
w.r.t. $y$.

Consider as an example the case of the vacuum state, $\langle
T_{\pm\pm}\rangle=0$. Consider any two spacelike separated points on the
boundary, denoted $x=(x^{+},x^{-})$, $y=(y^{+},y^{-})$ in lightcone
coordinates. They define a spacelike interval for which we compute the
entropy. With a few boosts, we can generalize the formula (4.15) to :

$\displaystyle S({\rm
vac},x,y)=\frac{c}{6}\log(\frac{|x^{+}-y^{+}||x^{-}-y^{-}|}{\epsilon^{2}})\ .$
(4.66)

One can easily verify that the bound of the QNEC is identically vanishing for
(4.66), for any choice of $y$. Thus, for the vacuum state, the QNEC is
saturated and becomes an equality. If one could determine whether the QNEC is
saturated without an explicit check, then one could determine the state given
the entanglement structure (or vice-versa) and obtain a sort of ”energy-
entanglement” equivalence. This is reminiscent of bulk reconstruction, where
in some sense the bulk geometry (and hence the bulk Stress-energy tensor)
emerges from the entanglement structure of the dual CFT[104].

From the saturation of the QNEC in the vacuum state, one can then easily show
that it will also saturate in any state that can be obtained by conformally
mapping the vacuum. This follows directly from the fact that (4.65) changes
covariantly with conformal transformations. For instance, one can verify
directly with (4.19) that the steady state, dual to the metric (4.53)
saturates the QNEC everywhere. With the same logic, one can push the reasoning
much further.

The QNEC will be saturated for any state, as long as the dual RT surfaces do
not cross bulk matter. Indeed, without matter, the bulk metric is locally
AdS3, so we can find a change of coordinates which brings us to the Poincaré
metric, at least for the portion of spacetime that the RT surface traverses.
Then the computation follows as in the vacuum case, and the QNEC is saturated.
In fact, one can quantify the deviation from saturation in the case of
perturbative matter in the bulk[218, 219]. To be more precise, the
aforementioned references look at the non-saturation of the inequality (4.65)
for some fixed $y$. However, we believe that the more interesting quantity to
inspect would actually be the best possible QNEC bound, namely :

$\displaystyle 2\pi\langle T_{\pm\pm}(x)\rangle\geq{\rm
Max}_{y}\left(\partial_{\pm}^{2}S(\rho,x,y)+\frac{6}{c}(\partial_{\pm}S(\rho,x,y))^{2}\right)\
.$ (4.67)

When considering the best bound as in (4.67), it is not clear anymore that
bulk matter necessarily invalidates the saturation. In that context, the
geometries that we have built-in chap. 2 and 3 are the perfect playground to
test such questions, as they provide a very simple form of matter in the bulk,
the thin membrane. Computing the QNEC in these geometries is also an important
check for the thin-brane approximation. Indeed, for intervals for which the
associated RT surfaces cross the membrane, it is not obvious that the QNEC
should hold generically. Were we to find cases where it is violated, it would
indicate some breakdown of the holographic bottom-up model we considered.

####  4.6.1 QNEC for static geometries

As we have learned throughout this chapter, computing (H)RT surfaces on our
stitched geometries is generically not an easy task. Thus it is not surprising
that computing the QNEC bound offers the same challenges. Once one has a
robust algorithm to compute HRT surfaces anchored at arbitrary boundary
points, the QNEC follows easily, but as of now, it is still lacking. We will
thus focus on geometries that can be brought back to the vacuum Poincaré case,
described in detail in sec.4.3.2.

Recall that for these geometries we have determined that the entanglement
entropy S associated with a crossing RT surface on an interval bounded by
$x=(\tau_{x},\sigma_{x})$ and $y=(\tau_{y},\sigma_{y})$ has the following
structure :

$\displaystyle S$
$\displaystyle=\frac{c_{i}}{6}\ln(\frac{\sigma_{x}}{\epsilon})+\frac{c_{i}}{6}\ln(\frac{\sigma_{y}}{\epsilon})+g(\xi)\
,$ (4.68) $\displaystyle\xi$
$\displaystyle=\frac{-(\tau_{x}-\tau_{y})^{2}+(\sigma_{x}-\sigma_{y})^{2}}{4\sigma_{x}\sigma_{y}}\
,$

where the $c_{i}$ and $g(\xi)$, should be chosen accordingly with the location
of the two points, see the formulas (4.40) and (4.32) for the specifics.

We exploit the form (4.68) to compute its lightlike derivatives without having
to compute any more RT surfaces. We specify to the case ”+” version of the
QNEC, as the staticity of the state guarantees that both directions will be
equivalent.

