# Phases of Holographic Interfaces

Constantin Bachas and Vassilis Papadopoulos

###### Abstract

We compute the phase diagram of the simplest holographic bottom-up model of
conformal interfaces. The model consists of a thin domain wall between three-
dimensional Anti-de Sitter (AdS) vacua, anchored on a boundary circle. We
distinguish five phases depending on the existence of a black hole, the
intersection of its horizon with the wall, and the fate of inertial observers.
We show that, like the Hawking-Page phase transition, the capture of the wall
by the horizon is also a first order transition and comment on its field-
theory interpretation. The static solutions of the domain-wall equations
include gravitational avatars of the Faraday cage, black holes with negative
specific heat, and an intriguing phenomenon of suspended vacuum bubbles
corresponding to an exotic interface/anti-interface fusion. Part of our
analysis overlaps with recent work by Simidzija and Van Raamsdonk but the
interpretation is different.

$\,{}^{1}$ Laboratoire de Physique de l’École Normale Supérieure,
CNRS, PSL Research University and Sorbonne Universités
24 rue Lhomond, 75005 Paris, France

###### Contents

  1. 1 Introduction
  2. 2 Finite-temperature AdS/CFT
    1. 2.1 Coordinates for the AdS3 black string
    2. 2.2 Hawking-Page transition
  3. 3 Topology of slices
  4. 4 Solving the wall equations
    1. 4.1 Matching conditions
    2. 4.2 Solution near the boundary
    3. 4.3 Critical tensions
    4. 4.4 Turning point and horizon
  5. 5 Phases: cold, hot $\&$ warm
  6. 6 Equations of state
    1. 6.1 High-$T$ phase
    2. 6.2 Low-$T$ phase(s)
    3. 6.3 Warm phases
  7. 7 Phase transitions
    1. 7.1 ICFT interpretation
    2. 7.2 Sweeping transitions
    3. 7.3 Warm-to-hot transitions
  8. 8 Exotic fusion and bubbles
  9. 9 Phase diagrams
    1. 9.1 Defect CFT
    2. 9.2 Non-degenerate vacua
    3. 9.3 Unstable black holes
  10. 10 Outlook
  11. A Renormalized on-shell action
  12. B Opening arcs as elliptic integrals
  13. C Sweeping is continuous
  14. D Bubbles exist

##  1 Introduction

Begining with the classic paper of Coleman and De Lucia [1] there have been
many studies of thin gravitating domain walls between vacua with different
values of the cosmological constant. Such walls figure in models of localized
gravity [2, 3, 4], in holographic duals of conformal interfaces [5, 6, 7], in
efforts to embed inflation in string theory by studying dynamical bubbles [8,
9, 10, 11], and more recently, following the ideas in refs. [12, 13, 14], in
toy models of black hole evaporation [15, 16, 17, 19, 18, 20, 21, 22, 23].
Besides being a simple form of matter coupled to gravity, domain walls are
also a key ingredient [24] in effective descriptions of the string-theory
landscape – see [25, 26, 27] for some recent discussions of domain walls in
this context.111The above list of references is nowhere nearly complete. It is
only meant as an entry to the vast and growing literature in these subjects.

In this paper we study a thin static domain wall between Anti-de Sitter (AdS)
vacua, anchored at the conformal boundary of spacetime. If a dual holographic
setup were to exist, it would have two conformal field theories, CFT1 and
CFT2, separated by a conformal interface [5, 6, 7]. We will calculate the
phase diagram of the system as function of the AdS radii, the tension of the
wall and the boundary data. Several parts of this analysis have appeared
before (see below) but the complete phase diagram has not, to the best of our
knowledge, been worked out. We will be interested in phenomena that are hard
to see at weak CFT coupling. A broader motivation, as in much of the AdS/CFT
literature, is understanding how the interior geometry is encoded on the
boundary and vice versa, but we will only briefly allude to this question in
the present work.

Our analysis is classical in gravity. Different phases are distinguished by
the presence/absence of a black hole and by the fate of inertial observers,
either those moving freely in the bulk or those bound to the wall. Inertial
observers are a guiding fixture of the analysis, not emphasized in earlier
works. In the high-temperature or ‘hot’ phase all inertial observers
eventually cross the black-hole horizon. In intermediate or ‘warm’ phases the
wall avoids the horizon, and may also shield bulk observers from falling
inside. Such two-center warm solutions are gravitational avatars of the
Faraday cage. Finally what differentiates ‘cold’ horizonless phases is whether
all timelike geodesics intersect inevitably the wall, or not.

Besides the domain wall and the black hole, the third actor in the problem is
the center of global AdS where an inertial observer may rest. The rich phase
diagram is the result of several competing forces: The attraction of the AdS
trap, with or without a black hole in its center, the tension of the wall, and
the repulsion between the domain wall and massive particles. In addition to
the first-order Hawking-Page transition [28] that signals the formation of a
black hole, new phase transitions occur when the wall sweeps an AdS rest point
or when part of it enters the horizon, see figure 1. One of our conclusions is
that the latter transition is always first-order.

Figure 1: A domain wall sweeping the center of the ‘false AdS vacuum’ where an
inertial observer could rest (left), or entering the horizon of a black hole
(right).

We work in 2+1 dimensions because calculations can be performed in closed
form. We expect, however, qualitative features of the phase diagram to carry
over to higher dimensions. For simplicity we consider a single type of non-
intersecting wall, and only comment briefly on extended models that allow
junctions of different types of wall.

The capture of the wall by the black hole is related to a transition analyzed
in a very interesting recent paper by Simidzija and Van Raamsdonk [29], see
also [30, 31]. These authors consider time-dependent spherically-symmetric
walls whose intersections with the conformal boundary describe Hamiltonian
quenches in the dual field theory. In this setting the boundary is the
infinite cylinder, with a stripe describing the evolution of the dual CFT
between the quench and ‘unquench’ times. By contrast, we are interested in
equilibrium configurations. This means that the domain-wall geometry is
static, and the stripes on the conformal boundary point in the time direction.
Furthermore, the boundary is not the cylinder but an orthogonal torus, adding
an extra parameter to the problem.

Although the interpretation is different, many of our formulae are
nevertheless related to those of refs. [30, 29, 31] by swapping the roles of
boundary space and time (thereby also swapping the BTZ geometry with thermal
AdS3, see section 2). This is fortuitous to 2+1 dimensions and does not carry
over to higher dimensions.

The gravitational action of the thin-wall model reads

$\displaystyle\,\hskip-42.67912ptI_{\rm
gr}=-\frac{1}{2}\int_{{\mathbb{S}}_{1}}d^{3}x\sqrt{g_{1}}\,(R_{1}+\frac{2}{\ell_{1}^{2}})\hskip-5.69054pt$
$\displaystyle-$
$\displaystyle\hskip-5.69054pt\frac{1}{2}\int_{{\mathbb{S}}_{2}}d^{3}x\sqrt{g_{2}}\,(R_{2}+\frac{2}{\ell_{2}^{2}})$
(1.1)
$\displaystyle\hskip-51.21495pt+\,\lambda\int_{\mathbb{W}}d^{2}s\sqrt{\hat{g}_{w}}\,+\,{\rm
GHY\ terms}\,+\,{\rm ct.}\ ,\ $

where $R_{j}(g_{j})$ are the Ricci scalars of the spacetime slices
$\mathbb{S}_{j}$ on either side of the wall, and $\hat{g}_{w}$ is the induced
metric on the wall’s worldvolume. The Gibbons-Hawking-York terms and
counterterms are given in appendix A. The action $I_{\rm gr}$ depends on three
parameters: the two AdS radii $\ell_{1},\ell_{2}$ and the wall tension
$\lambda$. The radii are related to the central charges of the dual CFTs [32],
and the tension to the entropy [33, 29] and to the energy-transport
coefficient [34] of the dual interface. Static solutions exist for

$\displaystyle\lambda_{\rm min}<\lambda<\lambda_{\rm max}\,,\qquad{\rm
with}\quad\lambda_{\rm max}={1\over\ell_{1}}+{1\over\ell_{2}}\,;\ \
\lambda_{\rm min}=\bigl{|}{1\over\ell_{1}}-{1\over\ell_{2}}\bigr{|}\ .$ (1.2)

The classical phase diagram depends on two dimensionless ratios of the above
(e.g. $\ell_{2}/\ell_{1}:=b\,$ and $\lambda\ell_{2}:=\kappa$) and on the two
parameters that determine the conformal class of the striped boundary torus,
e.g. $\tau_{1}:=TL_{1}$ and $\tau_{2}:=TL_{2}$, see figure 2. Without loss of
generality we assume henceforth that $\ell_{2}\geq\ell_{1}$, i.e. that
$\mathbb{S}_{1}$ is the true-vacuum slice.

An important question is how much of this analysis has a chance to carry over
to top-down holographic models, where back-reacting domain walls are not
thin.222Many examples of supergravity domain walls have been worked out in the
literature, a representative sample is [35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. None of these solutions depends,
however, on non-trivial (non-Lagrangian) boundary data, indeed all but one are
scale-invariant AdSn fibrations. The size of the horizon and the number of
stable rest points are order parameters that can be also defined for thick
walls, but a sharp criterion, that decides whether a thick domain wall enters
or avoids the horizon is hard to imagine. Nevertheless, the field-theory
interpretation of the transition suggests that such an order parameter may
exist, as we will explain in section 7.1.

It is worth stressing that the thin-wall model is a minimal gravity dual of
I(nterface) CFT in the same way that pure Einstein theory is a minimal dual
for homogeneous CFT. The model captures the two universal boundary operators –
the energy-momentum tensors on either side of the interface, as well as their
combination refered to as the displacement operator [54]. Top-down models have
many more operators, some of which correspond to internal excitations of the
domain wall.

Note also that boundaries, or end-of-the-world branes (EWBs), can be
considered as a limit of domain walls when one side becomes the zero-radius
AdS spacetime [55]. In this sense holographic B(oundary) CFT [56, 57] can be
recovered from holographic ICFT, though the limit is subtle and should be
handled with care.

A last remark concerns the Ryu-Takayanagi surfaces [58, 59] that delimit the
entanglement wedges of boundary subregions [60, 62, 61].333See e.g. [63, 64]
for reviews. It is clearly of interest to study if these surfaces intersect
the domain wall, as is done for BCFT in ref. [21, 22]. We hope to return to
this question elsewhere.

The plan of the paper and a summary of our results follows. In section 2 we
review some standard facts about AdS3/CFT2 at finite temperature. The wall
separates space in two slices that we color green (true-vacuum side) and pink
(false-vacuum side). Each of these comes in one of four topological types
described in section 3. In section 4 we solve the matching equations obeyed by
a thin static domain wall, which we parametrize conveniently by the blueshift
metric factor $g_{tt}$. This section overlaps substantially with ref. [29] via
double Wick rotation – a trick specific to 2+1 dimensions as earlier noted.

In section 5 we start analyzing the solutions. By studying the turning point
of the wall we classify the possible phases, i.e. the topologically distinct
solutions. We rule out in particular centerless geometries, in which no
inertial observer can avoid the wall, and solutions with two black holes whose
merging is prevented by the wall.

In section 6 we write down the equations of state that characterize these
phases. They relate the canonical variables $\tau_{1},\tau_{2}$ to
microcanonical variables that are natural for describing the interior
geometry. We also point out the relevance of a critical tension
$\lambda_{0}=\sqrt{\lambda_{\rm max}\lambda_{\rm min}}\,$ below which the hot
solution disappears from a region of parameter space.

In section 7 we compute the critical lines for sweeping transitions in both
the cold and the warm phases, and we show that the warm-to-hot transition is
always first-order – the domain wall cannot be lowered continuously to the
black horizon. The proof requires a detailed analysis of the region
$\mu\approx 1$ with $\lambda\leq\lambda_{0}$, where the hot and a warm
solution come arbitrarily close. We also point out some puzzles regarding the
ICFT interpretation of these phase transitions.

Section 8 presents a striking phenomenon: bubbles of the true vacuum suspended
from a point on the conformal boundary of the false vacuum. This is surprising
from the perspective of ICFT, since it implies that the fusion of an interface
and anti-interface does not produce the trivial (identity) defect, as expected
from free-field calculations [65], but an exotic defect that generates
spontaneously a new scale.

In section 9 we present numerical plots of the complete phase diagram in the
canonical ensemble, for different values of the Lagrangian parameters
$\lambda,\ell_{j}$. These plots confirm our earlier conclusions. We point out
a critical threshold $b=\ell_{2}/\ell_{1}=3$, probably an artifact of the
thin-wall approximation, above which black-hole solutions on the false-vacuum
side of the wall cease to exist. We also exhibit coexisting black-hole
solutions, including black holes with negative specific heat. This parallels
the discussion of black holes in deformed JT gravity in ref. [66].

Section 10 contains concluding remarks. In order not to interrupt the flow of
the arguments we relegate some detailed calculations to four appendices.

##  2 Finite-temperature AdS/CFT

For completeness we recall here some standard facts about AdS/CFT at finite
temperature in three spacetime dimensions. While doing this we will be also
setting notation and conventions.

###  2.1 Coordinates for the AdS3 black string

The metric of the static AdS3 black string in 2+1 dimensions is

$\displaystyle ds^{2}_{\rm BS}\,=\,{\ell^{2}dr^{2}\over
r^{2}-M\ell^{2}}-({r^{2}}-M\ell^{2})\,dt^{2}+r^{2}dx^{2}\,,$ (2.1)

where $M>0$, $\ell$ is the radius of AdS3 and the horizon at $r^{\rm
H}=\ell\sqrt{M}$ has temperature $T=\sqrt{M}/2\pi$. Length units on the
gravity side are such that $8\pi G=1$. The dual CFT lives on the AdS3
boundary, at $r={1/\epsilon}\to\infty$, with conformal coordinates
$x^{\pm}\equiv x\pm t\,\in\mathbb{R}^{2}$. Its central charge is $c=12\pi\ell$
[32].

The holographic dictionary becomes transparent in Fefferman-Graham
coordinates, in which any asymptotically-Poincaré AdS3 solution takes the
following form [67, 68]

$ds^{2}={\ell^{2}dz^{2}\over z^{2}}+{1\over z^{2}}\Bigl{(}dx^{+}+{\ell
z^{2}}h_{-}dx^{-}\Bigr{)}\Bigl{(}dx^{-}+{\ell z^{2}}h_{+}dx^{+}\Bigr{)}\ .$
(2.2)

Here $h_{\pm}=\langle T_{\pm\pm}\rangle$ are the expectation values of the
canonically-normalized energy-momentum tensor of the CFT. Note that it is a
special feature of 2+1 dimensions that the Fefferman-Graham expansion stops at
order $z^{2}$. For the static black string, $h_{+}=h_{-}=M\ell/4\,$ giving
$\langle T_{tt}\rangle=M\ell/2=(c/6)\pi T^{2}$. This is indeed the energy
density of finite-temperature CFT in two dimensions. The relation between $z$
and $r$ is

$\displaystyle{r}={1\over z}+{M\ell^{2}z\over 4}\ \ \Longleftrightarrow\ \
z={2\over M\ell^{2}}\left(r-\sqrt{{r^{2}}-M\ell^{2}}\,\right)\ ,$ (2.3)

and the black-string metric in the $(z,t,x)$ coordinates reads

$\displaystyle ds^{2}_{\rm BS}\,=\,\ell^{2}{dz^{2}\over z^{2}}-\Bigl{(}{1\over
z}-{M\ell^{2}z\over 4}\Bigr{)}^{2}dt^{2}+\Bigl{(}{1\over z}+{M\ell^{2}z\over
4}\Bigr{)}^{2}dx^{2}\ .$ (2.4)

Note that $z$ covers only the region outside the horizon ($r>r^{\rm H}$) and
that near the conformal boundary $z\approx r^{-1}$.

