aainstitutetext: Department of Physics, Ramakrishna Mission Vivekananda
Educational and Research Institute, Belur Math, Howrah 711202,
India.bbinstitutetext: Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Kolkata 700064, India.ccinstitutetext: Homi Bhabha National Institute,
Training School Complex, Anushaktinagar, Mumbai 400094, India.ddinstitutetext:
Institut für Theoretische Physik und Astrophysik, Julius-Maximilians-
Universität Würzburg, Am Hubland, 97074 Würzburg, Germany.

# Moving Mirrors, OTOCs and Scrambling

Parthajit Biswas a , Bobby Ezhuthachan b,c,d , Arnab Kundu a , Baishali Roy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore the physics of scrambling in the moving mirror models, in which a
two-dimensional CFT is subjected to a time-dependent boundary condition. It is
well-known that by choosing an appropriate mirror profile, one can model
quantum aspects of black holes in two-dimensions, ranging from Hawking
radiation in an eternal black hole (for an “escaping mirror”) to the recent
realization of Page curve in evaporating black holes (for a “kink mirror”). We
explore a class of OTOCs in the presence of such a boundary and explicitly
demonstrate the following primary aspects: First, we show that the dynamical
CFT data directly affect an OTOC and maximally chaotic scrambling occurs for
the escaping mirror for a large-$c$ CFT with identity block dominance. We
further show that the exponential growth of OTOC associated with the physics
of scrambling yields a power-law growth in the model for evaporating black
holes which demonstrates a unitary dynamics in terms of a Page curve. We also
demonstrate that, by tuning a parameter, one can naturally interpolate between
an exponential growth associated to scrambling and a power-law growth in
unitary dynamics. Our work explicitly exhibits the role of higher-point
functions in CFT dynamics as well as the distinction between scrambling and
Page curve. We also discuss several future possibilities based on this class
of models.

##  1 Introduction

It is well-known that a classical integrable dynamics can exhibit chaotic
features depending on the boundary conditions. The prototypical example of
this is the billiard ball problem, where a free particle can become ergodic
depending on the shape of the stadium and therefore the boundary conditions
imposed on the dynamics. The closest quantum analogue of diverging nearby
trajectories is given by measuring expectation values of non-commuting
observables in quantum mechanics. Consider two observers, $A$ and $B$ who
share a common physical state $|\psi\rangle$. If observer $A$ wants to
communicate one bit $a$ to observer $B$, she can apply a unitary operation to
the quantum state $|\psi\rangle$. Subsequently, the resulting state evolves in
time, and observer $B$ makes a measurement on the evolved state. The
probability of measuring a certain bit-information now can be written in terms
of an expectation value of the commutator of the two operators, in the state
$|\psi\rangle$.111more precisely, the probability is given by Xu:2022vko
$\displaystyle
P\left(B|A\right)=\left\langle\psi\right|U_{a}^{\dagger}[P_{b}(t),U_{a}]\left|\psi\right\rangle\
,$ (1) where $P_{b}$ is the measurement operator yielding the result $b$, as
measured by observer $B$ and  $P_{b}(t)\equiv e^{iHt}P_{b}e^{-iHt}\,,$ while
$U_{a}$ is a unitary operation performed on the state by observer $A.$ The
expectation value of the absolute value of commutator-squared of any two
operators can be written in terms of usual time-order correlators and out-of-
time-order correlators, or OTOCs in brief.

The dynamical behaviour of the OTOC quantitatively defines scrambling, which
relates to the transfer of local information to non-local degrees of freedom,
and its associated time-scale, see e.g. Yoshida:2017non .222Note that, as is
discussed in detail in Yoshida:2017non , the connection of OTOC to perfect
scrambling becomes precise when a Haar-random averaging is carried out over
the unitary evolution of operators. Such averaged OTOCs are then related to
Rényi $2$-mutual information. The decay of the corresponding OTOC is therefore
related to a faithful recovery of the state. Relatedly, scrambling of a
localized information across a given system is connected to thermalization and
a notion of “early time chaos”. It is further conjectured that black holes are
the fastest scramblers in Nature Sekino:2008he , in which the scrambling time
scales logarithmically in the number of degrees of freedom of the system:
$t_{\rm scramble}\sim\log N$; whereas for typical quantum mechanical and
quantum field theoretic systems, this scales as a power-law: $t_{\rm
scramble}\sim N^{\alpha}$, where $\alpha$ is a real number.333Note that,
exponential growth is still possible to obtain in weakly-coupled QFTs,
provided the contribution from an infinite diagrams can be resummed. These
infinite diagrams can be obtained as a result of a large-$N$ expansion, see
e.g. Stanford:2015owe . The exponent of the exponential growth is nonetheless
well below the chaos bound of Maldacena:2015waa . In this article, we will not
address the possibility of a resummation of perturbative contributions that
yield a power-law in time. It is an intriguing aspect to which we hope to
return in future.

Note that, while the notion of quantum chaos is multi-faceted and non-unique,
OTOCs have recently been studied extensively in the literature from a wide
perspective and within a very large class of models. While early studies of
OTOCs include Larkin:1964wok , more recently, an interesting bound, known as
the MSS bound Maldacena:2015waa , have been derived on the growth of such
correlators, based on unitarity and causality. Given this, an already
interesting aspect is to explore the role of boundary conditions on such
correlation functions and their respective bounds. In a general QFT, however,
this is a technically difficult task for two reasons: First, it is difficult
to calculate explicitly the Lyapunov growth of an OTOC in a perturbative
framework of QFT. Typically, one needs a large parameter (such as $N$ of a
large-$N$ theory) in which an expansion can be carried out and a resummation
of such a perturbation theory usually contains the information of chaotic
features. Note that, one needs to sum over an infinite class of diagrams and
therefore the OTOC growth is insensitive to a standard perturbative weak-
coupling calculation in QFT. See e.g. Stanford:2015owe ; Chowdhury:2017jzb
.444This aspect is somehow similar to computing non-trivial transport
coefficients in perturbative QFT, e.g. shear viscosity.

Conformal field theories (CFTs) on the other hand provide us with an ideal
framework for addressing this class of questions. While OTOCs are inherently
four-point correlation functions where the time-ordering of the operators need
to have a specific order and therefore are not completely fixed by the
symmetries of the CFT, it is also encouraging that dynamical CFT data (i.e.
OPE data) are inherently encoded in a CFT OTOC. In fact, OTOCs can be used to
diagnose and distinguish between the dynamics of an integrable CFT vs a
chaotic one Das:2021qsd ; Das:2022jrr  — a feature that cannot be captured by
a two-point or a three-point function which are universally fixed.

Furthermore, introducing a boundary in a two-dimensional CFT has a
particularly rich set of possibilities: such boundaries can be defects,
interfaces, or simply boundaries. Typically, these are distinguished by the
boundary condition on the energy-momentum tensor of the corresponding CFT, or,
more generally, on the Virasoro generators, at the location of the boundary.
Consequently, several physical phenomena can be addressed within the framework
of BCFT, including but not limited to critical quenches Calabrese:2006rx ;
Calabrese:2007mtj , universal reflection and transmission coefficients through
an interface Bachas:2020yxv ; Karch:2023evr ; Gillioz:2020mdd , etc. Most of
these examples are based on spatial boundaries (for interface CFTs) or
boundaries in the Euclidean time (e.g. for critical quench dynamics).

The class of models, known as the moving mirror models, on the other hand are
based on boundaries that are dynamically evolving. The framework is inherently
Lorentzian in which the mirrors are set on a timelike
trajectory.555Technically, there is no problem in considering a spacelike or a
null mirror trajectory, however, we do not consider these cases in the present
work. See e.g. Akal:2022qei  for more discussion on such trajectories. These
models have widely explored in the context of black hole Hawking radiation:
early works in Davies:1977yv ; PhysRevD.25.2569 ; PhysRevD.101.025012 , on
Bogoliubov coefficients in PhysRevD.36.2327 , on later refinements and
generalizations in Wilczek:1993jn , on a stochastic perspective in
Raval:1996vt , on partner particles in PhysRevD.91.124060 , on entropy and
information loss in Chen:2017lum ; Reyes:2021npy ; Akal:2020twv ; Akal:2021foz
. On Casimir effect: on radiation reaction in Jaekel:1997hr , on temperature
corrections in Plunien:1999ba , on time-dependence in PhysRevD.100.065022 . In
cosmological expansion: On cosmological radiation in Casadio:2002dj , on de-
Sitter moving mirror cosmology in Good:2020byh , production of entanglement
entropy from moving mirror in Cotler:2022weg ; entanglement extraction in
Cong:2018vqx ; Cong:2020nec , on moving interfaces in Biswas:2024xut  and many
such. An extensive literature on the dual holographic construction of the
moving mirror models also exists already Hirata:2008ka ; Akal:2020twv ;
Akal:2021foz ; Akal:2022qei . In brief, it is fair to state that the moving
mirror models are relevant in a broad class of interesting physical
situations, and therefore understanding the physics of scrambling of this
class of models is certainly a well-motivated and topical question.

In this article, we undertake this task focusing on instructive examples.
Since the mirrors can be thought of as boundaries, we will focus on a specific
class of three-point functions in the BCFT (referred to as the b-OTOC), which
has been introduced in Das:2019tga . While in a CFT without boundaries, three
point functions are completely fixed by conformal symmetries up to a constant,
in the presence of boundaries, this is not so. As was shown in Das:2019tga ,
in large-c CFTs, the b-OTOCs show the characteristic Lyapunov growth in time,
and thus provide a diagnostic for early-time chaos for these systems in the
presence of a boundary. Mathematically, the reason why the b-OTOC works as a
diagnostic for early-time chaos, despite being a three-point function is that
in the presence of a boundary these three-point functions behave as a
holomorphic four-point functions (using the doubling trick), under conformal
transformations. In particular, the “doubled” four-point function so obtained,
can be shown to be a OTOC. The Lyapunov behaviour then follows, from standard
assumptions of vacuum block dominance.

Using this as a diagnostic, we demonstrate, in keeping with previous
qualitative results in Das:2021qsd ; Das:2022jrr , that dynamical data of the
CFT does demarcate between the behaviour of its corresponding OTOCs. For
example, we explicitly demonstrate the following: In the “escaping mirror”
model, which captures features of Hawking radiation from an eternal black
hole, a large-$c$ CFT displays an exponentially growing OTOC with a maximal
Lyapunov exponent. However, a free boson or an Ising CFT does not exhibit any
such growth. Thus, while the two-point function of all two-dimensional CFTs
displays the same thermal feature in the escaping mirror model, there is a
clear distinction between them at the level of the OTOC.