Consider first the case where $\sigma_{x}$ is on side 1, and $\sigma_{y}$ is
on side 2. Expressing $\xi$ and $\sigma_{x}$ in terms of the lightlike
coordinates $w_{\pm}^{x}$, we find :

$\displaystyle\frac{\partial^{2}S}{(\partial
w_{+}^{x})^{2}}+\frac{6}{c}\frac{\partial S}{\partial
w_{+}^{x}}=\frac{\pi(w_{-}^{x}-w_{-}^{y})^{2}(w_{+}^{x}+w_{+}^{y})^{2}}{32\sigma_{x}^{4}\sigma_{y}^{2}}\left[g^{\prime\prime}(\xi)+\frac{12\pi}{c_{1}}g^{\prime}(\xi)^{2}\right]\
.$ (4.69)

In the case where $\sigma_{x}$ is on side 2, one simply has to change
$c_{1}\rightarrow c_{2}$. Notice that the bound is independent of the cutoff
$\epsilon$, and thus we can safely take the limit.

By scale invariance and time-translation, let us begin by fixing
$\sigma_{x}=1$, $\tau_{x}=0$ on side 1, and consider (to reduce
dimensionality) $y=(\tau_{y}=0,\sigma_{y})$ on side 2. In the notation of sec.
4.3.3, we have $\mu=\frac{\sigma_{y}}{\sigma_{x}}$. In our Euclidean
construction, the natural variable is the angle $\alpha$, and thus properly
speaking, we have $g(\xi)=g(\alpha(\xi))$. The formula (4.35) gives us
$\alpha(\mu)$, and we can express this for the cross-ratio by the equality
$\xi=\mu-2+\frac{1}{\mu}$. This gives :

$\displaystyle\mu(\xi)=1+2\xi\pm 2\sqrt{\xi(1+\xi)}\ .$ (4.70)

The two possible solutions will give the same conclusion, as they amount to
the inversion $\mu\rightarrow 1/\mu$ which is a symmetry of the problem. Thus
in the end, we have $g(\xi)=g(\alpha(\mu(\xi)))$ where each of the functions
is analytical, although extremely cumbersome.

In fig. 4.14 we plot the QNEC bound for the crossing geodesic, as we vary the
endpoint $y$ over the equal time slice.

(a) QNEC bound plot for $x$ lying on side 1. The bound is saturated for the
symmetric position, $y=x$.

(b) QNEC bound plot for $x$ lying on side 2. The bound is saturated for the
symmetric position, $y=x$.

Figure 4.14: Two plots of the QNEC bound, computed for boundary intervals that
contain the interface of the ICFT, in the vacuum state. They are plotted as as
function of varying $y$, having fixed $x$ (one can assume $x=1$ WLOG).

Re-assuringly, the bounds obtained while varying the endpoint $y$ are all
negative, which means that in the vacuum state the ICFT model is at least
consistent with the QNEC. The interesting point is that we find a specific $y$
for which the bound is 0, meaning (4.67) saturates and this is despite the
fact that the corresponding RT surface does cross bulk matter. The $y$ in
question is however rather special, as it is the point that lies symmetrically
to $x$ with respect to the interface (equivalently, the point for which
$\xi=0$).

The simple fact that (4.67) is saturated is actually unsurprising; indeed, in
the interface models we consider one could always take $y\rightarrow x$, for
which case the saturation is evident as we can always avoid the membrane.
Nonetheless, it is noteworthy that we managed to find a saturating point even
in the case where the RT surface crosses bulk matter. Whether this saturation
holds more generally, or it is just due to the increased symmetry of the
location of $y$, is yet to be determined.

Consider now the limits as the endpoint $y$ approaches the interface, from
side $2$ (x in side $1$). We see the QNEC approaching $0$, and this can be
understood assuming the smoothness of the entanglement entropy in this limit.
After $y$ crosses the interface, the two points lie on side $1$ and the RT
surface is the trivial one, giving the saturating bound. As $y$ approaches the
boundary, we see the QNEC reaching this bound.

The story is different when $y$ lies on side $1$ ($x$ on side $2$). As $y$
approaches the interface, we see the QNEC reaches a non-saturating bound.
Indeed, as $y$ crosses the interface, the dominating RT surface will not be
the trivial one, but rather the double-crossing one depicted in fig. 4.6. Such
RT surfaces will not in general saturate the QNEC bound, and as such neither
is the limit of the bound as $y$ tends to the interface.

To continue the analysis, the natural next step is to actually compute the
QNEC for the two points $x$, $y$ lying on side 2. The computation goes
similarly to what we have just shown, the only tricky step being the problems
with the analytic form of $\alpha(\mu)$, see fig. 4.9. Again, we unfortunately
haven’t finished the work needed to present clear results so this is added to
the list of works in progress. This also goes for the computation of the QNEC
in more complicated states, as it cannot be done without first having a firm
grasp on the RT construction.