A last change of coordinates worth recording, even though we will not use it
in this paper, is the one that maps $(z,t,x)$ to the standard Poincaré
parametrization of AdS3. Such a map is guaranteed to exist because all
constant-negative-curvature Einstein manifolds in three dimensions can be
obtained from AdS3 by identifications and excisions. For the case at hand
444The general transformation, for an arbitrary (conformally-flat) boundary
metric and vacuum expectation value $\langle T_{ab}\rangle$, is given in refs.
[69, 70, 71]. the transformation reads

$\displaystyle w^{\pm}=\zeta^{\pm}\left({4-M\ell^{2}z^{2}\over
4+M\ell^{2}z^{2}}\right)\,,\quad y={4z(M\zeta^{+}\zeta^{-})^{1/2}\over
4+M\ell^{2}z^{2}}\quad{\rm with}\quad\zeta^{\pm}=e^{\sqrt{M}(x\pm t)}\ .\ \ $
(2.5)

The reader can check that in these coordinates the metric (2.4) becomes

$\displaystyle ds^{2}_{\rm BS}\,=\,{\ell^{2}dy^{2}+dw^{+}dw^{-}\over y^{2}}\
,$ (2.6)

i.e. the standard Poincaré form of AdS3 as advertized. Outside the black
horizon ($M\ell^{2}z^{2}<4$) the coordinates $x^{\pm}\equiv x\pm t$ cover only
a Rindler wedge of the $w^{\pm}$ plane.

Since we will be refering to this later, let us verify the well-known fact
that no inertial observer can avoid crossing the horizon. In the proper-time
parametrization of the trajectory a simple calculation gives

$\displaystyle\ell^{2}\,{{\ddot{r}}\over r}=-1-M\ell^{2}{\dot{x}}^{2}$ (2.7)

where dots denote derivatives with respect to proper time. Since $M$ is
positive there is no centrifugal acceleration QED. Note that this is a
property of the asymptotically AdS black hole, not shared by asymptotically
flat black holes in higher dimension.

###  2.2 Hawking-Page transition

From the perspective of the CFT, the temperature $T$ is the only dimensionful
parameter of the infinite-black-string solution. By a scale transformation we
can always set it to one. Things get more interesting if the black string is
compactified, $x\sim x+L$, thereby converting the solution (2.1) to the non-
spinning BTZ black hole [72, 73]. In addition to the central charge $c$, there
is now a new dimensionless parameter $LT$. In the Euclidean geometry
$\tau=i\,LT$ is the complex-structure modulus of the boundary torus.

At the critical temperature $T_{\rm HP}=1/L$ the theory undergoes a Hawking-
Page phase transition [28, 74]. This is seen by comparing the action of the
two competing saddle points for the interior geometry: 555Thermal AdS3 and
Euclidean BTZ are part of an infinite SL(2,$\mathbb{Z}$) orbit of
gravitational instantons, [74, 75] but they are the only dominant ones for an
orthogonal torus. Their regularized Euclidean actions are $I_{\rm
TAdS}=-2\pi^{2}\ell/|\tau|$ and $I_{\rm BTZ}=-2\pi^{2}\ell|\tau|$, see below.
(i) the Euclidean BTZ black hole, and (ii) thermal AdS3, whose metric is the
same as (2.1) but with $M$ replaced by $\tilde{M}=-(2\pi/L)^{2}$. The
difference of free energies of these two saddle points reads

$\displaystyle F_{\rm BTZ}-F_{\rm TAdS}=-2\pi^{2}\ell\,\bigl{(}LT^{2}-{1\over
L}\bigr{)}\ .$ (2.8)

Thus thermal AdS3 is the dominant solution when $LT<1$, while the BTZ black
hole dominates when $LT>1$.

Thermal AdS3 and the Euclidean BTZ black hole differ in the choice of boundary
cycle that becomes contractible in the interior geometry. The periodicity
conditions, respectively $x\sim x+2\pi/|\tilde{M}|^{1/2}$ and $t_{E}\sim
t_{E}+2\pi/M^{1/2}$, ensure regularity when this contractible cycle
degenerates. Below we will encounter situations in which either the center of
AdS or the BTZ horizon are excised. In such cases the regularity conditions
can be relaxed.

One other comment in order here concerns the difference of free energies, eq.
(2.8). The renormalized gravitational action $I_{\rm gr}$ (where $I_{\rm
gr}=F/T$) is calculated for the general interface model in appendix A. In the
case of a homogeneous CFT one can, however, obtain the answer faster. Indeed,
from the Fefferman-Graham form of the metric, eq. (2.2), one reads the energy
of the CFT state,

$\displaystyle E=\,L\langle T_{tt}\rangle\,=\,{1\over 2}\ell ML\ .$ (2.9)

For $M=(2\pi T)^{2}$ this is the internal energy of the high-temperature
state, as previously noted, and for $M=-(2\pi/L)^{2}$ it is the Casimir energy
of the vacuum. The corresponding free energies obey the thermodynamic identity

$\displaystyle E=-T^{2}{\partial\over\partial T}\Bigr{(}{F\over T}\Bigl{)}\ .$
(2.10)

Eqs. (2.9) and (2.10) determine $F$ up to a term linear in $T$. This can be
argued to vanish both at low $T$, since the ground state has no entropy, and
at large $L$ since $F$ must be extensive. The final result is eq. (2.8).

Let us finally note that since in empty AdS the mass $M$ is negative, there is
a centrifugal contribution in eq.(2.7). An inertial observer may thus either
rest at, or orbit around the center $r=0$. But in the centerless slices that
we are about to discuss, all inertial observers hit the wall.

##  3 Topology of slices

Consider now two conformal field theories, CFT1 and CFT2, coexisting at
thermal equlibrium on a circle. This is illustrated in figure 1. The
horizontal and vertical axes parametrize space and Euclidean time. In addition
to the central charges $c_{1},c_{2}$, and to the properties of the interfaces
between the two CFTs, there are three more parameters in this system: the
sizes $L_{1},L_{2}$ of the regions in which each CFT lives, and the
equilibrium temperature $T$. This gives two dimensionless parameters, which we
can choose for instance to be $\tau_{1}:=TL_{1}$ and $\tau_{2}:=TL_{2}$.

Figure 2: The finite-temperature interface CFT at the AdS boundary. Both space
and Euclidean time are compact, so the depicted surface is an orthogonal
torus.

The gravity dual of this ICFT features domain walls, i.e. strings in 2+1
dimensions, 666We reserve the word “interface” for the CFT, and “domain wall”
or “string” for gravity. Interfaces are anchor points of domain walls on the
AdS boundary. The string of our bottom-up model should not be confused with
the black string responsible for the interior horizon. In top-down
supergravity embeddings the two types of string may be however
interchangeable.  anchored at the interfaces on the conformal boundary. We
will make the simplifying assumption that the two domain walls differ only in
orientation, and can join smoothly in the interior of spacetime. Extended
models allowing junctions of different domain walls are very interesting but
they are beyond our present scope. We will comment briefly on them in a later
section.

The green and pink boundary regions of fig. 2, in which CFT1 and CFT2 live,
extend in the interior to slices of gravitational solutions that belong to one
of several topological types. These are illustrated for the green slice in
figure 3. Each slice is either part of thermal AdS3 with the center, marked by
a grey flag, included (E1) or excised (E2), or part of the BTZ geometry with
the horizon excised (E2′), included (H1) or intersecting the domain wall (H2).
The same options are available for the pink spacetime slice.777 The Euclidean
manifold is a (thermal) circle fibration over the fixed-time slice drawn in
our figures. The fiber degenerates at the horizon, when one exists.

Figure 3: The different types of space-time slice described in the main text.
The actual slice is colored in green, the complementary region is excised. The
letters ‘E’ and ‘H’ stand for ‘empty’ and ‘horizon’, and the grey flag denotes
the rest point of an inertial observer. Note that since this is excised in E2,
a conical singularity in its place is permitted. The centerful slice E1 can
act as a gravitational Faraday cage.

As was explained in section 2, we may adopt the unified parametrization (2.1)
for all types of slice, with $M$ negative for the slices of type E1 and E2 of
global AdS3, and positive for the slices of type E2′, H1 and H2 of the BTZ
spacetime. We are interested in static configurations which are dual to
equilibrium CFT states, so time is globally defined and has fixed imaginary
period $t_{E}\sim t_{E}+1/T$. The coordinates $(x,r)$ on the other hand need
not be continuous across the wall. We therefore write the spacetime metric in
terms of two coordinate charts,

$\displaystyle ds^{2}\,=\,{\ell_{j}^{2}dr_{j}^{2}\over
r_{j}^{2}-M_{j}\ell_{j}^{2}}-({r_{j}^{2}}-M_{j}\ell_{j}^{2})\,dt^{2}+r_{j}^{2}dx_{j}^{2}\qquad{\rm
with}\quad(x_{j},r_{j})\in\Omega_{j}\,,\,\,$ (3.1)

where $\Omega_{1}$ is the range of coordinates for the green slice and
$\Omega_{2}$ the range of coordinates for the pink slice. These ranges are
delimited as follows:

  * •

by the embeddings of the static wall in the two coordinate systems,
$\\{x_{j}(\sigma),r_{j}(\sigma)\\}$, where $\sigma$ parametrizes the wall ;

  * •

by the horizon whenever the slice contains one, i.e. in cases H1 and H2 ;

  * •

by the cutoff surface $r_{j}\approx 1/\epsilon\to\infty$ .

The mass parameters of the slices, $M_{1}$ and $M_{2}$, are in general
different. Regularity requires however that

$\displaystyle M_{j}=(2\pi T)^{2}\qquad{\text{for\ slices\ \ {\small{H1,
H2}}}}$ (3.2)

that include a horizon, whereas $M_{j}$ is unconstrained for the other slice
types. Furthermore, for a slice of type E1 in which the spatial circle is
contractible, interior regularity fixes the periodicity of $x$,

$\displaystyle x_{j}\sim x_{j}+2\pi/\sqrt{-M_{j}}\qquad{\text{ in\ case\
{\small E1}}}\ .$ (3.3)

For E2, E2′ and H2 the coordinate $x_{j}$ is not periodic, while for H1 its
period, proportional to the horizon size, is unconstrained.

Since the horizon is a closed surface, a green slice of type H2 can only be
paired with a pink slice of the same type. This is the topology that dominates
at very high temperature when the black hole eats up most of the bulk
spacetime. As the temperature is lowered different pairs of the remaining
slice types dominate. The pairs that correspond to actual solutions of the
domain-wall equations will be determined in section 5.

For the time being let us comment on the differences between the horizonless
slices in the top row of fig. 3. What distinguishes E1 from the other two is
the existence of the AdS center (or ‘refuge’) where an inertial observer may
sit at rest. By contrast, in the slices of type E2 and E2′ all inertial
observers will inevitably hit the domain wall as explained in the previous
subsection. This discontinuous behavior differentiates the phases on either
side of a sweeping transition.

Note that there is no topological difference between the slices of type E2 and
E2′, which is why we distinguish them only by a prime. These slices differ
only in the sign of $M_{j}$, or equivalently the energy density per degree of
freedom in the boundary theory. Together E2 and E2′ describe a continuum
($-\infty<M_{j}<\infty$) of horizonless slices with no rest point.

##  4 Solving the wall equations

In this section we find the general solution of the domain wall equations in
terms of the mass parameters $M_{1},M_{2}$, the AdS radii $\ell_{1},\ell_{2}$,
and the tension of the wall $\lambda$. That a solution always exists for any
bulk geometries is a special feature of 2+1 dimensions, as is the double Wick
rotation that relates this part of our analysis to ref. [29].

###  4.1 Matching conditions

The matching conditions at a thin domain wall have appeared in numerous
studies of cosmology and AdS/CFT. They are especially simple in the case at
hand, where the wall/string is static and is characterized only by its
tension. Matching the induced worldsheet metric of the two charts (3.1) gives
one algebraic and one first-order differential equation for the embedding
functions $x_{1}(\sigma),r_{1}(\sigma),x_{2}(\sigma)$ and $r_{2}(\sigma)$:

$\displaystyle{r_{1}^{2}}-M_{1}\ell_{1}^{2}\,=\,{r_{2}^{2}}-M_{2}\ell_{2}^{2}\,\,\equiv\,f(\sigma)$
(4.1) $\displaystyle{\rm and}\qquad\ \
f^{-1}{\ell_{1}^{2}\,r_{1}^{\prime\,2}}+r_{1}^{2}\,x_{1}^{\prime\,2}=f^{-1}{\ell_{2}^{2}\,r_{2}^{\prime\,2}}+r_{2}^{2}\,x_{2}^{\prime\,2}\,\equiv\,g(\sigma),$
(4.2)

where the prime denotes a derivative with respect to $\sigma$. We have defined
the auxiliary functions $f$ and $g$ in terms of which the induced worldsheet
metric reads
$d{\hat{s}}^{2}|_{\mathbb{W}}=-f(\sigma)dt^{2}+g(\sigma)d\sigma^{2}$. A third
matching equation888The Israel-Lanczos matching conditions are matrix
equations,
$[K_{\alpha\beta}]-[\textrm{tr}K]\hat{g}_{\alpha\beta}=\lambda\,\hat{g}_{\alpha\beta}$,
where $K_{\alpha\beta}$ is the extrinsic curvature, $\hat{g}_{\alpha\beta}$
the induced metric, and brackets denote the discontinuity across the wall.
Only the trace part of this equation is non-trivial. The traceless part of $K$
is automatically continuous by virtue of the momentum constraints
$D^{\alpha}K_{\alpha\beta}-D_{\beta}K=0\,,$ where $D_{\alpha}$ is the
covariant derivative with respect to the induced metric. Equation (4.3) is the
$tt$ component of the matrix equation.  expresses the discontinuity of the
extrinsic curvature in terms of the tension, $\lambda$, of the wall [76, 77].
It can be written as follows :

$\displaystyle{r_{1}^{2}x_{1}^{\prime}\over\ell_{1}}+{r_{2}^{2}x_{2}^{\prime}\over\ell_{2}}=\lambda\,\sqrt{fg}\
.$ (4.3)

Our convention is that $\sigma$ increases as one circles $\Omega_{j}$ in the
$(x_{j},r_{j})$ plane clockwise. Other conventions introduce signs in front of
the two terms on the left-hand side of this equation.

Eqs. (4.1)–(4.3) are three equations for four unknown functions, but one of
these functions can be specified at will using the string-reparametrization
freedom. Furthermore the equations only involve first derivatives of $x_{j}$,
so the integration constants are irrelevant choices of the origin of the
$x_{j}$ axes. For given $\ell_{1},\ell_{2}$ and $\lambda$, the wall embedding
functions $x_{j}(r_{j})$, are thus uniquely determined by the parameteres
$M_{1}$ and $M_{2}$. Different choices of ($M_{1},M_{2}$) may correspond,
however, to the same boundary data ($L_{1},L_{2},T$). These are the competing
phases of the system.

###  4.2 Solution near the boundary

Near the conformal boundary, $r_{j}\to\infty$, the parameters $M_{j}$ can be
neglected and the worldsheet metric asymptotes to AdS2 by virtue of scale
invariance. Explicitly the solution reads [78]

$\displaystyle r_{1}\approx r_{2}\,,\qquad
x_{j}\,\approx\,-\ell_{j}\,(\tan\theta_{j})\,r_{j}^{-1}\ ,$ (4.4)

where $\theta_{j}$ is the angle in the $(x_{j}\,,\ell_{j}/r_{j})$ plane
between the normal to the boundary and the interface, see figure 4. The
matching eqs. (4.2) and (4.3) relate these angles to the bulk radii $\ell_{j}$
and to the string tension $\lambda$:

$\displaystyle{\ell_{1}\over\cos\theta_{1}}={\ell_{2}\over\cos\theta_{2}}\,\equiv\,\ell_{w}\qquad{\rm
and}\qquad\tan\theta_{1}+\tan\theta_{2}=\,{\lambda}\,\ell_{w}\ ,$ (4.5)

where $\ell_{w}$ is the radius of the AdS2 worldsheet, and
$-{\pi/2}<\theta_{j}<{\pi/2}$ .

Figure 4: Near the AdS boundary in the $(x_{j},\ell_{j}/r_{j})$ plane the
string is a straight line subtending an angle $\theta_{j}$ with the normal.

