NORDITA-2015-32

YITP-15-21

IPMU15-0033

Scrambling time from local perturbations
of the eternal BTZ black hole

Paweł Caputaa, Joan Simónb, Andrius Štikonasb,

Tadashi Takayanagic,d and Kento Watanabec

aNordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
b School of Mathematics and Maxwell Institute for Mathematical Sciences,
University of Edinburgh, King’s Buildings, Edinburgh EH9 3FD, UK
cYukawa Institute for Theoretical Physics (YITP), Kyoto University, Kyoto
606-8502,
dKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan

We compute the mutual information between finite intervals in two non-compact
2d CFTs in the thermofield double formulation after one of them has been
locally perturbed by a primary operator at some time $t_{\omega}$ in the large
$c$ limit. We determine the time scale, called the scrambling time, at which
the mutual information vanishes and the original entanglement between the
thermofield double gets destroyed by the perturbation. We provide a
holographic description in terms of a free falling particle in the eternal BTZ
black hole that exactly matches our CFT calculations. Our results hold for any
time $t_{\omega}$. In particular, when the latter is large, they reproduce the
bulk shock-wave propagation along the BTZ horizon description.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Single sided entropy
  3. 3 Two-sided entropies
    1. 3.1 Semi-infinite intervals
    2. 3.2 Mutual information for finite intervals
      1. 3.2.1 $S_{B}$
      2. 3.2.2 $S_{A\cup B}$
    3. 3.3 The evolution of the Mutual information
  4. 4 Scrambling time
  5. 5 Holographic description
    1. 5.1 Free falling particle in Kruskal coordinates
    2. 5.2 Back-reacted metric
  6. 6 Bulk mutual information
    1. 6.1 Geodesic in the left boundary
    2. 6.2 Geodesic in the right boundary
    3. 6.3 Geodesics across the horizon and Mutual Information
  7. A Twist operators in the TFD state
  8. B Details of the holographic model
  9. C Two sided 2-pt functions

##  1 Introduction and Summary

Consider a (1+1)d CFT in some state having non-trivial quantum correlations.
If the system is perturbed at some instant of time $t_{\omega}$ and evolve
unitarily afterwards, it is natural to ask whether there exists any time scale
when its subsystems become uncorrelated. Since the mutual information
$I_{A:B}=S_{A}+S_{B}-S_{A\cup B}$ between two such subsystems $A$ and $B$
provides an upper bound for the connected two-point functions of operators
acting on these subsystems111This bound is proved for finite dimensional
Hilbert spaces. We are not aware of an extension of this result to QFTs/CFTs,
but we expect it to hold when regulating and normalising appropriately the
relevant quantities in the continuum limit. [1]

$I_{A:B}\geq\frac{\left(\langle{\mathcal{O}}_{A}{\mathcal{O}}_{B}\rangle-\langle{\mathcal{O}}_{A}\rangle\langle{\mathcal{O}}_{B}\rangle\right)^{2}}{2\|{\mathcal{O}}_{A}\|^{2}\|{\mathcal{O}}_{B}\|^{2}}\,,$
(1)

it is natural to study the vanishing of this quantity to answer this question.
The study of the time dependence in this measure of entanglement can help us
to understand the time scales controlling how quantum systems get thermalized,
which is one of the most important problems in non-equilibrium physics.

A particular situation of the above scenario is when a perturbation acts on a
thermal state. In holographic theories, thermal states are believed to have a
gravity dual in terms of black holes [2, 3]. Black hole physics suggests that
the speed at which the system forgets initial conditions, i.e. the
perturbation, is the fastest among all physical diffusive processes. This gave
rise to the notion of fast scramblers and the scrambling conjecture [4, 5].
The main goal of this paper is to provide a first principle derivation for the
time scale at which this phenomenon occurs for 2d CFTs in the large $c$ limit
in a concrete setup which allows both CFT and holographic computations.

Recently, Shenker and Stanford considered an excellent and tractable setup to
study the fast scrambling phenomena in the context of an eternal black hole
[6, 7, 8]. This involves a pair of non-interacting CFTs in an entangled state,
the thermofield double state. Tracing any entire CFT Hilbert space, gives rise
to a thermal density matrix in the remaining CFT. The perturbation is
described by some boundary CFT operator and its gravity dual involved a shock-
wave propagating in the black hole background. No matter how small the
boundary perturbation is, the blue shift of energies when this perturbation
reaches the horizon suggests the existence of a non-trivial backreaction.

In this work, we study such a setup for a perturbation localized in a point-
like region, triggered by a primary operator in a given CFT. To obtain
analytical results, we consider 2d large $c$ CFTs and their gravity duals
given by a perturbation of the BTZ black hole [9]. Recent developments in the
calculation of 4-pt functions involving heavy and light operators in the large
$c$ limit of the dual 2d CFT [10, 11] (see also [12, 13]) allow us to
analytically test these ideas.

Computations of time evolutions of entanglement entropy after local
perturbations222Note that this setup looks similar to the local quenches in
CFTs [14]. However, in the latter the local excitations are triggered by
joining two semi-infinite lines and lead to local excitations in all sectors
of a given CFT. Thus their behaviours differ from each other in integrable
CFTs [15, 16, 17]. On the other hand, in large $c$ CFTs, they behave similarly
[18, 19, 20] in that both results for 2d CFTs show logarithmic time evolution
of entanglement entropy. by primary operators have been formulated in [15, 17]
and have been applied to many examples for CFTs at zero temperature in [16,
19, 20, 21, 22, 23]. Entanglement entropy and mutual information at finite
temperature CFTs has been analyzed for integral CFTs in [24]. On the other
hand, the holographic calculations of time-evolutions of entanglement entropy
after local perturbations have been analyzed in [18, 25, 26] at zero
temperature and in [24] at finite temperature. In this paper we will extend
the discussion of local excitations to the thermofield double formalism of
finite temperature CFTs in the large $c$ limit.

##### Summary of results :

Specifically, we perturb the thermofield double (TFD) state by a local primary
operator $\psi$ at time $t_{\omega}$ in the past and compute the mutual
information between regions $A$ and $B$ belonging to opposite boundaries. We
denote the two boundary times in the thermofield double by $t_{L}\equiv t_{-}$
and $t_{R}\equiv t_{+}$. When measuring the mutual information at
$t_{-}=t_{+}=0$, we ask for the time scale $t^{\star}_{\omega}$ when the
mutual information vanishes

$I_{A:B}(t^{\star}_{\omega})=0\,.$ (2)

Equivalently, we ask for the time scale $t^{\star}_{\omega}$ at which
correlations between $A$ and $B$ vanish. When both subsystems $A$ and $B$ are
the intervals $(0<)y\leq x\leq y+L$ and the perturbation is turned on at $x=0$
and time $t_{-}=-t_{\omega}$, we obtain the following analytical result for
$t^{\star}_{\omega}\gg\beta$

$t^{\star}_{\omega}=y+\frac{L}{2}-\frac{\beta}{2\pi}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)+\frac{\beta}{\pi}\log\left(2\sinh\frac{\pi
L}{\beta}\right),$ (3)

where $\beta$ is the inverse temperature and
$\alpha_{\psi}=\sqrt{1-24h_{\psi}/c}$ carries the information about the
primary operator perturbation of conformal dimension $h_{\psi}$. The parameter
$\epsilon$ represents a UV cut off for the local excitation, so that the
excited state is localized around a region of size $\epsilon$ of the operator
insertion. This makes the energy of the perturbation $E_{\psi}=\frac{\pi
h_{\Psi}}{\epsilon}$ finite. In the limit $h_{\psi}/c\ll 1$, which is the
relevant one to match the butterfly effect discussed in [6, 7], this reduces
to

$t^{\star}_{w}=f(L,\beta)+\frac{\beta}{2\pi}\log\left(\frac{\pi
S_{density}}{4E_{\psi}}\right),$ (4)

where $S_{density}=\frac{\pi c}{3\beta}$ is the entropy density of the
original thermal system. The $\log S$ behavior in (4) is consistent with the
fast scrambling conjecture [5, 6].

Given the bound (1), it should be possible to extract the same time scale from
the condition of vanishing two sided 2-pt functions. We explicitly show this
in appendix C, confirming the observation made in Shenker and Stanford [6]
that both scales are controlled by the same physics.

In the second part of this work, we derive the same time scale from bulk
holographic considerations and find a perfect matching between both
calculations. Our holographic model is based on the description of the local
boundary perturbation in terms of some free falling particle satisfying an
initial condition guaranteeing such particle carries the right amount of
energy from the CFT stress tensor perspective. This is done by generalizing
the model in [18, 24] to the two sided BTZ black hole, based on the back
reaction description of point particles as quotients of AdS3. Applying the
holographic entanglement entropy [28, 29] to evaluate the entanglement entropy
and mutual information in our set-up, leads to the same scrambling time (3).
At the same time, our setup and calculation may be useful for the interesting
question regarding the dual CFT interpretation of a particle falling into a
AdS black hole in future studies.

Our local boundary perturbation includes a regularization parameter $\epsilon$
describing its size. This parameter is holographically interpreted as the bulk
position (distance from the boundary) from which the massive particle falls
into the black hole. Our solution computes the back reacted geometry for any
$t_{\omega}$ and approaches a localised shock-wave in the limit of large
$t_{\omega}$ [27, 6].

The paper is organized as follows: In section 2, we will analyze the time
evolution of entanglement entropy in large $c$ 2d CFTs at finite temperature,
which agrees perfectly with a previous gravity dual computation. In section 3,
we study local perturbations in finite temperature CFTs by employing the
thermofield double formalism. We compute the mutual information from
entanglement entropies. In section 4, we compute the scrambling time for the
mutual information. In section 5, we introduce our holographic model. In
section 6, we present our holographic computations of mutual information in a
two sided $AdS_{3}$ black hole background with a local excitation. In appendix
A, we explained the details of treatment of twist operators in the replica
method computations of entanglement entropy in the thermofield double
formulation. In appendix B, we present some details of our holographic model.
In appendix C we describe a computation of two point function in our model.

Note Added: While finishing our main computations, the work of Roberts and
Stanford [30] appeared. The latter has a detailed account of two point
functions in the presence of localised excitations over thermal states and
briefly mentions the behaviour of the mutual information in the same set-up.
Thus, it has some overlap with our results. In our paper, we literally
evaluate the mutual information between the thermofield double in both 2d
large $c$ CFTs and their gravity duals independently and show their results
perfectly agree. Our gravity solutions explicitly have the regularization
parameter $\epsilon$ and our matching between gravity and CFT results holds
while keeping this parameter small but non-zero. We would also like to mention
that in the interesting recent paper [31] by Maldacena, Shenker and Stanford,
the fast scrambling behavior of the correlations functions has been
interpreted in terms of chaos.

##  2 Single sided entropy

To introduce our basic tools and fix the notation, we analyse the local
perturbation to a thermal state in a single 2d CFT at finite temperature in
this section.
Consider a thermal state $\rho_{\beta}$ locally perturbed by a primary
operator $\psi(0,-t_{\omega})$ inserted at $x=0$ at time $-t_{\omega}$. The
time evolution of the resulting density matrix is given by

$\rho(t)={\cal
N}\,e^{-iHt}\psi(0,-t_{\omega})\,\rho_{\beta}\,\psi^{\dagger}(0,-t_{\omega})\,e^{iHt}\,.$
(5)

where $H$ is the Hamiltonian of our system.
Denote by $\rho_{A}=\text{Tr}_{\bar{A}}\rho(t)$ the reduced density matrix on
a finite interval $A$ with endpoints333Notice that the perturbation is
originally inserted outside of the interval $A$. $y,\,y+L$ satisfying $y,L>0$.
Its entanglement entropy $S_{A}$ can be computed using the replica trick. We
first compute the Renyi entropies

$S^{(n)}_{A}\equiv\frac{1}{1-n}\log\text{Tr}\rho^{n}_{A}(t)\,.$ (6)

The entanglement entropy is obtained by taking the limit $S_{A}=\lim_{n\to
1}S^{(n)}_{A}$.
The trace of the reduced density matrix $\text{Tr}\rho^{n}_{A}(t)$ requires
the calculation of the normalised 4-point function

$\text{Tr}\rho^{n}_{A}(t)=\frac{\langle\Psi(x_{1},\bar{x}_{1})\sigma_{n}(x_{2},\bar{x}_{2})\tilde{\sigma}_{n}(x_{3},\bar{x}_{3})\Psi^{\dagger}(x_{4},\bar{x}_{4})\rangle}{\left(\langle\psi(x_{1},\bar{x}_{1})\psi^{\dagger}(x_{4},\bar{x}_{4})\rangle_{C_{1}}\right)^{n}}$
(7)

with the insertion points

$\displaystyle x_{1}=-i\epsilon,\quad x_{2}=y-t_{\omega}-t_{-},\quad
x_{3}=y+L-t_{\omega}-t_{-},\quad x_{4}=+i\epsilon$
$\displaystyle\bar{x}_{1}=+i\epsilon,\quad\bar{x}_{2}=y+t_{\omega}+t_{-},\quad\bar{x}_{3}=y+L+t_{\omega}+t_{-},\quad\bar{x}_{4}=-i\epsilon\,.$
(8)

The operator $\Psi$ stands for the product of the operators $\psi_{i}$ in each
of the i-th copies of the theory444This correlator is formally computed in the
cyclic orbifold $CFT^{n}/Z_{n}$.