###  4.7 Closing remarks

This section was dedicated to presenting partial results regarding the
entanglement structure of the various ICFT states that were obtained in the
previous chapters. The initial goal of exploiting the Euclidean geometry
construction of sec.4.3.2 and reducing all the entropy computations to this
case was unsuccessful, which forced us to consider numerical methods.
Nonetheless, we showed that this does allow full analytic control for static
geometries, in which the induced metric of the membrane is AdS2.

Efforts in the numerical direction were centered around the main idea to
exploit the AdS3-locality of the considered spacetimes. By locally mapping
them to Poincaré coordinates, we limit the necessity for numerical resolution
only at a few points. For now, the results obtained are still lacking, and the
priority for future work will be to iron out the details that prevent the
successful application of the algorithms. We believe that understanding the
entanglement structure of the ICFT NESS will not only offer insights into the
curious entropy production at the interface, but also regarding the properties
of non-killing, out-of-equilibrium horizons.

Finally, the recovery of the QNEC bound in these systems follows almost
automatically from the entanglement structure, so it is a natural quantity to
consider. Besides being a consistency check, of particular interest is the
question of its saturation. Indeed, some authors have claimed that for CFT in
$D\geq 3$, the QNEC is always saturated [220, 221]. This fails in $2$D as we
have noticed in our models, but it is a tantalizing question whether the
saturation still holds for the best bound (4.67). This question deserves
further study.

## Conclusion and outlook

The main focus of the work presented in this thesis was the study of a simple
model describing a minimal Interface CFT and its holographic dual, which in
the large N and ’t Hooft coupling limit reduces to Einstein gravity with a
gravitating membrane. The initial motivation to consider such a model was
their intimate connection with the (then) recent progress on the black hole
information paradox[93, 123]. Indeed, the examples in which the Island formula
can be applied usually involve Boundary CFT duals, which can be obtained as a
limiting case from ICFT.

The simpler case of the $3$-dimensional bulk was initially introduced as a
stepping stone in preparation for the study of full Supergravity solutions.
However, the model proved to be much richer than expected, such that it became
the main focus of this work. Furthermore, studying minimal models of this type
has the advantage of yielding more generally applicable results. This broadens
the scope of the results we obtain here to fields like condensed matter
theory[67, 68, 69, 66], where minimal holographic models are often used to
allow the computations at strong coupling, not easily accessible from the
field theory. Finally, having access to non-trivial holographic models which
allow for analytic results provides a nice playground to test and further our
understanding of AdS/CFT.

Aiming at applications in the black hole information problem, in chap. 2 we
considered the ICFT model at finite temperature whose gravity duals include
black holes. Surprisingly, the sole presence of the gravitating membrane
produced a very rich phase diagram for this system. While our analysis was
thorough within this minimal model, it identified several interesting
directions that deserve future study.

One key issue is the validity of the thin-brane approximation and how our
results would fare in full top-down ICFT/gravitating wall pairs. Another
approximation of our treatment is that we discarded bulk solutions involving
membrane fusion (fig. 2.2) even though such configurations sometimes appeared
naturally in cases where the membrane intersected itself. While they should
not alter the conclusions on the phase space drastically, this more complete
model could exhibit some interesting properties of the pair of interfaces.

A related question concerns the ”exotic fusion” geometries in which, as we
bring the two interfaces together, they do not fuse into the trivial defect
(fig. 2.10). Their mere existence is quite surprising[222, 223], and it might
be related to the ”membrane fusion” geometries mentioned above. This is
because deeper regions of the AdS bulk are supposed to probe the IR region of
the dual field theory. As we go to the IR the two interfaces approach one
another and eventually appear as one, ”fusing”. Thus, an allowable non-trivial
membrane fusion in the bulk might signal the presence of a potential exotic
interface fusion in the field theory.

One last problem that deserves future work is the elucidation of the meaning
of the sweeping transition (sec. 2.7.2) from the boundary theory. Our strong
feeling is that they must be connected to the entanglement structure of the
field theory, as it is the quantity that is related to the bulk
reconstruction. We have arrived at some conjectures in chap. 4, but they are
still very tentative. Related to this problem is the role of the critical
tension $\lambda_{0}$. It is still unclear if this is simply a quantity that
appears in the computations, or whether it carries a deeper meaning.

In chap.3 we looked at an extension of the static states of chap. 2 to more
general non-equilibrium stationary states. This is of interest both to confirm
the work that was already done from the field theory perspective [70, 133, 51]
and extend it, but also to generate tractable non-equilibrium black hole
solutions, which are still poorly understood. These goals were met. First, our
calculation confirmed the universality of the energy-transport coefficients
shown in ICFT in [51]. Secondly, it offers a much simpler way to compute these
coefficients on the gravity side, when compared to the original scattering
calculations of [153]. Lastly, perhaps the least expected result was the
discovery of the maximal entropy production at the interface and the
analytical recovery of the deformed, non-killing event horizon on the gravity
side.