Without loss of generality we assume that $\ell_{1}\leq\ell_{2}$, so that CFT1
has the smaller of the two central charges. Its gravity dual has the lower
vacuum energy, i.e. the green slice is the ‘true vacuum’ side of the domain
wall and the pink slice is the ‘false-vacuum’ side. The first eq. (4.5) then
implies that $|\tan\theta_{1}|\geq|\tan\theta_{2}|$ and, provided that the
tension is positive, the second eq. (4.5) implies that $\theta_{1}>0$. The
sign of $\theta_{2}$, on the other hand, depends on the precise value of
$\lambda$. Expressing the tangents in terms of cosines brings indeed this
equation to the form

$\displaystyle({1\over\ell_{1}^{\,2}}-{1\over\ell_{w}^{\,2}})^{1/2}\,+\,\varepsilon\,\sqrt{{1\over\ell_{2}^{\,2}}-{1\over\ell_{w}^{\,2}}}\,=\,\lambda\
\qquad{\rm with}\quad\varepsilon={\rm sign}(\theta_{2})\ .$ (4.6)

Since $\lambda$ is real we must have $\,\ell_{2}<\ell_{w}<\infty$. Furthermore
to each value of the worldsheet radius $\ell_{w}$ there correspond two values
of the tension $\lambda$, depending on sign($\theta_{2}$). Explicitly, 999It
was argued in ref. [79] that the walls in the $\lambda<\lambda_{0}$ range are
unstable. But the radius instability in this reference reduces the action by
an amount proportional to the infinite volume of AdS2 and does not correspond
to a normalizable mode. The only normalizable mode of the wall in the thin-
brane model corresponds to the displacement operator which is an irrelevant
(dimension = 2) operator [34].

$\displaystyle\lambda_{\rm min}<\lambda<\lambda_{0}\quad\ {\rm for}\
\varepsilon=-\qquad{\rm and}\qquad\lambda_{0}<\lambda<\lambda_{\rm
max}\quad{\rm for}\ \varepsilon=+\ ,$ (4.7)

where the three critical tensions read

$\displaystyle\lambda_{\rm min}={1\over\ell_{1}}-{1\over\ell_{2}}\
,\qquad\lambda_{\rm max}={1\over\ell_{1}}+{1\over\ell_{2}}\
,\qquad\lambda_{0}=\sqrt{\lambda_{\rm max}\lambda_{\rm min}}\ .$ (4.8)

Let us pause here to discuss the significance of these critical tensions.

###  4.3 Critical tensions

The meaning of the critical tensions $\lambda_{\rm min}$ and $\lambda_{\rm
max}$ has been understood in the work of Coleman-De Lucia [1] and Randall-
Sundrum [2]. Below $\lambda_{\rm min}$ the false vacuum is unstable to
nucleation of true-vacuum bubbles, so the two phases cannot coexist in
equilibrium.101010Ref. [1] actually computes the critical tension for a domain
wall separating Minkowski from AdS spacetime. Their result can be compared to
$\lambda_{\rm min}$ in the limit $\ell_{2}\to\infty$.  The holographic
description of such nucleating bubbles raises fascinating questions in its own
right, see e.g. refs.[8, 9, 10]. It has been also advocated that expanding
true-vacuum bubbles could realize accelerating cosmologies in string theory
[11]. Since our focus here is on equilibrium configurations, we will not
discuss these interesting issues any further.

The maximal tension $\lambda_{\rm max}$ is a stability bound of a different
kind.111111Both the $\lambda=\lambda_{\rm max}$ and the $\lambda=\lambda_{\rm
min}$ walls can arise as flat BPS walls in supergravity theories coupled to
scalars [80, 81]. These two extreme types of flat wall, called type II and
type III in [81], differ by the fact that the superpotential avoids,
respectively passes through zero as fields extrapolate between the AdS vacua
[82]. For $\lambda>\lambda_{\rm max}$ the two phases can coexist, but the
large tension of the wall forces this latter to inflate [4]. The phenomenon is
familiar for gravitating domain walls in asymptotically-flat spacetime [83],
i.e. in the limit $\ell_{1},\ell_{2}\to\infty$.

The meaning of $\lambda_{0}$ is less clear, its role will emerge later. For
now note that it is the turning point at which the worldsheet radius
$\ell_{w}(\lambda)$ reaches its minimal value $\ell_{2}$. Note also that the
range $\lambda_{\rm min}<\lambda<\lambda_{0}$ only exists for non-degenerate
AdS vacua, that is when $\ell_{1}$ is strictly smaller than $\ell_{2}$.

Since the wall in this minimal model is described by a single parameter, its
tension $\lambda$, all properties of the dual interface depend on it. These
include the interface entropy, and the energy-transport coefficients. The
entropy or $g$-factor, computed in [33, 29], reads 121212There are many
calculations of the boundary, defect, and interface entropy in a variety of
holographic models – a partial list is [84, 56, 57, 85, 86, 87]. The formula
for arbitrary left and right central charges, which we rederive below, was
found in ref.[29].

$\displaystyle{\rm log}\,g_{\rm I}\ =\ 2\pi\ell_{1}\ell_{2}\left[\lambda_{\rm
max}\,{\rm tanh}^{-1}\Bigl{(}{\lambda\over\lambda_{\rm
max}}\Bigr{)}-\lambda_{\rm min}\,{\rm tanh}^{-1}\Bigl{(}{\lambda_{\rm
min}\over\lambda}\Bigr{)}\right]\ .$ (4.9)

It varies monotonically between $-\infty$ and $\infty$ as $\lambda$ varies
inside its allowed range (4.7).

The fraction of transmitted energy for waves incident on the interface from
the CFT1 side, respectively CFT2 side, was computed in [34] with the result
(reexpressed here in terms of critical tensions)

$\displaystyle{\cal T}_{1\to 2}={\lambda_{\rm max}+\lambda_{\rm
min}\over\lambda_{\rm max}+\lambda}\,,\qquad{\cal T}_{2\to 1}={\lambda_{\rm
max}-\lambda_{\rm min}\over\lambda_{\rm max}+\lambda}\ .$ (4.10)

Note that using $\lambda_{\rm max}+\lambda_{\rm min}=2/\ell_{1}$ and
$\lambda_{\rm max}-\lambda_{\rm min}=2/\ell_{2}$, one can check that these
coefficients obey the detailed-balance condition $c_{1}{\cal T}_{1\to
2}=c_{2}{\cal T}_{2\to 1}$. The larger of the two transmission coefficients
reaches the unitarity bound when $\lambda=\lambda_{\rm min}$, and both
coefficients attain their minimum when $\lambda=\lambda_{\rm max}$. Total
reflection (from the false-vacuum to the true-vacuum side) is only possible if
$\ell_{1}/\ell_{2}\to 0$, i.e. when the “true-vacuum” CFT1 is almost entirely
depleted of degrees of freedom relative to CFT2.

Using eqs. (4.8) and the Brown-Henneaux formula one can express the central
charges $c_{1,2}$ in terms of the critical tensions $\lambda_{\rm min}$ and
$\lambda_{\rm max}$. As we just saw, $\lambda$ parametrizes two key properties
of the interface. The triplet ($\lambda_{\rm min},\,\lambda_{\rm
max},\,\lambda)$ of parameters in the gravitational action defines therefore
the basic data of the putative dual ICFT.

###  4.4 Turning point and horizon

We will now derive the general solution of the equations (4.1) - (4.3), and
then relate the geometric parameters $M_{j}$ to the data $(T,L_{j})$ of the
boundary torus shown in fig. 2.

A convenient parametrization of the string outside any black horizons is in
terms of the blueshift factor of the worldsheet metric, eq.(4.1),

$\displaystyle f(\sigma)=\sigma\ \ \Longrightarrow\ \
r_{j}=\sqrt{\sigma+M_{j}\ell_{j}^{\,2}}\ .$ (4.11)

In this parametrization
$d{\hat{s}}^{2}|_{\mathbb{W}}=-\sigma\,dt^{2}+g(\sigma)d\sigma^{2}$. Let
$\sigma_{+}$ correspond to the minimal value of the blueshift, this is either
zero or positive. If $\sigma_{+}=0$ the string enters the horizon. If on the
other hand $\sigma_{+}>0$ then, as we will confirm in a minute, this is the
turning point of $r_{j}(\sigma)$ where both $x_{1}^{\prime}$ and
$x_{2}^{\prime}$ diverge.

A static string has (at most) one turning point, and is symmetric under
reflection in the axis that passes through the centers of the boundary arcs,
131313In ref.[29] this corresponds to the time-reflection symmetry of the
instanton solutions. as illustrated in figure 5. It follows that the
parametrization is one-to-two. Henceforth we focus on the half string with
positive $x_{j}$ (at least near the conformal boundary). The other half string
is obtained by $x_{j}\to-x_{j}$.

Figure 5: Schematic drawing of a low-temperature and a high-temperature
solution, corresponding to pairs of type [E1,E2] and [H2,H2]. The broken line
is the axis of reflection symmetry. The blueshift parameter $|\sigma|$
decreases monotonically until the string reaches either the turning point or
the black-hole horizon.

Eqs. (4.11) imply that $2r_{j}r_{j}^{\prime}=1$. Inserting in eq. (4.2) gives

$\displaystyle(x_{j}^{\prime})^{2}=r_{j}^{-2}\Bigl{(}g(\sigma)-{\ell_{j}^{2}\over
4\sigma r_{j}^{2}}\Bigr{)}\ .$ (4.12)

Squaring now twice eq. (4.3) and replacing $(x_{j}^{\prime})^{2}$ from the
above expressions leads to a quadratic equation for $g(\sigma)$, the
${\tiny\sigma\sigma}$ component of the worldsheet metric. This equation has a
singular solution $g=0$, and a non-trivial one

$g(\sigma)=\lambda^{2}\Biggl{[}\Bigl{(}\frac{2r_{1}r_{2}}{\ell_{1}\ell_{2}}\Bigr{)}^{2}\hskip-1.13809pt-\Bigl{(}\frac{r_{1}^{2}}{\ell_{1}^{\,2}}+\frac{r_{2}^{2}}{\ell_{2}^{\,2}}-\lambda^{2}\sigma\Bigr{)}^{2}\,\Biggr{]}^{-1}\hskip-5.406pt={\lambda^{2}\over
A\sigma^{2}+2B\sigma+C}\,,$ (4.13)

where in the second equality we used eqs. (4.11), and

$A=(\lambda_{\rm max}^{2}-\lambda^{2})(\lambda^{2}-\lambda_{\rm min}^{2})\ ;$
$\displaystyle B=\lambda^{2}(M_{1}+M_{2})-\lambda_{0}^{2}(M_{1}-M_{2})\ ;\quad
C=-(M_{1}-M_{2})^{2}\ .$ (4.14)

We expressed the quadratic polynomial appearing in the denominator of (4.13)
in terms of $M_{j},\lambda$ and the critical tensions, eqs. (4.8), in order to
render manifest the fact that for $\lambda$ in the allowed range,
$\lambda_{\rm min}<\lambda<\lambda_{\rm max}$, the coefficient $A$ is
positive. This is required for $g(\sigma)$ to be positive near the boundary
where $\sigma\to\infty$. In addition, $AC\leq 0$ which ensures that the two
roots of the denominator in eq. (4.13)

$\displaystyle\sigma_{\pm}={-B\pm(B^{2}-AC)^{1/2}\over A}$ (4.15)

are real, and that the larger root $\sigma_{+}$ is non-negative. Inserting eq.
(4.13) in eq. (4.12) and fixing the sign of the square root near the conformal
boundary gives after a little algebra

$\displaystyle\frac{x_{1}^{\prime}}{\ell_{1}}\,=\,-\frac{\sigma\,(\lambda^{2}+\lambda_{0}^{2})+M_{1}-M_{2}}{{2(\sigma+M_{1}\ell_{1}^{\,2})}\,\sqrt{A\sigma(\sigma-\sigma_{+})(\sigma-\sigma_{-})}}\
\ ,$ (4.16a)
$\displaystyle\frac{x_{2}^{\prime}}{\ell_{2}}\,=\,-\frac{\sigma\,(\lambda^{2}-\lambda_{0}^{2})+M_{2}-M_{1}}{{2(\sigma+M_{2}\ell_{2}^{\,2})}\,\sqrt{A\sigma(\sigma-\sigma_{+})(\sigma-\sigma_{-})}}\
\ .$ (4.16b)

We may now confirm our earlier claim that if $\sigma_{+}>0$ then both
$x_{1}^{\prime}\propto dx_{1}/dr_{1}$ and $x_{2}^{\prime}\propto
dx_{2}/dr_{2}$ diverge at this point. Furthermore, since
$\sigma+M_{j}\ell_{j}^{\,2}=r_{j}^{2}$ is positive,141414Except for the
measure-zero set of solutions in which the string passes through the center of
global AdS3. the $x_{j}^{\prime}$ are finite at all $\sigma>\sigma_{+}$. Thus
$\sigma_{+}$ is the unique turning point of the string, as advertized.

Eqs. (4.11) and (4.16) give the general solution of the string equations for
arbitrary mass parameters $M_{1},M_{2}$ of the green and pink slices. These
must be determined by interior regularity, and by the Dirichlet conditions at
the conformal boundary. Explicitly, the boundary conditions for the different
slice types of figure 3 read:

$L_{j}\
=\,2\int_{\,\sigma_{+}}^{\infty}d\sigma\,x_{j}^{\prime}\,\qquad\qquad{\text{for
\ \ {\small E2, E2${}^{\prime}$}}}\ ;$ (4.17a) $L_{j}\ =\
nP_{j}+\,2\int_{\,\sigma_{+}}^{\infty}d\sigma\,x_{j}^{\prime}\,\qquad{\text{
for \ \ {\small E1, H1}}}\ ;$ (4.17b) $L_{j}\ =\ \Delta x_{j}\bigl{|}_{\rm
Hor}\,+\,2\int_{\,\sigma_{+}}^{\infty}d\sigma\,x_{j}^{\prime}\,\qquad\quad{\text{
for \ \ {\small H2}}}\ .$ (4.17c)

The integrals in these equations are the opening arcs, $\Delta x_{j}$, between
the two endpoints of a half string. They can be expressed as complete elliptic
integrals of the first, second and third kind, see appendix B. For the slices
E1, H1 where $x_{j}$ is a periodic coordinate, we have denoted by $P_{j}>0$
its period, and by $n$ the string winding number. Finally for strings entering
the horizon we denote by $\Delta x_{j}|_{\rm Hor}$ the opening arc between the
two horizon-entry points in the $j$th coordinate chart.

Possible phases of the ICFT for given torus parameters $T,L_{j}$ must be
solutions to one pair of conditions (4.17). Apart from interior regularity, we
will also require that the string does not self-intersect. In principle, two
string bits intersecting at an angle $\not=\pi$ could join into another
string. Such string junctions would be the gravitational counterparts of
interface fusion [65], and allowing them would make the holographic model much
richer.151515Generically the intersection point in one slice will correspond
to two points that must be identified in the other slice; this may impose
further conditions. To keep, however, our discussion simple we will only allow
a single type of domain wall in this work.

The reader can easily convince herself that to avoid string intersections we
must have $P_{j}>L_{j}$ and $n=1$ in (4.17b), and $\Delta x_{j}\bigl{|}_{\rm
Hor}>0$ in (4.17c).

##  5 Phases: cold, hot $\&$ warm

Among the five slice types of figure 3, H2 stands apart because it can only
pair with itself. This is because a horizon is a closed surface, so it cannot
end on the domain wall.161616Except possibly in the limiting case where the
wall is the boundary of space. We will now show that the matching equations
actually rule out several other pairs among the remaining slice types.

One pair that is easy to exclude is [H1,H1], i.e. solutions that describe two
black holes sitting on either side of the wall. Interior regularity would
require in this case $M_{1}=M_{2}=(2\pi T)^{2}$. But eqs. (4.14) and (4.15)
then imply that $\sigma_{+}=0$, so the wall cannot avoid the horizon leading
to a contradiction.

This gives our first no-go lemma:

Two black holes on either side of a static domain wall are ruled out.

Note by contrast that superheavy domain walls ($\lambda>\lambda_{\rm max}$)
inflate and could thus prevent the black holes from
coalescing.171717Asymptotically-flat domain walls, which have been studied a
lot in the context of Grand Unification [83], are automatically in this range.

A second class of pairs one can exclude are the ‘centerless geometries’
[E2,E2], [E2,E2′], [E2′,E2] and [E2′,E2′]. We use the word ‘centerless’ for
geometries that contain neither a center of global AdS, nor a black hole in
its place (see fig. 3). If such solutions existed, all inertial observers
would necessarily hit the domain wall since there would be neither a center
where to rest, nor a horizon where to escape.181818In the double-Wick rotated
context of Simidzija and Van Raamsdonk the [E2,E2] geometries give traversible
wormholes [29].