Furthermore, this result also suggests two intriguing complementary
possibilities related to the quantum features of a black hole: (i) Fast
scramblers such as black holes are not able to scramble strongly integrable
dynamics of free bosons or the Ising CFT. There is a clear competition between
the chaotic features of a black hole and the integrable structure of such
examples. Note that, usually, it is believed that a black hole will act as a
fast scrambler for any degree of freedom. However, strongly constrained
systems (e.g. integrable systems with an infinite number of conserved charges)
may not scramble even in the presence of a black hole.666This feature has a
qualitative resemblance to the “scar-states” in an otherwise strongly
thermalizing system. It will be very interesting to explore a more precise
description of this connection. (ii) Alternatively, the moving mirrors are
unable to capture finer dynamical aspects of quantum black holes. While
possibility (i) will have important implications in sharpening the scrambling
features of a black hole, possibility (ii) is strongly suggestive of a
limitation of the moving mirror models. At this point, however, we do not have
an argument in favour of either of the possibilities above and we hope to
address this in the near future.

Furthermore, in the “kink” mirror model, we are able to explicitly demonstrate
that the exponential growth of OTOC cannot set in. This model captures the
dynamics of an evaporating black hole and it has been previously shown that
the kink mirror model reproduces a unitary Page curve for the evolution of the
fine-grained entropy Akal:2020twv . It is thus expected that information
recovery in this model should somehow be explicit. It is possible to obtain
the “escaping mirror” profile as a limit of the “kink mirror”, since the
eternal black hole can also be obtained from the evaporating black hole in the
limit when evaporation disappears. The mirror model captures a simple
realization of this limit, in terms of one free parameter denoted by $u_{0}$.
The escaping mirror is obtained in the limit $u_{0}\to\infty$. We have
explicitly demonstrated that in the escaping mirror limit the exponential
growth of OTOC gets squeezed and eventually yields for a power-law growth for
the kink mirror.

There is a simple picture that emerges from these models. In the eternal black
hole, when no information recovery is possible by definition, we only observe
the physics of information scrambling, i.e. an exponential growth of the OTOC
for a large-$c$ CFT. In other words, when the Page time, $t_{\rm Page}$, has
been pushed to infinity, the scrambling time-scale, $t_{\rm scramble}$, is
clearly visible777Evidently, a more precise statement is that scrambling is
visible in the limit $t_{\rm Page}\gg t_{\rm scramble}$. and $t_{\rm
scramble}\sim\log(c)$. However, when evaporation of the black hole is
introduced into the system, the time-dynamics of OTOCs are power-laws and
therefore no natural scale can be associated with a fast-scrambling.888in
other words, they are slow scramblers. Furthermore, we also observe that in
the $t_{\rm Page}\gg t_{\rm scramble}$ limit, the exponential growth is
recovered; however, as the system is observed at a time-scale $t_{\rm
observation}\sim t_{\rm Page}$, there is only a power law behaviour.

There is an intriguing parallel of the above observation to the scrambling and
exponential behaviour of OTOCs in a weakly coupled QFT. In a weakly coupled
perturbative framework, keeping only a few terms in the Feynman perturbation
series generally do not reveal any exponentially growing piece in the OTOC. To
observe the exponential behaviour, one needs to keep the contribution from
several orders of the coupling constant Stanford:2015owe . The parameter
$u_{0}$ above is similar to the coupling constant in QFT: When $u_{0}$ is
large, i.e. the coupling is strong, we observe a fast scrambling behaviour. On
the other hand, when $u_{0}\sim{\cal O}(1)$, the scrambling becomes slow or is
absent from the dynamics. Thus, we have a simple model that can interpolate
between a fast scrambling and a slow scrambling behaviour.

This article is divided into the following sections. After a brief review of
the moving mirror setup in section 2, we present the main computation of the
b-OTOC in the escaping mirror and the kink mirror profiles in section 2.1 and
2.2 respectively. We show that for the former mirror profile, a large-c CFT
shows an exponential growth in the b-OTOC, while it shows a power law
behaviour in the case of a kink profile. As we further show in section 2.1,
this exponential growth in the escaping mirror profile, is not universal, in
that, a similar computation in the Ising model and an orbifold CFT model, does
not show an exponential growth. This is in contrast to the two-point function
computation in these models, presented in section 3 which is universal,
showing a thermal behaviour for any CFTs in the escaping mirror background. In
section 2.3, we further show, that by taking a particular limit of a parameter
in the kink profile, one can reproduce the exponential behaviour in the
b-OTOC. In section 2.4, we discuss another generalization/deformation of the
escaping mirror profile, where once again, the deformation parameter helps to
interpolate between a scrambling and non-scrambling dynamics. We end with a
summary of our results and a discussion on open issues and future directions
to explore in section 4. Some details of the computations of the main draft
are presented in appendix A, while in appendix B we generalize the b-OTOC
computations of the main draft to other mirror profiles.

##  2 Moving mirror: preliminary remarks

We begin with a brief review of the framework which has been discussed and
analyzed extensively in the literature, see e.g. Akal:2020twv ; Akal:2022qei .
In this part, we will discuss the essential ingredients which are relevant for
the latter part where we discuss our results. Consider a two-dimensional CFT
in flat Minkowski geometry:

$\displaystyle ds^{2}=-dt^{2}+dx^{2}=-dudv\ ,\quad u=t-x\ ,\quad v=t+x\ .$ (2)

The ground-state of this CFT has a vanishing energy-momentum tensor
expectation value. A mirror is a physical boundary from which the CFT degrees
of freedom will “bounce-off”, preserving conformal invariance. This is
achieved by imposing conformal boundary conditions on the CFT stress-tensor,
in terms of a boundary CFT (BCFT) framework. Such boundary conditions can
equivalently be thought of as boundary states in the CFT.999Note that, it is
generally difficult to explicitly construct boundary states in a CFT, even
though in examples such as free fields, Ising CFT, explicit results are known
(see e.g. Brehm:2015lja ; Brehm:2015plf ). The simplest mirror is described by
$x=0$ and therefore the corresponding physical space is determined by $x>0$.
In the ground state of this BCFT, the conformal boundary condition is
trivially satisfied, since $\langle T_{uu}\rangle=\langle T_{vv}\rangle=0$.

On the other hand, a time-dependent boundary, by definition, would be
described by a profile of the type: $x=Z(t)$. In light cone coordinates, this
can be expressed as $F(u,v)=0$ or equivalently as $v=p(u)$. Since the
motivation behind the moving mirror profile is to mimic a black hole
background, in which the ground state contains radiation from the black hole,
instead of imposing conformal boundary condition, one imposes the so-called
radiative boundary conditions, in which the out-going radiation $\langle
T_{uu}\rangle\neq 0$, while $\langle T_{vv}\rangle$ vanishes. A simple way to
achieve this boundary condition is to first start with a BCFT in its ground
state described by coordinates $\\{\tilde{t},\tilde{x}\\}$, where the usual
conformal boundary condition is imposed on the static mirror at
$\tilde{x}=0$,101010Correspondingly, the physical space is described by
$\tilde{x}>0$. which corresponds to $\tilde{u}=\tilde{v}$, where:

$\displaystyle\left(\tilde{u},\tilde{v}\right)=\left(\tilde{t}-\tilde{x},\tilde{t}+\tilde{x}\right)\
,$ (3)

To obtain the radiative boundary condition, one then relates $u$ and $v$ with
$\tilde{u}$ and $\tilde{v}$ via a chiral conformal transformation

$\displaystyle\tilde{u}=p(u)\ ,\quad\tilde{v}=v\quad\implies\quad
t+Z(t)=p\left(t-Z(t)\right)\ ,$ (4)

where, as alluded above, $x=Z(t)$ is the profile of the mirror. As remarked in
Akal:2022qei , the above equation is equivalent to the following:

$\displaystyle\frac{p(u)-u}{2}=Z\left(\frac{u+p(u)}{2}\right)\ .$ (5)

As mentioned above, eqn. (4) is a chiral conformal transformation.111111In
terms of the transformations defined on the complex-plane, using the standard
$\\{z,{\bar{z}}\\}$ coordinates, chirality is realized by e.g. transforming
$z$ while keeping ${\tilde{z}}$ same. Under this transformation the energy-
momentum changes by a Schwarzian derivative contribution to:

$\displaystyle
T_{uu}=\frac{c}{24\pi}\left(\frac{3}{2}\left(\frac{p^{\prime\prime}(u)}{p^{\prime}(u)}\right)^{2}-\frac{p^{\prime\prime\prime}(u)}{p^{\prime}(u)}\right)\
,\quad T_{vv}=0\ ,\quad T_{uv}=0\ ,$ (6)

which yields a non-vanishing contribution for a generic choice of the function
$p(u)$, as desired. Also,

$\displaystyle T_{tt}=T_{uu}+T_{vv}-2T_{uv}=T_{uu}\ ,$ (7)

using eqn. (6). Thus, a generic function $p(u)$ yields a non-vanishing
$T_{tt}$.

Note that, when $p(u)\in SL(2,R)$, $T_{uu}=0$. The corresponding conformal
transformations are of three types:

$\displaystyle p(u)=u+\alpha\ ,\quad p(u)=\lambda u\ ,\quad
p(u)=-\frac{\kappa}{u}\ ,$ (8)

where $\\{\alpha,\lambda,\kappa\\}$ are real constants and the three choices
above correspond to translation, dilatation and inversion. Using eqn. (4),
these cases correspond to:

$\displaystyle Z(t)=\frac{\alpha}{2}\ ,\quad
Z(t)=\frac{\lambda-1}{\lambda+1}t\ ,\quad Z(t)=\pm\sqrt{t^{2}+\kappa}\ .$ (9)

These correspond to mirrors with a uniform position (i.e. completely static),
a uniform velocity, and a uniform acceleration- the Rindler profile,
respectively. Furthermore, in the limit $t\gg\kappa$, $Z(t)$ approaches
uniform velocity, and in the limit $t\ll\kappa$, $Z(t)$ approaches a constant
mirror profile. Therefore, in order to observe a non-trivial energy flux, one
must consider a non-uniform accelerating mirror.121212Note that, it is now
easy to construct mirror trajectories that correspond to a non-uniform
acceleration. For example, a profile of the following type:  $\displaystyle
Z(t)=\left(\sum_{i=0}^{N}t^{i}\right)^{a}\ ,$ (10) where $a$ and $N$ are real
constants, takes us out of the $SL(2,R)$-class.