These results naturally lead to the fascinating question of the relation of
the deformed horizon with entanglement entropy. We touched on this issue in
chap.4 and hope to complete our partial results in the near future. The well-
known Bekenstein-Hawking formula, which has been re-derived with the RT
prescription in AdS/CFT, has been mainly applied to black holes with static
killing horizons. In this model, we have an example of an (analytically
controlled) event horizon that is neither Killing, nor coinciding with its
”apparent” counterpart. It is the perfect setup to elucidate the role played
by these various horizons in holography.

One other question that needs elucidation concerns the thermal transmission of
the interface pairs. For a single interface, we recovered results previously
obtained perturbatively[24], but in the case of more than one interface, there
appeared two distinct phases. When the membrane avoids the black hole and
curves back to the boundary, the conductance suddenly becomes perfect. This
behavior certainly pertains to some sort of coherent scattering, which allows
all the thermal flux through as if there were no interfaces. The precise way
in which this happens in the field theory is however unclear and deserves
further study.

The final chapter 4 ties the project together as it brings us back to the
initial goal of connecting our models to the Island computations. We have
already mentioned that the computations of RT surfaces in the obtained
geometries was the obvious next step in their study, and that is what is
undertaken in this chapter. Based on the vacuum computations of [20] which we
present, we outline different angles (analytical and numerical) to tackle
computations of the sought-out RT surfaces.

While some interesting results were obtained, we haven’t yet managed to reach
the initial goal: showing the existence of horizon probing RT surfaces in the
non-equilibrium case, and understanding the entropy production at the
interface. The main focus of future work will be the completion of these
goals. Finally, another tangent direction that was mentioned is the
computation of the QNEC bound for these same geometries. Its application to
the ICFT model will cement its consistency, while potentially providing
interesting insights on the necessary conditions for its saturation. The QNEC
becomes much more interesting when one considers the Wick-rotated version of
our models [77, 205], in which the interface now plays the role of a quench.
It could be interesting to see which restrictions it imposes if any, and
whether it tells us something about the non Wick-rotated version.

Although much was accomplished, much still remains to be worked out, as is
always the case in physics.

##  Appendix A Misc

###  A.1 Extrinsic curvature

Consider a surface of codimension one embedded in a manifold of dimension $D$.
It is parametrized by (D-1) parameters denoted $s^{a}$. In a Lorentzian
manifold, one should differentiate according to the nature of the hypersurface
(lightlike, spacelike or timelike). Unless otherwise stated, we assume here
that the surface is timelike. Generally, its equation can be written as

$\displaystyle x^{\mu}=x^{\mu}(s^{a})\ ,$ (A.1)

where $x^{\mu}(s^{a})$ are $D$ functions specifying the surface’s shape. We
can then naturally define a set of $D-1$ tangent vectors to the surface :

$\displaystyle t_{a}^{\mu}=\frac{\partial x^{\mu}}{\partial s^{a}}\ ,$ (A.2)

which allows us to define the induced metric :

$\displaystyle h_{ab}=g_{\mu\nu}t_{a}^{\mu}t_{b}^{\nu}\ ,$ (A.3)

where $g_{\mu\nu}$ is the metric of the ambient spacetime. To define the
normal covector $n_{\mu}$, one can use the generalized cross-product :

$\displaystyle
n_{\mu}=\frac{1}{2}\epsilon_{\mu\nu_{1}\ldots\nu_{D-1}}\epsilon^{a_{1}\ldots
a_{D-1}}t^{\nu_{1}}_{a_{1}}\ldots t^{\nu_{D-1}}_{a_{D-1}}\ ,$ (A.4)

where the $\epsilon$ symbols are the levi-civita tensors corresponding to the
metric $g$ and $h$ (with the appropriate normalisation with $\sqrt{|g|}$).
Using (A.4), the normal vector is normalized, $n^{\mu}n_{\mu}=1$. (For a
lightlike surface, $n^{\mu}n_{\mu}=0$, and for a spacelike surface,
$n^{\mu}n_{\mu}=-1$).

In the case where the surface can be defined in terms of an equation
$f(x^{\mu})=0$, there is a much simpler way to obtain the normal co-vector :

$\displaystyle
n_{\mu}=\frac{\partial_{\nu}f}{\sqrt{\partial_{\nu}f\partial^{\nu}f}}\ ,$
(A.5)

which is obtained by varying the defining equation along tangent directions.
The normal vector of course satisfies :

$\displaystyle n_{\mu}t^{\mu}_{a}=0\ .$ (A.6)

Using it, we can define the projection operator on the surface :

$\displaystyle\Pi^{\mu}_{\nu}=\delta^{\mu}_{\nu}-n_{\nu}n^{\mu}$ (A.7)

Note that formally, $n_{\mu}$ and hence $\Pi^{\mu}_{\nu}$ are defined only on
the surface $x^{\mu}(s^{a})$. Thus they are not vector fields on the ambient
manifold. One can however arbitrarily extend them to all of the spacetime;
this allows us to more easily define operations involving them, and in the
end, it will not matter since we will evaluate everything on the surface.