The argument excluding such solutions is based on a simple observation: What
distinguishes the centerless slices E2 and E2′ from those with a AdS center
(E1) or a black hole (H1) is the sign of $x_{j}^{\prime}$ at the turning
point,

$\displaystyle{\rm
sign}\bigl{(}x_{j}^{\prime}\bigl{|}_{\sigma\approx\sigma_{+}}\bigr{)}\
=\begin{cases}+\quad{\rm for\ {\small E2},{\small E2}}^{\prime}\,,\\\
-\quad{\rm for\ {\small E1},{\small H1}}\ .\end{cases}$ (5.1)

Now from eqs. (4.16) one has

$\displaystyle(\sigma+M_{1}\ell_{1}^{\,2})\,\frac{x_{1}^{\prime}}{\ell_{1}}+(\sigma+M_{2}\ell_{2}^{\,2})\,\frac{x_{2}^{\prime}}{\ell_{2}}\,<\,0\
,$ (5.2)

so both $x_{j}^{\prime}$ cannot be simultaneously positive. This holds for all
$\sigma$, and hence also near the turning point QED. This is our second no-go
lemma:

‘Centerless’ static spacetimes in which all inertial observers would
inevitably hit the domain wall are ruled out.

We can actually exploit this argument further. As is clear from eq. (4.16a),
if $M_{1}>M_{2}$ then $x^{\prime}_{1}$ is manifestly negative, i.e. the green
slice is of type E1 or H1. The pairs [E2′,E1] and [E2′,E2] for which the above
inequality is automatic are thus ruled out. One can also show that
$x^{\prime}_{2}|_{\sigma\approx\sigma_{+}}$ is negative if $M_{2}>0>M_{1}$.
This is obvious from eq. (4.16b) in the range $\lambda>\lambda_{0}$, and less
obvious but also true as can be checked by explicit calculation for
$\lambda<\lambda_{0}$.191919 The tedious algebra is straightforward and not
particularly instructive, so we chose not to present it here. We did it with
mathematica but also tested it numerically. The pairs [E1,E2′] and [E2,E2′]
for which the above mass inequality is automatic, are thus also excluded.

Recall that the energy density of the $j$th CFT reads $\langle
T_{tt}\rangle={1\over 2}\ell_{j}M_{j}$. Ruling out all pairs of E2′ with E1 or
E2 implies therefore that in the ground state the energy density must be
everywhere negative. When one $L_{j}$ is much smaller than the other, the
Casimir energy scales like $E_{0}\sim\\#/L_{j}$. The fact that the coefficient
$\\#$ is negative means that the Casimir force is attractive, in agreement
with general theorems [88, 89]. This is the third no-go lemma:

Figure 6: Phases of the domain-wall spacetime. The type of the green slice
labels the rows of the table, and that of the pink slice the columns. In the
hot (red) phase the wall enters the black-hole horizon, while in the warm
(yellow) phases it avoids it. The cold (blue) phases have no black hole.
Geometries in which an inertial observer is attracted to two different centers
are indicated by a different shade (light yellow or darker blue).

A slice of global AdS3 cannot be paired with a horizonless BTZ slice. This
implies that in the ground state of the putative dual ICFT the energy density
is everywhere negative.

We have collected for convenience all these conclusions in fig. 6. The table
shows the eligible slice pairs, or the allowed topologies of static-domain-
wall spacetimes. It also defines a color code for phase diagrams.

The light yellow phases that feature a wall between the black hole and an AdS
restpoint are the gravitational avatars of the Faraday cage. Such solutions
are easier to construct for larger $\lambda$. Domain walls lighter than
$\lambda_{0}$, in particular, can never shield from a black hole in the ‘true-
vacuum’ side. Indeed, as follows easily from eq. (4.16b), for
$\lambda<\lambda_{0}$ and $M_{1}>0>M_{2}$, the sign of
$x_{2}^{\prime}|_{\sigma\approx\sigma_{+}}$ is positive, so geometries of type
[H1,E1] are excluded.

##  6 Equations of state

The different colors in figure 6 describe different phases of the system,
since the corresponding geometries are topologically distinct. They differ in
how the wall, the horizon (if one exists) and inertial observers intersect or
avoid each other.

Let us now think thermodynamics. For fixed Lagrangian parameters
$\lambda,\ell_{j}$, the canonical variables that determine the state of the
system are the temperature $T$ and the volumes $L_{1},L_{2}$. Because of scale
invariance only two dimensionless ratios matter:

$\displaystyle\tau_{1}:=TL_{1}\ ,\quad\tau_{2}:=TL_{2}\qquad{\rm
or}\quad\quad\gamma:={L_{1}\over L_{2}}={\tau_{1}\over\tau_{2}}\ .$ (6.1)

The microcanonical variables, the energy density and the entropy of each
subsystem, read (see section 2, and recall that $8\pi G=1$)

$\displaystyle{E_{j}\over L_{j}}={\ell_{j}\over 2}M_{j}\qquad{\rm and}\qquad
S_{j}={r^{\rm H}_{j}\Delta x_{j}|_{\rm Hor}\over
4G}=2\pi\ell_{j}\sqrt{M_{j}}\,\Delta x_{j}|_{\rm Hor}\ .\ \ $ (6.2)

These are the natural parameters of the interior geometry. The entropies are
scale invariant. The other key dimensionless variable is the mass ratio, viz.
the ratio of energy densities per degree of freedom in the two CFTs

$\displaystyle\mu:={M_{2}\over M_{1}}\ .$ (6.3)

When several phases coexist the dominant one is the one with the lowest free
energy $F=\sum_{j}(E_{j}-TS_{j})$. As a sanity check, we rederive $F$ from the
renormalized on-shell gravitational action in appendix A.

The Dirichlet conditions, eqs. (4.17), give for each type of geometry two
relations among the above variables that play the role of equations of
state.202020In homogeneous systems there is a single equation of state. Here
we have one equation for each subsystem. They relate the natural interior
parameters $S_{j}$ and $\mu$ to the variables $\tau_{j}$ and $\gamma$ of the
boundary torus. Note that in each phase of the system since for horizonless
slices $S_{j}=0$ and for slices with horizon $M_{j}=(2\pi T)^{2}$. In
computing the phase diagram we will have to invert these equations of state.

###  6.1 High-$T$ phase

For fixed $L_{j}$ and very high temperature the black hole grows so large that
it eats away a piece of the domain wall and the AdS rest points. The dominant
solution is thus of type [H2,H2] and regularity fixes the mass parameters in
both slices, $M_{1}=M_{2}=(2\pi T)^{2}$. The boundary conditions (4.17c)
reduce in this case to simple equations for the opening horizon arcs $\Delta
x_{j}|_{\rm Hor}$. Performing explicitly the integrals (see appendix B) gives

$\displaystyle L_{1}-\Delta x_{1}\bigl{|}_{\rm Hor}\ =\ -{1\over\pi T}\,{\rm
tanh}^{-1}\left({\ell_{1}(\lambda^{2}+\lambda^{2}_{0})\over 2\lambda}\right)\
,$ (6.4a) $\displaystyle L_{2}-\Delta x_{2}\bigl{|}_{\rm Hor}\ =\ -{1\over\pi
T}\,{\rm tanh}^{-1}\left({\ell_{2}(\lambda^{2}-\lambda^{2}_{0})\over
2\lambda}\right)\ .$ (6.4b)

For consistency we must have $\Delta x_{j}|_{\rm Hor}>0$, which is automatic
if $\lambda>\lambda_{0}$. If $\lambda<\lambda_{0}$, on the other hand,
positivity of $\Delta x_{2}|_{\rm Hor}$ puts a lower bound on $\tau_{2}$,

$\displaystyle\tau_{2}\,\geq\,{1\over\pi}\,{\rm
tanh}^{-1}\left({\ell_{2}(\lambda_{0}^{2}-\lambda^{2})\over 2\lambda}\right)\
:=\,\tau_{2}^{*}\ .$ (6.5)

We see here a first interpretation of the critical tension $\lambda_{0}$
encountered in section 4.3. For walls lighter than $\lambda_{0}$ there is a
region of parameter space where the hot solution ceases to exist, even as a
metastable phase.

The total energy and entropy in the high-$T$ phase read

$\displaystyle E_{\rm[hot]}\,=\,{1\over
2}(\ell_{1}L_{1}M_{1}+\ell_{2}L_{2}M_{2})\,=\,2\pi^{2}T^{2}\,(\ell_{1}L_{1}+\ell_{2}L_{2})\
,$ (6.6) $\displaystyle S_{\rm[hot]}=4\pi^{2}T\bigl{(}\ell_{1}\Delta
x_{1}\bigl{|}_{\rm Hor}+\,\ell_{2}\Delta x_{2}\bigl{|}_{\rm
Hor}\bigr{)}=4\pi^{2}T^{2}(\ell_{1}L_{1}+\ell_{2}L_{2})+2\log g_{I}\ ,\ \ \ $
(6.7)

where $\log g_{\rm I}\,$ is given by eq. (4.9) and the rightmost expression of
the entropy follows from eqs. (6.4) and a straightforward reshuffling of the
arctangent functions. This is a satisfying result. Indeed, the first term in
the right-hand side of (6.7) is the thermal entropy of the two CFTs (being
extensive these entropies cannot depend on the ratio $L_{1}/L_{2}$), while the
second term is the entropy of the two interfaces on the circle. The
Bekenstein-Hawking formula captures nicely both contributions.

Eqs.(6.4) and (6.7) show that shifting the $L_{j}$ at fixed $T$ does not
change the entropy if and only if $\ell_{1}\,\delta L_{1}=-\ell_{2}\,\delta
L_{2}$. Moving in particular a defect (for which $\ell_{1}=\ell_{2}$) without
changing the volume $L_{1}+L_{2}$ is an adiabatic process, while moving a more
general interface generates/absorbs entropy by modifying the density of
degrees of freedom.

###  6.2 Low-$T$ phase(s)

Consider next the ground state of the system, at $T=0$. The dual geometry
belongs to one of the three horizonless types: the double-center geometry [E1,
E1], or the single-center ones [E1, E2] and [E2, E1] (see fig. 6). Here the
entropies $S_{j}=0$, and the only relevant dimensionless variables are the
volume and energy-density ratios, $\gamma$ and $\mu$. Note that they are both
positive since $L_{j}>0$ and $M_{j}<0$ for both $j$.

The Dirichlet conditions (4.16) for horizonless geometries read

$\displaystyle\sqrt{|M_{1}|}\,L_{1}=2\pi\,\delta_{{\mathbb{S}}_{1},{\rm
E1}}-f_{1}(\mu)\
,\qquad\sqrt{|M_{1}|}\,L_{2}={2\pi\over\sqrt{\mu}}\,\delta_{{\mathbb{S}}_{2},{\rm
E1}}-f_{2}(\mu)\ ,$ (6.8)

where $\delta_{{\mathbb{S}}_{j},{\rm E1}}=1$ if the $j$th slice is of type E1
and $\delta_{{\mathbb{S}}_{j},{\rm E1}}=0$ otherwise, and

$\displaystyle
f_{1}(\mu)\,=\,{\ell_{1}\over\sqrt{A}}\int_{s_{+}}^{\infty}ds{s(\lambda^{2}+\lambda_{0}^{2})-1+\mu\over(s-\ell_{1}^{\,2})\sqrt{s(s-s_{+})(s-s_{-})}}\
,\ \ \ $ (6.9a) $\displaystyle
f_{2}(\mu)\,=\,{\ell_{2}\over\sqrt{A}}\int_{s_{+}}^{\infty}ds{s(\lambda^{2}-\lambda_{0}^{2})+1-\mu\over(s-\mu\ell_{2}^{\,2})\sqrt{s(s-s_{+})(s-s_{-})}}\
,\ \ \ $ (6.9b)

with

$\displaystyle A\,s_{\pm}=\lambda^{2}(1+\mu)-\lambda_{0}^{2}(1-\mu)\pm
2\lambda\sqrt{{1-\mu\over\ell_{2}^{2}}+{\mu^{2}-\mu\over\ell_{1}^{2}}+\mu\lambda^{2}}\
\ .$ (6.10)

The dummy integration variable $s$ is the appropriately rescaled blueshift
factor of the string worldsheet, $s=\sigma/|M_{1}|$ .

Dividing the two sides of eqs. (6.8) gives $\gamma$ as a function of $\mu$ for
each of the three possible topologies.212121The functions $f_{j}(\mu)$ are
combinations of complete elliptic integrals of the first, second and third
kind, see appendix B. The value $\mu=1$ gives $\gamma=1$, corresponding to the
scale-invariant AdS2 string worldsheet. The known supersymmetric top-down
solutions live at this special point in phase space.  If the ground state of
the putative dual quantum-mechanical system was unique, we should find a
single slice-pair type and value of $\mu$ for each value of $\gamma$.
Numerical plots show that this is indeed the case. Specifically, we found that
$\gamma(\mu)$ is a monotonically-increasing function of $\mu$ for any given
slice pair, and that it changes continuously from one type of pair to another.
We will return to these branch-changing ‘sweeping transitions’ in section 7.
Let us stress that the uniqueness of the cold solution did not have to be
automatic in classical gravity, nor in the dual large-$N$ quantum mechanics.

For most of the $(\ell_{j},\lambda)$ parameter space, as $\gamma$ ranges in
$(0,\infty)$ the mass ratio $\mu$ covers also the entire range $(0,\infty)$.
However, if $\ell_{1}<\ell_{2}$ (strict inequality) and for sufficiently light
domain walls, we found that $\gamma$ vanishes at some positive
$\mu=\mu_{0}(\lambda,\ell_{j})$. Below this critical value $\gamma$ becomes
negative signaling that the wall self-intersects and the solution must be
discarded. This leads to a striking phenomenon that we discuss in section 8.

###  6.3 Warm phases

The last set of solutions of the model are the yellow- or orange-coloured ones
in fig. 6. Here the string avoids the horizon, so the slice pair is of type
[H1,X] or [X,H1] with X one of the horizonless types: E1, E2 or E2′.

Assume first that the black hole is on the green side of the wall, so that
$M_{1}=(2\pi T)^{2}$. In terms of $\mu$ the Dirichlet conditions (4.17a,
4.17b) read:

$\displaystyle 2\pi T\Delta x_{1}\bigl{|}_{{\rm
Hor}}-\,2\pi\tau_{1}=\tilde{f}_{1}(\mu)\ ,\qquad
2\pi\tau_{2}={2\pi\over\sqrt{-\mu}}\,\delta_{{\mathbb{S}}_{2},{\rm
E1}}-\tilde{f}_{2}(\mu)\ ,$ (6.11)

where

$\displaystyle\tilde{f}_{1}(\mu)\,=\,{\ell_{1}\over\sqrt{A}}\int_{\tilde{s}_{+}}^{\infty}ds{s(\lambda^{2}+\lambda_{0}^{2})+1-\mu\over(s+\ell_{1}^{\,2})\sqrt{s(s-\tilde{s}_{+})(s-\tilde{s}_{-})}}\
,\ \ \ $ (6.12a)
$\displaystyle\tilde{f}_{2}(\mu)\,=\,{\ell_{2}\over\sqrt{A}}\int_{\tilde{s}_{+}}^{\infty}ds{s(\lambda^{2}-\lambda_{0}^{2})-1+\mu\over(s+\mu\ell_{2}^{\,2})\sqrt{s(s-\tilde{s}_{+})(s-\tilde{s}_{-})}}\
,\ \ \ $ (6.12b)

and the roots $\tilde{s}_{\pm}=\sigma_{\pm}/M_{1}$ inside the square root are
given by

$\displaystyle
A\,\tilde{s}_{\pm}=-\lambda^{2}(1+\mu)+\lambda_{0}^{2}(1-\mu)\pm
2\lambda\sqrt{{1-\mu\over\ell_{2}^{2}}+{\mu^{2}-\mu\over\ell_{1}^{2}}+\mu\lambda^{2}}\
\ .$ (6.13)

In the first condition (6.11) we have used the fact that the period of the
green slice that contains the horizon is $P_{1}=\Delta x_{1}|_{{\rm Hor}}$.