A few comments are in order. Using the mirror position described by
$\tilde{u}=\tilde{v}$, on the mirror $2t=u+\tilde{u}=u+p(u)$ and
$2x=\tilde{u}-u=p(u)-u$. Thus, the induced metric on the mirror is given by
Akal:2022qei

$\displaystyle ds_{\rm
mirror}^{2}=-p^{\prime}(u)\left(\frac{2}{1+p^{\prime}(u)}\right)^{2}dt^{2}\ .$
(11)

Thus, the mirror is spacelike if $p^{\prime}(u)<0$, timelike if
$p^{\prime}(u)>0$ and null if $p^{\prime}(u)=0$, or
$p^{\prime}(u)\rightarrow\infty$. It is now easy to check that for the
constant static profile, the mirror is always timelike. For the uniform
velocity profile, it is timelike, spacelike, or null if $\lambda>0$,
$\lambda<0$ or $\lambda=0$. For the constant acceleration profile, when
$\kappa>0$ the mirror is timelike but asymptotes to a null curve in the limit
$u\to\infty$. These statements are also nicely visible in the corresponding
profile functions $Z(t)$ given in eqn (9). On physical grounds, we should
demand $p^{\prime}(u)>0$.

###  2.1 Escaping mirror

Let us begin with the “escaping mirror” profile Akal:2020twv ; Akal:2021foz .
The profile is given by

$\displaystyle v=p(u)=-\beta\log\left(1+e^{-u/\beta}\right)\ ,$ (12)

where $\beta$ is a real positive parameter. Note that, the following limits
are instructive:

$\lim_{u\to\infty}p(u)=-\beta e^{-u/\beta}\,$ (13)
$\lim_{u\to-\infty}p(u)=u\,$ (14)

where the latter ($u\to-\infty$ limit) corresponds to a static mirror. Thus,
the early time behaviour corresponds to an $SL(2,R)$-class mirror profile and
will contribute a vanishing energy flux; while the $u\to\infty$ limit will
contribute a non-vanishing value. The corresponding mirror profile is shown in
fig. 1.

$t$$x$$v$$u$ Figure 1: Escaping mirror profile, described by the function
$p(u)=-\beta\log\left(1+e^{-u/\beta}\right)$.

Note, furthermore, that $p^{\prime}(u)=(1+e^{u/\beta})^{-1}$. This is a
strictly positive function and therefore the mirror is timelike. Also, note
that $\lim_{u\to-\infty}p^{\prime}(u)\to 1$ and
$\lim_{u\to\infty}p^{\prime}(u)\to e^{-u/\beta}$. Thus, at past infinity, the
mirror is timelike while at the future infinity, the mirror asymptotes to a
null curve. Both these features are explicitly visible in fig. 1.

Let us now consider the conformal transformation that maps the moving mirror
to a static mirror. As we reviewed above, under $\tilde{u}=p(u)$ and
$\tilde{v}=v$, the escaping mirror $x=Z(t)$ gets mapped to a static mirror
$\tilde{x}=0$. This is pictorially demonstrated in fig. 2. Additionally, in
these figures, we have explicitly inserted three primary operators.
Specifically, we placed the same operator $V$ at two points $P_{1}$ and
$P_{2}$ on the mirror boundary, and another operator $W$ in the bulk at the
point $P$. We will discuss the correlation function of these operators and
will connect the result to an out-of-time-order correlator (OTOC) momentarily.

$t$$x$$v$$u$$\tilde{t}$$\tilde{x}$$\tilde{v}$$\tilde{u}$$\beta\log
2$$P_{1}$$P_{2}$$P(u,v)$$P_{1}$$P_{2}$$P(\tilde{u},\tilde{v})$$P_{r}(\tilde{v},\tilde{u})$
Figure 2: Static mirror from escaping mirror using conformal mapping. In the
figure, we have explicitly depicted three operator insertions: two identical
operators, denoted by $V$, inserted on the boundary of the mirror at points
$P_{1}$ and $P_{2}$, and another operator, denoted by $W$, inserted in the
bulk of the physical space at point $P$. The bulk operator $W$ creates an
image in the resulting BCFT and therefore a three-point function of $V$ and
$W$ becomes a four-point function in the CFT.

Before delving into the OTOCs, let us review the basic features of the moving
mirror. The corresponding energy-momentum can be calculated using (6) and the
specific function of $p(u)$. This yields Akal:2021foz :

$\displaystyle
T_{uu}=\frac{c}{48\pi\beta^{2}}\left(1-\frac{1}{\left(1+e^{u/\beta}\right)^{2}}\right)\
,$ (15)

which has the following instructive limits: $\lim_{u\to-\infty}T_{uu}\to 0$
and $\lim_{u\to+\infty}T_{uu}\to c/(48\pi\beta^{2})$. The latter is precisely
the energy-momentum expectation value in a thermal state of inverse
temperature $(2\pi\beta)$.131313Note that, the large and the small $u$ limits
are precisely defined in terms of the parameter $u/\beta$. For example, it is
easy to observe that in the limit $u\to\infty$ and $\beta\to\infty$, such that
$u/\beta$ is fixed, the energy flux $T_{uu}$ never settles to the thermal
value.  Physically, this thermal energy flux is identified with the constant
Hawking radiation from an eternal black hole in two dimensions. As further
evidenced in Akal:2021foz , by considering a massless free CFT in the moving
mirror background, one obtains a finer matching of the system with radiation
from an eternal black hole in terms of Bogoliubov coefficients and thermal
spectrum.

We will now discuss the calculation of a certain class of OTOCs, which are
referred to as $b$-OTOCs in Das:2019tga . While the free CFT thermal result is
very encouraging, it is a purely kinematic answer based on a two-point
function in a CFT and therefore does not capture dynamical information of the
system. By analyzing the OTOCs, we would like to probe finer details of the
dynamics of the corresponding CFT.

Correlation function in a Lorentzian quantum field theory can be obtained from
its Euclidean counterpart through analytic continuation. To achieve this
operationally, one begins with a Euclidean correlator with operators inserted
at small and purely imaginary times $i\epsilon_{i}$. Then analytically
continue the time coordinate to the desired Lorentzian value
$t_{i}+i\epsilon_{i}$ and finally take $\epsilon_{i}\rightarrow 0$. Different
operators in the Lorentzian correlator will be ordered according to the order
in which $\epsilon_{i}$’s are taken to zero, e.g. $\langle
O(t_{1})O(t_{2})\dots O(t_{n})\rangle$ will correspond to the ordering
$\epsilon_{1}<\epsilon_{2}<\dots<\epsilon_{n}$ Osterwalder:1973dx ;
Haag:1992hx .

In CFTs, the two and three-point functions exhibit a simple structure
determined by conformal symmetry, up to an overall constant. However, in the
context of BCFTs, the reduced symmetry leads to nontrivial and interesting
features in these correlation functions. Specifically, the conformal Ward
identities imply that the $n$-point function in a BCFT behaves like a
$2n$-point function of a holomorphic bulk CFT under conformal transformations
Cardy:1984bb . While the two-point function in a BCFT resembles a four-point
function in a holomorphic bulk CFT, it can not be an OTOC and therefore not of
much use to us. With this motivation, we will consider the analytic structure
of a three-point OTOC with two operators on the boundary and one in the bulk.

We begin with operator insertions shown in fig. 2. In terms of
$(\tilde{u},\tilde{v})$, the coordinates of the boundary points satisfy
$\tilde{u}=\tilde{v}$. Let us assign the boundary points $P_{1}$ and $P_{2}$
the following coordinates : $P_{1}\equiv(\tilde{v}_{1},\tilde{v}_{1})$ and
$P_{2}\equiv(\tilde{v}_{2},\tilde{v}_{2})$. Let us also assign the point in
the bulk $P$ the following coordinates $P\equiv(\tilde{u},\tilde{v})$. Using
Cardy’s doubling trick Cardy:1984bb , the above three-point function will be
the same as that of a four-point function with one extra operator $W$ at the
reflected point $P_{r}$ with coordinates $P_{r}\equiv(\tilde{v},\tilde{u})$.
The conformal cross ratio is

$\begin{split}\eta&=\frac{(\tilde{u}-\tilde{v})(\tilde{v}_{1}-\tilde{v}_{2})}{(\tilde{u}-\tilde{v}_{1})(\tilde{v}-\tilde{v}_{2})}\
.\end{split}$ (16)

As discussed above, we will now analytically continue to
$\tilde{t}\rightarrow\tilde{t}+i\,\tilde{\epsilon}$. We will be doing this
analytic continuation in $(\tilde{u},\tilde{v})$ coordinates (see appendix A
for discussion of analytic continuation in $(u,v)$ coordinates). Therefore, in
$(\tilde{u},\tilde{v})$ coordinates, we have:

$\begin{split}P_{1}&\equiv(\tilde{v}_{1}+i\,\tilde{\epsilon}_{1},\tilde{v}_{1}+i\,\tilde{\epsilon}_{1})\
,\qquad
P_{2}\equiv(\tilde{v}_{2}+i\,\tilde{\epsilon}_{2},\tilde{v}_{2}+i\,\tilde{\epsilon}_{2})\
,\\\
P&\equiv(\tilde{u}+i\,\tilde{\epsilon}_{0},\tilde{v}+i\,\tilde{\epsilon}_{0})\
,\quad~{}\quad~{}P_{r}\equiv(\tilde{v}+i\,\tilde{\epsilon}_{0},\tilde{u}+i\,\tilde{\epsilon}_{0})\
.\end{split}$ (17)

The conformal cross-ratio is given by

$\begin{split}\eta&=\frac{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}-i\,\tilde{\epsilon}_{0})(\tilde{v}_{1}+i\,\tilde{\epsilon}_{1}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{1}-i\,\tilde{\epsilon}_{1})(\tilde{v}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}\
.\end{split}$ (18)

We will take the two boundary points $P_{1}$ and $P_{2}$ at a same point
$P_{1}\equiv P_{2}$ which implies $\tilde{v}_{1}=\tilde{v}_{2}$. The conformal
cross ratio becomes:

$\begin{split}\eta&=\frac{(\tilde{u}-\tilde{v})i\,\tilde{\epsilon}_{12}}{(\tilde{u}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{01})(\tilde{v}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=i\,\tilde{\epsilon}_{12}\,\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\
.\end{split}$ (19)

We will set $v=0$ for the bulk operator $W$. In terms of $(t,x)$ it implies
$x=-t$, therefore $u=2t$ for the bulk operator. Setting $v=0$, we will study
behaviour of b-OTOC as we vary $u$ from $0$ to $\infty$.