The projection operator can be applied to any tensor to isolate its components
tangent to the surface. For instance, applied on the metric :

$h_{\mu\nu}=\Pi^{\alpha}_{\mu}\Pi^{\beta}_{\nu}g_{\alpha\beta}=\Pi_{\mu\nu}\
.$ (A.8)

This is another way to see the metric (A.3), as expressed in the embedding
coordinates. One can verify we have of course
$h_{ab}=t_{a}^{\mu}t_{b}^{\nu}h_{\mu\nu}$. Thus, the projection tensor and the
surface metric are essentially interchangeable.

We have all we need to define Extrinsic curvature :

$\displaystyle
K_{\mu\nu}=\Pi^{\alpha}_{\mu}\Pi^{\beta}_{\nu}\nabla_{\alpha}n_{\beta}(=\Pi^{\alpha}_{\mu}\nabla_{\alpha}n_{\nu})\
.$ (A.9)

Although it is not immediately obvious, one can show that $K_{\mu\nu}$ is
symmetric. This comes from the fact that locally, we can always write (A.5) by
the implicit function theorem. In this form, $n_{\mu}$ is locally hypersurface
orthogonal, namely it satisfies $n_{[\rho}\nabla_{\mu}n_{\nu]}=0$. Contracting
this with $n^{\rho}$ yields $K_{[\mu\nu]}=0$ as required.

Since $K_{\mu\nu}$ is a tensor defined only on the surface, it is easier to
work in the $s^{a}$ coordinates :

$\displaystyle
K_{ab}=t^{\mu}_{a}t^{\nu}_{b}K_{\mu\nu}=t^{\mu}_{a}t^{\nu}_{b}\nabla_{\mu}n_{\nu}\
.$ (A.10)

To compute $\nabla_{\mu}n_{\nu}$, one has to extend the vector field $n_{\nu}$
to the full embedding space. Alternatively, expanding (A.10) :

$\displaystyle
K_{ab}=t^{\mu}_{a}t^{\nu}_{b}(\partial_{\mu}n_{\nu}+\Gamma^{\rho}_{\mu\nu}n_{\rho})=t^{\nu}_{b}\partial_{a}n_{\nu}+t^{\mu}_{a}t^{\nu}_{b}\Gamma^{\rho}_{\mu\nu}n_{\rho}\
.$ (A.11)

With the notation (A.11), one does not need to go through the trouble of
extending $n_{\nu}$; indeed, the embedding space derivative has been
”absorbed” by $t_{a}^{\mu}$, so that we need only take derivatives in the
tangent directions, in which $n_{\mu}$ is well-defined.

The trace of the extrinsic curvature that will enter the Gibbons-Hawking
action is then simply $K=K_{ab}h^{ab}=K_{\mu\nu}h^{\mu\nu}$.

##  Appendix B Thermal holographic interfaces

###  B.1 Extrinsic curvature formulas

In this section we collect some formulas for the extrinsic curvature of the
membrane, in the parametrization:

$\displaystyle
x^{\mu}_{m}(\sigma,\tau)=(\tau,\sqrt{\sigma-M\ell^{2}},x_{m}(\sigma))\ ,$
(B.1)

where the order of coordinates is $(t,r,x)$, and the metric (2.20). The
tangent and normal (co)vector then are :

$\displaystyle t_{\tau}^{\mu}$ $\displaystyle=(1,0,0)\ ,$ (B.2) $\displaystyle
t_{\sigma}^{\mu}$
$\displaystyle=(0,\frac{1}{2\sqrt{\sigma-M\ell^{2}}},x_{m}^{\prime}(\sigma))\
,$ $\displaystyle n_{\mu}$
$\displaystyle=\frac{1}{\mathcal{N}}(0,-x_{m}^{\prime}(\sigma),\frac{1}{2\sqrt{\sigma-M\ell^{2}}})\
,$

where
$\mathcal{N}=\frac{1}{\ell}\sqrt{\frac{(r^{\prime})^{2}}{r^{2}}+(-M\ell^{2}+r^{2})(x^{\prime})^{2}}$
is the normalisation of the normal vector. Note that we write
$\sqrt{\sigma-M\ell^{2}}=r(\sigma)\equiv r$ where useful.