If the black hole is on the pink side of the wall, the conditions take a
similar form in terms of the inverse mass ratio
$\hat{\mu}=\mu^{-1}=M_{1}/M_{2}$,

$\displaystyle 2\pi T\Delta x_{2}\bigl{|}_{{\rm
Hor}}-\,2\pi\tau_{2}={\hat{f}}_{2}(\hat{\mu})\ ,\quad
2\pi\tau_{1}={2\pi\over\sqrt{-\hat{\mu}}}\,\delta_{{\mathbb{S}}_{1},{\rm
E1}}-\hat{f}_{1}(\hat{\mu})\ ,$ (6.14)

where here

$\displaystyle\hat{f}_{1}(\hat{\mu})\,=\,{\ell_{1}\over\sqrt{A}}\int_{\hat{s}_{+}}^{\infty}ds{s(\lambda^{2}+\lambda_{0}^{2})+\hat{\mu}-1\over(s+\hat{\mu}\ell_{1}^{\,2})\sqrt{s(s-\hat{s}_{+})(s-\hat{s}_{-})}}\
,\ \ \ $ (6.15a)
$\displaystyle\hat{f}_{2}(\mu)\,=\,{\ell_{2}\over\sqrt{A}}\int_{\hat{s}_{+}}^{\infty}ds{s(\lambda^{2}-\lambda_{0}^{2})-\hat{\mu}+1\over(s+\ell_{2}^{\,2})\sqrt{s(s-\hat{s}_{+})(s-\hat{s}_{-})}}\
.\ \ \ $ (6.15b)

and the roots $\hat{s}_{\pm}=\sigma_{\pm}/M_{2}$ inside the square root are
given by

$\displaystyle
A\,\hat{s}_{\pm}=-\lambda^{2}(\hat{\mu}+1)+\lambda_{0}^{2}(\hat{\mu}-1)\pm
2\lambda\sqrt{{\hat{\mu}^{2}-\hat{\mu}\over\ell_{2}^{2}}+{1-\hat{\mu}\over\ell_{1}^{2}}+\hat{\mu}\lambda^{2}}\
\ .$ (6.16)

The functions $\tilde{f}_{j}$ and $\hat{f}_{j}$, as well as the $f_{j}$ of the
cold phase, derive from the same basic formulae (4.16a, 4.16b) and differ only
by a few signs. We chose to write them out separately because these signs are
important. Note also that while in cold solutions $\mu$ is always positive,
here $\mu$ and its inverse $\hat{\mu}$ can have either sign.

All the values of $\mu$ and $\hat{\mu}$ do not, however, correspond to
admissible solutions. For a pair of type [H1,X] we must demand (i) that the
right-hand sides in (6.11) be positive – the non-intersection requirement, and
(ii) that $x_{1}^{\prime}|_{\sigma\approx\sigma_{+}}$ be negative – the
turning point condition (5.1). Likewise for solutions of type [X, H1] we must
demand that the right-hand sides in (6.14) be positive and that
$x_{2}^{\prime}|_{\sigma\approx\sigma_{+}}$ be negative.

The turning-point requirement is easy to implement. In the [H1,X] case,
$x_{1}^{\prime}|_{\sigma\approx\sigma_{+}}$ is negative when the numerator of
the integrand in (6.12a), evaluated at at $s=\tilde{s}_{+}$, is positive.
Likewise for the [X,H1] pairs, $\,x_{2}^{\prime}|_{\sigma\approx\sigma_{+}}$
is negative when the numerator of the integrand in (6.15b), evaluated at at
$s=\hat{s}_{+}$, is positive. After a little algebra these conditions take a
simple form

$\displaystyle{\rm for}\ \ {\rm[H1,X]}\quad\mu\in(-\infty,1]\ ;\qquad{\rm
for}\ \ {\rm[X,H1]}\quad\hat{\mu}=\mu^{-1}\in(-\infty,1]\ .\ \ \ $ (6.17)

Recalling that $\mu=\hat{\mu}^{-1}=M_{2}/M_{1}$, we conclude that in all the
cases the energy density per degree of freedom in the horizonless slice is
lower than the corresponding density in the black hole slice.

This agrees with physical intuition: the energy density per degree of freedom
in the cooler CFT is less than the thermal density $\pi T^{2}/6$ – the
interfaces did not let the theory thermalize. When $\mu\to 1$ or $\hat{\mu}\to
1$, the wall enters the horizon and the energy is equipartitioned.

This completes our discussion of the equations of state. To summarize, these
equations relate the parameters of the interior geometry ($\mu,S_{j}$) to
those of the conformal boundary ($\gamma,\tau_{j}$). The relation involves
elementary functions in the hot phase, and was reduced to a single function
$\gamma(\mu)$, that can be readily plotted, in the cold phases. Furthermore at
any given point in parameter space the hot and cold solutions, when they
exist, are unique. The excluded regions are
$\tau_{2}<\tau_{2}^{*}(\lambda,\ell_{j})$ for the hot solutions, and
$\mu>\mu_{0}(\lambda,\ell_{j})$ for the cold solution with $\mu_{0}$ the point
where $\gamma=0$.

In warm phases the story is richer since more than one solutions typically
coexist at any given value of $(\gamma,\tau_{j})$. Some solutions have
negative specific heat, as we will discuss later. To find the parameter
regions where different solutions exist requires inverting the relation
between ($\gamma,\tau_{j}$) and ($\mu,S_{j}$). We will do this analytically in
some limiting cases, and numerically to compute the full phase diagram in
section 9.

##  7 Phase transitions

The transitions between different phases are of three kinds:

  * •

Hawking-Page transitions describing the formation of a black hole. These
transitions from the cold to the hot or warm phases of fig. 6 are always first
order;

  * •

Warm-to-hot transitions during which part of the wall is captured by the
horizon. We will show that these transitions are also first-order;

  * •

Sweeping transitions where the wall sweeps away a center of global AdS, i.e. a
rest point for inertial observers. These are continuous transitions between
the one- and two-center phases of fig. 6.

It is instructive to picture these transitions by plotting the metric factor
$g_{tt}$ while traversing space along the axis of reflection symmetry. The
curve changes qualitatively as shown in figure 9, illustrating the topological
nature of the transitions on the gravity side.

Figure 7: Curves of the blueshift factor $g_{tt}$ as one traverses space along
the $Z_{2}$ symmetry axis. The color code is the same as in fig. 6. The wall
is located at the turning point $g_{tt}=\sigma_{+}$ where the curve is
discontinuous. The grey arrows indicate possible transitions. The blackened
parts of the curves are regions behind the horizon.

Before embarking in numerical plots, we will first do the following things:
(i) Comment on the ICFT interpretation of these transitions; (ii) Compute the
sweeping transitions analytically; and (iii) Prove that the warm-to-hot
transitions are first order, i.e. that one cannot lower the wall to the
horizon continuously by varying the boundary data.

###  7.1 ICFT interpretation

When a holographic dual exists, Witten has argued that the appearance of a
black hole at the Hawking-Page (HP) transition signals deconfinement in the
gauge theory [90]. Assuming this interpretation 222222There is an extensive
literature on the subject including [91, 92], studies specific to two
dimensions [93, 94], and recent discussions in relation with the
superconformal index in $N=4$ super Yang Mills [95, 96, 97, 98]. For an
introductory review see [99].  leads to the conclusion that in warm phases a
confined theory coexists with a deconfined one. We will see below that such
coexistence is easier when the confined theory is CFT2, i.e. the theory with
the larger central charge.232323Even though for homogeneous 2-dimensional CFTs
the critical temperature, $\tau_{\rm HP}=1$, does not depend on the central
charge by virtue of modular invariance. This is natural from the gravitational
perspective. Solutions of type [H1, X] are more likely than solutions of type
[X, H1] because a black hole forms more readily on the ‘true-vacuum’ side of
the wall. We will actually provide some evidence later that if $c_{2}>3c_{1}$
there are no equilibrium phases at all in which CFT2 is deconfined while CFT1
stays confined.

The question that jumps to one’s mind is what happens for thick walls, where
one expects a warm-to-hot crossover rather than a sharp transition. One
possibility is that the coexistence of confined and deconfined phases is
impossible in microscopic holographic models. Alternatively, an appropriately
defined Polyakov loop [90] could provide a sharp order parameter for this
transition.

For sweeping transitions the puzzle is the other way around. Here a sharp
order parameter exists in classical gravity – it is the number of rest points
for inertial observers. This can be defined both for thin- and for thick-wall
geometriess. The interpretation on the field theory side is however unclear.
The transitions could be related to properties of the low-lying spectrum at
infinite $N$, or to the entanglement structure of the ground state.

We leave these questions open for future work.

###  7.2 Sweeping transitions

Sweeping transitions are continuous transitions that happen at fixed values of
the mass ratio $\mu$. We will prove these statements here.

Assume for now continuity, and let the $j$th slice go from type E1 to type E2.
The transition occurs when the string turning point and the center of the
$j$th AdS slice coincide, i.e. when

$\displaystyle r_{j}(\sigma_{+})=\sqrt{\sigma_{+}+M_{j}\ell_{j}^{2}}\,=\,0\ .$
(7.1)

Clearly this has a solution only if $M_{j}<0$. Inserting in (7.1) the
expressions (4.15) - (4.14) for $\sigma_{+}$ gives two equations for the
critical values of $\mu$ with the following solutions

$\displaystyle\mu_{1}^{*}={1-\ell_{2}^{\,2}\lambda^{2}\over\ell_{2}^{2}/\ell_{1}^{2}}\quad\
{\rm and}\quad\ \mu_{2}^{*}={\ell_{1}^{2}/\ell_{2}^{2}\over
1-\ell_{1}^{\,2}\lambda^{2}}\ \ .$ (7.2)

In the low-$T$ phases both $M_{j}$ are negative and $\mu$ is positive.
Furthermore, a little algebra shows that for all $\lambda\in(\lambda_{\rm
min},\,\lambda_{\rm max})$ the following is true

$\displaystyle x_{1}^{\prime}\bigl{|}_{\sigma\approx\sigma_{+}}<0\quad{\rm
at}\ \ \mu\gg 1\ \quad{\rm and}\quad
x_{2}^{\prime}\bigl{|}_{\sigma\approx\sigma_{+}}<0\quad{\rm at}\ \ \mu\ll
1\,.\ \ $ (7.3)

This means that for $\mu\gg 1$ the green slice is of type E1, and for $\mu\ll
1$ the pink slice is of type E1. A sweeping transition can occur if the
critical mass ratios (7.2) are in the allowed range. We distinguish three
regimes of $\lambda$:

  * •

Heavy ($\lambda>1/\ell_{1}$): None of the $\mu_{j}^{*}\,$ is positive, so the
solution is of type [E1,E1] for all $\mu$, i.e. cold solutions are always
double-center ;

  * •

Intermediate (${1/\ell_{1}}>\lambda>1/\ell_{2}$): Only $\mu_{2}^{*}\,$ is
positive. If this is inside the range of non-intersecting walls, the solution
goes from [E1,E2] at large $\mu\,$, to [E1,E1] at small $\mu$. Otherwise the
geometry is always of the single-center type [E1,E2] ;

  * •

Light ($\lambda<1/\ell_{2}$): Both $\mu_{1}^{*}\,$ and $\mu_{2}^{*}\,$ are
positive, so there is the possibility of two sweeping transitions: from
[E2,E1] at small $\mu$ to [E1,E2] at large $\mu$ passing through the double-
center type [E1,E1] . Note that since $\lambda_{\rm
min}=1/\ell_{1}-1/\ell_{2}$, this range of $\lambda$ only exists if
$\ell_{2}<2\ell_{1}$, i.e. when CFT2 has no more than twice the number of
degrees of freedom of the more depleted CFT1.

We can now confirm that sweeping transitions are continuous, not only in terms
of the mass ratio $\mu$ but also in terms of the ratio of volumes $\gamma$. To
this end we expand the relations (6.8) around the above critical points and
show that the $L_{j}$ vary indeed continuously across the transition. The
calculations can be found in appendix C .

For the warm phases we proceed along similar lines. One of the two $M_{j}$ is
now equal to $(2\pi T)^{2}>0$, so sweeping transitions may only occur for
negative $\mu$. Consider first warm solutions of type [H1,X] with the black
hole in the ‘true vacuum’ side. A little calculation shows that
$\,x_{2}^{\prime}|_{\sigma\approx\sigma_{+}}$ is negative, i.e. X=E1, if and
only if

$\displaystyle\lambda>{1\over\ell_{1}}\qquad{\rm and}\qquad\mu<\mu_{2}^{*}<0\
.$ (7.4)

Recall that when X=E1 some inertial observers can be shielded from the black
hole by taking refuge at the restpoint of the pink slice. We see that this is
only possible for heavy walls ($\lambda>1/\ell_{1}$) and for
$\mu<\mu_{2}^{*}$. A sweeping transition [H1,E1] $\to$ [H1,E2] takes place at
$\mu=\mu_{2}^{*}$.

Consider finally a black hole in the ‘false vacuum’ side, namely warm
solutions of type [X,H1]. Here $\,x_{1}^{\prime}|_{\sigma\approx\sigma_{+}}$
is negative, i.e. $X$ has a rest point, if and only if the following
conditions are satisfied

$\displaystyle\lambda>{1\over\ell_{2}}\qquad{\rm
and}\qquad\hat{\mu}:=\mu^{-1}<(\mu_{1}^{*})^{-1}<0\ .$ (7.5)

Shielding from the black hole looks here easier, both heavy and intermediate-
tension walls can do it. In reality, however, we have found that solutions
with the black hole in the ‘false vacuum’ side are rare, and that the above
inequality pushes $\hat{\mu}$ outside the admissible range.

The general trend emerging from the analysis is that the heavier the wall the
more likely are the two-center geometries. A suggestive calculation actually
shows that

$\displaystyle{\partial\sigma_{+}\over\partial\lambda}\Bigl{|}_{M_{j}\ {\rm
fixed}}\ \ {\rm is}\ \begin{cases}{\textrm{ positive\ \ for \ two-center\
solutions}}\\\ \,{\textrm{negative\ \ for \ single-center\
solutions}}\end{cases}$ (7.6)

where the word ‘center’ here includes both an AdS restpoint and a black hole.
At fixed energy densities a single center is therefore pulled closer to a
heavier wall, while two centers are instead pushed away. It might be
interesting to also compute ${\partial\sigma_{+}/\partial\lambda}$ and
${\partial V/\partial\lambda}$ at $L_{j}$ fixed, where $V$ is the regularized
volume of the interior space. In the special case of the vacuum solution with
an AdS2 wall, the volume (and the associated complexity [100]) can be seen to
grow with the tension $\lambda$.

###  7.3 Warm-to-hot transitions

In warm-to-hot transitions the thin domain wall enters the black-hole horizon.
One may have expected this to happen continuously, i.e. to be able to lower
the wall to the horizon smoothly, by slowly varying the boundary data
$L_{j},T$. We will now show that, if the tension $\lambda$ is fixed, the
transition is actually always first order.

Note first that in warm solutions the slice that contains the black hole has
$M_{j}=(2\pi T)^{2}$. If the string turning point approaches continuously the
horizon, then $\sigma_{+}\to 0$. From eqs. (4.14, 4.15) we see that this can
happen if and only if $(M_{1}-M_{2})\to 0$, which implies in passing that the
solution must necessarily be of type [H1,E2′] or [E2′,H1]. Expanding around
this putative point where the wall touches the horizon we set

$\displaystyle{M_{1}-M_{2}\over M_{1}+M_{2}}:=\delta\quad{\rm
with}\quad|\delta|\ll 1\ \ \Longrightarrow\ \ \sigma_{+}\approx\biggl{(}{2\pi
T\over\lambda}\biggr{)}^{2}\delta^{2}\,.$ (7.7)

Recalling that the horizonless slice has the smaller $M_{j}$ we see that for
positive $\delta$ the black hole must be in the green slice and
$\mu=1-2\delta+{\cal O}(\delta^{2})$, while for negative $\delta$ the black
hole is in the pink slice and $\hat{\mu}=1+2\delta+{\cal O}(\delta^{2})$.