$\Psi=\psi_{1}\cdot\psi_{2}\cdots\psi_{n}$ (9)

and has conformal dimension $h_{\Psi}=n\,h_{\psi}$, where $h_{\psi}$ is the
conformal dimension of the original perturbation $\psi$. Notice $\epsilon$ is
a parameter smearing the local operator perturbation and all the time
evolution is carried by the twist operators $\sigma_{n},\,\tilde{\sigma}_{n}$
which are initially inserted at both ends of the interval when cyclically
gluing the different cylinder copies that give rise to the manifold $C_{n}$.
Finally, the conformal dimension $\Delta_{\sigma}=2H_{\sigma}$ of the twist
operators is

$H_{\sigma}=\frac{c}{24}\left(n-\frac{1}{n}\right)\,.$ (10)

We compute (7) analogously to [20] but with an additional composition of a map
from the cylinder to the plane

$w(x)=e^{\frac{2\pi}{\beta}x},$ (11)

to take care of the thermal nature of the original state, as well as the map

$z(w)=\frac{(w_{1}-w)w_{34}}{w_{13}(w-w_{4})}$ (12)

that brings the points $w_{1}\to 0$, $w_{2}\to z$, $w_{3}\to 1$,
$w_{4}\to\infty$.
The transformation properties of primary operators determine the resulting
trace to be555We already used the regularised twist operators so that
$\varepsilon_{UV}$ is the standard UV cut-off.

$\text{Tr}\rho_{A}^{n}(t)=\left|\frac{\beta}{\pi\varepsilon_{UV}}\sinh\left(\frac{\pi
x_{23}}{\beta}\right)\right|^{-4H_{\sigma}}\left|1-z\right|^{4H_{\sigma}}G(z,\bar{z})$
(13)

where we used the 2-pt function on the cylinder $C_{1}$

$\langle\psi(x_{1},\bar{x}_{1})\psi(x_{4},\bar{x}_{4})\rangle_{C_{1}}=\left|\frac{\beta}{\pi}\sinh\left(\frac{\pi
x_{14}}{\beta}\right)\right|^{-4h_{\psi}}$ (14)

and introduced the canonical 4-point function

$\displaystyle G(z,\bar{z})$
$\displaystyle\equiv\lim_{z_{4}\to\infty}|z_{4}|^{4h_{\Psi}}\langle\psi(z_{4},\bar{z}_{4})\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\psi(0,0)\rangle$
(15)
$\displaystyle\equiv\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)|\psi\rangle$

defined in terms of the cross-ratio

$z=\frac{w_{12}w_{34}}{w_{13}w_{24}},\qquad
1-z=\frac{w_{14}w_{23}}{w_{13}w_{24}}\,.$ (16)

We defined $w_{ij}=w_{i}-w_{j}$ in all the above formulas and the same
conventions hold for $x_{ij}$ and $z_{ij}$.
The Renyi entropies are computed by inserting (13) into (6)

$S^{(n)}_{A}=\frac{c(n+1)}{6}\log\left(\frac{\beta}{\pi\varepsilon_{UV}}\sinh\frac{\pi\,L}{\beta}\right)+\frac{1}{n-1}\log(|1-z|^{4H_{\sigma}}G(z,\bar{z}))\,,$
(17)

where $\varepsilon_{UV}$ is the UV cut off of the CFT i.e. the lattice
spacing. The first term is the standard Renyi entropy for an interval $L$ in a
2d CFT at finite temperature $T=1/\beta$; the second term captures the extra
contribution due to the local operator insertion. In particular, the
dependence on the conformal dimension of the local operator $h_{\psi}$ is
encoded in $G(z,\bar{z})$.

In general, the extra contribution to the Renyi entropies requires the
knowledge of the full four-point function $G(z,\bar{z})$, i.e. the dynamical
details of the particular 2d CFT under consideration. To make further
progress, we consider the large $c$ limit666We assume a class of CFTs allowing
such a limit.. Notice that in the limit $n\to 1$, the twist operators
$\sigma_{n},\,\tilde{\sigma}_{n}$ become light

$H_{\sigma}/c=\frac{1}{24}\left(n-\frac{1}{n}\right)\to 0\quad\text{as}\quad
n\to 1$ (18)

If $h_{\psi}/c$ remains fixed in the large $c$ limit, the 4-pt function (15)
becomes a 4-pt function involving two heavy and two light operators. This is
precisely the set-up considered in [10, 11] to compute the dominant (vacuum)
contribution to $G(z,\bar{z})$777We refer the readers to [32, 33] for the
description of the regularisation of twist operators used to compute
entanglement entropy.. Using their results, one derives [20]

$\log
G(z,\bar{z})\simeq-\frac{c(n-1)}{6}\log\left(\frac{z^{\frac{1}{2}(1-\alpha_{\psi})}\bar{z}^{\frac{1}{2}(1-\bar{\alpha}_{\psi})}(1-z^{\alpha_{\psi}})(1-\bar{z}^{\bar{\alpha}_{\psi}})}{\alpha_{\psi}\bar{\alpha}_{\psi}}\right)+{\cal
O}((n-1)^{2})$ (19)

where

$\alpha_{\psi}=\sqrt{1-\frac{24h_{\psi}}{c}}\,,$ (20)

encodes all the dependence on the conformal dimension of the local operator
$h_{\psi}$. Finally, we can compute the variation in the entanglement entropy
due to the insertion of the local primary operator to be

$\Delta
S_{A}=\frac{c}{6}\log\left(\frac{z^{\frac{1}{2}(1-\alpha_{\psi})}\bar{z}^{\frac{1}{2}(1-\bar{\alpha}_{\psi})}(1-z^{\alpha_{\psi}})(1-\bar{z}^{\bar{\alpha}_{\psi}})}{\alpha_{\psi}\bar{\alpha}_{\psi}(1-z)(1-\bar{z})}\right)\,,$
(21)

where we subtracted the entanglement entropy of the interval $L$ at finite
temperature $T=1/\beta$

$S_{\text{thermal}}=\frac{c}{3}\log\left(\frac{\beta}{\pi\varepsilon_{UV}}\sinh\frac{\pi\,L}{\beta}\right)\,.$
(22)

As explained in [24], to extract a non-trivial contribution to the
entanglement entropy in the CFT at finite temperature we must take the
smearing parameter $\epsilon$ small but finite. This still allows us to work
with completely analytic formulas at order $\epsilon$. Then, in the small
$\epsilon/\beta$ limit888We are assuming that $y-t_{-}-t_{\omega}$ and
$y+L-t_{-}-t_{\omega}$ are larger than the smearing parameter $\epsilon$ in
units of $\beta$., the cross-ratios are

$\displaystyle z$ $\displaystyle\simeq 1+\frac{2\pi
i\epsilon}{\beta}\frac{\sinh\frac{\pi\,L}{\beta}}{\sinh\frac{\pi(y+L-t_{-}-t_{\omega})}{\beta}\sinh\frac{\pi(y-t_{-}-t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})$ (23) $\displaystyle\bar{z}$ $\displaystyle\simeq
1-\frac{2\pi
i\epsilon}{\beta}\frac{\sinh\frac{\pi\,L}{\beta}}{\sinh\frac{\pi(y+L+t_{-}+t_{\omega})}{\beta}\sinh\frac{\pi(y+t_{-}+t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})$

Due to the non-trivial monodromy properties of $G(z,\bar{z})$, we must
carefully deal with the sign of the imaginary part of the cross-ratios [20,
30]999We choose a reference phase to be consistent with causality and make
entanglement entropies real and non-negative.. Notice the imaginary part of
$\bar{z}$ never changes sign, for $t_{-}+t_{\omega}\geq 0$. Thus, we conclude
$\bar{z}\simeq 1$ for all such times. On the other hand, the imaginary part of
$z$ does flip sign whenever $t_{-}+t_{\omega}\in(y,\,y+L)$. Thus, we either
have $(z,\bar{z})\to(1,1)$ for $t_{-}+t_{\omega}<y$ and $t_{-}+t_{\omega}>y+L$
or $(z,\bar{z})\to(e^{2\pi i},1)$ for $y<t+t_{\omega}<y+L$. Using these phases
in (21), we reach our first important result

$\boxed{\begin{aligned} \Delta S_{A}&=0\,,\quad
t_{-}+t_{\omega}<y\,\,\text{and}\,\,t_{-}+t_{\omega}>y+L\\\ \Delta
S_{A}&=\frac{c}{6}\log\left[\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\frac{\sinh\left(\frac{\pi(y+L-t_{-}-t_{\omega})}{\beta}\right)\sinh\left(\frac{\pi(t_{-}+t_{\omega}-y)}{\beta}\right)}{\sinh\left(\frac{\pi
L}{\beta}\right)}\right]\quad\quad y<t_{-}+t_{\omega}<y+L\,.\end{aligned}}$
(24)

Thus, there is no variation in the entanglement entropy $S_{A}$ either till
the perturbation reaches region $A$ $(t_{-}+t_{\omega}<y)$ or when it leaves
region $A$ $(t_{-}+t_{\omega}>y+L)$. While the perturbation can causally be in
region $A$, the variation in entanglement reaches a maximum at
$t_{-}+t_{\omega}=y+\frac{L}{2}$

$(\Delta
S_{A})_{\text{max}}=\frac{c}{6}\log\left[\frac{\beta}{2\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\tanh\frac{\pi\,L}{2\beta}\right]\,.$
(25)

In the high temperature limit (or large interval $L$), the increase in
entanglement due to the perturbation equals

$\Delta
S_{A}\simeq\frac{c}{6}\log\left[\frac{\beta}{2\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right]\qquad\beta\to
0\,,\,\,\frac{\beta}{\epsilon}\gg 1\,.$ (26)

In section 6.1 we will match this result with the gravity dual computation
using the holographic entanglement entropy. Let us now proceed with the
entanglement entropies that involve intervals in both CFTs.

##  3 Two-sided entropies

Entanglement entropy and mutual information in the thermofield double state
involving intervals in both CFTs were discussed in detail in [35] and their
time evolution in [36, 37]. Here we only briefly review this setup and extend
by the insertion of a local operator to one of the CFTs.

Consider two non-interacting 2d CFTs, $CFT_{L}$ and $CFT_{R}$ with isomorphic
Hilbert spaces $\mathcal{H}_{L}$ and $\mathcal{H}_{R}$. The thermofield double
(TFD) state is a particular entangled state in the total Hilbert space
$\mathcal{H}_{tot}=\mathcal{H}_{L}\otimes\mathcal{H}_{R}$.

$|\Psi_{\beta}\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_{n}e^{-\frac{\beta}{2}E_{n}}|n\rangle_{L}|n\rangle_{R},$
(27)

where $|n\rangle_{L,R}\in\mathcal{H}_{L,R}$ are the eigenstates in each
Hilbert space and $Z(\beta)$ is a partition function in total Hilbert space
(and also in each Hilbert space).

$Z(\beta)=\sum_{n}e^{-\beta E_{n}}$ (28)

If we have the total Hamiltonian $H_{tot}=H_{L}+H_{R}$, this partition
function is a partition function on a cylinder $C_{1}$ with circumference
$\beta$

$Z(\beta)=\sum_{n,m}\langle n|_{L}\langle
n|_{R}e^{-\frac{\beta}{2}(H_{L}+H_{R})}|m\rangle_{L}|m\rangle_{R}=\mathrm{Tr}_{tot}\left[e^{-\frac{\beta}{2}(H_{L}+H_{R})}\right]$
(29)

Tracing out the Hilbert space $\mathcal{H}_{R}$ from the pure state density
matrix, we can get the thermal density matrix in $CFT_{L}$ with temperature
$\beta$

$\rho_{L}=\mathrm{Tr}_{R}|\Psi_{\beta}\rangle\langle\Psi_{\beta}|=\frac{1}{Z(\beta)}\sum_{n}e^{-\beta
E_{n}}|n\rangle_{L}\langle n|_{L}$ (30)

A general time evolution of the TFD state is obtained by applying the
evolution operator to both CFTs

$|\Psi_{\beta}(t_{-},t_{+})\rangle=e^{-it_{-}H_{L}+it_{+}H_{R}}|\Psi_{\beta}\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_{n}e^{-i(t_{+}-t_{-}-i\frac{\beta}{2})E_{n}}|n\rangle_{L}|n\rangle_{R}$
(31)

One can immediately check that setting $t_{-}=t_{+}=t$, that can be seen as
evolving with Hamiltonian $H_{L}-H_{R}$, leaves the TFD state invariant (the
symmetry of the TFD state). On the other hand, setting $t_{-}=-t_{+}=t$ yields
the time dependent state corresponding to evolution with $H_{L}+H_{R}$ as in
[36]. These two configurations should be kept in mind since we leave general
$t_{-}$ and $t_{+}$ in our formulas so that our formalism can be used to
extract the evolution after a local excitation with any of the two total
Hamiltonians101010In fact our results also hold for evolution of the TFD state
with $H_{L}$ or $H_{R}$ only and one can extract these formulas by setting
$t_{-}=t$ and $t_{+}=0$ or $t_{-}=0$ and $t_{+}=t$ respectively..