Formula (A.11) gives us the Extrinsic curvature :

$\displaystyle K_{\tau\tau}$
$\displaystyle=\frac{r(r^{2}-M\ell^{2})x^{\prime}(\sigma)}{\ell\sqrt{\frac{r^{\prime}{}^{2}\ell^{2}}{r^{2}}+(r^{2}-M\ell^{2})x^{\prime
2}}}=\frac{r^{2}x^{\prime}}{\ell}\times\sqrt{\frac{\sigma}{g(\sigma)}}\ ,$
(B.3) $\displaystyle K_{\sigma\tau}$ $\displaystyle=0\ ,$ $\displaystyle
K_{\sigma\sigma}$
$\displaystyle=\frac{x^{\prime}r^{\prime}{}^{2}\ell^{2}(3\sigma+M\ell^{2})+r\sigma
x^{\prime}(r\sigma
x^{\prime}{}^{2}+\ell^{2}r^{\prime\prime})+\frac{1}{2}\ell^{2}\sigma
x^{\prime\prime}}{\ell\sigma\sqrt{\sigma g(\sigma)}}\ ,$

where all $r$ should be seen as function of $\sigma$. Note we used the
shorthand $g(\sigma)$ defined in (2.22). Then writing down the traced-reversed
Israel condition (2.18), for the $\tau\tau$ component, we find :

$\displaystyle\frac{r^{2}_{1}x^{\prime}_{1}}{\ell_{1}}+\frac{r^{2}_{2}x^{\prime}_{2}}{\ell_{2}}=-\lambda\sqrt{\sigma
g(\sigma)}\ ,$ (B.4)

which is indeed the equation (2.25) used in the main text. The $\sigma\sigma$
Israel condition is much more complex as seen from the lengthy formula (B.3)
for $K_{\sigma\sigma}$. Nonetheless, one can write it down and plug in the
solution (2.37), and after a computer-aided simplification, it indeed
evaluates to $0=0$, such that it is redundant with the other equation.

###  B.2 Renormalized on-shell action

We reproduce here the Euclidean action of the holographic-interface model, in
units $8\pi G=1$. It is the sum of bulk, brane, boundary and corner
contributions, (see [224] for an explanation of the corner term)

$\displaystyle S_{\rm gr}=-\frac{1}{2}$
$\displaystyle\int_{{\mathbb{S}}_{1}}d^{3}x\sqrt{g_{1}}(R_{1}+\frac{2}{\ell_{1}^{2}})-\frac{1}{2}\int_{{\mathbb{S}}_{2}}d^{3}x\sqrt{g_{2}}(R_{2}+\frac{2}{\ell_{2}^{2}})+\lambda\int_{\mathbb{W}}d^{2}s\sqrt{h_{m}}$
$\displaystyle-\int_{\partial{\mathbb{S}}_{1}}d^{2}s\sqrt{h_{1}}K_{1}-\int_{\partial{\mathbb{S}}_{2}}d^{2}s\sqrt{h_{2}}K_{2}+\int_{\rm
C}(\theta-\pi)\sqrt{h_{c}}+{\rm c.t.}\ ,$

where the counterterms, abbreviated above by c.t., read [225]

$\displaystyle{\rm
c.t.}=\frac{1}{\ell_{1}}\int_{\mathbb{B}_{1}}\sqrt{h_{1}}+\frac{1}{\ell_{2}}\int_{\mathbb{B}_{2}}\sqrt{h_{2}}-\int_{\mathbb{B}_{1}\cap\mathbb{B}_{2}}(\theta_{1}+\theta_{2})\sqrt{h_{c}}\
.$ (B.5)

We remind that $\mathbb{S}_{j}$ are the spacetime slices whose boundary is the
sum of the asymptotic boundary $\mathbb{B}_{j}$ and of the string worldsheet
$\mathbb{M}$, i.e. $\partial\mathbb{S}_{j}=\mathbb{B}_{j}\cup\mathbb{M}$.

The induced metrics are denoted by the letter $h$. The $K_{j}$ are the traces
of the extrinsic curvatures on each slice computed with the outward-pointing
normal vector. Finally, in addition to the standard Gibbons-Hawking-York
boundary terms, one must add the Hayward term [226, 224] at corners of
$\partial\mathbb{S}_{j}$ denoted by C. 111These play no role here, but they
can be important in the case of string junctions.

There is at least one such corner at the cutoff surface,
$\mathbb{B}_{1}\cap\mathbb{B}_{2}$, where $\theta-\pi$ is the sum of the
angles $\theta_{j}$ defined in figure 2.5.