### Case-1: $v=0,u\rightarrow\infty$

$\eta=i\tilde{\epsilon}_{12}\frac{-\beta\,e^{-u/\beta}}{v_{1}^{2}}\ .$ (20)

### Case-2: $v=0,u=0$

$\eta=i\tilde{\epsilon}_{12}\frac{\beta\log 2}{(\beta\log 2+v_{1})(-v_{1})}\
.$ (21)

$-v_{1}>0$, and let’s assume $\beta\log 2+v_{1}>0$. The above assumptions
imply that the points $P_{1}$ and $P_{2}$ have to lie in the quadrant
$(u>0,v<0)$ in the $(u,v)$ plane. So we see that $\eta$ approaches $0$ from
the opposite directions for Case-1 and Case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)}{\left[-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)-v_{1}+i\,\tilde{\epsilon}_{01}\right][-v_{1}]}\\\
&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\
.\end{split}$ (22)

The cross ratio $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $u\rightarrow\infty$. For the intermediate point
$u=u_{1}$, in the out-of-time-ordered case,
$\tilde{\epsilon}_{2}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{1}$ or,
$\tilde{\epsilon}_{1}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{2}$, it crosses
the real axis after $\eta=1$ branch point. But for the time-ordered case, it
crosses the real axis before the $\eta=1$ branch point. This behaviour is
similar to what was obtained in Das:2019tga , with one interesting difference,
which is that the ordering here, refers to the ordering of operators in the
static frame. As we discuss in detail in appendix A, in the moving mirror
frame i.e. in the ($u,v$) coordinates, the ordering becomes
$\frac{\epsilon_{2}}{p^{\prime}(u_{1})}>\epsilon_{0}>\frac{\epsilon_{1}}{p^{\prime}(u_{1})}$.
Here $p^{\prime}(u)$ is the Weyl scaling factor of the metric at the location
of mirror $(d\tilde{u}d\tilde{v}=\frac{1}{p^{\prime}(u)}dudv)$ and $u_{1}$, as
mentioned previously, is the location of the boundary operators on the mirror.
So, the operator ordering, for boundary operators, in the mirror frame, is
effectively decided by $\frac{\epsilon_{i}}{p^{\prime}(u)}$. It would be nice
to better understand the physical reason behind this scaling of the boundary
$\epsilon$’s. One should note that such scale factors also appear in the
presence of a moving interface profile, where the transmission and reflection
coefficients, as computed in the moving frame, are off from their static frame
counterparts by such scaling factors (see e.g. eqn. (3.23) of Biswas:2024xut
).

So far, the discussion is purely kinematical, i.e. holds for any two-
dimensional CFT. To determine the explicit form of the four-point function, we
will now need to invoke dynamical input of the specific theory. Let us begin
with large-$c$ CFTs, assuming the identity block dominance in the
corresponding four-point function. The explicit form of the identity block is
given, as a function of the cross-ration, by

$\mathcal{F}_{0}(\eta)\approx\left(\frac{\eta}{1-(1-\eta)^{1-\frac{12h_{w}}{c}}}\right)^{2h_{v}}\
.$ (23)

The above formula is valid at large central charge $c$, with $\frac{h_{w}}{c}$
fixed and small, and $h_{v}$ fixed and large. We take our bulk operator to be
$W$ with scaling dimension $h_{w}$ and the boundary operator to be $V$ with
scaling dimension $h_{v}$.

In the OTO case, $\eta$ crosses the branch point at $\eta=1$ and therefore, in
the second Riemann sheet $1-\eta\to(1-\eta)e^{-2\pi i}$. Taking $\eta$ small,
we obtain:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{\eta}{1-(1-\eta)^{1-\frac{12h_{w}}{c}}e^{-2\pi
i\left(1-\frac{12h_{w}}{c}\right)}}\right)^{2h_{v}}\\\
&=\left(\frac{\eta}{1-\left[1-\left(1-\frac{12h_{w}}{c}\right)\eta\right]e^{\frac{24\pi
ih_{w}}{c}}}\right)^{2h_{v}}\\\
&=\left(\frac{\eta}{1-\left[1-\eta\right]\left[1+\frac{24\pi
ih_{w}}{c}\right]}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\beta}e^{\frac{u}{\beta}}}\right)^{2h_{v}}\approx
1-\frac{48\pi
h_{v}h_{w}v_{1}^{2}}{c\tilde{\epsilon}_{12}\beta}e^{u/\beta}+\cdots\,.\end{split}$
(24)

We see exponential behaviour with Lyapunov index $\lambda=\frac{1}{\beta}=2\pi
T$. This is the maximal Lyapunov exponent and is therefore maximally chaotic
as described by OTOCs.

On the other hand, Ising CFT and Orbifold CFT do not display any chaotic
features at all. For Ising CFT (see appendix B of Roberts:2014ifa )

$f_{\sigma\epsilon}(\eta,\bar{\eta})=\left(\frac{2-\eta}{2\sqrt{1-\eta}}\right)\left(\frac{2-\bar{\eta}}{2\sqrt{1-\bar{\eta}}}\right)\,.$
(25)

As we take $u\rightarrow\infty$, both $\eta,\bar{\eta}\rightarrow 0$. $\eta$
crosses the real axis after the brunch point at $\eta=1$. From (25), we get

$\begin{split}f_{\sigma\epsilon}(\eta,\bar{\eta})&=\frac{2-\eta}{2(1-\frac{1}{2}\eta)e^{-i\pi}}\,\frac{2-\bar{\eta}}{2(1-\frac{1}{2}\bar{\eta})}\\\
&=-1\,.\end{split}$ (26)

For the orbifold CFT $(T^{2})^{2}/Z_{2}$ (see eqn. (3.29) of Caputa:2017rkm ),

$\begin{split}F_{2}(\eta,\bar{\eta})&\simeq\frac{i\pi}{2\log\left(\frac{\eta}{16}\right)}\,.\\\
\end{split}$ (27)

Therefore, in the $u\rightarrow\infty$ limit, we will get

$\begin{split}F_{2}(\eta,\bar{\eta})&=\frac{i\pi}{2\log\left(-i\frac{\tilde{\epsilon}_{12}\beta}{16v_{1}^{2}}e^{-\frac{u}{\beta}}\right)}\approx-\frac{i\pi\beta}{2u}\,.\end{split}$
(28)

Thus for the orbifold CFTs, the dependence on $u$ is polynomial.

###  2.2 Kink mirror

Let us now consider the kink mirror Akal:2020twv ; Akal:2021foz  which models
an evaporating black hole formed from collapse. The corresponding mirror
profile is given by

$v=p(u)=-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\beta\log\left(1+e^{\frac{u-u_{0}}{\beta}}\right)\
,$ (29)

where, we now have two real positive parameters $u_{0}$ and $\beta$. As
before, let us inspect some instructive limits:

$\displaystyle\lim_{u\to\pm\infty}p(u)=u\ ,$ (30) $\displaystyle\lim_{u\to
u_{0}}p(u)=-\beta\log\left(1+e^{-u_{0}/\beta}\right)+\beta\log 2\ ,$ (31)

which implies that in the $u\to\pm\infty$ limit, the mirror profile belongs to
an $SL(2,R)$-class and therefore the corresponding energy flux
vanishes.141414In fact, in both these limits, the mirror becomes a static one.
In the $u\to u_{0}$ limit, however, non-trivial flux is created. Given the
profile, we also obtain:

$\displaystyle
p^{\prime}(u)=\left(1+e^{u/\beta}\right)^{-1}+\left(1+e^{(u-u_{0})/\beta}\right)^{-1}>0\
,$ (32)

and therefore the mirror is timelike. Unlike the escaping mirror, there is no
limit where $p^{\prime}(u)$ can asymptotically vanish. It is now
straightforward to obtain the corresponding expectation value of $T_{uu}$.
Furthermore, the kink mirror models the formation and evaporation of a black
hole, which can be observed by comparing the Bogoliubov coefficients of a free
scalar theory in a two-dimensional collapsing and evaporating black hole with
the same in a kink mirror geometry Akal:2020twv . Note, furthermore, that in
Akal:2020twv , a fine-grained entropic computation has been shown to exhibit
the Page curve associated to the unitary evolution of a quantum black hole.

Our goal here is to analyze the corresponding $b$-OTOCs and understand the
physics of scrambling. Towards this, we arrange for a similar operator
insertion that we have discussed in the previous section. This is pictorially
shown in fig. 3. We perform the usual conformal transformation
$\tilde{u}=p(u)$ and $\tilde{v}=v$ to obtain the static mirror configuration.

$t$$x$$v$$u$$P_{1}$$P_{2}$$P(u,v)$$\frac{u_{0}}{2}$$\tilde{t}$$\tilde{x}$$\tilde{v}$$\tilde{u}$$P_{1}$$P_{2}$$P(\tilde{u},\tilde{v})$$P_{r}(\tilde{v},\tilde{u})$
Figure 3: Static mirror from kink mirror using conformal mapping. We have also
explicitly shown operator insertions in the corresponding CFT. These are
primary operators inserted at two boundary points $P_{1,2}$ and at one bulk
point denoted by $P$.

Let us take the bulk point to be $P\equiv(\tilde{u},\tilde{v})$, the reflected
point would be $P_{r}\equiv(\tilde{v},\tilde{u})$. Boundary points are
$P_{1}\equiv(\tilde{v}_{1},\tilde{v}_{1})$ and
$P_{2}\equiv(\tilde{v}_{2},\tilde{v}_{2})$. We will fuse the two boundary
points $P_{1}$ and $P_{2}$ to coincide, $P_{1}\equiv P_{2}$, which implies
$\tilde{v}_{1}=\tilde{v}_{2}$. The cross ratio is defined as:

$\begin{split}\eta&=\frac{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}-i\,\tilde{\epsilon}_{0})(\tilde{v}_{1}+i\,\tilde{\epsilon}_{1}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{1}-i\,\tilde{\epsilon}_{1})(\tilde{v}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}\\\
&=\frac{(\tilde{u}-\tilde{v})i\,\tilde{\epsilon}_{12}}{(\tilde{u}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{01})(\tilde{v}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\beta\log\left(1+e^{\frac{u-u_{0}}{\beta}}\right)-v}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\beta\log\left(1+e^{\frac{u-u_{0}}{\beta}}\right)-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\
.\end{split}$ (33)

### Case-1: $v=0,u=0$

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{\beta\log
2-\beta\log\left(1+e^{-\frac{u_{0}}{\beta}}\right)}{\left[\beta\log
2-\beta\log\left(1+e^{-\frac{u_{0}}{\beta}}\right)+v_{1}\right]\left[-v_{1}\right]}\,.\end{split}$
(34)

We assume $v_{1}<0$,
$\beta\log\left(\frac{2}{1+e^{-\frac{u_{0}}{\beta}}}\right)>|v_{1}|$. If we
set $u=0$ in the kink mirror profile (29) we get

$v=-\beta\log\left(\frac{2}{1+e^{-\frac{u_{0}}{\beta}}}\right)\,.$ (35)

So, the above assumptions imply that the points $P_{1}$ and $P_{2}$ have to
lie in the quadrant $(u>0,v<0)$ in the $(u,v)$ plane.