In the TFD formalism, we can also relate one-sided and two-sided correlators
by the analytical continuation of $t$. For example, consider the following
one-sided correlator

$\langle\Psi_{\beta}|\mathcal{O}_{L}(x_{1},0)\mathcal{O}_{L}^{\dagger}(x_{2},t)|\Psi_{\beta}\rangle=\sum_{n,m}e^{-\beta
E_{n}+it(E_{n}-E_{m})}\langle
n|_{L}\mathcal{O}_{L}(x_{1},0)|m\rangle_{L}\langle
m|_{L}\mathcal{O}_{L}^{\dagger}(x_{2},0)|n\rangle_{L}$ (32)

On the other hand, we can deform the two-sided correlator as follows

$\displaystyle\langle\Psi_{\beta}|\mathcal{O}_{L}(x_{1},0)\mathcal{O}_{R}(x_{2},t)|\Psi_{\beta}\rangle$
$\displaystyle=\sum_{n,m}e^{-\frac{\beta}{2}(E_{n}+E_{m})+it(E_{n}-E_{m})}\langle
n|_{L}\mathcal{O}_{L}(x_{1},0)|m\rangle_{L}\langle
n|_{R}\mathcal{O}_{R}(x_{2},0)|m\rangle_{R}$
$\displaystyle=\sum_{n,m}e^{-\beta
E_{n}+i(t-i\frac{\beta}{2})(E_{n}-E_{m})}\langle
n|_{L}\mathcal{O}_{L}(x_{1},0)|m\rangle_{L}\langle
m|_{R}\mathcal{O}_{R}^{\dagger}(x_{2},0)|n\rangle_{R}$ (33)

Therefore, the one-sided and two-sided correlators can be related through the
analytical continuation $t\to t+i\frac{\beta}{2}$

$\langle\Psi_{\beta}|\mathcal{O}_{L}(x_{1},0)\mathcal{O}_{L}(x_{2},t)|\Psi_{\beta}\rangle=\langle\Psi_{\beta}|\mathcal{O}_{L}(x_{1},0)\mathcal{O}_{R}^{\dagger}(x_{2},t-i\frac{\beta}{2})|\Psi_{\beta}\rangle$
(34)

If the operators $\mathcal{O}_{L}$ in $CFT_{L}$ are located at $\tau=0$, the
operators $\mathcal{O}_{R}$ are located at $\tau=\frac{\beta}{2}$, or at the
opposite side on the cylinder. We can express the correlators in the TFD state
as the correlators on the cylinder $C_{1}$ (see Fig. 1).

Figure 1: Our setup in the computation of the mutual information. We have two
intervals $A$ and $B$ of size $L_{2}-L_{1}=L$ in each CFT and a local operator
inserted at time $t_{w}$ in the past. The operators are separated by distance
$2\epsilon$ and in the CFT formulas we use $L_{1}=y$ and $L_{2}=y+L$.

Let us also mention a simple fact related to the symmetry of the TFD state.
Namely, in the CFT, we compute the entanglement entropies as well as the
mutual information in a state

$\tilde{\left|\psi\right>}=e^{-iH_{L}t_{\omega}}\mathcal{O}_{L}(x)e^{iH_{L}t_{\omega}}\left|\psi_{\beta}\right>$
(35)

with the TFD state $\left|\psi_{\beta}\right>$. Since $H_{L}-H_{R}$ leaves the
TFD invariant, the above state is equivalent to

$\tilde{\left|\psi\right>}=e^{-i(H_{L}-H_{R})t_{\omega}}\mathcal{O}_{L}(x)\left|\psi_{\beta}\right>\,.$
(36)

Thus, the mutual information computed at $t_{-}=t_{+}=0$ in these two states
has exactly the same functional dependence on $t_{\omega}$. This can be
confirmed from our explicit formulas for $I_{A:B}$ in this section.

Notice that in the previous single sided entanglement entropy calculations, we
used translation invariance to write the time dependence of the operator
insertions in (8) as a function of $t_{-}+t_{\omega}$. When computing two-
sided observables, the same shift will be applied on the CFT time $t_{+}$ in
the opposite boundary. This is consistent with the TFD path integral
construction [3] and it also appears naturally in our holographic dual model
as it can explicitly be seen in the embedding equations (160) and (161)
appearing in appendix B.

###  3.1 Semi-infinite intervals

Before proceeding with finite entangling regions, consider $A$ and $B$ to be
semi-infinite intervals $x\in[0,\infty)$. We want to clarify the difference
between previous results for the second ($n=2$) Renyi mutual information in
this setup [24] and our current mutual information $(n=1)$ discussion. In the
large central charge $c$ limit, and after the insertion of a local operator,
the second Renyi entanglement entropy of the union $S^{(2)}_{A\cup B}$ grows
linearly with time. Equivalently, the change in the second Renyi mutual
information for semi-infinite intervals decreases linearly with time [24]

$\Delta I^{(2)}_{A\cup B}\simeq-\frac{8\pi h_{\psi}}{\beta}t\,.$ (37)

This holds for late times in the regime where $1\ll h_{\psi}\ll c$. Below, we
want to compare this behavior with a large $c$ computation of the mutual
information ($n\to 1$) with twist operators and for heavy local operators
$h_{\psi}\sim\mathcal{O}(c)$ (as in [30]).

To compute the entanglement entropy $S_{A\cup B}$ between two semi-infinite
intervals $A$ and $B$ with starting point $L_{1}=y>0$ on each boundary CFT, we
must calculate

$\text{Tr}\rho^{n}_{A\cup
B}(t)=\frac{\langle\Psi(x_{1},\bar{x}_{1})\sigma_{n}(x_{2},\bar{x}_{2})\tilde{\sigma}_{n}(x_{6},\bar{x}_{6})\Psi^{\dagger}(x_{4},\bar{x}_{4})\rangle}{\left(\langle\psi(x_{1},\bar{x}_{1})\psi^{\dagger}(x_{4},\bar{x}_{4})\rangle_{C_{1}}\right)^{n}}$
(38)

with the insertion points

$\displaystyle x_{1}$ $\displaystyle=-i\epsilon,\quad
x_{2}=y-t_{\omega}-t_{-},\quad x_{6}=y+i\frac{\beta}{2}-t_{+}-t_{\omega},\quad
x_{4}=+i\epsilon$ $\displaystyle\bar{x}_{1}$
$\displaystyle=+i\epsilon,\quad\bar{x}_{2}=y+t_{\omega}+t_{-},\quad\bar{x}_{6}=y-i\frac{\beta}{2}+t_{+}+t_{\omega},\quad\bar{x}_{4}=-i\epsilon\,.$
(39)

Notice the edge of region B (the location $x_{6}$) was shifted by
$i\frac{\beta}{2}$, in accordance with (34) and the dependence on $t_{+}$ is
also through $t_{+}+t_{\omega}$.

We follow the same strategy as before : after mapping the cylinder to a plane
by $w=e^{\frac{2\pi}{\beta}x}$, it is the cross-ratio $z,\bar{z}$ on the plane
that controls the relevant 4-pt function

$\displaystyle z=\frac{w_{12}w_{64}}{w_{16}w_{24}}$ $\displaystyle\simeq
1+\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y-t_{-}-t_{\omega})}{\beta}\cosh\frac{\pi(y-t_{+}-t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$ (40)
$\displaystyle\bar{z}=\frac{\bar{w}_{12}\bar{w}_{64}}{\bar{w}_{16}\bar{w}_{24}}$
$\displaystyle\simeq 1-\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y+t_{-}+t_{\omega})}{\beta}\cosh\frac{\pi(y+t_{+}+t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,.$ (41)

As before, the sign of the imaginary part of $z$ flips for
$t_{\omega}+t_{-}>y$ from positive to negative, whereas that of $\bar{z}$ is
negative for all $t_{\omega}$ and $t_{\pm}$. To the first order in $\epsilon$,
we find $(z,\bar{z})\to(1,1)$ for $t_{\omega}+t_{-}<y$ and
$(z,\bar{z})\to(e^{2\pi i},1)$ for $t_{\omega}+t_{-}>y$. Using the same method
as for the single sided case, we obtain the entanglement entropy $S_{A\cup B}$
for the semi-infinite intervals to be

$S_{A\cup
B}=\left\\{\begin{array}[]{ll}\frac{c}{3}\log\left[\frac{\beta}{\pi\varepsilon_{UV}}\cosh\frac{\pi\Delta
t}{\beta}\right]&(t_{-}+t_{\omega}<y)\\\
\frac{c}{6}\log\left[\left(\frac{\beta}{\pi\varepsilon_{UV}}\right)^{2}\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\cosh\frac{\pi\Delta
t}{\beta}\sinh\frac{\pi(t_{\omega}+t_{-}-y)}{\beta}\cosh\frac{\pi(t_{+}+t_{\omega}-y)}{\beta}\right]&(y<t_{-}+t_{\omega})\end{array}\right.$
(42)

where $\Delta t=t_{-}-t_{+}$. In particular, when $t_{-}=t_{+}=0$ and
$t_{\omega}$ is very large ($y\ll t_{\omega}$), $S_{A\cup B}$ grows linearly
with $t_{\omega}$

$\displaystyle S_{A\cup B}\sim\frac{\pi
c}{3\beta}(t_{\omega}-y)+\frac{c}{6}\log\left[\left(\frac{\beta}{\pi\varepsilon_{UV}}\right)^{2}\frac{\beta}{4\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right].$
(43)

We can find that the first term behaves like thermal entropy which is
proportional to $1/\beta$. Equivalently the mutual information decreases
linearly with $t_{w}$ but now with a coefficient proportional to the central
charge $c$. As explained in [24], this behavior is interpreted as the
destruction of the entanglement between $CFT_{L}$ and $CFT_{R}$ and the broken
"entanglement bond" reconnects between the subsystem $A\cup B$ and its
complement. The unit cost of this reconnection process is proportional to
$\frac{\pi c}{3\beta}$ (which is the entropy density $S_{density}$).

###  3.2 Mutual information for finite intervals

In this section we compute the mutual information between finite regions $A$
and $B$ in opposite boundaries in the TFD state at large central charge. The
setup is depicted on Fig. 1.
The mutual information is defined as

$I_{A:B}=S_{A}+S_{B}-S_{A\cup B}$ (44)

where $S_{A\cup B}$ stands for the entanglement entropy of the union of the
two intervals. Each of the three entropies is computed using the replica trick
in terms of the correlators of the local operators $\Psi$ (9) and twist fields
inserted at the endpoints of the entangling regions. Since we already computed
$S_{A}$, we focus on $S_{B}$ and $S_{A\cup B}$.

####  3.2.1 $S_{B}$

The calculation of the entanglement entropy $S_{B}$ in the second CFTR is
analogous to the one for $S_{A}$. It involves the same type of normalised
correlation function

$\text{Tr}\rho^{n}_{B}(t)=\frac{\langle\Psi(x_{1},\bar{x}_{1})\sigma_{n}(x_{5},\bar{x}_{5})\tilde{\sigma}_{n}(x_{6},\bar{x}_{6})\Psi^{\dagger}(x_{4},\bar{x}_{4})\rangle}{\left(\langle\psi(x_{1},\bar{x}_{1})\psi^{\dagger}(x_{4},\bar{x}_{4})\rangle_{C_{1}}\right)^{n}}$
(45)

but with the insertion points for the twist operators conveniently shifted by
$\pm i\frac{\beta}{2}$ as reviewed above

$\displaystyle x_{1}=-i\epsilon,\quad
x_{5}=y+L+i\frac{\beta}{2}-t_{+}-t_{\omega},\quad
x_{6}=y+i\frac{\beta}{2}-t_{+}-t_{\omega},\quad x_{4}=+i\epsilon$
$\displaystyle\bar{x}_{1}=+i\epsilon,\quad\bar{x}_{5}=y+L-i\frac{\beta}{2}+t_{+}+t_{\omega},\quad\bar{x}_{6}=y-i\frac{\beta}{2}+t_{+}+t_{\omega},\quad\bar{x}_{4}=-i\epsilon\,.$
(46)

As stated earlier, $t_{+}+t_{\omega}$ is the shifted time being used in the
right CFT. To compute the 4-pt correlator, we compose the map (11) with

$z(w)=\frac{(w_{1}-w)(w_{6}-w_{4})}{(w_{1}-w_{6})(w-w_{4})}\,.$ (47)

The corresponding cross-ratios equal

$\displaystyle z=z_{5}$ $\displaystyle\simeq 1-\frac{2\pi
i\epsilon}{\beta}\frac{\sinh\frac{\pi\,L}{\beta}}{\cosh\frac{\pi(y-t_{+}-t_{\omega})}{\beta}\cosh\frac{\pi(y+L-t_{+}-t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$ (48) $\displaystyle\bar{z}=\bar{z}_{5}$
$\displaystyle\simeq 1+\frac{2\pi
i\epsilon}{\beta}\frac{\sinh\frac{\pi\,L}{\beta}}{\cosh\frac{\pi(y+t_{+}+t_{\omega})}{\beta}\cosh\frac{\pi(y+L+t_{+}+t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,.$

Notice that the signs of the imaginary parts are the same for all $t_{+}$ and
$t_{w}$. Thus, in the small $\epsilon$ limit, $(z,\bar{z})\to(1,1)$ for all
$t_{+}$. This reflects the intuition that the local perturbation turned on on
the left CFT has no effect, at lowest order in $\epsilon$, in the quantum
entanglement measured in the right CFT. Using the expansion in these cross-
ratios, we derive

$\boxed{S_{B}=\frac{c}{3}\log\left(\frac{\beta}{\pi\varepsilon_{UV}}\sinh\frac{\pi
L}{\beta}\right)\quad\quad\forall\,t_{+}}$ (49)

Thus, quantum entanglement in the region $B$ remains thermal for all $t_{+}$
at lowest order in $\epsilon$, i.e. $\Delta S_{B}=0$.