Let us break the action into an interior and a conformal boundary term,
$S_{\rm gr}=S_{\rm int}+S_{\mathbb{B}}$, with the former including
contributions from the worldsheet $\mathbb{M}$. Using the field equations
$R_{j}=-{6/\ell_{j}^{2}}$ and
$K_{1}|_{\mathbb{M}}+K_{2}|_{\mathbb{M}}=2\lambda$, and the volume elements
that follow from (2.20) and (2.21, 2.22),

$\sqrt{g_{j}}d^{3}x=\ell_{j}r_{j}dr_{j}dx_{j}dt\quad{\rm
and}\quad\sqrt{h_{m}}d^{2}s=\sqrt{fg}d\sigma dt\ ,$ (B.6)

We can write the interior on-shell action as follows :

$S_{\rm
int}=\frac{2}{\ell_{1}}\int_{{\Omega}_{1}}r_{1}\,dr_{1}dx_{1}dt+\frac{2}{\ell_{2}}\int_{{\Omega}_{2}}r_{2}dr_{2}dx_{2}dt-\lambda\int_{\mathbb{M}}\sqrt{fg}d\sigma
dt\ .$ (B.7)

We have been careful to distinguish the spacetime slice Sj from the coordinate
chart $\Omega_{j}$, because we will now use Stoke’s theorem treating
$\Omega_{j}$ as part of flat Euclidean space,

$\displaystyle\sum_{j=1,2}\frac{2}{\ell_{j}}\int_{{\Omega}_{j}}r_{j}dr_{j}dx_{j}=\sum_{j=1,2}\frac{1}{\ell_{j}}\oint_{{\partial\Omega}_{j}}r_{j}^{2}(\hat{r}_{j}\cdot
d\hat{n}_{j})\ ,$ (B.8)

with $d\hat{n}_{j}dt$ the surface element on the boundary
$\partial\Omega_{j}$, and $\hat{r}_{j}$ the unit vector in direction of
increasing $r_{j}$. Crucially, the boundary of $\Omega_{j}$ may include a
horizon which is a regular interior submanifold of the Euclidean spacetime and
is not therefore part of $\partial\mathbb{S}_{j}$. In particular, there is no
Gibbons-Hawking-York contribution there.

The boundary integral in (B.8) receives contributions from the three pieces of
${\partial\Omega}_{1,2}$ : the asymptotic cutoff surface
$\mathbb{B}_{1}\cup\mathbb{B}_{2}$, the horizon if there is one, and the
worldsheet $\mathbb{M}$.

Consider first the contributions from the asymptotic cutoff and horizon. The
normal vector $\hat{n}_{j}$, in this case, is simply $\pm\hat{r}_{j}$, so the
contribution is proportional to the size $\Delta x$ of the boundary. For the
membrane, the surface element in the Euclidean space $\Omega_{j}$ is :

$\displaystyle d\hat{n}_{j}=(-x_{j}^{\prime}(\sigma),r_{j}^{\prime}(\sigma))\
,$ (B.9)

where here this is the outgoing normal vector for one of the two membrane
pieces. Both of them will contribute equally to the integral, we integrate on
them in opposing directions, while also having opposing normal vectors.
Combining this together yields :

$\displaystyle\frac{1}{\ell_{j}}\oint_{{\partial\Omega}_{j}}r_{j}^{2}(\hat{r}_{j}\cdot
d\hat{n}_{j})=\frac{r_{j}^{2}L_{j}}{\ell_{j}}-\frac{r_{h\,j}^{2}\Delta
x_{j}\Bigl{|}_{Hor}}{\ell_{j}}-\int_{\sigma\in\Sigma}\frac{r^{2}_{j}(\sigma)x_{j}^{\prime}(\sigma)}{\ell_{j}}\
.$ (B.10)

In the last term, $\Sigma$ denotes the two ”half-membranes”, i.e. twice the
interval $\sigma\in[\sigma_{+},\infty]$.

Very conveniently, this last term precisely cancels the third term in (B.7) by
virtue of the Israel-Lanczos equation (2.25). In this way, we fortunately do
not have to bother with integrals on the brane worldvolume.

Thus, after all the dust has settled, the action can be written as the sum of
terms evaluated either at the black-hole horizon or at the cutoff. After
integrating over periodic time (which simply contributes to a prefactor of
$\frac{1}{T}$ in front of the expressions) the interior part of the action,
(B.7), reads

$\displaystyle S_{\rm
int}=\frac{r_{1}^{2}L_{1}}{\ell_{1}T}-\frac{M_{1}\ell_{1}\Delta x_{1}|_{\rm
Hor}}{T}+\frac{r_{2}^{2}L_{2}}{\ell_{2}T}-\frac{M_{2}\ell_{2}\Delta
x_{2}|_{\rm Hor}}{T}\ .$ (B.11)

If the slice $\mathbb{S}_{j}$ does not contain a horizon the corresponding
contribution is absent, $\Delta x_{1}|_{\rm Hor}=0$. We wrote $r_{j}$ for the
cutoff radius, which is to be sent to infinity at the end. Note that we
replaced $r_{hj}=M_{j}\ell_{j}^{2}$, the horizon radius.