The second option can be immediately ruled out since it is impossible to
satisfy the boundary conditions (6.14). Indeed, $\hat{f}_{1}(\hat{\mu}\approx
1)$ is manifetsly positive, as is clear from eq. (6.15a), and we have assumed
that $\mathbb{S}_{1}$ is of type E2′. Thus the second condition (6.14) cannot
be obeyed. By the same reasonning we see that for $\delta$ positive, and since
now $\mathbb{S}_{2}$ is of type E2′, we need that $\tilde{f}_{2}(\mu\approx
1)$ be negative. As is clear from the expression (6.12b) this implies that
$\lambda<\lambda_{0}$.

The upshot of the discussion is that a warm solution arbitrarily close to the
hot solution may exist only if $\lambda<\lambda_{0}$ and if the black hole is
on the true-vacuum side.

It is easy to see that under these conditions the two branches of solution
indeed meet at $\mu=1$, $\Delta x_{2}|_{{\rm Hor}}=0$ and hence from (6.4b)

$\displaystyle\tau_{2}\,=\,{1\over\pi}\,{\rm
tanh}^{-1}\left({\ell_{2}(\lambda_{0}^{2}-\lambda^{2})\over
2\lambda}\right)\,:=\,\tau_{2}^{*}\,.$ (7.8)

Recall from section 6.1 that this is the limiting value for the existence of
the hot solution – the solution ceases to exist at $\tau_{2}<\tau_{2}^{*}\,.$
The nearby warm solution could in principle take over in this forbidden range,
provided that $\tau_{2}(\delta)$ decreases as $\delta$ moves away from zero.
It actually turns out that $\tau_{2}(\delta)$ initially increases for small
$\delta$, so this last possibility for a continuous warm-to-hot transition is
also ruled out.

To see why this is so, expand (6.11) and (6.12b) around $\mu=1$,

$\tilde{s}_{+}={\delta^{2}\over\lambda^{2}}+{\cal
O}(\delta^{3})\,,\quad\tilde{s}_{-}=-{4\lambda^{2}\over
A}\Bigl{(}1-\delta(1+{\lambda_{0}^{2}\over\lambda^{2}})\Bigr{)}+{\cal
O}(\delta^{2})\,,\quad$

and shift the integration variable $s:=y+\tilde{s}_{+}$ so that (6.12b) reads

$\displaystyle
2\pi\,\tau_{2}(\delta)={\ell_{2}\over\sqrt{A}}\int_{0}^{\infty}dy\left[{y(\lambda_{0}^{2}-\lambda^{2})+2\delta\over(y+\mu\ell_{2}^{2})\sqrt{y(y+\tilde{s}_{+})(y-\tilde{s}_{-})}}+{\cal
O}(\delta^{2})\right]\ .$ (7.9)

We neglected in the integrand all contributions of ${\cal O}(\delta^{2})$
except for the $\tilde{s}_{+}$ in the denominator that regulates the
logarithmic divergence of the ${\cal O}(\delta\log\delta)$ correction. Now use
the inequalities

${y(\lambda_{0}^{2}-\lambda^{2})+2\delta\over\sqrt{(y+\tilde{s}_{+})(y-\tilde{s}_{-})}}>{y(\lambda_{0}^{2}-\lambda^{2})+2\delta\over\sqrt{(y+\delta^{2}/\lambda^{2})(y+4\lambda^{2}/A)}}>{\sqrt{y}(\lambda_{0}^{2}-\lambda^{2})\over\sqrt{(y+4\lambda^{2}/A)}}\
,$

where the second one is equivalent to
$2\delta>(\lambda_{0}^{2}/\lambda^{2}-1)\delta^{2}$, which is true for small
enough $\delta$. Plugging in (7.9) shows that $\tau_{2}(\delta)>\tau_{2}(0)$
at the leading order in $\delta$, proving our claim.

Figure 8: The function $\tau_{2}(\mu)$ in the [H1,E2] and [H1,E2′] branches of
solutions, for $2\ell_{2}=3\ell_{1}$ and
$\lambda={3/5\ell_{2}}<\lambda_{0}={\sqrt{5}/2\ell_{2}}$ . The red line
indicates the bound $\tau_{2}^{*}$ below which the hot solution ceases to
exist.

A typical $\tau_{2}(\mu)$ in the [H1,E2] and [H1,E2′] branch of solutions, and
for $\lambda<\lambda_{0}$, is plotted in figure 8. The function grows
initially as $\mu$ moves away from 1, reaches a maximum value and then turns
around and goes to zero as $\mu\to-\infty$. The red line indicates the
limiting value $\tau_{2}^{*}$ below which there is no hot solution. For
$\tau_{2}$ slightly above $\tau_{2}^{*}$ we see that there are three
coexisting black holes, the hot and two warm ones. For
$\tau_{2}<\tau_{2}^{*}$, on the other hand, only one warm solution survives,
but it describes a wall at a finite distance from the horizon. Whether this is
the dominant solution or not, the transition is therefore necessarily first
order.

##  8 Exotic fusion and bubbles

Before proceeding to the phase diagram, we pause here to discuss the peculiar
phenomenon announced earlier, in section 6.2. This arises in the limits
$\gamma=L_{1}/L_{2}\to 0$ or $\gamma\to\infty$, with $L_{1}+L_{2}$ and $T$
kept fixed. In these limits the conformal boundary of one slice shrinks to a
point.

Consider for definiteness the limit $L_{1}\to 0$. In the language of the dual
field theory the interface and anti-interface fuse in this limit into a defect
of CFT2. The naive expectation, based on free-field calculations [65, 101,
102], is that this is the trivial (or identity) defect. Accordingly, the green
interior slice should recede to the conformal boundary, leaving as the only
remnant a (divergent) Casimir energy.

We have found that this expectation is not always borne out as we will now
explain.

Suppose first that the surviving CFT2 is in its ground state, and that the
result of the interface-antiinterface fusion is the expected trivial defect.
The geometry should in this case approach global AdS3 of radius $\ell_{2}$,
with $M_{2}$ tending to $-(2\pi/L_{2})^{2}$, see section 2. Furthermore,
$\sigma_{+}$ should go to infinity in order for the green slice to shrink
towards the ultraviolet region. As seen from eqs. (4.15, 4.14) this requires
$M_{1}\to-\infty$, so that $\mu\,$ should vanish together with $\gamma$. This
is indeed what happens in much of the $(\lambda,\ell_{1},\ell_{2})$ parameter
space. One finds $\mu\sim\gamma^{2}\to 0$, a scaling compatible with the
expected Casimir energy $\sim\\#/L_{1}$.

Nevertheless, sometimes $\gamma$ vanishes at finite $\mu_{0}$. In such cases,
as $\mu\to\mu_{0}$ the green slice does not disappear even though its
conformal boundary has shrunk to a point. This is illustrated by the left
figure 9, which shows a static bubble of ‘true vacuum’ suspended from a point
on the boundary of the ‘false vacuum’.242424 These are static solutions, not
to be confused with ‘bags of gold’ which are cosmologies glued onto the
backside of a Schwarzschild-AdS spacetime, see e.g. [103, 30]. The phenomenon
is reminiscent of spacetimes that realize ‘wedge’ or codimension-2 holography,
like those in refs. [104, 105, 106].  To convince ourselves that the
phenomenon is real, we give an analytic proof in appendix D of the existence
of such suspended bubbles in at least one region of parameters
($\ell_{2}>\ell_{1}$ and $\lambda\approx\lambda_{\rm min}>0$). Furthermore,
since the vacuum solution for a given $\gamma$ is unique, there is no other
competing solution. In the example of appendix D, in particular, $\gamma$ is
finite and negative at $\mu=0$.

Figure 9: Left: A bubble of true vacuum that survives inside the false vacuum
despite the fact that its conformal boundary shrinks to a point. Right: A
bubble of false vacuum with $\lambda=\lambda_{0}$ inscribed between the
boundary and the horizon of a black hole.

In the language of field theory this is a striking phenomenon. It implies that
interface and anti-interface do not annihilate, but fuse into an exotic defect
generating spontaneously a new scale in the process. This is the blueshift at
the tip of the bubble, $\sigma_{+}(\mu_{0},L_{2})$, or better the
corresponding frequency scale $r_{2}(\sigma_{+})$ in the D(efect)CFT.

The phenomenon is not symmetric under the exchange $1\leftrightarrow 2$.
Static bubbles of the false vacuum (pink) spacetime inside the true (green)
vacuum do not seem to exist. We proved this analytically for
$\lambda<\lambda_{0}$, and numerically for all other values of the tension. We
have also found that the suspended green bubble can be of type E1, i.e. have a
center. The redshift factor $g_{tt}$ inside the bubble can even be lower than
in the surrounding space, so that the bubble hosts the excitations of lowest
energy. We did not show this analytically, but the numerical evidence is
compeling.

Do suspended bubbles also exist when the surrounding spacetime contains a
black hole ? The answer is affirmative as one can show semi-analytically by
focussing on the region $\lambda\approx\lambda_{0}$. We have seen in the
previous section that near this critical tension there exist warm solutions of
type [H1,E2′] with the wall arbitrarily close the horizon. Let us consider the
function $\tau_{2}(\mu,\lambda)$ given in this branch of solutions by eqs.
(6.12b) and (6.11) (with $\mathbb{S}_{2}\not=$E1). This is a continuous
function in both arguments, so as $\lambda$ increases past $\lambda_{0}$,
$\tau_{2}(1)$ goes from positive to negative with the overall shape of the
function varying smoothly. This is illustrated in figure 10, where we plot
$\tau_{2}(\mu)$ for $\lambda$ slightly below and slightly above $\lambda_{0}$.
It should be clear from these plots that for $\lambda>\lambda_{0}$ (the plot
on the right) $\tau_{2}$ vanishes at a finite $\mu\approx 1$. This is a warm
bubble solution, as advertized.

Figure 10: Plots of the function $\tau_{2}(\mu)$ in the [H1,E2] or [H1,E2′]
branch of solutions for $\ell_{2}/\ell_{1}=1.5$. The critical tension is
$\lambda_{0}\ell_{2}\approx 1.12$. The curve on the left is for
$\lambda_{0}\ell_{2}=1.05$, and the curve on the right for
$\lambda_{0}\ell_{2}=1.35$.

We have found more generally that warm bubbles can be also of type E1, thus
acting as a suspended Faraday cage that protects inertial observers from
falling towards the horizon of the black hole. Contrary, however, to what
happened for the ground state, warm bubble solutions are not unique. There is
always a competing solution at $\mu\to-\infty$, and it is the dominant one by
virtue of its divergent negative Casimir energy. A stability analysis would
show if warm bubble solutions can be metastable and long-lived, but this is
beyond our present scope.

As for warm bubbles of type [X, H1], that is with the black hole in the false-
vacuum slice, these also exist but only if $\ell_{2}<3\ell_{1}$. Indeed, as we
will see in a moment, when $\ell_{2}>3\ell_{1}$ the wall cannot avoid a
horizon located on the false-vacuum side.

Finally simple inspection of fig. 10 shows that by varying the tension, the
bubble solutions for $\lambda>\lambda_{0}$ go over smoothly to the hot
solution at $\lambda=\lambda_{0}$. At this critical tension the bubble is
inscribed between the horizon and the conformal boundary, as in figure 9. This
gives another meaning to $\lambda_{0}$: Only walls with this tension may touch
the horizon without falling inside.

##  9 Phase diagrams

In this last section of the paper we present numerical plots of the phase
diagram of the model. We work in the canonical ensemble, so the variables are
the temperature and volumes, or by scale invariance two of the dimensionless
ratios defined in (6.1). We choose these to be
$\,\tau=\tau_{1}+\tau_{2}=T(L_{1}+L_{2})$ and $\,\gamma=L_{1}/L_{2}$. The
color code is as in fig. 6.

We plot the phase diagram for different values of the action parameters
$\ell_{1},\ell_{2},\lambda$. Since our analysis is classical in gravity,
Newton’s constant $G$ plays no role. Only two dimensionless ratios
matter,252525Dimensionless in gravity, not in the dual ICFT.  for instance

$\displaystyle b:={\ell_{2}\over\ell_{1}}={c_{2}\over c_{1}}\geq 1\qquad{\rm
and}\quad\kappa:=\lambda\ell_{2}\in(b-1,b+1)\ .$ (9.1)

The value $b=1$ corresponds to a defect CFT, while $b\gg 1$ is the opposite
“near void” limit in which the degrees of freedom of CFT2 ovewhelm those of
CFT1. The true vacuum approaches in this limit the infinite-radius AdS, and/or
the false vacuum approaches flat spacetime. The critical tension $\lambda_{0}$
corresponds to $\kappa_{0}=\sqrt{b^{2}-1}$.

To plot the phase diagrams we solved numerically for $\mu$ in terms of the
boundary data $(\gamma,\tau)$ and for all types of slice pair, and compared
their free energies when solutions of different type coexist. As explained in
the introduction, although the interpretation is different, our diagrams are
related to the ones of Simidzija and Van Raamsdonk [29] by double-Wick
rotation (special to 2+1 dimensions). Since time in this reference is non-
compact, only the boundaries of our phase diagrams, at $\gamma=0$ or
$\gamma=\infty$, can be compared. The roles of thermal AdS and BTZ are also
exchanged

###  9.1 Defect CFT

Consider first $b=1$. By symmetry, we may restrict in this case to $\gamma\geq
1$. Figure 11 presents the phase diagram in the $(\gamma,\tau)$ plane for a
very light ($\kappa=0.03$) and a very heavy ($\kappa=1.8$) domain wall. For
the light, nearly tensionless, wall the phase diagram approaches that of a
homogeneous CFT. The low-$T$ solution is single-center, and the Hawking-Page
(HP) transition occurs at $\tau\approx 1$. Light domain walls follow closely
geodesic curves, and avoid the horizon in a large region of parameter space.
262626One can compute this phase diagram analytically by expanding in powers
of $\lambda$.

Figure 11: Phase diagrams of a very light (left) and a very heavy (right)
domain wall between degenerate vacua ($b=1$). The horizontal and vertical axes
are $\gamma$ and $\tau$. The broken line in the left diagram separates
solutions of type [H1, E2′] and [H1,E2] that only differ in the sign of the
energy of the horizonless slice. The color code is as in fig. 6.

Comparing the left with the right figure 11 shows that heavy walls facilitate
the formation of the black hole and have a harder time staying outside.
Indeed, in the right figure the HP transition occurs at lower $T$, and the
warm phase recedes to $L_{1}\gg L_{2}$. Furthermore, both the cold and the
warm solutions have now an additional AdS restpoint. This confirms the
intuition that heavier walls repel probe masses more strongly, and can shield
them from falling inside the black hole.

The transition that sweeps away this AdS restpoint is shown explicitly in the
phase diagrams of figure 12. Recall from the analysis of section 7.2 that in
the low-$T$ phase such transitions happen for
$\lambda<1/\ell_{1}\Longrightarrow\kappa<b=1$. Furthermore, the transitions
take place at the critical mass ratios $\mu_{j}^{*}\,$, given by eq. (7.2).
Since in cold solutions the relation between $\mu$ and $\gamma$ is one-to-one,
the dark-light blue critical lines are lines of constant $\gamma$. These
statements are in perfect agreement with the findings of fig. 12.

Figure 12: Phase diagrams for intermediate-tension walls exhibiting sweeping
transitions. On the left a restpoint of the vacuum solution is swept away as
$\gamma$ increases beyond a critical value. On the right the same happens in
the warm solution but for decreasing $\gamma$. In these diagrams, the one-
center warm solution is always [H1,E2].

Warm solutions of type [H1,E1], respectively [E1,H1], exist for tensions
$\lambda>1/\ell_{1}\Longrightarrow\kappa>b$, respectively
$\lambda>1/\ell_{2}\Longrightarrow\kappa>1$. In the case of a defect these two
ranges coincide. The stable black hole forms in the larger of the two slices,
i.e. for $\gamma>1$ in the $j=1$ slice. The sweeping transition occurs at the
critical mass ratio $\mu_{2}^{*}=(b^{2}-\kappa^{2})^{-1}$, which through eqs.
(6.11) and (6.12a) corresponds to a fixed value of $\tau_{2}$. Since
$\tau=\tau_{2}(1+\gamma)$, the critical orange-yellow line is a straight line
in the $(\gamma,\tau)$ plane, in accordance again with the findings of fig.
12.