####  3.2.2 $S_{A\cup B}$

The most interesting piece in the mutual information is $S_{A\cup B}$.
Following [35], this requires the calculation of the 6-pt function

$\text{Tr}\rho^{n}_{A\cup
B}(t)=\frac{\langle\psi(x_{1},\bar{x}_{1})\sigma_{n}(x_{2},\bar{x}_{2})\tilde{\sigma}_{n}(x_{3},\bar{x}_{3})\sigma_{n}(x_{5},\bar{x}_{5})\tilde{\sigma}_{n}(x_{6},\bar{x}_{6})\psi^{\dagger}(x_{4},\bar{x}_{4})\rangle}{\left(\langle\psi(x_{1},\bar{x}_{1})\psi^{\dagger}(x_{4},\bar{x}_{4})\rangle_{C_{1}}\right)^{n}}$
(50)

where the different insertion points correspond to the different interval
endpoints

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle-i\epsilon,\quad
x_{2}=y-t_{-}-t_{\omega},\quad x_{3}=y+L-t_{-}-t_{\omega},\quad
x_{4}=+i\epsilon$ $\displaystyle\bar{x}_{1}$ $\displaystyle=$
$\displaystyle+i\epsilon,\quad\bar{x}_{2}=y+t_{-}+t_{\omega},\quad\bar{x}_{3}=y+L+t_{-}+t_{\omega},\quad\bar{x}_{4}=-i\epsilon$
$\displaystyle x_{5}$ $\displaystyle=$ $\displaystyle
y+L+i\frac{\beta}{2}-t_{+}-t_{\omega},\quad
x_{6}=y+i\frac{\beta}{2}-t_{+}-t_{\omega},$ $\displaystyle\bar{x}_{5}$
$\displaystyle=$ $\displaystyle
y+L-i\frac{\beta}{2}+t_{+}+t_{\omega},\quad\bar{x}_{6}=y-i\frac{\beta}{2}+t_{+}+t_{\omega}\,.$
(51)

See Appendix A, for further comments on the ordering of the twist operators
appearing in (50). Following the same strategy as before, we compose the two
maps

$w(x)=e^{\frac{2\pi}{\beta}x}\quad\text{and}\quad
z(w)=\frac{(w_{1}-w)w_{34}}{w_{13}(w-w_{4})}\,,$ (52)

and use the transformation properties of primary operators, to write the trace
(50) as

$\text{Tr}\rho_{A\cup
B}^{n}=\left|\frac{\beta}{\pi\varepsilon_{UV}}\sinh\left(\frac{\pi
L}{\beta}\right)\right|^{-8H_{\sigma}}|1-z|^{4H_{\sigma}}\left|z_{56}\right|^{4H_{\sigma}}\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle$
(53)

where the cross-ratios $(z,\bar{z})$ are given in (23), and $z_{i}\equiv
z(w_{i})$.

In the following, we discuss two different CFT channels: S and T-channel,
where we compute this 6-pt function on the plane in the large $c$ limit (see a
detailed discussion in [12]). The corrections to the particular channel choice
are suppressed by $e^{-O(c)}$ factors. We will explicitly see how these
channels match the two different bulk geodesics determining the holographic
entanglement entropy in our holographic discussions. The upshot is that S and
T-channel correspond to the disconnected and connected geodesics for the
holographic calculation of $S_{A\cup B}$, respectively.

##### S-channel :

Let us introduce a resolution of the identity

$\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle=\sum_{\alpha}\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\left|\alpha\right>\left<\alpha\right|\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle$
(54)

where the sum runs over all possible intermediate states.

Consider the first 4-pt function
$\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\left|\alpha\right>$.
As we have seen in (23), the relevant limit corresponding to $\epsilon\to 0$,
is either $(z,\bar{z})\to(1,1)$ or $(z,\bar{z})\to(e^{2\pi i},1)$. Thus, in
either limit, the correlation function can be computed using the OPE of twist
operators [38, 39]

$\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\sim\mathbb{I}+{\cal{O}}\left((z-1)^{r}\right)\quad\quad
r\in\mathbb{Z}^{+}$ (55)

Ignoring the terms proportional to $(z-1)$ and focusing in the dominant
contribution due to the identity operator, we reach the important conclusion
that the summation over the entire set of intermediate steps is restricted to
$\left|\alpha\right>=\left|\psi\right>$ due to the orthogonality of 2-pt
functions in any CFT. We stress that we could have reached the same conclusion
in the limit of small $L_{2}-L_{1}$, but this is not required in our set-up.

Thus, our 6-pt function can then be approximated by

$\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle\simeq\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\left|\psi\right>\,\left<\psi\right|\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle$
(56)

The first 4-pt function equals $G(z,\bar{z})$ in (15), whereas the second 4-pt
function factor will be proportional to the same function but evaluated at a
different cross-ratio. To see this, consider the map taking $z_{1}\to 0$ and
$z_{4}\to\infty$

$\tilde{z}(x)=\frac{(z_{1}-x)(z_{6}-z_{4})}{(z_{1}-z_{6})(x-z_{4})}\,.$ (57)

This allows us to write the desired correlator as

$\left<\psi\right|\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle=\left|1-\tilde{z}_{5}\right|^{4H_{\sigma}}|z_{56}|^{-4H_{\sigma}}\left<\psi\right|\sigma_{n}(\tilde{z}_{5},\bar{\tilde{z}}_{5})\tilde{\sigma}_{n}(1,1)|\psi\rangle\,.$
(58)

Thus, the leading contribution in this channel is

$\displaystyle\text{Tr}\rho_{A\cup
B}^{n}\simeq\left|\frac{\beta}{\pi\varepsilon_{UV}}\sinh\left(\frac{\pi
L}{\beta}\right)\right|^{-8H_{\sigma}}|1-z|^{4H_{\sigma}}\left|1-\tilde{z}_{5}\right|^{4H_{\sigma}}G(z,\bar{z})G(\tilde{z}_{5},\bar{\tilde{z}}_{5})+...$
(59)

where the dots stand for the contributions coming from the subleading terms in
the OPE of the twist operators (55). Interestingly, since the cross-ratio
$\tilde{z}_{5}$ equals $z_{5}$, the cross-ratio determining $S_{B}$, we reach
the conclusion that

$S_{A\cup B}=S_{A}+S_{B},\qquad\text{and}\qquad I_{A:B}=0\,.$ (60)

This channel reproduces the bulk expectation coming from geodesics joining
points in the same boundary, leading to a vanishing mutual information.

##### T-channel :

We could also introduce the resolution of the identity as follows

$\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(1,1)\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})|\psi\rangle=\sum_{\alpha}\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})\left|\alpha\right>\left<\alpha\right|\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(1,1)|\psi\rangle\,.$
(61)

Notice the correlations involve twist operators inserted in different
boundaries. Thus, we expect this channel to reproduce the bulk contribution
from geodesics connecting both boundaries. Remember that in the small
$\epsilon$ limit, we already argued that $z_{5}\to 1$. Thus, we can use the
same OPE argument as above to conclude that the dominant contribution comes
from $\left|\alpha\right>=\left|\psi\right>$. By definition, this gives

$\left<\psi\right|\sigma_{n}(z_{5},\bar{z}_{5})\tilde{\sigma}_{n}(1,1)|\psi\rangle=G(z_{5},\bar{z}_{5})$
(62)

The remaining correlation is again proportional to the same function, but
evaluated at a different cross-ratio. This is proved by considering the map

$\tilde{z}(x)=\frac{(z_{1}-x)(z_{6}-z_{4})}{(z_{1}-z_{6})(x-z_{4})}\,.$ (63)

which allows us to derive

$\langle\psi|\sigma_{n}(z,\bar{z})\tilde{\sigma}_{n}(z_{6},\bar{z}_{6})\left|\psi\right>=|1-\tilde{z}_{2}|^{4H_{\sigma}}|z_{26}|^{-4H_{\sigma}}G(\tilde{z}_{2},\bar{\tilde{z}}_{2})$
(64)

where $z_{2}=z(w_{2})=z$ as in (23) and $\tilde{z}_{2}=\tilde{z}(z_{2})$.
Thus, after some manipulations we have

$\displaystyle\text{Tr}\rho_{A\cup B}^{n}$
$\displaystyle\simeq\left|\frac{\beta}{\pi\varepsilon_{UV}}\sinh\left(\frac{\pi
L}{\beta}\right)\right|^{-8H_{\sigma}}\left|\frac{x}{1-x}\right|^{4H_{\sigma}}|1-z_{5}|^{4H_{\sigma}}|1-\tilde{z}_{2}|^{4H_{\sigma}}G(\tilde{z}_{2},\bar{\tilde{z}}_{2})G(z_{5},\bar{z}_{5})+...$
(65)

where $(x,\bar{x})$ are the cross-ratios computed out of the insertion points
of the four twist operators

$x=\frac{z_{23}z_{56}}{z_{25}z_{36}}=\frac{w_{23}w_{56}}{w_{25}w_{36}}=\frac{2\sinh^{2}\frac{\pi\,L}{\beta}}{\cosh\frac{2\pi\,L}{\beta}+\cosh\frac{2\pi(t_{-}-t_{+})}{\beta}}=\bar{x}\,,$
(66)

what allows us to write the dominant contribution from the T-channel as

$\text{Tr}\rho^{n}_{A\cup
B}\simeq\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|^{-8H_{\sigma}}|1-\tilde{z}_{2}|^{4H_{\sigma}}G(\tilde{z}_{2},\bar{\tilde{z}}_{2})|1-z_{5}|^{4H_{\sigma}}G(z_{5},\bar{z}_{5})\,+...,$
(67)

where $\Delta t=t_{-}-t_{+}$ and the cross-ratios

$\displaystyle z_{5}$ $\displaystyle=1-\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y+L-t_{-}-t_{\omega})}{\beta}\cosh\frac{\pi(y+L-t_{+}-t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$ (68) $\displaystyle\bar{z}_{5}$
$\displaystyle=1+\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y+L+t_{-}+t_{\omega})}{\beta}\cosh\frac{\pi(y+L+t_{+}+t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$ $\displaystyle\tilde{z}_{2}$ $\displaystyle=1+\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y-t_{-}-t_{\omega})}{\beta}\cosh\frac{\pi(y-t_{+}-t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$ $\displaystyle\tilde{\bar{z}}_{2}$
$\displaystyle=1-\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(t_{-}-t_{+})}{\beta}}{\sinh\frac{\pi(y+t_{-}+t_{\omega})}{\beta}\cosh\frac{\pi(y+t_{+}+t_{\omega})}{\beta}}+{\cal
O}(\epsilon^{2})\,,$

Now, using that at large central charge and for two heavy and two light
operators we have the identity [10, 11]

$|1-z|^{4h}G(z,\bar{z})\simeq\left(\frac{z^{\frac{1-\alpha}{2}}(1-z^{\alpha})\bar{z}^{\frac{1-\alpha}{2}}(1-\bar{z}^{\alpha})}{\alpha^{2}(1-z)(1-\bar{z})}\right)^{-2h}$
(69)

as well as (68) we can extract the behaviour of $S_{A\cup B}$ for any time
regime. Let us analyse this carefully below assuming as before that $0<y<y+L$.

It is clear that the monodromies of the correlator are determined depending on
the relation of $t_{-}+t_{\omega}$ with $y$ and $y+L$. From (68) the signs of
the imaginary parts of $\bar{z}_{5}$ and $\tilde{\bar{z}}_{2}$ do not change
with time and we have $\bar{z}_{5}\simeq 1$ and $\tilde{\bar{z}}_{2}\simeq 1$.
On the other hand $\tilde{z}_{2}\simeq e^{2\pi i}$ when $t_{-}+t_{\omega}>y$
and $z_{5}\simeq e^{-2\pi i}$ when $t_{-}+t_{\omega}>y+L$. This gives us three
possible contributions:

$\boxed{\begin{aligned} S_{A\cup
B}&\simeq\frac{2c}{3}\log\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|\quad t_{-}+t_{\omega}<y\\\ S_{A\cup
B}&\simeq\frac{2c}{3}\log\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|\\\
&+\frac{c}{6}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\frac{\sinh\frac{\pi(t_{-}+t_{w}-y)}{\beta}\cosh\frac{\pi(t_{+}+t_{w}-y)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)\quad y<t_{-}+t_{\omega}<y+L\end{aligned}}$ (70)

and for $t_{-}+t_{\omega}>y+L$ we can rewrite our trace as

$\displaystyle\text{Tr}\rho^{n}_{A\cup
B}\simeq\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|^{-4H_{\sigma}}\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\frac{\sinh\frac{\pi(t_{-}+t_{w}-y)}{\beta}\cosh\frac{\pi(t_{+}+t_{w}-y)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)^{-2H_{\sigma}}$
$\displaystyle\times\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|^{-4H_{\sigma}}\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\frac{\sinh\frac{\pi(t_{-}+t_{w}-y-L)}{\beta}\cosh\frac{\pi(t_{+}+t_{w}-y-L)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)^{-2H_{\sigma}}\,.$ (71)

The entanglement entropy $S_{A\cup B}$ in this time regime can then be written
as

$\boxed{\begin{aligned} S_{A\cup
B}&\simeq\frac{c}{6}\log\left(\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-y)}{\beta}\cosh\frac{\pi(t_{+}+t_{\omega}-y)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-y-L)}{\beta}\cosh\frac{\pi(t_{+}+t_{\omega}-y-L)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)\\\
&+\frac{2c}{3}\log\left|\frac{\beta}{\pi\varepsilon_{UV}}\cosh\left(\frac{\pi\Delta
t}{\beta}\right)\right|+\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)\quad\quad
t_{-}+t_{\omega}>y+L\end{aligned}}$ (72)

Notice that (71) resembles the contributions from two different pieces. In the
holographic part, these will be interpreted as the contributions from two bulk
geodesics connecting points in opposite boundaries.