### Case-2: $v=0$, $u=\frac{u_{0}}{2}-\alpha$, $\alpha\rightarrow 0$

We can not take $u\rightarrow\infty$ limit in the kink mirror case. Instead,
we will take $u=\frac{u_{0}}{2}-\alpha$ where $\alpha$ is a small positive
number and will take $\alpha\rightarrow 0$ limit. We get

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{1}{\beta}\Big{(}\frac{u_{0}}{2}-\alpha\Big{)}}\right)+\beta\log\left(1+e^{\frac{1}{\beta}\Big{(}-\frac{u_{0}}{2}-\alpha\Big{)}}\right)}{\left[-\beta\log\left(1+e^{-\frac{1}{\beta}\Big{(}\frac{u_{0}}{2}-\alpha\Big{)}}\right)+\beta\log\left(1+e^{\frac{1}{\beta}\Big{(}-\frac{u_{0}}{2}-\alpha\Big{)}}\right)-v_{1}\right]\left[-v_{1}\right]}\\\
&=-i\,\tilde{\epsilon}_{12}\frac{2\alpha}{\Big{(}1+e^{\frac{u_{0}}{2\beta}}\Big{)}v_{1}^{2}}\,.\end{split}$
(36)

So, we see that $\eta$ approaches $0$ from the opposite directions for case-1
and case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)+\beta\log\left(1+e^{\frac{u_{1}-u_{0}}{\beta}}\right)}{\left[-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)+\beta\log\left(1+e^{\frac{u_{1}-u_{0}}{\beta}}\right)-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[-v_{1}\right]}\\\
&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\
.\end{split}$ (37)

The cross ratio $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $\alpha\rightarrow 0$. For the intermediate point
$u=u_{1}$, in the out of time order case,
$\tilde{\epsilon}_{2}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{1}$ or,
$\tilde{\epsilon}_{1}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{2}$, it crosses
the real axis after $\eta=1$ branch point. But for time-ordered case, it
crosses the real axis before the $\eta=1$ branch point. As before, the
discussion up to this point is completely general and applies to any two-
dimensional CFT.

For a large-$c$ CFT, as before, we can again use the Identity block result.
From (24), we can very easily see that as $\alpha\rightarrow 0$:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}}{c\,\tilde{\epsilon}_{12}2\alpha}\left(1+e^{\frac{u_{0}}{2\beta}}\right)v_{1}^{2}}\right)^{2h_{v}}\\\
&=1-\frac{24\pi
h_{v}h_{w}}{c\tilde{\epsilon}_{12}}\left(1+e^{\frac{u_{0}}{2\beta}}\right)v_{1}^{2}\left(\frac{1}{\alpha}\right)\,.\end{split}$
(38)

In this case, we get polynomial behaviour instead of exponential behaviour and
therefore does not contain a scrambling time-scale.

###  2.3 Escaping mirror from kink mirror

It is possible to obtain the escaping mirror from the kink mirror profile.
Relatedly, this is possible if we can switch off the evaporation part of the
black hole. This is certainly possible since the additional parameter $u_{0}$
controls the evaporation time-scale, or more precisely, the Page time. If we
push away the Page time to a parametrically large distance, the eternal black
hole dynamics remains and we expect to capture a scrambling dynamics.

Escaping mirror profile is given by $(\beta>0,u_{0}>0)$:

$v=p(u)=-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\beta\log\left(1+e^{\frac{u-u_{0}}{\beta}}\right)\
.$ (39)

In the limit $u_{0}>>u$, the second term becomes small, so the kink mirror
profile becomes a small perturbation around the escaping mirror:

$v=p(u)=-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\delta\,\beta\,e^{-\frac{u_{0}-u}{\beta}}+\mathcal{O}\left(\delta\right)^{2}\
.$ (40)

Where, in the above, $\delta$ is a bookkeeping device introduced by hand to
keep track of the perturbative expansion. In the end, we will set $\delta=1$.
Now, we will do the same analysis as in §2.1 assuming the mirror profile (40).

In $(\tilde{u},\tilde{v})$ coordinates (see fig. 3), we take the bulk point to
be $P\equiv(\tilde{u},\tilde{v})$, the reflected point would be
$P_{r}\equiv(\tilde{v},\tilde{u})$. Let’s assume $P_{1}$ and $P_{2}$ are two
boundary points with coordinates $P_{1}\equiv(\tilde{v}_{1},\tilde{v}_{1})$
and $P_{2}\equiv(\tilde{v}_{2},\tilde{v}_{2})$. We will take the two boundary
points $P_{1}$ and $P_{2}$ at a same point $P_{1}\equiv P_{2}$ which implies
$\tilde{v}_{1}=\tilde{v}_{2}$. The cross ratio is defined as:

$\begin{split}\eta&=\frac{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}-i\,\tilde{\epsilon}_{0})(\tilde{v}_{1}+i\,\tilde{\epsilon}_{1}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}{(\tilde{u}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{1}-i\,\tilde{\epsilon}_{1})(\tilde{v}+i\,\tilde{\epsilon}_{0}-\tilde{v}_{2}-i\,\tilde{\epsilon}_{2})}\\\
&=\frac{(\tilde{u}-\tilde{v})i\,\tilde{\epsilon}_{12}}{(\tilde{u}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{01})(\tilde{v}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\delta\,\beta
e^{-\frac{u_{0}-u}{\beta}}-v}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)+\delta\,\beta
e^{-\frac{u_{0}-u}{\beta}}-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\\\
&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\\\
&+i\delta\,\tilde{\epsilon}_{12}\frac{\beta
e^{-\frac{u_{0}-u}{\beta}}}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v_{1}+i\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\tilde{\epsilon}_{02}\right]}\left[1-\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v}{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-v_{1}+i\tilde{\epsilon}_{01}}\right]+\mathcal{O}(\delta)^{2}\,.\\\
\end{split}$ (41)

In the analysis of kink mirror in §2.2, where we kept $u_{0}$ to be fixed
$\mathcal{O}(1)$ number it was not possible to set $v=0$ and then take
$u\rightarrow\infty$ for the bulk operator because the kink mirror profile
crosses the $u$-axis at $u_{0}/2$. In this section, we are always keeping
$u_{0}>>u$. If we take $u_{0}>2u$, then it is possible to set $v=0$ and then
take $u\rightarrow\infty$ for the bulk operator.

For example, let us now consider $v=0,u\rightarrow\infty$. In this limit,

$\begin{split}\eta&=-i\,\tilde{\epsilon}_{12}\,\frac{\beta\,e^{-u/\beta}}{v_{1}^{2}}+i\delta\,\tilde{\epsilon}_{12}\frac{\beta
e^{-\frac{u_{0}-u}{\beta}}}{v_{1}^{2}}\left[1-\frac{\beta
e^{-\frac{u}{\beta}}}{v_{1}}\right]+\mathcal{O}(\delta)^{2}\\\
&=-i\,\tilde{\epsilon}_{12}\,\frac{\beta\,e^{-u/\beta}}{v_{1}^{2}}+i\delta\,\tilde{\epsilon}_{12}\frac{\beta
e^{-\frac{u_{0}-u}{\beta}}}{v_{1}^{2}}-i\delta\,\tilde{\epsilon}_{12}\frac{\beta
e^{-\frac{u_{0}-u}{\beta}}}{v_{1}^{2}}\frac{\beta
e^{-\frac{u}{\beta}}}{v_{1}}+\mathcal{O}(\delta)^{2}\\\
&=-i\,\tilde{\epsilon}_{12}\,\frac{\beta\,e^{-u/\beta}}{v_{1}^{2}}+i\delta\,\tilde{\epsilon}_{12}\frac{\beta
e^{-\frac{u_{0}-u}{\beta}}}{v_{1}^{2}}+\mathcal{O}(\delta)^{2}\,.\end{split}$
(42)

Going from second line to third line, we have neglected the last term which is
exponentially suppressed with respect to the second term. The above
perturbative expansion will break down when

$\begin{split}&i\,\tilde{\epsilon}_{12}\,\frac{\beta\,e^{-u/\beta}}{v_{1}^{2}}\approx
i\,\tilde{\epsilon}_{12}\frac{\beta e^{-\frac{u_{0}-u}{\beta}}}{v_{1}^{2}}\\\
\Rightarrow~{}&u\approx u_{0}/2\,.\end{split}$ (43)

The leading order term in (42) is non-zero, therefore $\mathcal{O}(\delta)$
and higher order terms would not be able to change the behaviour of $\eta$. In
other words, $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $u\rightarrow\infty$, and for some intermediate point
$u=u_{1}$ in the out-of-time-order case it crosses the real axis after
$\eta=1$ branch point, whereas for time-ordered case, it crosses the real axis
before the $\eta=1$ branch point.

Now, for a large-$c$ CFT, using the Virasoro Identity conformal block, we
obtain:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}\,v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\,\beta}e^{\frac{u}{\beta}}\left(1-\delta
e^{-\frac{u_{0}-2u}{\beta}}\right)^{-1}}\right)^{2h_{v}}\,.\end{split}$ (44)

Defining $\Lambda\equiv\frac{24\pi
h_{w}\,v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\,\beta}e^{\frac{u}{\beta}}$, we
obtain:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1+\Lambda\left(1+\delta
e^{-\frac{u_{0}-2u}{\beta}}\right)}\right)^{2h_{v}}\\\
&=\left(\frac{1}{1+\Lambda}\right)^{2h_{v}}\left(\frac{1}{1+\delta\frac{\Lambda}{1+\Lambda}e^{-\frac{u_{0}-2u}{\beta}}}\right)^{2h_{v}}\\\
&\sim\left(\frac{1}{1+\Lambda}\right)^{2h_{v}}-\delta\,2h_{v}\left(\frac{\Lambda}{1+\Lambda}\right)\left(\frac{1}{1+\Lambda}\right)^{2h_{v}}e^{-\frac{u_{0}-2u}{\beta}}\
.\end{split}$ (45)

It is straightforward to observe that the exponential growth in $u$ is
drastically cut-off by $u_{0}$. In the limit $u_{0}\gg u$, the exponential
function has a marked existence, however for $u_{0}\sim{\cal O}(1)$, this
exponential growth is terminated.