We now turn to the conformal-boundary contributions from the lower line in the
action (B.2). For a fixed-$r_{j}$ surface, the outward-pointing unit normal
expressed as a 1-form is n${}_{j}=dr_{j}/\sqrt{r_{j}^{2}-M_{j}\ell_{j}^{2}}$.
Dropping the index $j$ for simplicity, one finds after a little algebra
(virtually identical to the computations of sec.1.3.2) :

$\displaystyle
K_{xx}=K_{tt}=-\frac{r}{\ell}\sqrt{r^{2}-M\ell^{2}}\Longrightarrow\sqrt{\hat{g}}K=-\frac{1}{\ell}(2r^{2}-M\ell^{2})\
.$ (B.12)

Combining the Gibbons-Hawking-York terms and the counterterms gives

$\displaystyle
S_{\mathbb{B}}=\frac{1}{\ell_{1}T}(r_{1}\sqrt{r_{1}^{2}-M_{1}\ell_{1}^{2}}-2r_{1}^{2}+M_{1}\ell_{1}^{2})\Delta
x_{1}\Bigl{|}_{\mathbb{B}_{1}}+(1\leftrightarrow 2)\ .$ (B.13)

Expanding for large cutoff radius, $r_{j}|_{\mathbb{B}_{j}}\to\infty$, and
dropping the terms that vanish in the limit we obtain

$\displaystyle
S_{\mathbb{B}}=\frac{1}{\ell_{1}T}(-r_{1}^{2}+\frac{1}{2}M_{1}\ell_{1}^{2})L_{1}+(1\rightarrow
2)\ .$ (B.14)

Upon adding up (B.11) and (B.14) the leading divergent term cancels, giving
the following result for the renormalized on-shell action:

$\displaystyle S_{\rm gr}=\frac{M_{1}\ell_{1}}{2T}\bigl{(}L_{1}-2\Delta
x_{1}\bigl{|}_{{\rm
Hor}}\bigr{)}+\frac{M_{2}\ell_{2}}{2T}\bigl{(}L_{2}-2\Delta
x_{2}\bigl{|}_{{\rm Hor}}\bigr{)}\ .$ (B.15)

We used here the fact that $\Delta x_{j}|_{{\mathbb{B}}_{j}}=L_{j}$, and that
$r_{j}^{2}=M_{j}\ell_{j}^{2}$ at the horizon when one exists. We also used
implicitly the fact that for smooth strings the Hayward term receives no
contribution from the interior and is removed by the counterterm at the
boundary.

As a check of this on-shell action let us compute the entropy. Using our
formula for the internal energy $\langle
E\rangle=\frac{1}{2}(M_{1}\ell_{1}L_{1}+M_{2}\ell_{2}L_{2})$, and $S_{\rm
gr}=F/T=\langle E\rangle/T-S$ we find

$\displaystyle S$ $\displaystyle=\frac{1}{T}\bigl{(}M_{1}\ell_{1}\Delta
x_{1}\bigl{|}_{{\rm Hor}}+M_{2}\ell_{2}\Delta x_{2}\bigl{|}_{{\rm
Hor}}\bigr{)}\ ,$ (B.16) $\displaystyle=4\pi^{2}T\bigl{(}\ell_{1}\Delta
x_{1}\bigl{|}_{{\rm Hor}}+\ell_{2}\Delta x_{2}\bigl{|}_{{\rm
Hor}}\bigr{)}=\frac{\mathcal{A}({\rm horizon})}{4G}\ .$

In the lower line we used the fact that $M_{j}=(2\pi T)^{2}$ and $r_{j}^{\rm
H}=2\pi T\ell_{j}$ for slices with horizon, plus our choice of units $8\pi
G=1$. The calculation thus reproduces correctly the Bekenstein-Hawking
entropy.

###  B.3 Opening arcs as elliptic integrals

In this appendix we express the opening arcs, (2.38-2.40), in terms of
complete elliptic integrals of the first, second and third kind,

$\displaystyle{\rm\bf K}(\nu)$
$\displaystyle=\int_{0}^{1}\frac{dy}{\sqrt{(1-y^{2})(1-\nu y^{2})}}\ ,$ (B.17)
$\displaystyle{\bf E}(\nu)$ $\displaystyle=\int_{0}^{1}\frac{\sqrt{1-\nu
y^{2}}\,dy}{\sqrt{1-y^{2}}}\ ,$ (B.18) $\displaystyle{\bf\Pi}(u,\nu)$
$\displaystyle=\int_{0}^{1}\frac{dy}{(1-uy^{2})\sqrt{(1-y^{2})(1-\nu y^{2})}}\
.$ (B.19)

Consider the boundary conditions(2.38). The other conditions (2.39,2.40)
differ only by the constant periods or horizon arcs, $P_{j}$ or $\Delta
x_{j}|_{\rm hor}$.