###  3.3 The evolution of the Mutual information

The evolution of the mutual information after turning on the local excitation
can now be computed in the large central charge limit. At early times
$t_{-}+t_{\omega}<y$, the single sided entropies are thermal $S_{A}\simeq
S_{B}=S_{\text{thermal}}$. Thus, the mutual information equals

$I^{0}_{A:B}\equiv\frac{2c}{3}\log\left(\frac{\sinh\frac{\pi\,L}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)\,.$ (73)

This is clearly finite and depends on the Hamiltonian driving the evolution.
If we use the bulk isometry $H_{L}-H_{R}$, then $t_{-}=t_{+}$ and $\Delta
t=0$, giving rise to a time independent mutual information, as it should.
Notice that positivity of the mutual information in this case requires $\pi
L/\beta\gtrsim 1$. Whereas for the $H_{L}+H_{R}$ Hamiltonian, we recover the
mutual information time decrease discussed in [36].

These results can be understood using causality considerations : for
$t_{-}+t_{\omega}<y$, the perturbation did not enter into region $A$ and could
not possibly disturb the original thermal entanglement. Once the excitation
reaches region $A$ $(y<t_{-}+t_{\omega}<y+L)$, using (24), (49) and (70), the
mutual information evolves as

$I_{A:B}\simeq
I^{0}_{A:B}+\frac{c}{6}\log\left[\frac{\sinh\frac{\pi(y+L-t_{-}-t_{\omega})}{\beta}\cosh\frac{\pi\Delta
t}{\beta}}{\cosh\frac{\pi(t_{+}+t_{w}-y)}{\beta}\sinh\frac{\pi\,L}{\beta}}\right]$
(74)

Note how the dependence on the conformal dimension $h_{\psi}$ of the
perturbation cancels, between the contributions in $S_{A}$ and $S_{A\cup B}$,
in this regime. It would be interesting to understand the mechanism behind
this large $c$ behaviour of the mutual information.

In the last region $t_{-}+t_{\omega}>y+L>y$ the mutual information equals

$\boxed{\begin{aligned} I_{A:B}&\simeq
I^{0}_{A:B}-\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)\\\
&-\frac{c}{6}\log\left(\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-y)}{\beta}\cosh\frac{\pi(t_{+}+t_{\omega}-y)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-y-L)}{\beta}\cosh\frac{\pi(t_{+}+t_{\omega}-y-L)}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right)\end{aligned}}$ (75)

It is important to stress that when extracting the answer for the mutual
information for various times $t_{\mp}$ as well as $t_{\omega}$, one has to
maximise the mutual information between the $S$ and the $T$ channel answers so
that it is always non-negative.

##  4 Scrambling time

Shenker and Stanford [6] defined the scrambling time $t^{\star}_{\omega}$ as
the time scale at which the perturbation has destroyed all the preexistent
correlations. In our notation, their condition reduces to setting
$t_{-}=t_{+}=0$ and to study the vanishing of the mutual information

$I_{A:B}(t^{\star}_{\omega})=0\,.$ (76)

Evaluating our previous results (73), (74) and (75) for $t_{-}=t_{+}=0$, we
obtain

$\displaystyle I_{A:B}$ $\displaystyle\simeq$
$\displaystyle\frac{2c}{3}\log\sinh\frac{\pi\,L}{\beta}\,,\quad t_{\omega}<y$
(77) $\displaystyle I_{A:B}$ $\displaystyle\simeq$
$\displaystyle\frac{c}{6}\log\left[\frac{\left(\sinh\frac{\pi\,L}{\beta}\right)^{3}\sinh\frac{\pi(y+L-t_{\omega})}{\beta}}{\cosh\frac{\pi(t_{\omega}-y)}{\beta}}\right]\,,\quad
y<t_{\omega}<y+L$ (78) $\displaystyle I_{A:B}$ $\displaystyle\simeq$
$\displaystyle\frac{2c}{3}\log\sinh\frac{\pi\,L}{\beta}-\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)$
(79)
$\displaystyle-\frac{c}{6}\log\left(\frac{\sinh\frac{2\pi(t_{\omega}-y)}{\beta}\,\sinh\frac{2\pi(t_{\omega}-y-L)}{\beta}}{4}\right)\,,\quad
t_{\omega}>y+L$

Notice the mutual information is a monotonically decreasing function of
$t_{\omega}$. Thus, starting with a positive mutual information, i.e. $\pi
L/\beta\gtrsim 1$, there is a single root $t^{\star}_{\omega}$ where (76)
holds. After that, by switching channels, the mutual information remains zero.

The first question to answer is whether $t^{\star}_{\omega}\in(y,\,y+L)$ or
whether $t^{\star}_{\omega}>y+L$. Clearly, the second condition can only hold
if the mutual information is positive at the transition. This requirement
gives rise to the constraint

$t^{\star}_{\omega}>y+L\,\,\Rightarrow\,\,I_{A:B}(y+L+\epsilon)>0\quad\Rightarrow\quad\frac{\beta}{\pi\epsilon}\left(\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)^{2}<\frac{\sinh^{3}\frac{\pi
L}{\beta}}{\cosh\frac{\pi L}{\beta}}\,,$ (80)

where we already used $\epsilon\ll\beta$, as in our previous CFT
analysis111111If $\epsilon\ll\beta$ breaks down, then condition (80) is
modified.. Because of working in this region of parameter space
$(\epsilon\ll\beta)$, we conclude that only small perturbations
$(\alpha_{\psi}\to 1)$ allow scrambling time scales $t^{\star}_{\omega}>y+L$.
Since the function $\sin(\pi\alpha_{\psi})/\alpha_{\psi}$ is monotonically
decreasing in $\alpha_{\psi}$ (or increasing in $h_{\psi}$), the smaller the
perturbation is, the easier it is to fulfil condition (80) for generic values
of $L/\beta$. Since this is the regime considered in [6], we will study the
scrambling time under these circumstances121212One can equally study the
conditions under which $t^{\star}_{\omega}\in(y,\,y+L)$. These generically
require heavier perturbations. We do not understand this regime, which appears
precisely when the mutual information (74) is $\alpha_{\psi}$ independent..

For $t_{\omega}>y+L$ and $\Delta t=0$, the mutual information (75) becomes

$I_{A:B}(t_{\omega})\simeq\frac{c}{6}\log\frac{\sinh^{4}\frac{\pi
L}{\beta}\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)^{-2}}{\cosh\left(\frac{\pi(t_{\omega}-y)}{\beta}\right)\sinh\frac{\pi(t_{\omega}-y)}{\beta}\sinh\frac{\pi(t_{\omega}-y-L)}{\beta}\cosh\frac{\pi(t_{\omega}-y-L)}{\beta}}$
(81)

This vanishes when

$\left(\frac{2\pi\alpha_{\psi}\epsilon\sinh^{2}\left(\frac{\pi
L}{\beta}\right)}{\beta\sin(\pi\alpha_{\psi})}\right)^{2}=\sinh^{2}\frac{2\pi(t_{\omega}^{\star}-y)}{\beta}\cosh\frac{2\pi
L}{\beta}\left(1-\frac{\tanh\frac{2\pi
L}{\beta}}{\tanh\frac{2\pi(t_{\omega}^{\star}-y)}{\beta}}\right)\,.$ (82)

Notice that $t_{\omega}^{\star}-y>L$ guarantees the positivity of the left
hand side. Condition (82) gives rise to a quadratic equation in
$\sinh^{2}\frac{2\pi(t_{\omega}^{\star}-y)}{\beta}$. It can be shown that
there is a unique consistent root, in agreement with our previous arguments.
In the limit $t^{\star}_{\omega}/\beta\gg 1$, this root reduces the scrambling
time $t_{\omega}^{\star}$ to

$\boxed{t^{\star}_{\omega}=y+\frac{L}{2}-\frac{\beta}{2\pi}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\right)+\frac{\beta}{\pi}\log\left(2\sinh\frac{\pi
L}{\beta}\right)}$ (83)

Due to the non-compactness of the 2d CFT, no recurrences were expected to be
seen in our calculation. Working in the small $h_{\psi}/c$ limit, as required
by our analysis, then

$\frac{\beta}{\pi\epsilon}\frac{\sin\pi\alpha_{\psi}}{\alpha_{\psi}}\sim\frac{E_{\psi}}{S_{\text{density}}},\qquad\text{where}\qquad
S_{\text{density}}=\frac{\pi c}{3\beta},$ (84)

and $E_{\psi}=\frac{\pi h_{\psi}}{\epsilon}$ is the total energy of our local
excitation given by integrating the energy density as in [19]. In this limit,
the scrambling time reduces to

$t^{\star}_{\omega}=y+\frac{L}{2}+\frac{\beta}{2\pi}\log\left(\frac{\pi
S_{density}}{4E_{\psi}}\right)+\frac{\beta}{\pi}\log\left(2\sinh\frac{\pi
L}{\beta}\right)\,.$ (85)

The $\log S$ dependence is indeed consistent with the original scrambling
conjecture [5, 6].

##  5 Holographic description

In this section we compute the mutual information using the AdS3 gravity dual
of the 2d CFT set-up discussed in the previous section. The starting thermal
state $\rho_{\beta}$ in the quantum theory is described in the semiclassical
approximation of the correspondence by a black hole in the gravity side, the
BTZ black hole [9]. Observables involving a single Hilbert space are described
by the metric outside of the event horizon, whereas observables depending on
both Hilbert spaces require the Kruskal-like extension of the BTZ black hole,
as described in [3].

To compute the time evolution of the holographic entanglement entropy, we
approximate the local CFT perturbation with conformal dimension
$\Delta(=2h_{\psi})$ by a bulk free falling massive point particle with mass
$m=\Delta/R$ [18]. Next, we compute its back-reaction on the BTZ background in
Kruskal coordinates using the coordinate transformation in [40]. Finally, we
compute the entanglement entropy and the mutual information using the
holographic prescription [28, 29].

In three dimensions, the back-reacted metric of a point particle at $r=0$ in
global coordinates is known

$ds^{2}=-\left(r^{2}+R^{2}-\mu\right)d\tau^{2}+\frac{R^{2}dr^{2}}{r^{2}+R^{2}-\mu}+r^{2}d\varphi^{2}\,,$
(86)

where the mass of the point particle is related to $\mu$ by
$\mu=8G_{N}R^{2}m=\frac{24h_{\psi}}{c}R^{2}$ and $R$ is the radius of AdS3.
Depending on the mass of the particle, the background describes a conical
singularity or a BTZ black hole.

Now, the holographic entanglement entropy is given by
$\frac{R}{4G_{N}}=\frac{c}{6}$ times the length $L_{\gamma}$ of geodesic
$\gamma$ which connects the boundaries of the subsystem $A$ for which we
define the entanglement entropy $S_{A}$. In the above metric, the entanglement
entropy of the boundary region A with endpoints
$(r^{(1)}_{\infty},\tau^{(1)}_{\infty},\varphi^{(1)}_{\infty})$ and
$(r^{(2)}_{\infty},\tau^{(2)}_{\infty},\varphi^{(2)}_{\infty})$ is [18]

$S_{A}=\frac{c}{6}\log\left[\frac{2r^{(1)}_{\infty}\cdot
r^{(2)}_{\infty}}{R^{2}}\frac{\cos\left(|\Delta\tau_{\infty}|a\right)-\cos\left(|\Delta\varphi_{\infty}|a\right)}{a^{2}}\right],$
(87)

where $a\equiv\sqrt{1-\frac{\mu}{R^{2}}}=\alpha_{\psi}$ carries the
information on the perturbation, as in the CFT discussion,
$\Delta\tau_{\infty}=\tau^{(2)}_{\infty}-\tau^{(1)}_{\infty}$ and
$\Delta\varphi_{\infty}=\varphi^{(2)}_{\infty}-\varphi^{(1)}_{\infty}$
satisfies $0<|\Delta\varphi_{\infty}|<\pi$.

Mapping the static $r=0$ geodesic to one starting at some distance $\epsilon$
from the boundary and falling into the horizon afterwards can approximate the
local perturbation turned on in the boundary theory. This is precisely the
approach followed in [24] to describe the time dependent evolution of
entanglement entropy in the bulk for locally perturbed thermal states. To
describe the evolution across the horizon, which is required to study two
sided correlation functions, one must use Kruskal coordinates. This is one of
the tasks we will undertake in this section.

###  5.1 Free falling particle in Kruskal coordinates

The geodesic of a free falling particle in the AdS-Schwarzschild patch of the
BTZ black hole

$ds^{2}=\frac{R^{2}}{z^{2}}\left[-\left(1-Mz^{2}\right)dt^{2}_{-}+\frac{dz^{2}}{1-Mz^{2}}+d\theta^{2}\right]\,,\quad\quad\theta\sim\theta+2\pi$
(88)

was already computed in [24] to be

$t_{-}=\tilde{\tau},\quad\theta=0,\quad
1-Mz^{2}=(1-M\epsilon^{2})\cosh^{-2}\left(\sqrt{M}(\tilde{\tau}+t_{\omega})\right)\,.$
(89)

The only addition in the expression above is the shift
$\tilde{\tau}\to\tilde{\tau}+t_{\omega}$ to account for the initial boundary
condition $z(-t_{\omega})=\epsilon$ guaranteeing the particle’s energy

$E=\frac{m\,R}{\epsilon}\,\sqrt{1-M\epsilon^{2}}\,,$ (90)

matches the energy of the CFT perturbation in the small $\epsilon$
limit131313In the absence of matter fields and working in the large $c$ limit,
as we do, the satisfaction of this condition guarantees that our holographic
model should capture the bulk description of the identity conformal block,
which in 2d includes the stress tensor.. We extend this result to the entire
eternal black hole by working in Kruskal coordinates.