A noteworthy fact is the rapidity of these transitions as function of
$\kappa$. For $\kappa$ a little below or above the critical value the single-
center cold, respectively warm phases almost disappear. Note also the cold-to-
warm transitions are always near $\tau\approx 1$. This is the critical value
for Hawking-Page transitions in the homogeneous case, as expected at large
$\gamma$ when the $j=1$ slice covers most of space.

The critical curves for the cold-to-hot and warm-to-hot transitions also look
linear in the above figures, but this is an illusion. Since the transitions
are first order we must compare free energies. Equating for example the hot
and cold free energies gives after some rearrangements (and with
$\ell_{1}=\ell_{2}:=\ell$)

$\displaystyle 2\pi^{2}\tau+{2\over\ell}\log g_{I}={1\over
2\tau_{1}}|M_{1}|L_{1}^{2}\,(1+{\mu\over\gamma})\ .$ (9.2)

Now $|M_{1}|L_{1}^{2}$ can be expressed in terms of $\mu$ through eq. (6.8,
6.9a), and $\mu$ in the cold phase is a function of $\gamma$. Furthermore
$\log g_{I}/\ell=4\pi{\rm tanh}^{-1}(\kappa/2)$ is constant, see eq. (4.9),
and $\tau_{1}=\tau/(1+\gamma^{-1})$. Thus (9.2) can be written as a relation
$\tau=\tau_{\rm hc}(\gamma)$, and we have verified numerically that $\tau_{\rm
hc}$ is not a linear function of $\gamma$.

###  9.2 Non-degenerate vacua

Figure 13 presents the phase diagram in the case of non-degenerate AdS vacua,
$b=\ell_{2}/\ell_{1}=c_{2}/c_{1}=3$, and for different values of the tension
in the allowed range, $\kappa\in(2,4)$. Since there is no
$\gamma\to\gamma^{-1}$ symmetry, $\gamma$ here varies between 0 to $\infty$.
To avoid squeezing the $\gamma\in(0,1)$ region, we use for horizontal axis
$\alpha:=\gamma-\gamma^{-1}$. This is almost linear in the larger of $\gamma$
or $\gamma^{-1}$, when either of these is large, but the region $\gamma\approx
1$ is distorted compared to figs. 11 and 12 of the previous section.

Figure 13: Phase diagrams for $b=3$, and values of the tension that increase
from the top-left figure clockwise. The horizontal and vertical axes are
$\alpha:=\gamma-\gamma^{-1}$ and $\tau$. The broken red curve is the bound
$\tau=\tau_{2}^{*}(1+\gamma)$ below which the hot solution does not exist
(there is no such bound in the lower panels in which the tension
$\lambda>\lambda_{0}$)

. Note the absence of a warm phase in the left ($\gamma<1$) region of the
diagrams. For the heaviest wall all non-hot solutions are double-center.

The most notable new feature in these phase diagrams is the absence of a warm
phase in the region $\gamma<1$. This shows that it is impossible to keep the
wall outside the black hole when the latter forms on the false-vacuum side.
From the perspective of the dual ICFT, see section 7.1, the absence of
[X,H1]-type solutions means that no interfaces, however heavy, can keep CFT1
in the confined phase if CFT2 (the theory with larger central charge) has
already deconfined.

We suspect that this is a feature of the thin-brane model which does not allow
interfaces to be perfectly-reflecting [34].

Warm solutions with the horizon in the pink slice appear to altogether
disappear above the critical ratio of central charges $b_{c}=3$.272727This
critical value was also noticed in ref. [29], who also note that multiple
branes can evade the bound confirming the intuition that it is a feaure
specific to thin branes. As a matter of fact, although [X,H1] solutions do
exist for $b<3$ as we show below, they have very large $\gamma$, outside the
range of our numerical plots, unless $b$ is very close to 1.  The boundary
conditions corresponding to topologies of type [X,H1] are given by eqs.
(6.14). We plotted the right-hand side of the second condition (6.14) for
different values of $\lambda$ and $\mu$ in their allowed range, and found no
solution with positive $\tau_{1}$ for $b>3$. Analytic evidence for the
existence of a strict $b_{c}=3$ bound can be found by considering the limit of
a maximally isolating wall, $\lambda\approx\lambda_{\rm max}$, and of a
shrinking green slice $\hat{\mu}\to-\infty$. In this limit, the right-hand
side of (6.14) can be computed in closed form with the result

$\displaystyle\tau_{1}(\hat{\mu})={\pi\over\sqrt{-\hat{\mu}}}\biggl{(}2-\sqrt{1+{\ell_{2}\over\ell_{1}}}\,\,\biggr{)}+{\rm
subleading}\ .$ (9.3)

We took X=E1 as dictated by the analysis of sweeping transitions, see section
7.2 and in particular eq. (7.5). This limiting $\tau_{1}(\hat{\mu})$ is
negative for $b>3$, and positive for $b<3$ where warm [E1,H1] solutions do
exist, as claimed.

An interesting corollary is that end-of-the-world branes cannot avoid the
horizon of a black hole, since the near-void limit, $\ell_{1}\ll\ell_{2}$, is
in the range that has no [X,H1] solutions.

###  9.3 Unstable black holes

The phase diagrams in figs. 11, 12, 13 show the solution with the lowest free
energy in various regions of parameter space. Typically, this dominant phase
coexists with solutions that describe unstable or metastable black-holes which
are ubiquitous in the thin-wall model.282828For a similar discussion of
deformed JT gravity see ref.[66]. Note that in the absence of a domain wall,
the only static black hole solution of pure Einstein gravity in 2+1 dimensions
is the non-spinning BTZ black hole.

Figure 14 shows the number of black hole solutions in the degenerate case,
$b=1$, for small, intermediate and large wall tension, and in different
regions of the $(\tau,\gamma)$ parameter space. The axes are the same as in
figs. 11 and 12 but the range of $\gamma$ is halved. At sufficiently high
temperature the growing horizon captures the wall, and the only solution is
the hot solution. We see

Figure 14: The number of independent black hole solutions in the
$(\gamma,\tau)$ parameter space for $b=1$, and three values of the tension
$(\kappa=0.2;\ 1.1;{\rm and}\ 1.8)$. The darker the shade the larger the
number of black holes.

however that in a large region of intermediate temperatures the hot solution
coexists with two warm solutions. Finally at very low temperature the hot
solution coexists with four other black-hole solutions, two on either side of
the wall. The dominant phase in this region is vacuum, so the black holes play
no role in the canonical ensemble.

The hot solution exists almost everywhere, except when $\lambda<\lambda_{0}$
and $\tau=\tau_{2}(1+\gamma)<\tau_{2}^{*}(1+\gamma)$ with $\tau_{2}^{*}$ given
by eq. (6.5). It has positive specific heat even when it is not the dominant
phase. For warm black holes, on the other hand, the specific heat can have
either sign. One can see this semi-analytically by focussing once again on our
favourite near-critical region $\lambda\approx\lambda_{0}$. Simple inspection
of fig. 8 shows that in some range $\tau_{2}^{*}<\tau_{2}<\tau_{2}^{\rm max}$
the hot solution coexists with two nearby warm solutions. At the maximum
$\tau_{2}^{\rm max}$, where $d\tau_{2}/d\mu=0$, the warm solutions merge and
then disappear. Since the black hole is in the $j=1$ slice, $M_{1}=(2\pi
T)^{2}$ and their energy reads

$\displaystyle E_{\rm[warm]}\,=\,{1\over
2}(\ell_{1}M_{1}L_{1}+\ell_{2}M_{2}L_{2})\,=\,2\pi^{2}T^{2}L_{2}\,\bigl{(}\ell_{1}\gamma+\ell_{2}{\mu}\,\bigr{)}\
.$ (9.4)

Taking a derivative with respect to $T$ with $L_{1},L_{2}$ kept fixed we
obtain

$\displaystyle{d\over dT}E_{\rm[warm]}\,=\,{2\over
T}E_{\rm[warm]}+2\pi^{2}T^{2}\,L_{2}^{\,2}\,\ell_{2}\,{d\mu\over d\tau_{2}}\
.$ (9.5)

Near $\tau_{2}^{\rm max}$ the dominant contribution to this expression comes
from the derivative ${d\mu/d\tau_{2}}$ which jumps from $-\infty$ to
$+\infty$. It follows that the warm black hole with the higher mass has
negative specific heat, and should decay to its companion black hole either
classically or in the quantum theory.292929We have verified numerically that
the black holes with negative specific heat are never the ones with lowest
free energy, a conclusion similar to the one reached in deformed JT gravity in
ref. [66].

It would be very interesting to calculate this decay process, but we leave
this for future work.

One last comment concerns transitions from the double-center vacuum
geometries, of type [E1,E1], to warm solutions where the wall avoids the
horizon. One can ask what side of the wall does the black hole choose. A
natural guess is that it forms in the deepest of the two AdS wells. The
relative depth is the ratio of blueshift factors at the two rest points,

$\displaystyle{\mathfrak{R}}:=\sqrt{g_{tt}|_{r_{1}=0}\over
g_{tt}|_{r_{2}=0}}\,=\,{\ell_{2}\over\ell_{1}\sqrt{\mu(\gamma)}}\,\ .$ (9.6)

One expects the black hole to form in the $j=1$ (green) slice if
${\mathfrak{R}}<1$ and in the $j=2$ (red) slice if ${\mathfrak{R}}>1$. Our
numerical plots confirmed in all cases this expectation.

##  10 Outlook

One urgent question, already noted in the introduction, is how much of this
analysis will survive in top-down interface models, where gravitating domain
walls are typically thick. The order parameters of the Hawking-Page and
sweeping transitions – the area of the horizon and the number of rest points
for inertial observers, do not depend on the assumption of a thin wall and
could go through. The warm-to-hot transition, on the other hand, may be
replaced by a crossover, since there is no sharp criterion to decide if a
thick wall enters or avoids the horizon. As discussed, however, in section 7.1
a sharp order parameter, such as a Polyakov loop, may be suggested by the
field theory side of the correspondence.

One other question left open in the present work is the entanglement structure
of the equilibrium states. Indeed, a guiding thread of our paper were the
intersections of the domain wall with the black hole horizon and the
trajectories of inertial observers. The Ryu-Takayanagi (RT) surfaces [58, 59]
are another natural class of curves whose intersection with the wall should be
studied along lines similar to ref. [21, 22] for BCFT.

Simple extensions of the minimal model, such as the addition of a Chern-Simons
field (see e.g. [107]) might also be worth exploring.

Last but not least, the simplicity of the model and its rich spectrum of black
holes make it a promising ground where to try to shed some more light on the
recent exciting developments related to black hole evaporation, islands and
the Page curve [12, 13, 14]. We hope to return to some of these questions in
the near future.

Aknowledgements

We are grateful to Mark Van Raamsdonk for his critical reading of a
preliminary draft of this paper and for many useful comments. Many thanks also
to Panos Betzios, Shira Chapman, Dongsheng Ge, Elias Kiritsis, Ioannis Lavdas,
Bruno Le Floch, Emil Martinec, Olga Papadoulaki and Giuseppe Policastro for
discussions during the course of this work.

##  Appendix A Renormalized on-shell action

The Euclidean action of the holographic-interface model, in units $8\pi G=1$,
is the sum of bulk, brane, boundary and corner contributions, see e.g.[108]

$\displaystyle I_{\rm gr}=-\frac{1}{2}$
$\displaystyle\hskip-7.11317pt\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{\hat{g}_{w}}\
\ \ \ \ \ $ (A.1)
$\displaystyle\hskip-17.07164pt+\int_{\partial{\mathbb{S}}_{1}}d^{2}s\sqrt{\hat{g}_{1}}\,K_{1}+\int_{\partial{\mathbb{S}}_{2}}d^{2}s\sqrt{\hat{g}_{2}}\,K_{2}\
+\ \int_{\rm C}(\theta-\pi)\sqrt{\hat{g}_{c}}+{\rm c.t.}\ $

where the counterterms, abbreviated above by c.t., read [109]

$\displaystyle{\rm c.t.}=\,{1\over\ell_{1}}\int_{{\rm
B}_{1}}\sqrt{\hat{g}_{1}}\,+{1\over\ell_{2}}\int_{{\rm
B}_{2}}\sqrt{\hat{g}_{2}}\,-\,\int_{{\rm B}_{1}\cap{\rm
B}_{2}}(\theta_{1}+\theta_{2})\sqrt{\hat{g}_{c}}\ .$ (A.2)

Here $\mathbb{S}_{j}$ are the spacetime slices whose boundary is the sum of
the cutoff surface Bj and of the string worldsheet $\mathbb{W}$, i.e.
$\partial\mathbb{S}_{j}=$B${}_{j}\cup\mathbb{W}$ . The induced metrics are
denoted by hats. The $K_{j}$ are traces of the extrinsic curvatures on each
slice computed with the inward-pointing normal vector. Finally, in addition to
the standard Gibbons-Hawking-York boundary terms, one must add the Hayward
term [110, 108] at corners of $\partial\mathbb{S}_{j}$ denoted by C.
303030These 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,
B${}_{1}\cap$B2, where $\theta-\pi$ is the sum of the angles $\theta_{j}$
defined in figure 4.

Let us break the action into an interior and a conformal boundary term,
$I_{\rm gr}=I_{\rm int}+I_{\rm B}$, with the former including contributions
from the worldsheet W. Using the field equations $R_{j}=-{6/\ell_{j}^{2}}$ and
$K_{1}|_{\rm W}+K_{2}|_{\rm W}=-2\lambda$, and the volume elements that follow
from eqs. (3.1) and (4.1, 4.2),

$\sqrt{g_{j}}\,d^{3}x=\ell_{j}r_{i}dr_{j}dx_{j}dt\,\quad{\rm
and}\quad\,\sqrt{\hat{g}_{w}}\,d^{2}s=\sqrt{fg}\,d\sigma dt\ ,$

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

$I_{\rm
int}\,=\,\frac{2}{\ell_{1}}\int_{{\Omega}_{1}}r_{1}\,dr_{1}dx_{1}dt+\frac{2}{\ell_{2}}\int_{{\Omega}_{2}}r_{2}\,dr_{2}dx_{2}dt-\lambda\int_{\mathbb{W}}\sqrt{fg}\,d\sigma
dt\ .$ (A.3)

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,

$\displaystyle\sum_{j=1,2}\,{2\over\ell_{j}}\int_{{\Omega}_{j}}r_{j}dr_{j}dx_{j}dt\
=\
\sum_{j=1,2}\,{1\over\ell_{j}}\oint_{{\partial\Omega}_{j}}r_{j}^{2}(\hat{r}_{j}\cdot
d\hat{n}_{j})dt\ ,$ (A.4)

with $d\hat{n}_{j}dt$ the surface element on the boundary
$\partial\Omega_{j}$. 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 eq. (A.4) receives contributions from the three
pieces of ${\partial\Omega}_{1,2}$ : the cutoff surface B${}_{1}\cup$B2, the
horizon if there is one, and the worldsheet $\mathbb{W}$. Conveniently, this
last term precisely cancels the third term in (A.3) by virtue of the Israel-
Lanczos equation (4.3). 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 the interior part of the
action, eq. (A.3) , reads

$\displaystyle I_{\rm int}\,=\ {1\over\ell_{1}T}\,\Bigl{[}r_{1}^{2}\,\Delta
x_{1}\Bigr{]}_{\rm Hor}^{{\rm
B}_{1}}+\,{1\over\ell_{2}T}\,\Bigl{[}r_{2}^{2}\,\Delta x_{2}\Bigr{]}_{\rm
Hor}^{{\rm B}_{2}}$ (A.5)

where we employ the shorthand notation $[X]_{a}^{b}=X|_{b}-X|_{a}$ , and
$X|_{a}$ for $X$ evaluated at $a$ . If the slice $\mathbb{S}_{j}$ does not
contain a horizon the corresponding contribution is absent.