Physically, this offers an intuitive understanding. In the limit when
$u_{0}\gg u$, the kink mirror profiles asymptotes to the escaping mirror
profile, since the location of the kink is set by the parameter $u_{0}$.
Pictorially, this is easily seen by sliding $u_{0}$ in fig. 3 further and
further away to obtain the fig. 2. In this limit, the $b$-OTOC displays an
exponential growth and therefore a scrambling dynamics. Once $u_{0}$ slides
back into the picture, the range of $u$ get appropriately squeezed and for
$u_{0}\sim{\cal O}(1)$, the exponential growth yields to a power-law
behaviour. Correspondingly, no scrambling is seen. Note that, it is in the
latter limit where a Page curve is obtained Akal:2020twv ; Akal:2021foz .

###  2.4 Generalized escaping mirror

We now consider another example of a model with a tunable parameter, such that
at large values of the parameter, the system develops scrambling dynamics. In
this case, the profile takes the form:

$p(u)=-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-\beta\left[\log\left(e^{\frac{u_{0}}{\beta}}+e^{\frac{u}{\beta}}\right)\right]^{-n}\,,$
(46)

which is similar to Type-B mirrors considered in Akal:2022qei . Notice that in
the asymptotic limit of $n$, the above profile reduces to that of the escaping
mirror.

To analyze the b-OTOC, we Consider a similar set-up as discussed in §2.1. From
eqn. (19) we get conformal cross ratio

$\begin{split}\eta&=\frac{(\tilde{u}-\tilde{v})i\,\tilde{\epsilon}_{12}}{(\tilde{u}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{01})(\tilde{v}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=i\,\tilde{\epsilon}_{12}\,\frac{-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-\beta\left[\log\left(e^{\frac{u_{0}}{\beta}}+e^{\frac{u}{\beta}}\right)\right]^{-n}-v}{\left[-\beta\log\left(1+e^{-\frac{u}{\beta}}\right)-\beta\left[\log\left(e^{\frac{u_{0}}{\beta}}+e^{\frac{u}{\beta}}\right)\right]^{-n}-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\,.\end{split}$
(47)

### Case-1: $v=0,u\rightarrow\infty$

In the asymptotic $u$ limit, the profile (46) takes the form:

$p(u)\sim-\frac{\beta^{n+1}}{u^{n}}+\mathcal{O}\left(e^{-\frac{u}{\beta}}\right)\,.$
(48)

Therefore in this limit, the cross-ratio takes the form:

$\eta=i\tilde{\epsilon}_{12}\frac{-\beta^{n+1}}{u^{n}\,v_{1}^{2}}\,.$ (49)

### Case-2: $v=0,u=0$

$\eta=i\tilde{\epsilon}_{12}\frac{\beta\log
2+\beta\left[\log\left(1+e^{\frac{u_{0}}{\beta}}\right)\right]^{-n}}{\left(\beta\log
2+\beta\left[\log\left(1+e^{\frac{u_{0}}{\beta}}\right)\right]^{-n}+v_{1}\right)(-v_{1})}\,.$
(50)

$-v_{1}>0$, and let’s assume $\beta\log
2+\beta\left[\log\left(1+e^{\frac{u_{0}}{\beta}}\right)\right]^{-n}+v_{1}>0$.
The above assumptions imply that the points $P_{1}$ and $P_{2}$ have to lie in
the quadrant $(u>0,v<0)$ in the $(u,v)$ plane. So we see that $\eta$
approaches $0$ from the opposite directions for Case-1 and Case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)-\beta\left[\log\left(e^{\frac{u_{0}}{\beta}}+e^{\frac{u_{1}}{\beta}}\right)\right]^{-n}}{\left[-\beta\log\left(1+e^{-\frac{u_{1}}{\beta}}\right)-\beta\left[\log\left(e^{\frac{u_{0}}{\beta}}+e^{\frac{u_{1}}{\beta}}\right)\right]^{-n}-v_{1}+i\,\tilde{\epsilon}_{01}\right][-v_{1}]}\\\
&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\,.\end{split}$
(51)

Cross-ratio shows similar behaviour as that of the escaping mirror. From (24),
we can very easily see that as $u\rightarrow\infty$

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}u^{n}v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\beta^{n+1}}}\right)^{2h_{v}}\\\
&\approx 1-\frac{48\pi
h_{v}h_{w}v_{1}^{2}}{c\tilde{\epsilon}_{12}\beta^{n+1}}u^{n}+\cdots\,.\end{split}$
(52)

For all finite $n\geq 0$ it shows polynomial behaviour.

##  3 Two-point function

It is well-known that the thermal two-point function decays exponentially at
late times. From the analysis of the previous section, we have seen that the
escaping mirror profile effectively induces a temperature set by the parameter
$\beta$ though this is not the case for the kink mirror profile. In this
section, we will compute two-point function with one operator on the boundary
and another in the bulk subjected to the escaping mirror and kink mirror
boundary conditions.

###  3.1 Escaping mirror

We compute two-point function with one operator $V$ on the boundary at the
point $P_{0}\equiv(\tilde{v}_{0},\tilde{v}_{0})$ and another operator $W$ in
the bulk at the point $P\equiv(\tilde{u},\tilde{v})$ subjected to the escaping
mirror boundary condition. Using Cardy’s doubling trick, this two-point
function will be the same as three-point function without boundary with one
extra operator $W$ at the reflected point $P_{r}\equiv(\tilde{v},\tilde{u})$.
Therefore, we get

$\begin{split}\langle
W(u,v)V(u_{0},v_{0})\rangle&\sim\frac{\gamma}{(\tilde{u}-\tilde{v}_{0})^{h_{w}+h_{v}-h_{w}}(\tilde{v}-\tilde{u})^{2h_{w}-h_{v}}(\tilde{v}-\tilde{v}_{0})^{h_{w}+h_{v}-h_{w}}}\\\
&=\frac{\gamma}{[p(u)-v_{0}]^{h_{v}}[v-p(u)]^{2h_{w}-h_{v}}[v-v_{0}]^{h_{v}}}\,.\end{split}$
(53)

Where $\gamma$ is some constant. Now, if for our bulk operator we set $v=0$
and then take $u\rightarrow\infty$, we get

$\begin{split}\langle
W(u,v)V(u_{0},v_{0})\rangle&\sim\frac{\gamma}{v_{0}^{2h_{v}}\left[\beta\log\Big{(}1+e^{-\frac{u}{\beta}}\Big{)}\right]^{2h_{w}-h_{v}}}\\\
&=\frac{\gamma}{v_{0}^{2h_{v}}\beta^{2h_{w}-h_{v}}}e^{-\frac{u}{\beta}(h_{v}-2h_{w})}\,.\end{split}$
(54)

It decays exponentially fast as $u\rightarrow\infty$.

###  3.2 Kink mirror

Similarly, we can compute two-point function in kink mirror profile (29). We
take the operator $V$ on the boundary at a point
$P_{0}\equiv(\tilde{v}_{0},\tilde{v}_{0})$ and another operator $W$ in the
bulk at a point $P\equiv(\tilde{u},\tilde{v})$. The reflected point is
$P_{r}\equiv(\tilde{v},\tilde{u})$. The two-point function is given by

$\begin{split}\langle
W(u,v)V(u_{0},v_{0})\rangle&\sim\frac{\gamma}{(\tilde{u}-\tilde{v}_{0})^{h_{w}+h_{v}-h_{w}}(\tilde{v}-\tilde{u})^{2h_{w}-h_{v}}(\tilde{v}-\tilde{v}_{0})^{h_{w}+h_{v}-h_{w}}}\\\
&=\frac{\gamma}{[p(u)-v_{0}]^{h_{v}}[v-p(u)]^{2h_{w}-h_{v}}[v-v_{0}]^{h_{v}}}\,.\end{split}$
(55)

Now, if for our bulk operator we set $v=0$ and then take
$u=\frac{u_{0}}{2}-\alpha$, $\alpha\rightarrow 0$, we get

$\begin{split}\langle
W(u,v)V(u_{0},v_{0})\rangle&\sim\frac{\gamma}{v_{0}^{2h_{v}}\left(\frac{2\alpha}{1+e^{\frac{u_{0}}{2\beta}}}\right)^{2h_{w}-h_{v}}}\\\
&=\frac{\gamma}{v_{0}^{2h_{v}}}\left(\frac{2\alpha}{1+e^{\frac{u_{0}}{2\beta}}}\right)^{h_{v}-2h_{w}}\,.\end{split}$
(56)

It decays polynomially as $\alpha\rightarrow 0$.

##  4 Discussions

In this article, we have explored the physics of scrambling in a wide class of
moving mirror models and within a two-dimensional CFT. We have observed that
scrambling, defined in terms of OTOCs, exhibits a class of desired properties
ranging from (maximally) chaotic dynamics to a unitary evolution of black
holes, within the framework of moving mirror models. We have also observed
that most mirror profiles yield a power-law behaviour of OTOC and therefore do
not exhibit an exponential-in-time scrambling physics. The power of the
polynomial-in-time behaviour is mostly governed by the conformal dimension of
the heavy operator of the Virasoro identity block. It is therefore possible to
approximate an exponential behaviour for a range of reasonably large-time
dynamics by such a power-law. Nonetheless, such a scrambling quickly yields
for a unitary physics.151515Roughly, we can always write:  $\displaystyle
t^{\Delta}=e^{\Delta\log t}=e^{\Delta t\left(\frac{\log
t}{t}\right)}=e^{\Delta(t)t}\ ,$ (57) where $\Delta(t)=\frac{\log t}{t}$ can
be identified with a time-dependent Lyapunov exponent. The function
$\Delta(t)$ varies reasonably slowly near $t\approx e$ and therefore the
exponential behaviour can be interpreted as a Lyapunov growth.  Note that,
this transient exponential behaviour is associated with a Lyapunov exponent
which can be arbitrarily large in principle. This is in no contradiction with
the chaos-bound of Maldacena:2015waa  since the latter strictly applies for
time-independent thermal state.

It is worthwhile to emphasize again the escaping mirror and the kink mirror
models. While the former captures physics of an eternal black hole and
correspondingly displays a fast scrambling, maximally chaotic behaviour, the
latter demonstrates how the exponential growth yields to a power law behaviour
near the Page time and therefore becomes a slow scrambler towards the later
part of the dynamics. Qualitatively, it is connected to unitarity of the
dynamics, since near the Page time information begins being retrieved and thus
a fast scrambling behaviour is expected to disappear. Furthermore, this
provides a simple and tractable model where an interpolation from fast
scrambling to a slow scrambling dynamics can be realized as a function of a
single parameter.