One way to achieve this goal is to map the global AdS3 description (86) to
Kruskal coordinates. A second one is to solve the geodesic equation directly
in Kruskal coordinates. We check below that, as expected, both approaches
agree.

##### Free falling particle in Kruskal coordinates :

The Kruskal extension of the BTZ metric (88) is given by

$ds^{2}=R^{2}\frac{-4dudv+(-1+uv)^{2}d\phi^{2}}{(1+uv)^{2}}=R^{2}\frac{-4dT^{2}+4dX^{2}+\left(1-T^{2}+X^{2}\right)^{2}d\phi^{2}}{\left(1+T^{2}-X^{2}\right)^{2}}\,,$
(91)

where $u=T-X\in\mathbb{R}$, $v=T+X\in\mathbb{R}$ with their range satisfying
$-1<uv<1$ and $\phi\sim\phi+4\pi^{2}/\beta$. The conformal boundary, horizons
and singularities are at $uv=-1$, $uv=0$ and $uv=1$, respectively, with the
left and right Kruskal regions defined by

$\displaystyle\text{Left:}\quad R_{-}$ $\displaystyle=\\{0\leq u,-1\leq uv\leq
0\\}$ (92) $\displaystyle\text{Right:}\quad R_{+}$ $\displaystyle=\\{u\leq
0,-1\leq uv\leq 0\\}$

Both coordinate systems are related to the AdS-Schwarzschild patches via

$\displaystyle u$
$\displaystyle=\pm\sqrt{\frac{z_{H}-z}{z_{H}+z}}e^{t_{\mp}/z_{H}}\qquad\qquad\qquad\quad
v=\mp\sqrt{\frac{z_{H}-z}{z_{H}+z}}e^{-t_{\mp}/z_{H}}$ (93) $\displaystyle T$
$\displaystyle=\pm\sqrt{\frac{1-\sqrt{M}z}{1+\sqrt{M}z}}\sinh\left(\sqrt{M}t_{\mp}\right)\qquad
X=\mp\sqrt{\frac{1-\sqrt{M}z}{1+\sqrt{M}z}}\cosh\left(\sqrt{M}t_{\mp}\right)$

Using these maps (93), we can rewrite the geodesic (89) in the parametric form

$\displaystyle X(\tilde{\tau})$
$\displaystyle=-\frac{\sqrt{1-M\epsilon^{2}}\frac{\cosh\left(\sqrt{M}\tilde{\tau}\right)}{\cosh\left(\sqrt{M}(\tilde{\tau}+t_{\omega})\right)}}{1+\sqrt{1-(1-M\epsilon^{2})\cosh^{-2}\left(\sqrt{M}(\tilde{\tau}+t_{\omega})\right)}}\,,$
(94) $\displaystyle T(\tilde{\tau})$
$\displaystyle=\frac{\sqrt{1-M\epsilon^{2}}\frac{\sinh\left(\sqrt{M}\tilde{\tau}\right)}{\cosh\left(\sqrt{M}(\tilde{\tau}+t_{\omega})\right)}}{1+\sqrt{1-(1-M\epsilon^{2})\cosh^{-2}\left(\sqrt{M}(\tilde{\tau}+t_{\omega})\right)}}\,,$

The initial condition $(t_{-},z)=(-t_{\omega},\epsilon)$ is mapped to

$\displaystyle(T_{0},X_{0})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1-\sqrt{M}\epsilon}{1+\sqrt{M}\epsilon}}\left(-\sinh\left(\sqrt{M}t_{\omega}\right),-\cosh\left(\sqrt{M}t_{\omega}\right)\right)\,.$
(95)

This allows us to determine $t_{\omega}=t_{\omega}(T_{0},X_{0})$. Similarly,
$\tilde{\tau}=\tilde{\tau}(T,X)$ can be determined from (93). Altogether, we
can solve for $T(X)$ as

$\displaystyle T(X)$ $\displaystyle=$
$\displaystyle-\frac{\sinh\left(\sqrt{M}t_{\omega}\right)}{\sqrt{1-M\epsilon^{2}}}\pm\sqrt{\left(X+\frac{\cosh\left(\sqrt{M}t_{\omega}\right)}{\sqrt{1-M\epsilon^{2}}}\right)^{2}-\frac{M\epsilon^{2}}{1-M\epsilon^{2}}}$
(96)

Proceeding in a similar way, we can obtain the geodesic $v=v(u)$ that reduces
to

$v(u)=-\frac{a_{1}u-1}{u+a_{2}},$ (97)

with

$a_{1}=\frac{1-u_{0}v_{0}}{2u_{0}}=\frac{e^{\sqrt{M}t_{\omega}}}{\sqrt{1-M\epsilon^{2}}},\quad
a_{2}=\frac{1-u_{0}v_{0}}{2v_{0}}=-\frac{e^{-\sqrt{M}t_{\omega}}}{\sqrt{1-M\epsilon^{2}}}\,.$
(98)

##### Checking equations of motion :

Consider the relativistic action for a particle of mass $m$ moving in the
background metric (91) at constant $\phi$. Working in the gauge where the
parameter along the curve equals $u$, this effective action reduces to

$S=-2mR\int\frac{\sqrt{v^{\prime}}du}{1+uv}\,.$ (99)

Its equation of motion

$v^{\prime\prime}(uv+1)-2v^{\prime}(uv^{\prime}-v)=0\,,$ (100)

has general solution

$v(u)=\frac{C_{2}+\left(C_{1}+C^{2}_{2}\right)u}{1+C_{2}u},\qquad
v^{\prime}(u)=\frac{C_{1}}{(1+C_{2}u)^{2}}\,.$ (101)

This agrees with the geodesic solution (97) if the integration constants are
matched as

$C_{1}=-\frac{v_{0}(u_{0}v_{0}+1)^{2}}{u_{0}(u_{0}v_{0}-1)^{2}}=M\epsilon^{2}e^{2\sqrt{M}t_{\omega}},\qquad
C_{2}=\frac{2v_{0}}{1-u_{0}v_{0}}=-\sqrt{1-M\epsilon^{2}}e^{\sqrt{M}t_{\omega}}\,.$
(102)

In $T,X$ coordinates these extended geodesics have two branches which meet at
the point

$X_{m}=\frac{\sqrt{M}\epsilon-\cosh\left(\sqrt{M}t_{\omega}\right)}{\sqrt{1-M\epsilon^{2}}},\qquad
T_{m}=-\frac{\sinh\left(\sqrt{M}t_{\omega}\right)}{\sqrt{1-M\epsilon^{2}}}\,.$
(103)

They cross the future and past horizons at

$X_{h^{\pm}}=-\frac{1}{2}\sqrt{1-M\epsilon^{2}}e^{\mp\sqrt{M}t_{\omega}}$
(104)

and hit the past and future singularities at

$X_{s^{\pm}}=\pm\sinh\left(\sqrt{M}t_{\omega}\right)$ (105)

All these features can be seen in the sample geodesic plotted on the $T-X$
Kruskal diagram (Fig.2).

Figure 2: Plot shows our time-like geodesic on Kruskal diagram. The red part
is given by (94) and the full geodesic (blue) by (96). Plot for $M=10$,
$\epsilon=0.01$ and $t_{\omega}=0.25$

###  5.2 Back-reacted metric

The back reaction of the free falling particle in Kruskal coordinates can be
obtained by following [40] and rewriting the metric (86) in Kruskal
coordinates, but taking into the account the specific initial conditions
discussed above. To solve this problem when $t_{\omega}=0$, in [24], we
considered a boost in the plane $X_{1}-X_{3}$ with rapidity
$\lambda_{2}=\lambda_{2}(M,\epsilon)$ . In order to introduce the further
parameter $t_{\omega}$, we will consider a preliminary boost in $X_{0}-X_{3}$,
since this corresponds to the natural boost action on the light-like
coordinates $u-v$ which captures the blue-shift of energy near the horizon
stressed in [6]. In practice we then apply two particular boosts into the
embedding coordinates of $AdS_{3}$ such that the identification between global
and Kruskal coordinates becomes

$\displaystyle\sqrt{R^{2}+r^{2}}\sin\tau$
$\displaystyle=\cosh\lambda_{1}\,X_{0}+\sinh\lambda_{1}\,X_{3}$ (106)
$\displaystyle=R\frac{e^{\lambda_{1}}u+e^{-\lambda_{1}}v}{1+uv}$
$\displaystyle\sqrt{R^{2}+r^{2}}\cos\tau$
$\displaystyle=\cosh\lambda_{2}\,X_{1}-\sinh\lambda_{2}\,\left(\sinh\lambda_{1}\,X_{0}+\cosh\lambda_{1}\,X_{3}\right)$
$\displaystyle=\frac{R\cosh\lambda_{2}(1-uv)}{1+uv}\left(\cosh\phi-\tanh\lambda_{2}\frac{e^{\lambda_{1}}u-e^{-\lambda_{1}}v}{1-uv}\right)$
$\displaystyle r\sin\varphi$ $\displaystyle=X_{2}=R\frac{1-uv}{1+uv}\sinh\phi$
$\displaystyle r\cos\varphi$
$\displaystyle=-\sinh\lambda_{2}\,X_{1}+\cosh\lambda_{2}\left(\sinh\lambda_{1}\,X_{0}+\cosh\lambda_{1}\,X_{3}\right)$
$\displaystyle=\frac{R\cosh\lambda_{2}(1-uv)}{1+uv}\left(\frac{e^{\lambda_{1}}u-e^{-\lambda_{1}}v}{1-uv}-\tanh\lambda_{2}\cosh\phi\right)$

Solving for $r=r(u,v,\phi)$

$r=\left|\frac{R(1-uv)\cosh\lambda_{2}}{1+uv}\right|\sqrt{\frac{\sinh^{2}\phi}{\cosh^{2}\lambda_{2}}+\left(\frac{e^{\lambda_{1}}u-e^{-\lambda_{1}}v}{1-uv}-\tanh\lambda_{2}\cosh\phi\right)^{2}}$
(107)

we can determine $\lambda_{1}$ and $\lambda_{2}$ requiring that the location
of the static particle in global AdS3, $r=0$, gets mapped into the free
falling geodesic $v(u)$ in (97). This fixes both boost parameters to be

$\lambda_{1}=\sqrt{M}t_{\omega}\,,\quad\quad\tanh\lambda_{2}=\sqrt{1-M\epsilon^{2}}\,.$
(108)

These boost parameters determine the explicit map between global AdS3 and a
free falling particle in Kruskal coordinates :

$\displaystyle
r=\frac{R}{\sqrt{M}\epsilon}\left|\frac{1-uv}{1+uv}\right|\sqrt{M\epsilon^{2}\sinh^{2}\phi+\left(\frac{e^{\sqrt{M}t_{\omega}}\,u-e^{-\sqrt{M}t_{\omega}}\,v}{1-uv}-\sqrt{1-M\epsilon^{2}}\cosh\phi\right)^{2}}\,$
(109)

and also

$\displaystyle\tan\tau$ $\displaystyle=$
$\displaystyle\sqrt{M}\epsilon\frac{\frac{e^{\sqrt{M}t_{\omega}}u+e^{-\sqrt{M}t_{\omega}}v}{1-uv}}{\cosh\phi-\sqrt{1-M\epsilon^{2}}\,\frac{e^{\sqrt{M}t_{\omega}}\,u-e^{-\sqrt{M}t_{\omega}}\,v}{1-uv}}\,,$
(110) $\displaystyle\tan\varphi$ $\displaystyle=$
$\displaystyle\sqrt{M}\epsilon\frac{\sinh\phi}{\frac{e^{\sqrt{M}t_{\omega}}u-e^{-\sqrt{M}t_{\omega}}v}{1-uv}-\sqrt{1-M\epsilon^{2}}\cosh\phi}\,.$
(111)

Using this map, we can compute the exact back-reacted metric corresponding to
a free falling particle in the eternal black hole satisfying the initial
condition $(u_{0},\,v_{0})$.

Our analysis is valid for any value of $t_{\omega}$. This allows us to compare
with some approaches in the literature where the back reaction of the local
perturbation in the CFT was approximated by a shock-wave, i.e. a BTZ spacetime
in the presence of some non-trivial stress tensor localised at the horizon.
Since our approach in 3d was based on computing the explicit backreaction of
some point particle moving in some geodesic into the BTZ geometry, we can
study the limit $t_{\omega}/\beta\gg 1$ in our geodesic analysis in subsection
5.1. In particular, figure 2 illustrates how our particle geodesic approaches
a null geodesic on the horizon for such large $t_{\omega}$. Thus, our back-
reacted metric in this particular limit should indeed correspond to a shock-
wave propagating in the BTZ background as originally described in [6]. The
advantage of the shock-wave description is that it also applies in higher
dimensions, whereas our finite $t_{\omega}$ results show the agreement between
CFT and bulk computations also hold beyond this regime.