We now turn to the conformal-boundary contributions from the lower line in the
action (A.1). For a fixed-$r_{j}$ surface, the inward-pointing unit normal
expressed as a 1-form is n${}_{j}=-dr_{j}/\sqrt{r_{j}^{2}-M_{j}\ell_{j}^{2}}$
. One finds after a little algebra (we here drop the index $j$ for simplicity)

$\displaystyle
K_{xx}=K_{tt}=-{r\over\ell}\sqrt{r^{2}-M\ell^{2}}\,\Longrightarrow\,\sqrt{\hat{g}}\,K=\,-{1\over\ell}(2r^{2}-M\ell^{2})\
.$ (A.6)

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

$\displaystyle I_{\rm
B}={1\over\ell_{1}T}(r_{1}\sqrt{r_{1}^{2}-M_{1}\ell_{1}^{2}}-2r_{1}^{2}+M_{1}\ell_{1}^{2})\,\Delta
x_{1}\,\Bigl{|}_{{\rm B}_{1}}\,+\,(1\rightarrow 2)\ .$ (A.7)

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

$\displaystyle I_{\rm B}={1\over\ell_{1}T}(-r_{1}^{2}+{1\over
2}M_{1}\ell_{1}^{2})\,\Delta x_{1}\,\Bigl{|}_{{\rm B}_{1}}\,+\,(1\rightarrow
2)\ .$ (A.8)

Upon adding up (A.5) and (A.8) the leading divergent term cancels, giving the
following result for the renormalized on-shell action :

$\displaystyle I_{\rm gr}\ =\ {M_{1}\ell_{1}\over 2T}\bigl{(}L_{1}-2\Delta
x_{1}\bigl{|}_{{\rm Hor}}\bigr{)}\,+\,{M_{2}\ell_{2}\over
2T}\bigl{(}L_{2}-2\Delta x_{2}\bigl{|}_{{\rm Hor}}\bigr{)}\,\ .$ (A.9)

We used here the fact that $\Delta x_{j}|_{{\rm B}_{j}}=L_{j}$, 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 $\langle E\rangle={1\over
2}(M_{1}\ell_{1}L_{1}+M_{2}\ell_{2}L_{2})$, see section 2, and $I_{\rm
gr}=\langle E\rangle/T-S$ we find

$\displaystyle S={1\over T}$
$\displaystyle\hskip-28.45274pt\bigl{(}M_{1}\ell_{1}\Delta x_{1}\bigl{|}_{{\rm
Hor}}+M_{2}\ell_{2}\Delta x_{2}\bigl{|}_{{\rm Hor}}\bigr{)}$ (A.10)
$\displaystyle=$ $\displaystyle\hskip-5.69054pt4\pi^{2}T\bigl{(}\ell_{1}\Delta
x_{1}\bigl{|}_{{\rm Hor}}+\ell_{2}\Delta x_{2}\bigl{|}_{{\rm
Hor}}\bigr{)}\,=\,{A({\rm horizon})\over 4G}\ .$

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.

##  Appendix B Opening arcs as elliptic integrals

In this appendix we express the opening arcs, eqs. (4.17), in terms of
complete elliptic integrals of the first, second and third kind,

$\displaystyle{\rm\bf K}(\nu)=\int_{0}^{1}\frac{dy}{\sqrt{(1-y^{2})(1-\nu
y^{2})}}$ (B.1) $\displaystyle{\bf E}(\nu)=\int_{0}^{1}\frac{\sqrt{1-\nu
y^{2}}\,dy}{\sqrt{1-y^{2}}}\ .$ (B.2)
$\displaystyle{\bf\Pi}(u,\nu)=\int_{0}^{1}\frac{dy}{(1-uy^{2})\sqrt{(1-y^{2})(1-\nu
y^{2})}}\ .$ (B.3)

Consider the boundary conditions (4.17a). The other conditions (4.17b, 4.17c)
differ only by the constant periods or horizon arcs, $P_{j}$ or $\Delta
x_{j}|_{\rm hor}$. Inserting the expression (4.16) for $x_{1}^{\prime}$ gives

$\displaystyle
L_{1}=-\int_{\sigma_{+}}^{\infty}\frac{\ell_{1}\,d\sigma}{(\sigma+M_{1}\ell_{1}^{2})}\,\frac{(\lambda^{2}+\lambda_{0}^{2})\,\sigma+M_{1}-M_{2}}{\sqrt{A\sigma(\sigma-\sigma_{+})(\sigma-\sigma_{-})}}\
,$ (B.4)

and likewise for $L_{2}$. The roots $\sigma_{\pm}$ are given by eqs. (4.14,
4.15). We assume that we are not in the case [H2, H2] where $M_{1}=M_{2}>0$,
nor in the fringe case $\sigma_{+}=-M_{j}\ell_{j}^{2}$ when the string goes
through an AdS center. These cases will be treated separately.

Separating the integral in two parts, and trading the integration variable
$\sigma$ for $y$, with $y^{2}:=\sigma_{+}/\sigma$, we obtain

$L_{1}=-\frac{2\ell_{1}}{\sqrt{A\,\sigma_{+}}}\bigg{[}{M_{1}-M_{2}\over
M_{1}\ell_{1}^{2}}\int_{0}^{1}\frac{dy}{\sqrt{(1-y^{2})(1-\nu y^{2})}}$
$\displaystyle+\Bigl{(}(\lambda^{2}+\lambda_{0}^{2})-{M_{1}-M_{2}\over
M_{1}\ell_{1}^{2}}\Bigr{)}\,\int_{0}^{1}\frac{y^{2}dy}{(1-u_{1}y^{2})\sqrt{(1-y^{2})(1-\nu
y^{2})}}\,\bigg{]}\ \ \ $ (B.5)

where $\nu=\sigma_{-}/\sigma_{+}$ and $u_{1}=-M_{1}\ell_{1}^{2}/\sigma_{+}$ .
Identifying the elliptic integrals finally gives

$\displaystyle
L_{1}=-\frac{2\ell_{1}}{\sqrt{A\,\sigma_{+}}}\bigg{[}\frac{M_{1}-M_{2}}{M_{1}\ell_{1}^{\,2}}\,\Bigl{(}{\rm\bf
K}(\nu)-{\bf\Pi}(u_{1},\nu)\Bigr{)}+(\lambda^{2}+\lambda_{0}^{2})\,{\bf\Pi}(u_{1},\nu)\bigg{]}\
,\ \ \ $ (B.6)

and a corresponding expression for $L_{2}$

$\displaystyle
L_{2}=-\frac{2\ell_{2}}{\sqrt{A\,\sigma_{+}}}\bigg{[}\frac{M_{2}-M_{1}}{M_{2}\ell_{2}^{\,2}}\,\Bigl{(}{\rm\bf
K}(\nu)-{\bf\Pi}(u_{2},\nu)\Bigr{)}+(\lambda^{2}-\lambda_{0}^{2})\,{\bf\Pi}(u_{2},\nu)\bigg{]}\
\ \ $ (B.7)

with $u_{2}=-M_{2}\ell_{2}^{2}/\sigma_{+}$ . The prefactors in (B.6) diverge
when $M_{1}\to 0$ but the singilarity is removed by expanding
${\bf\Pi}(u_{1},\nu)$ around $u_{1}=0$. In this limit

$\displaystyle
L_{1}(M_{1}=0)=-\frac{2\ell_{1}}{\sqrt{A\,\sigma_{+}}}\bigg{[}\frac{M_{2}}{\sigma_{-}}({\rm\bf
E}(\nu)-{\rm\bf K}(\nu))+(\lambda^{2}+\lambda_{0}^{2}){\rm\bf K}(\nu)\bigg{]}$
(B.8a) and similarly $\displaystyle
L_{2}(M_{2}=0)=-\frac{2\ell_{2}}{\sqrt{A\,\sigma_{+}}}\bigg{[}\frac{M_{1}}{\sigma_{-}}({\rm\bf
E}(\nu)-{\rm\bf K}(\nu))+(\lambda^{2}-\lambda_{0}^{2}){\rm\bf K}(\nu)\bigg{]}$
(B.8b)

with ${\rm\bf E}(\nu)$ the complete elliptic integral of the second kind.

The $M_{1}=M_{2}>0$ geometries correspond to the high-temperature phase where
$M_{j}=(2\pi T)^{2}$, $\sigma_{+}=0$ and $\sigma_{-}=-(4\pi T\lambda)^{2}/A$.
The integrals (4.17c) simplify to elementary functions in this case:

$L_{1}-\Delta_{1}^{\rm Hor}\ =\
-{\ell_{1}(\lambda^{2}+\lambda_{0}^{2})\over\sqrt{A\,|\sigma_{-}|\,}}\underbrace{\int_{0}^{\infty}{ds\over(s+a)\,\sqrt{s+1}\,}}_{\textstyle\begin{array}[]{c}={2\over\sqrt{1-a}}{\rm
arctanh}({\sqrt{1-a}})\end{array}}$

with $a=A\ell_{1}^{2}/4\lambda^{2}$. Using the expression (4.14) for $A$, and
going through the same steps for $j=2$, gives after a little algebra

$\displaystyle L_{1}-\Delta_{1}^{\rm Hor}\ =\ -{1\over\pi T}\,{\rm
tanh}^{-1}\left({\ell_{1}(\lambda^{2}+\lambda^{2}_{0})\over 2\lambda}\right)\
,$ (B.9a) $\displaystyle L_{2}-\Delta_{2}^{\rm Hor}\ =\ -{1\over\pi T}\,{\rm
tanh}^{-1}\left({\ell_{2}(\lambda^{2}-\lambda^{2}_{0})\over 2\lambda}\right)\
.$ (B.9b)

Interestingly, since $\Delta_{2}^{\rm Hor}$ must be positive, $TL_{2}$ is
bounded from below in the range $\lambda<\lambda_{0}$ as discussed in section
6.2.

In the high-temperature phase the on-shell action, eq. (A.9), reads

$\displaystyle I_{\rm gr}^{\rm(high-T)}=4\pi^{2}T\Bigl{[}-{1\over
2}(\ell_{1}L_{1}+\ell_{2}L_{2})+\ell_{1}(L_{1}-\Delta_{1}^{\rm
Hor})+\ell_{2}(L_{2}-\Delta_{2}^{\rm Hor})\Bigr{]}\,.\ \ $ (B.10)

Using the expressions (B.9) and rearranging the arc-tangent functions gives

$\displaystyle I_{\rm gr}^{\rm(high-T)}:={E\over
T}-S=-2\pi^{2}T(\ell_{1}L_{1}+\ell_{2}L_{2})-\log\,g_{I}$ (B.11)

where the interface entropy $S=\log\,g_{I}$ is given by eq. (4.9).

##  Appendix C Sweeping is continuous

In this appendix we show that sweeping transitions are continuous.

We focus for definiteness on the sweeping of the $j=2$ AdS center at zero
temperature (all other cases work out the same). The transition takes place
when $\mu$ crosses the critical value $\mu_{2}^{*}$ given by eq. (7.2).
Setting $\mu=\mu_{2}^{*}(1-\delta)$ in expression (6.9b) gives

$f_{2}(\mu)=\,\frac{\ell_{2}}{\sqrt{A}}\int_{s_{+}}^{\infty}ds\,\frac{(\lambda^{2}-\lambda_{0}^{2})(s-\mu\ell_{2}^{2})\,+\delta}{(s-\mu\ell_{2}^{2})\,\sqrt{As(s-s_{+})(s-s_{-})}}$
$\displaystyle=\,\frac{2\ell_{2}(\lambda^{2}-\lambda_{0}^{2})}{\sqrt{As_{+}}}\,{\bf
K}\bigl{(}\frac{s_{-}}{s_{+}}\bigr{)}\,+\,\frac{\ell_{2}\,\delta}{\sqrt{A}}\,\underbrace{\int_{s_{+}}^{\infty}\frac{ds}{(s-\mu\ell_{2}^{2})\sqrt{s(s-s_{+})(s-s_{-})}}}_{J}\
.\ \ $ (C.1)

The first term is continuous at $\delta=0$, but the second requires some care
because the integral $J$ diverges. This is because for small $\delta$

$\displaystyle
s_{+}-\mu\ell_{2}^{2}\,=\,\frac{\delta^{2}}{4\lambda^{2}\mu_{2}^{*}}+\mathcal{O}(\delta^{3})\
,$ (C.2)

as one finds by explicit computation of the expression (6.10). If we set
$\delta=0$, $J$ diverges near the lower integration limit. To bring the
singular behavior to $0$ we perform the change of variable $u^{2}=s-s_{+}$, so
that

$\displaystyle
J=\int_{0}^{\infty}\frac{2du}{(u^{2}+\delta^{2}/4\lambda^{2}\mu_{2}^{*})\sqrt{(u^{2}+s_{+}^{*})(u^{2}+s_{+}^{*}-s_{-}^{*})}}\
,$ (C.3)

where we kept only the leading order in $\delta$, and $s_{\pm}^{*}$ are the
roots at $\mu=\mu_{2}^{*}$. Since $s_{+}^{*}$ and $s_{+}^{*}-s_{-}^{*}$ are
positive and finite, the small-$\delta$ behavior of the integral is (after
rescaling appropriately $u$)

$\displaystyle
J={4\lambda|\mu_{2}^{*}|\over|\delta|\sqrt{s_{+}^{*}(s_{+}^{*}-s_{-}^{*})}}\,\underbrace{\int_{0}^{\infty}{du\over
u^{2}+1}}_{\pi/2}\ +\ {\rm finite}\ .$ (C.4)

Inserting in expression (C.1) and doing some tedious algebra leads finally to
a discontinuity of the function $f_{2}(\mu)$ equal to
sign$(\delta)\,\pi/\sqrt{\mu_{2}^{*}}$ . This is precisely what is required
for $L_{2}$, eq. (6.8), to be continuous when the red ($j=2$) slice goes from
type E1 at negative $\delta$ to type E2 at positive $\delta$.

##  Appendix D Bubbles exist

We show here that the bubble phenomenon of section 8 is indeed realized in a
region of the parameter space of the holographic model.

This is the region of non-degenerate gravitational vacua ($\ell_{2}$ strictly
bigger than $\ell_{1}$) and a sufficiently light domain wall. Specifically, we
will show that for $\lambda$ close to its minimal value, $\lambda_{\rm min}$,
the arc $L_{1}(\mu=0)$ is negative, so the wall self-intersects and $\mu_{0}$
is necessarily finite.

Let $\lambda=\lambda_{\rm min}(1+\delta)$ with $\delta\ll 1$. Setting $\mu=0$
and expanding eqs. (6.10) for small $\delta$ gives

$A={8\lambda_{\rm
min}^{2}\,\delta\over\ell_{1}\ell_{2}}+\mathcal{O}(\delta^{2})\ ,\quad
s_{+}={\ell_{2}\over 4\lambda_{\rm min}}+\mathcal{O}(\delta)\ ,\quad
s_{-}=-{\ell_{1}\over\ 2\lambda_{\rm min}\delta}+\mathcal{O}(1)\ .$

Plugging into eq. (B.6) with $M_{2}=\mu M_{1}\approx 0$ we find :

$\sqrt{|M_{1}|}\,L_{1}=-\frac{2}{\ell_{1}\sqrt{As_{+}}}\left[{\bf
K}\left({s_{-}\over
s_{+}}\right)+(1-\frac{2\ell_{1}}{\ell_{2}}){\bf\Pi}\left({\ell_{1}^{2}\over
s_{+}},\,{s_{-}\over s_{+}}\right)\right]$ (D.1)

where we have only kept leading orders in $\delta$. Now we need the asymptotic
form of the elliptic integrals when their argument diverges

${\bf
K}\left[-\frac{a}{\delta}\right]\approx{\bf\Pi}\left[u,-\frac{a}{\delta}\right]\approx-\frac{\ln(\delta)\sqrt{\delta}}{2\sqrt{a}}+\mathcal{O}(\sqrt{\delta})$
(D.2)

for $\delta\to 0_{+}$ with $u,a$ fixed. Using $a=2\ell_{1}/\ell_{2}$ finally
gives

$\displaystyle\sqrt{|M_{1}|}\,L_{1}\,\approx\,\bigl{(}\frac{\ell_{2}}{\ell_{1}}-1\bigr{)}^{1/2}\ln(\delta)\,+\,{\rm
subleading}\ .$ (D.3)

For $\delta\ll 1$ this is negative, proving our claim. Note that we took the
green slice to be of type E2, as follows from our analysis of the sweeping
transitions for light domain walls – see section 7.2.

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