Our results are in qualitative agreement with the physics discussed in
Cotler:2022weg , where it was shown that an escaping mirror profile gives rise
to a non-unitary dynamics at the future infinity. It was argued that this is
arising from throwing away details beyond a UV cut-off which is essential for
the computation of entanglement entropy. At very late times, these UV-modes
get Doppler shifted back and contribute to the entanglement entropy which
should yield for a unitary dynamics. This physics has a very close parallel to
the physics of scrambling in terms of the kink-mirror, as a function of the
location of the kink. In the escaping mirror limit, we observe a perfect
exponential growth with a maximal Lyapunov exponent which yields for a power-
law and therefore unitary dynamics when the kink is located at a finite
distance. In a qualitative sense, the escaping mirror limit allows for degrees
of freedom to escape from the system which are brought back when the kink
appears at a finite distance.

Note further that, in the strict “escaping mirror” limit, the density matrix
corresponds to a mixed thermal density matrix and an emergent KMS-condition is
expected. This happens in the strict $u_{0}\to\infty$ limit. For any finite
$u_{0}$, a KMS-condition can only be a transient approximate relation. This is
reminiscent of a type I∞ von Neumann algebra approximating a type III von
Neumann algebra. In recent times, there has been a surge of activities to
better understand aspects of operator algebra and factorization, including
possible transitions between them, in quantum mechanical as well as quantum
field theoretic models, see e.g. Witten:2021jzq ; Witten:2021unn ;
Witten:2023qsv ; Banerjee:2023eew ; Banerjee:2023liw ; Basteiro:2024cuh . It
will be extremely interesting to explicitly realize such transitions/crossover
of algebras within a simple model like CFT. We hope to address this aspect in
near future.

There are several other future directions and we will briefly discuss some of
them now. First, by now there is a standard literature on Holographic
descriptions of such mirror models as well as models with defects and
interfaces in the CFT. It will be interesting to realize the physics from a
purely bulk geometric perspective. This is not just for reassurance, several
technical questions often have an intriguing geometric understanding in the
Holographic dual, e.g. global and local quenches (see e.g. Das:2011nk  for a
review), OTOCs and shockwave geometries (see e.g. Shenker:2013pqa ;
Shenker:2014cwa ) and many more.

The Holographic dual of the kink mirror is particularly intriguing in this
respect. It is well-known that the exponential growth of an OTOC originates
from the blue-shift near the horizon and from a corresponding shockwave
geometry. The kink mirror model somehow ensures that this blue-shift will
disappear as a result of the dynamical evaporation of the black hole. It will
be very interesting to understand this effect in explicit details. In a
precise sense, all essential ingredients are already available, and we hope to
address this issue in near future.

Generally speaking, the presence of boundaries in a CFT is realized by
inserting branes and hypersurfaces in the bulk Holographic dual. Sometimes,
these branes are the so-called End-of-World (EOW) branes. There is a broader
question on how the presence of such branes may affect the dynamics of e.g. a
probe field in a given asymptotically AdS-geometry. Recent studies in
Das:2022evy ; Das:2023ulz ; Das:2023xjr ; Krishnan:2023jqn ; Burman:2023kko ;
Banerjee:2024dpl  suggest that a non-generic form of chaos may appear as a
function of the location of a geometric brane. It is, however, not understood
how a corresponding OTOC may behave in these cases which will be a very
interesting aspect to address in future.

It is worthwhile to pause at this point and reflect back on the possibilities
that the moving mirror models can capture in a broad physics sense. It is
clear that this framework allows us to explicitly calculate and catalogue OTOC
behaviours of two-dimensional CFTs in these backgrounds and therefore it will
be highly relevant to understand in detail the possible ramifications of such
explicit results on the physical aspects that the moving mirror models can
represent.

## Acknowledgements

We thank Souvik Banerjee, Suchetan Das, Anirban Dinda, Johanna Erdmenger, Jani
Kastikainen, Dominik Neuenfeld, and Somnath Porey for useful discussions
related to this work. PB is supported by the grant CRG/2021/004539. BE
acknowledges the support provided by the grant CRG/2021/004539. AK is
partially supported by CEFIPRA 6304 - 3, DAE-BRNS 58/14/12/2021-BRNS and
CRG/2021/004539 of Govt. of India. AK further acknowledges the support of
Humboldt Research Fellowship for Experienced Researchers by the Alexander von
Humboldt Foundation and for the hospitality of the Theoretical Physics III,
Department of Physics and Astronomy, University of Wurzburg during the course
of this work.

##  Appendix A Analytic continuation in $(u,v)$ coordinates

Different operators in OTOCs get ordered according to their Lorentzian
continuation. This appendix will explain the analytic continuation in $(u,v)$
coordinates. We take our bulk operator $W$ with scaling dimension $h_{w}$ at
the point $P$ and the boundary operator $V$ with scaling dimension $h_{v}$ at
points $P_{1}$ and $P_{2}$.

$\begin{split}P_{1}&\equiv(v_{1}+i\epsilon_{1},v_{1}+i\epsilon_{1})\,,\qquad
P_{2}\equiv(v_{1}+i\epsilon_{2},v_{1}+i\epsilon_{2})\,,\\\
P&\equiv(u+i\epsilon_{0},v+i\epsilon_{0})\,,\quad\quad~{}~{}P_{r}\equiv(v+i\epsilon_{0},u+i\epsilon_{0})\,.\end{split}$
(58)

We will take the two boundary points $P_{1}$ and $P_{2}$ at the same point
$P_{1}\equiv P_{2}$ which implies $v_{1}=v_{2}$. The conformal cross ratio is
given by

$\begin{split}\eta&=\frac{(\tilde{u}-\tilde{v})(\tilde{v}_{1}-\tilde{v}_{2})}{(\tilde{u}-\tilde{v}_{1})(\tilde{v}-\tilde{v}_{2})}\\\
&=\frac{\Big{[}p(u+i\epsilon_{0})-v-i\epsilon_{0}\Big{]}\Big{[}v_{1}+i\epsilon_{1}-v_{1}-i\epsilon_{2}\Big{]}}{\Big{[}p(u+i\epsilon_{0})-v_{1}-i\epsilon_{1}\Big{]}\Big{[}v+i\epsilon_{0}-v_{1}-i\epsilon_{2}\Big{]}}\\\
&=i\epsilon_{12}\frac{-\beta\log\left(1+e^{-\frac{u+i\epsilon_{0}}{\beta}}\right)-v-i\epsilon_{0}}{\left[-\beta\log\left(1+e^{-\frac{u+i\epsilon_{0}}{\beta}}\right)-v_{1}-i\epsilon_{1}\right]\Big{[}v-v_{1}+i\epsilon_{02}\Big{]}}\,.\end{split}$
(59)

### Case-1: $v=0,u\rightarrow\infty$

$\eta=-\frac{i\epsilon_{12}\,\beta\,e^{-u/\beta}}{v_{1}^{2}}\,.$ (60)

### Case-2: $v=0,u=0$

$\eta=\frac{i\epsilon_{12}\,\beta\log 2}{(\beta\log 2+v_{1})(-v_{1})}\,.$ (61)

$-v_{1}>0$, and let’s assume $\beta\log 2+v_{1}>0$. The above assumption
implies that the points $P_{1}$ and $P_{2}$ have to lie in the quadrant
$(u>0,v<0)$ in the $(u,v)$ plane. So we see that $\eta$ approaches $0$ from
the opposite directions for Case-1 and Case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\epsilon_{12}\frac{-\beta\log\left(1+e^{-\frac{u_{1}+i\epsilon_{0}}{\beta}}\right)}{\left[-\beta\log\left(1+e^{-\frac{u_{1}+i\epsilon_{0}}{\beta}}\right)-v_{1}-i\epsilon_{1}\right][-v_{1}]}\\\
&=-\frac{i\epsilon_{12}}{-\beta\log\left[\left(1+e^{-\frac{u_{1}}{\beta}}\right)\left(1-\frac{i\epsilon_{0}}{\beta}\frac{e^{-\frac{u_{1}}{\beta}}}{1+e^{-\frac{u_{1}}{\beta}}}\right)\right]-v_{1}-i\epsilon_{1}}\\\
&=-\frac{i\epsilon_{12}}{i\epsilon_{0}\frac{e^{-\frac{u_{1}}{\beta}}}{1+e^{-\frac{u_{1}}{\beta}}}-i\epsilon_{1}}\,.\end{split}$
(62)

Let’s define
$\epsilon_{\tilde{0}}=\epsilon_{0}\frac{e^{-\frac{u_{1}}{\beta}}}{1+e^{-\frac{u_{1}}{\beta}}}$.
Under this redefinition the cross-ratio becomes

$\begin{split}\eta&=-\frac{\epsilon_{12}}{\epsilon_{\tilde{0}1}}=1+\frac{\epsilon_{2\tilde{0}}}{\epsilon_{\tilde{0}1}}\,.\end{split}$
(63)

The cross ratio $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $u\rightarrow\infty$. It crosses the real axis after
$\eta=1$ branch point if we choose
$\epsilon_{2}>\epsilon_{\tilde{0}}>\epsilon_{1}$ (or,
$\epsilon_{1}>\epsilon_{\tilde{0}}>\epsilon_{2}$). We can choose $\epsilon$’s
in such a way that
$\epsilon_{2}>\epsilon_{0}>\epsilon_{1}\Rightarrow\epsilon_{2}>\epsilon_{\tilde{0}}>\epsilon_{1}$
but there are choices for which
$\epsilon_{2}>\epsilon_{0}>\epsilon_{1}\nRightarrow\epsilon_{2}>\epsilon_{\tilde{0}}>\epsilon_{1}$.
In other words, in $(u,v)$ plane a subset of OTOCs would show the above
behaviour.

In all the previous computations in the literature, OTOCs are always
determined by the analytic continuation in the static case. Now, in our case,
static mirror profile is conformally related to the moving mirror profile. The
above ordering issue is due to the fact that different operators that are
ordered in a certain fashion in static case can become differently ordered
because of conformal transformation.

##  Appendix B Other mirror profiles

###  B.1 Collapsing null shell

The mirror profile PhysRevD.36.2327 ; Akal:2021foz

$v=p(u)=-\frac{1}{\kappa}e^{-\kappa u}$ (64)

mimics black hole that is formed from a collapsing null shell.

$t$$x$$v$$u$$\tilde{t}$$\tilde{x}$$\tilde{v}$$\tilde{u}$ Figure 4: Conformal
mapping for moving mirror corresponding to a collapsing null shell.