##  6 Bulk mutual information

The mutual information $I_{A:B}$ between regions $A$ and $B$ in the left and
right boundaries, respectively

$I_{A:B}=S_{A}+S_{B}-S_{A\cup B}\,,$ (112)

can now be computed by applying (87) to the three different bulk geodesics
providing the relevant minimal surface computing entanglement entropy in the
bulk. All we need to know are the locations of the endpoints in the limits of
small $\epsilon$ and $z_{\infty}$ that we will insert into (87).
To keep the gravity formulas compact, the endpoints of the intervals will be
denoted by $L_{i}$, $i\in\\{1,2\\}$, where $0<L_{1}<L_{2}$. To compare with
the CFT formulas one can substitute $L_{1}=y$ and $L_{2}=y+L$.

###  6.1 Geodesic in the left boundary

The two endpoints of the entanglement region $A$ in the left boundary are
$(t_{-},z_{1},\theta_{1})=(t_{-},z_{\infty},L_{1})$ and
$(t_{-},z_{2},\theta_{1})=(t_{-},z_{\infty},L_{2})$. It is convenient to
compute their image in global AdS3 using the asymptotic maps to the right and
left regions (see Appendix B). Proceeding this way, their radial coordinates
in global AdS3 satisfy

$r^{(1)}r^{(2)}\simeq\left(\frac{R}{M\epsilon
z_{\infty}}\right)^{2}D_{1}D_{2}$ (113)

where

$D_{i}=|\cosh\sqrt{M}L_{i}-\cosh\sqrt{M}(t_{-}+t_{\omega})|\quad i=1,2$ (114)

whereas the other coordinates are

$\displaystyle\tan\tau^{(i)}$ $\displaystyle\simeq$
$\displaystyle\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}(t_{-}+t_{\omega})\right)}{\cosh\left(\sqrt{M}L_{i}\right)-\cosh\left(\sqrt{M}(t_{-}+t_{\omega})\right)}$
(115) $\displaystyle\tan\varphi^{(i)}$ $\displaystyle\simeq$
$\displaystyle\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}L_{i}\right)}{\cosh\left(\sqrt{M}(t_{-}+t_{\omega})\right)-\cosh\left(\sqrt{M}L_{i}\right)}$
(116)

with $i=1,2$.

The length of the geodesics depends on the value of the time argument
$t_{-}+t_{w}$. When $t_{-}+t_{w}<L_{1}<L_{2}$, then the boundary points equal

$\tau^{(i)}\simeq\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}(t_{-}+t_{\omega})\right)}{D_{i}},\qquad\varphi^{(i)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}L_{i}\right)}{D_{i}}\,.$
(117)

These determine the coordinate intervals to be

$\displaystyle|\Delta\tau|$
$\displaystyle\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}|D_{2}-D_{1}|\sinh\sqrt{M}(t_{-}+t_{\omega})\,,$
(118) $\displaystyle|\Delta\varphi$
$\displaystyle|\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left|D_{1}\sinh\sqrt{M}L_{2}-D_{2}\sinh\sqrt{M}L_{1}\right|\,.$

Due to the identity

$D_{1}D_{2}(|\Delta\varphi|^{2}-|\Delta\tau|^{2})=4M\epsilon^{2}\sinh^{2}\frac{\pi\Delta
L}{\beta}\,,$ (119)

the geodesic length is

$\displaystyle L_{\gamma}$
$\displaystyle\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\cos\left(a|\Delta\tau|\right)-\cos\left(a|\Delta\varphi|\right)}{a^{2}}\right]\simeq\log\left[\frac{r^{(1)}r^{(2)}}{R^{2}}(|\Delta\varphi|^{2}-|\Delta\tau|^{2})\right]$
(120) $\displaystyle\simeq 2\log\left(\frac{\beta}{\pi
z_{\infty}}\sinh\frac{\pi L}{\beta}\right)\,.$

This reproduces the thermal entanglement entropy (22) computed in the CFT in
the same time interval once both UV cut-offs are identified
$\varepsilon_{UV}=z_{\infty}$.
Similarly, when $t_{-}+t_{w}>L_{2}>L_{1}$, the boundary points equal

$\tau^{(i)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}(t_{-}+t_{\omega})\right)}{D_{i}},\qquad\varphi^{i}\simeq\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}L_{i}\right)}{D_{i}}\,.$
(121)

These are different from (117), but give rise to the same intervals (118).
Thus, the length of the bulk geodesic joining them equals (120). This matches
our CFT again.
Finally, when $L_{2}>t_{-}+t_{\omega}>L_{1}$, the boundary points are

$\displaystyle\tau^{(1)}$
$\displaystyle\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{-}+t_{\omega})}{D_{1}}\,,\quad\varphi^{(1)}\simeq\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{1}}{D_{1}}\,,$
(122) $\displaystyle\tau^{(2)}$
$\displaystyle\simeq\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{-}+t_{\omega})}{D_{2}}\,,\quad\varphi^{(2)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{2}}{D_{2}}\,.$

From them we can easily get the absolute values of the intervals

$\displaystyle|\Delta\tau|$
$\displaystyle\simeq\pi-\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}(D_{1}+D_{2})\sinh\sqrt{M}(t_{-}+t_{\omega})\,,$
(123) $\displaystyle|\Delta\varphi|$
$\displaystyle\simeq\pi-\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left(D_{1}\sinh\sqrt{M}L_{2}+D_{2}\sinh\sqrt{M}L_{1}\right)\,.$

Notice that in the small $\epsilon$ limit we are working on, $|\Delta\tau|$
and $|\Delta\varphi|$ are close to each other

$\delta=|\Delta\varphi|-|\Delta\tau|=\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left[(D_{1}+D_{2})\sinh\sqrt{M}(t_{-}+t_{\omega})-D_{1}\sinh\sqrt{M}L_{2}-D_{2}\sinh\sqrt{M}L_{1}\right]\,.$
(124)

This allows us to write the length of the bulk geodesic between these two
boundary points as

$\displaystyle L_{\gamma}$
$\displaystyle\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\cos\left(a|\Delta\tau|\right)-\cos\left(a|\Delta\varphi|\right)}{a^{2}}\right]\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\sin\pi
a}{a}\delta\right]$ (125) $\displaystyle\simeq\log\left[\left(\frac{\beta}{\pi
z_{\infty}}\sinh\frac{\pi\Delta
L}{\beta}\right)^{2}\frac{\beta}{\pi\epsilon}\frac{\sin\pi
a}{a}\frac{\sinh\frac{\pi(t_{\omega}+t_{-}-L_{1})}{\beta}\sinh\frac{\pi(L_{2}-t_{\omega}-t_{-})}{\beta}}{\sinh\frac{\pi\Delta
L}{\beta}}\,.\right]$

where $\Delta L=L_{2}-L_{1}$. This also perfectly matches our CFT result (24)
after employing the Ryu-Takayanagi formula.

###  6.2 Geodesic in the right boundary

The two endpoints of the entanglement region $B$ in the right boundary are
$(t_{+},z_{1},\theta_{1})=(t_{+},z_{\infty},L_{1})$ and
$(t_{+},z_{2},\theta_{1})=(t_{+},z_{\infty},L_{2})$. Their radial coordinates
satisfy

$r^{(1)}r^{(2)}\simeq\left(\frac{R}{M\epsilon
z_{\infty}}\right)^{2}D_{1}D_{2}$ (126)

where

$D_{i}=|\cosh\sqrt{M}L_{i}+\cosh\sqrt{M}(t_{+}+t_{\omega})|\quad i=1,2$ (127)

whereas the other coordinates are

$\tan\tau^{(i)}\simeq-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}(t_{+}+t_{\omega})\right)}{D_{i}}\,,\quad\tan\varphi^{(i)}\simeq-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}L_{i}\right)}{D_{i}}$
(128)

In this case, no matter what the value of $t_{+}$ is, the boundary points are
identified as

$\tau^{(i)}\simeq-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}(t_{+}+t_{\omega})\right)}{D_{i}}\,,\quad\varphi^{(i)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\left(\sqrt{M}L_{i}\right)}{D_{i}}\,.$
(129)

These give rise to the intervals

$\displaystyle|\Delta\tau|$
$\displaystyle\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}|D_{1}-D_{2}|\sinh\sqrt{M}(t_{+}+t_{\omega})\,,$
(130) $\displaystyle|\Delta\varphi|$
$\displaystyle\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left|D_{1}\sinh\sqrt{M}L_{2}-D_{2}\sinh\sqrt{M}L_{1}\right|\,.$

Using the identity

$D_{1}D_{2}(|\Delta\varphi|^{2}-|\Delta\tau|^{2})=4M\epsilon^{2}\sinh^{2}\frac{\pi\Delta
L}{\beta}\,,$ (131)

the geodesic length equals

$\displaystyle L_{\gamma}$
$\displaystyle\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\cos\left(a|\Delta\tau|\right)-\cos\left(a|\Delta\varphi|\right)}{a^{2}}\right]\simeq\log\left[\frac{r^{(1)}r^{(2)}}{R^{2}}(|\Delta\varphi|^{2}-|\Delta\tau|^{2})\right]$
(132) $\displaystyle\simeq 2\log\left(\frac{\beta}{\pi
z_{\infty}}\sinh\frac{\pi\Delta L}{\beta}\right)\,.$

This reproduces the well-known thermal answer obtained in the CFT [32]

$S_{B}\simeq\frac{c}{3}\log\left(\frac{\beta}{\pi
z_{\infty}}\sinh\frac{\pi\Delta L}{\beta}\right)=S_{\text{thermal}},$ (133)

which also agrees with the CFT expression for $S_{B}$ in (49).

###  6.3 Geodesics across the horizon and Mutual Information

We want to compute the geodesic length between two opposite boundary points
located at the same space like location but with different time labels
$t_{\mp}$. We will describe the calculation once and apply it to the two cases
of interest afterwards. The product of the radial coordinates equals

$r^{(1)}r^{(2)}\simeq\left(\frac{R}{M\epsilon
z_{\infty}}\right)^{2}D_{1}D_{2}$ (134)

where

$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle|\cosh\sqrt{M}L_{i}-\cosh\sqrt{M}(t_{-}+t_{\omega})|$ (135)
$\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle|\cosh\sqrt{M}L_{i}+\cosh\sqrt{M}(t_{+}+t_{\omega})|$ (136)

where $L_{i}$ labels the space like location in both boundaries, i.e. either
$L_{1}$ or $L_{2}$. The other coordinates for the left boundary point are

$\displaystyle\tan\tau^{(1)}$ $\displaystyle\simeq$
$\displaystyle\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{-}+t_{\omega})}{\cosh\sqrt{M}L_{i}-\cosh\sqrt{M}(t_{-}+t_{\omega})}$
(137) $\displaystyle\tan\varphi^{(1)}$ $\displaystyle\simeq$
$\displaystyle\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{i}}{\cosh\sqrt{M}(t_{-}+t_{\omega})-\cosh\sqrt{M}L_{i}}\,.$
(138)

At early times, $L_{i}>t_{\omega}$, these are given by

$\displaystyle\tau^{(1)}\simeq\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{-}+t_{\omega})}{\cosh\sqrt{M}L_{i}-\cosh\sqrt{M}(t_{-}+t_{\omega})}$
(139)
$\displaystyle\varphi^{(1)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{i}}{\cosh\sqrt{M}L_{i}-\cosh\sqrt{M}(t_{-}+t_{\omega})}$
(140)

whereas at late times,

$\displaystyle\tau^{(1)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{-}+t_{\omega})}{\cosh\sqrt{M}(t_{-}+t_{\omega})-\cosh\sqrt{M}L_{1}}$
(141)
$\displaystyle\varphi^{(1)}\simeq\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{1}}{\cosh\sqrt{M}(t_{-}+t_{\omega})-\cosh\sqrt{M}L_{1}}$
(142)

The remaining coordinates for the right boundary point are

$\displaystyle\tan\tau^{(2)}$ $\displaystyle\simeq$
$\displaystyle-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{+}+t_{\omega})}{\cosh\sqrt{M}L_{i}+\cosh\sqrt{M}(t_{+}+t_{\omega})}$
(143) $\displaystyle\tan\varphi^{(2)}$ $\displaystyle\simeq$
$\displaystyle-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{i}}{\cosh\sqrt{M}(t_{+}+t_{\omega})+\cosh\sqrt{M}L_{i}}$
(144)

In this case, they are always given by

$\displaystyle\tau^{(2)}\simeq-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{+}+t_{\omega})}{\cosh\sqrt{M}L_{i}+\cosh\sqrt{M}(t_{+}+t_{\omega})}=-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}(t_{+}+t_{\omega})}{D_{2}}$
(145)
$\displaystyle\varphi^{(2)}\simeq\pi-\sqrt{M}\epsilon\frac{\sinh\sqrt{M}L_{i}}{D_{2}}$
(146)

Let us compute the length of the geodesic in the early time regime
$L_{i}>t_{\omega}$. In this case, the interval differences are

$\displaystyle|\Delta\tau|$
$\displaystyle=|\tau^{(1)}-\tau^{(2)}|\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left|D_{2}\sinh\sqrt{M}(t_{-}+t_{\omega})+D_{1}\sinh\sqrt{M}\,(t_{+}+t_{\omega})\right|\,,$
(147) $\displaystyle|\Delta\varphi|$
$\displaystyle=|\varphi^{(1)}-\varphi^{(2)}|\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left|D_{2}\sinh\sqrt{M}L_{i}-D_{1}\sinh\sqrt{M}L_{i}\right|$