In the limit $u\rightarrow\infty$ the escaping mirror becomes (13)

$\lim_{u\to\infty}p(u)=-\beta\,e^{-u/\beta}\,.$ (65)

In the $u\rightarrow\infty$ limit both the escaping mirror (12) and the
collapsing mirror(64) give same behaviour. which is also evident from figure 2
and figure 4. But, in the limit $u\rightarrow-\infty$ the escaping mirror
becomes

$\lim_{u\to-\infty}p(u)=u\,,$ (66)

this behaviour is different for collapsing and escaping mirrors.

Considering similar set-up as discussed in §2.1, we can very easily compute
conformal cross ratios

### Case-1: $v=0,u\rightarrow\infty$

$\eta=-i\,\tilde{\epsilon}_{12}\frac{1}{\kappa}\frac{e^{-\kappa
u}}{v_{1}^{2}}\ .$ (67)

### Case-2: $v=0,u=0$

$\eta=i\tilde{\epsilon}_{12}\frac{1/\kappa}{(1/\kappa+v_{1})(-v_{1})}\ .$ (68)

$-v_{1}>0$, and let’s assume $1/\kappa+v_{1}>0$. The above assumption implies
that the points $P_{1}$ and $P_{2}$ have to lie in the quadrant $(u>0,v<0)$ in
the $(u,v)$ plane. So we see that $\eta$ approaches $0$ from the opposite
directions for Case-1 and Case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\
.\end{split}$ (69)

For the computation of the Virasoro identity block the $u\rightarrow\infty$
result is important and this is the same as that of the escaping mirror. This
is because both the escaping mirror and the collapsing mirror behave similarly
in the limit $u\rightarrow\infty$. From (24), our final result for the
Virasoro identity block is given by

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1+\frac{24\pi
h_{w}\,\kappa\,v_{1}^{2}}{c\,\tilde{\epsilon}_{12}}e^{\kappa\,u}}\right)^{2h_{v}}\approx
1-\frac{48\pi
h_{v}h_{w}\,\kappa\,v_{1}^{2}}{c\,\tilde{\epsilon}_{12}}e^{\kappa\,u}+\cdots\,.\end{split}$
(70)

Which again shows exponential behaviour at large $u$.

###  B.2 Rindler mirror

The mirror profile is given by Akal:2021foz

$v=p(u)=-\frac{\sigma^{2}}{u}\,.$ (71)

$t$$x$$v$$u$$\tilde{t}$$\tilde{x}$$\tilde{v}$$\tilde{u}$ Figure 5: Conformal
mapping for Rindler mirror.

Considering similar set-up as discussed in §2.1, from (19) we get conformal
cross ratio

$\begin{split}\eta&=\frac{(\tilde{u}-\tilde{v})i\,\tilde{\epsilon}_{12}}{(\tilde{u}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{01})(\tilde{v}-\tilde{v}_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\\\
&=i\,\tilde{\epsilon}_{12}\,\frac{-\frac{\sigma^{2}}{u}-v}{\left[-\frac{\sigma^{2}}{u}-v_{1}+i\,\tilde{\epsilon}_{01}\right]\left[v-v_{1}+i\,\tilde{\epsilon}_{02}\right]}\,.\end{split}$
(72)

### Case-1: $v=0,u\rightarrow\infty$

$\eta=-i\,\tilde{\epsilon}_{12}\frac{\sigma^{2}}{v_{1}^{2}\,u}\ .$ (73)

### Case-2: $v=0,u=0$

$\eta=i\tilde{\epsilon}_{12}\frac{1}{(-v_{1})}\ .$ (74)

$-v_{1}>0$. So we see that $\eta$ approaches $0$ from the opposite directions
for Case-1 and Case-2.

### Case-3: $v=0,u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\tilde{\epsilon}_{12}\frac{-\sigma^{2}/u_{1}}{[-\sigma^{2}/u_{1}-v_{1}+i\tilde{\epsilon}_{01}][-v_{1}]}\\\
&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\
.\end{split}$ (75)

The cross ratio $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $u\rightarrow\infty$. For the intermediate point
$u=u_{1}$, in the out of time order case,
$\tilde{\epsilon}_{2}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{1}$ or,
$\tilde{\epsilon}_{1}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{2}$, it crosses
the real axis after $\eta=1$ branch point.

For a large-$c$ CFT, as before, we can again use the Identity block result. We
can very easily see that as $u\rightarrow\infty$:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\sigma^{2}}u}\right)^{2h_{v}}\\\
&\approx 1-\frac{48\pi
h_{v}h_{w}v_{1}^{2}}{c\,\tilde{\epsilon}_{12}\sigma^{2}}u+\cdots\,.\end{split}$
(76)

In this case, we get polynomial behaviour.

###  B.3 Receding mirror

In this appendix, we consider the following mirror profile Cotler:2022weg

$\begin{split}v=p(u)&=u,\quad u<0,\\\ &=\frac{1}{2\pi T_{H}}\left(1-e^{-2\pi
T_{H}u}\right),\quad 0<u<L,\\\ &=e^{-2\pi T_{H}L}(u-L)+\frac{1}{2\pi
T_{H}}\left(1-e^{-2\pi T_{H}L}\right),\quad u>L\,.\end{split}$ (77)

$t$$x$$v$$u$$\tilde{t}$$\tilde{x}$$\tilde{v}$$\tilde{u}$ Figure 6: Conformal
mapping for receding mirrors.

In all the previous computations, we took our bulk point at $(u,0)$ and then
analyzed behaviour of OTOC’s as a function of $u$. Here, instead of taking
$v=0$ for the bulk point we will take $v=\frac{1}{2\pi T_{H}}\left(1-e^{-2\pi
T_{H}L}\right)\equiv c_{0}$ which is the value of $p(L)$.

We perform the usual conformal transformation $\tilde{u}=p(u)$ and
$\tilde{v}=v$ to map the dynamical mirror profile to a static one. Let us take
the bulk point to be $P\equiv\left(\tilde{u},\tilde{v}\right)$, the reflected
point would be $P_{r}\equiv(\tilde{v},\tilde{u})$. We take our boundary points
to be $P_{1}\equiv\left(\tilde{v}_{1},\tilde{v}_{1}\right)$ and
$P_{2}\equiv\left(\tilde{v}_{2},\tilde{v}_{2}\right)$. We will fuse the two
boundary points at a same point $P_{1}\equiv P_{2}$. The cross-ratio becomes

$\begin{split}\eta&=i\tilde{\epsilon}_{12}\frac{\left(\tilde{u}-\tilde{v}\right)}{\left(\tilde{u}-\tilde{v}_{1}+i\tilde{\epsilon}_{01}\right)\left(\tilde{v}-\tilde{v}_{1}+i\tilde{\epsilon}_{02}\right)}\\\
&=\frac{\big{(}p(u)-v\big{)}i\,\tilde{\epsilon}_{12}}{\big{(}p(u)-v_{1}+i\,\tilde{\epsilon}_{01}\big{)}\big{(}v-v_{1}+i\,\tilde{\epsilon}_{02})}\,.\end{split}$
(78)

### Case-1: $v=c_{0},u=0$

$\begin{split}\eta&=i\,\tilde{\epsilon}_{12}\frac{-c_{0}}{(-v_{1})(c_{0}-v_{1})}\\\
&=i\,\tilde{\epsilon}_{12}\frac{c_{0}}{(v_{1})(c_{0}-v_{1})}\,.\end{split}$
(79)

As in the previous cases, we take $v_{1}>0$ and $v_{1}<c_{0}$.

### Case-2: $v=c_{0},~{}~{}u=L-\alpha,~{}~{}\alpha\rightarrow 0$

In this case, we can’t take $u\rightarrow\infty$ limit. Instead, we will take
$u=L-\alpha$ where $\alpha$ is a small positive number and we will take
$\alpha\rightarrow 0$.

$\begin{split}\eta&=i\tilde{\epsilon}_{12}\frac{\frac{1}{2\pi
T_{H}}\left(1-e^{-2\pi T_{H}(L-\alpha)}\right)-c_{0}}{\left(\frac{1}{2\pi
T_{H}}\left(1-e^{-2\pi
T_{H}(L-\alpha)}\right)-v_{1}\right)\left(c_{0}-v_{1}\right)}\\\
&=i\tilde{\epsilon}_{12}\frac{-\frac{1}{2\pi T_{H}}e^{-2\pi T_{H}L}2\pi
T_{H}\alpha}{\left(c_{0}-v_{1}\right)\left(c_{0}-v_{1}\right)}\\\
&=i\tilde{\epsilon}_{12}\frac{-e^{-2\pi
T_{H}L}\alpha}{(c_{0}-v_{1})^{2}}\,.\end{split}$ (80)

So we see that $\eta$ approaches $0$ from the opposite directions for Case-1
and Case-2.

### Case-3: $v=c_{0},u=u_{1}=p^{-1}(v_{1})$

$\begin{split}\eta&=i\tilde{\epsilon}_{12}\frac{p(u_{1})-c_{0}}{[p(u_{1})-v_{1}+i\tilde{\epsilon}_{01}][c_{0}-v_{1}+i\tilde{\epsilon}_{02}]}\\\
&=-\frac{\tilde{\epsilon}_{12}}{\tilde{\epsilon}_{01}}=1+\frac{\tilde{\epsilon}_{20}}{\tilde{\epsilon}_{01}}\
.\end{split}$ (81)

The cross ratio $\eta$ goes to zero from the opposite direction in the limit
$u\rightarrow 0$ and $\alpha\rightarrow 0$. For the intermediate point
$u=u_{1}$, in the out of time order case,
$\tilde{\epsilon}_{2}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{1}$ or,
$\tilde{\epsilon}_{1}>\tilde{\epsilon}_{0}>\tilde{\epsilon}_{2}$, it crosses
the real axis after $\eta=1$ branch point.

For a large-$c$ CFT, as before, we can again use the Identity block result. We
can very easily see that as $\alpha\rightarrow 0$:

$\begin{split}\mathcal{F}_{0}(\eta)&=\left(\frac{1}{1-\frac{24\pi
ih_{w}}{c\eta}}\right)^{2h_{v}}\\\ &=\left(\frac{1}{1+\frac{24\pi
h_{w}(c_{0}-v_{1})^{2}}{c\,\tilde{\epsilon}_{12}e^{-2\pi
T_{H}L}\alpha}}\right)^{2h_{v}}\\\ &\approx 1-\frac{48\pi
h_{v}h_{w}(c_{0}-v_{1})^{2}}{c\,\tilde{\epsilon}_{12}e^{-2\pi
T_{H}L}}\left(\frac{1}{\alpha}\right)+\cdots\,.\end{split}$ (82)

In this case, we again get polynomial behaviour.

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