Plugging this into the geodesic length (87), we obtain

$\displaystyle L_{\gamma}$ $\displaystyle\simeq$
$\displaystyle\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\cos\left(a|\Delta\tau|\right)-\cos\left(a|\Delta\varphi|\right)}{a^{2}}\right]\simeq\log\left[\frac{r^{(1)}r^{(2)}}{R^{2}}(|\Delta\varphi|^{2}-|\Delta\tau|^{2})\right]$
(148) $\displaystyle\simeq$ $\displaystyle 2\log\left[\frac{\beta}{\pi
z_{\infty}}\cosh\frac{\pi\Delta t}{\beta}\right]$

In the late time regime, the interval differences equal

$\displaystyle|\Delta\tau|\simeq\pi-\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left(D_{2}\sinh\sqrt{M}(t_{-}+t_{\omega})-D_{1}\sinh\sqrt{M}(t_{+}+t_{\omega})\right)$
(149)
$\displaystyle|\Delta\varphi|\simeq\pi-\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left(D_{1}\sinh\sqrt{M}L_{i}+D_{2}\sinh\sqrt{M}L_{i}\right)\,.$
(150)

Since they are very close, we have

$|\Delta\tau|\simeq|\Delta\tau|-\delta$ (151)

where

$\delta\simeq\frac{\sqrt{M}\epsilon}{D_{1}D_{2}}\left[D_{2}(\sinh\sqrt{M}(t_{-}+t_{\omega})-\sinh\sqrt{M}L_{i})-D_{1}(\sinh\sqrt{M}(t_{+}+t_{\omega})+\sinh\sqrt{M}L_{i})\right]$
(152)

This allows to write the geodesic length as

$\displaystyle L_{\gamma}$
$\displaystyle\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\cos\left(a|\Delta\tau|\right)-\cos\left(a|\Delta\varphi|\right)}{a^{2}}\right]\simeq\log\left[\frac{2r^{(1)}r^{(2)}}{R^{2}}\frac{\sin\pi
a}{a}\delta\right]$ (153)
$\displaystyle\simeq\log\left[\frac{\beta^{2}\frac{1}{2}\left(1+\cosh\frac{2\pi\Delta
t}{\beta}\right)}{\pi^{2}z^{2}_{\infty}}\frac{2}{\sqrt{M}\epsilon}\frac{\sin\pi
a}{a}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-L_{i})}{\beta}\cosh\frac{\pi(L_{i}-t_{+}-t_{\omega})}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right]$

where $\Delta t=t_{-}-t_{+}$. These geodesics can now be used to compute the
entanglement entropy of the union $S_{A\cup B}$.
In particular we will be interested in large $t_{\omega}>L_{2}>L_{1}$ when
$S_{A}=S_{B}=S_{\text{thermal}}$. In this case, there is a competition between
the two geodesics connecting points in opposite boundaries and the geodesics
connecting points in the same boundary giving rise to $2S_{B}$. The length of
the new geodesics is

$\displaystyle L^{1}_{\gamma}$ $\displaystyle\simeq$
$\displaystyle\log\left[\left(\frac{\beta\cosh\frac{\pi\Delta t}{\beta}}{\pi
z_{\infty}}\right)^{2}\frac{\beta}{\pi\epsilon}\frac{\sin\pi
a}{a}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-L_{1})}{\beta}\cosh\frac{\pi(L_{1}-t_{+}-t_{\omega})}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right]$ (154) $\displaystyle L^{2}_{\gamma}$ $\displaystyle\simeq$
$\displaystyle\log\left[\left(\frac{\beta\cosh\frac{\pi\Delta t}{\beta}}{\pi
z_{\infty}}\right)^{2}\frac{\beta}{\pi\epsilon}\frac{\sin\pi
a}{a}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-L_{2})}{\beta}\cosh\frac{\pi(L_{2}-t_{+}-t_{\omega})}{\beta}}{\cosh\frac{\pi\Delta
t}{\beta}}\right]$ (155)

where again $\Delta t=t_{-}-t_{+}$.
Summarizing, the holographic entanglement entropy of the union of two
intervals on the left and the right boundary is given by

$S_{A\cup B}\simeq\frac{c}{6}\left(L^{1}_{\gamma}+L^{2}_{\gamma}\right),$
(156)

which matches with the CFT result (72).

Finally from the above holographic results, we obtain the holographic mutual
information $I_{A:B}=S_{A}+S_{B}-S_{A\cup B}$ and this again reproduces the
CFT result (75) perfectly. As a consequence the scrambling time derived in the
CFT (83) also holds as a result in gravity replacing $\alpha_{\psi}\to a$, as
stressed below (87).

### Acknowledgements

We would like to thank Joan Camps, Tomoki Nosaka and K. Skenderis for
discussions. PC is supported by the Swedish Research Council (VR) grant
2013-4329. TT is supported by JSPS Grant-in-Aid for Scientific Research (B)
No.25287058 and by JSPS Grant-in-Aid for Challenging Exploratory Research
No.24654057. TT is also supported by World Premier International Research
Center Initiative (WPI Initiative) from the Japan Ministry of Education,
Culture, Sports, Science and Technology (MEXT). The work of JS and AS is
supported by the Science and Technology Facilities Council (STFC) [grant
number ST/L000458/1]. PC would like to thank the Galileo Galilei Institute for
Theoretical Physics for the hospitality and the INFN for partial support
during the completion of this work.

##  Appendix A Twist operators in the TFD state

In this appendix we explain the proper ordering prescription of twist operator
insertions when computing the two sided entanglement entropy $S_{A\cup B}$ (or
mutual information $I_{A:B}$) in the thermofield double state.

When computing $S_{A\cup B}$, with regions $A$ and $B$ belonging to opposite
boundaries, the replica trick instructs us to insert twist operators
$\sigma_{n},\tilde{\sigma}_{n}$ on each boundary. The order of these
insertions is important, because different orderings can give rise to
different replica geometries. There are two kinds of insertion orders. One,
where both boundaries have the same order, such as

$(\sigma_{n},\tilde{\sigma}_{n})_{L},\ \
(\sigma_{n},\tilde{\sigma}_{n})_{R}\,.$ (157)

In this case, going around the replica n-sheeted cylinder by passing through
the cuts between the twist operators on each boundary, one returns to the
starting point after going around $n$ cylinders (see the right picture in
Fig.3). There exists a second insertion order in which both boundaries have
opposite twist operator orders, such as

$(\sigma_{n},\tilde{\sigma}_{n})_{L},\ \
(\tilde{\sigma}_{n},\sigma_{n})_{R}\,.$ (158)

The same operation as above returns to the same point after going once around
a standard cylinder (see the left picture in Fig.3).

Figure 3: In the left picture (the second choice (158)), one returns to the
starting point after going around two standard cylinders. In the right picture
(the first choice (157)), this operation involves going around $n$ standard
cylinders.

The question is what the right order prescription is when we consider the TFD
state. To answer this question, we provide two CFTs and one holographic bulk
arguments.

It is convenient to remember the TFD state can be constructed by Euclidean
time evolution from the pure state $\sum_{n}|n\rangle_{L}|n\rangle_{R}$ with
Euclidean time $\pm\beta/4$ for each boundary. To compute the density matrix
$\rho_{A\cup B}$, one glues partially by imposing boundary conditions to each
sheet (half cylinder) . Finally, to construct $\mathrm{Tr}\rho_{A\cup B}^{n}$,
one glues the cylinders through their cuts on the cylinders, giving rise to a
partition function on the n-sheeted cylinder. If one goes around this
n-sheeted cylinder through the cuts, one returns to the starting point after
going once around a standard cylinder because one should obtain the original
state in the limit $\beta\to 0$. Thus, the second insertion order (158) for
twist operators is selected in this way.

A further argument to confirm this choice is as follows. The n-sheeted replica
geometry is constructed by gluing single-sheets together along the cuts. If
one enters the cuts from the negative region, one emerges in the upper sheet,
whereas if one enters from positive region, one ends in the lower sheet. From
this gluing condition, one obtains the n-sheeted replica geometry
corresponding to the second type of insertion order of twist operators (see
Fig. 4).

Figure 4: In the replica geometry which we obtain by gluing n cylinders along
the cuts, we can go around 2 standard cylinders, for example, $i$-th and
$(i+1)$-th cylinders (dotted line).

Our final argument is holographic. In the bulk, one is instructed to consider
geodesics connecting the different edges of the subsystems. Twist operators
are inserted in these edges. For geodesics connecting edges on the same
boundary, the inserted twist operators must have the same monodromy (or
charge) properties, whereas for geodesics connecting edges in opposite
boundaries, twist operators must have opposite monodromy (or charge). For
example, if $\sigma_{n}$ is on an edge, $\tilde{\sigma}_{n}$ must appear at
the other edge. This consistency condition chooses the same twist operator
insertion order as in the CFT side.

##  Appendix B Details of the holographic model

Here we collect several useful formulas and conventions that we used in the
part with holographic computations.

The relation between Kruskal and AdS-Schwarzschild coordinates can be obtained
by referring both descriptions to the $\mathbb{R}^{2,2}$ where AdS3 becomes
the quadratic surface

$-X_{0}^{2}-X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=-R^{2}$ (159)

in which we have

$\displaystyle\pm\frac{R\sqrt{1-Mz^{2}}}{\sqrt{M}z}\sinh\left(\sqrt{M}t_{\mp}\right)$
$\displaystyle=X_{0}=R\frac{u+v}{1+uv}=R\frac{2T}{1+T^{2}-X^{2}}\,,$ (160)
$\displaystyle\frac{R}{\sqrt{M}z}\cosh\left(\sqrt{M}\theta\right)$
$\displaystyle=X_{1}=R\frac{1-uv}{1+uv}\cosh\phi=R\frac{1-T^{2}+X^{2}}{1+T^{2}-X^{2}}\,\cosh\phi\,,$
$\displaystyle\frac{R}{\sqrt{M}z}\sinh\left(\sqrt{M}\theta\right)$
$\displaystyle=X_{2}=R\frac{1-uv}{1+uv}\sinh\phi=R\frac{1-T^{2}+X^{2}}{1+T^{2}-X^{2}}\,\sinh\phi\,,$
$\displaystyle\pm\frac{R\sqrt{1-Mz^{2}}}{\sqrt{M}z}\cosh\left(\sqrt{M}t_{\mp}\right)$
$\displaystyle=X_{3}=R\frac{u-v}{1+uv}=-R\frac{2X}{1+T^{2}-X^{2}}\,.$

This formulas fix our conventions for the appropriate signs on the gravity
side.

In order to derive lengths of the geodesic in the back-reacted metric we only
need to know the asymptotic form of the map from the left and the right wedge
to $AdS_{3}$ in global coordinates. This is just the map that takes the
trajectory of our massive point particle in Kruskal coordinates to the $r=0$
particle in global $AdS_{3}$. In the left and right exteriors the map becomes

$\displaystyle\sqrt{R^{2}+r^{2}}\sin\tau$
$\displaystyle=\pm\frac{R\sqrt{1-Mz^{2}_{\infty}}}{\sqrt{M}z_{\infty}}\sinh\left(\sqrt{M}(t_{\mp}+t_{\omega})\right)\,,$
(161) $\displaystyle\sqrt{R^{2}+r^{2}}\cos\tau$
$\displaystyle=\frac{R}{M\epsilon
z_{\infty}}\left(\cosh\left(\sqrt{M}\theta\right)\mp\sqrt{1-M\epsilon^{2}}\sqrt{1-Mz^{2}_{\infty}}\cosh\left(\sqrt{M}(t_{\mp}+t_{\omega})\right)\right)\,,$
$\displaystyle r\sin\varphi$
$\displaystyle=\frac{R}{\sqrt{M}z_{\infty}}\sinh\left(\sqrt{M}\theta\right)\,,$
$\displaystyle r\cos\varphi$ $\displaystyle=\frac{R}{M\epsilon
z_{\infty}}\left(\pm\sqrt{1-Mz^{2}_{\infty}}\cosh\left(\sqrt{M}(t_{\mp}+t_{\omega})\right)-\sqrt{1-M\epsilon^{2}}\cosh\left(\sqrt{M}\theta\right)\right)\,.$

The points in the main text are extracted to the first order in $\epsilon$.
Note that the map from the right wedge also depends on $\epsilon$ as well as
$t_{\omega}$.

##  Appendix C Two sided 2-pt functions

Given the bound (1) that the mutual information provides on the amount of
correlation, it is natural to study the two sided two-point correlation
function in the TFD after a local perturbation is turned on in one of the
boundaries. This is the calculation described in this appendix.

Given a local primary probe operator $O_{h}(x,\bar{x})$ of conformal dimension
$h$ and a perturbation described by a different primary $O_{h_{w}}$, the
appropriate normalised 4-point two sided correlation is

$C_{4}=\frac{\langle
O_{h_{w}}(x_{1},\bar{x}_{1})O_{h}(x_{2},\bar{x}_{2})O_{h}(x_{3},\bar{x}_{3})O_{h_{w}}(x_{4},\bar{x}_{4})\rangle}{\langle
O_{h_{w}}(x_{1},\bar{x}_{1})O_{h_{w}}(x_{4},\bar{x}_{4})\rangle}$ (162)

where the insertion points are

$\displaystyle x_{1}=-i\epsilon\qquad\qquad
x_{2}=L_{1}-t_{-}-t_{\omega},\qquad
x_{3}=L_{2}-t_{+}-t_{\omega}+i\frac{\beta}{2}\qquad x_{4}=i\epsilon$
$\displaystyle\bar{x}_{1}=i\epsilon\qquad\bar{x}_{2}=L_{1}+t_{-}+t_{\omega},\qquad\bar{x}_{3}=L_{2}+t_{+}+t_{\omega}-i\frac{\beta}{2}\qquad\bar{x}_{4}=-i\epsilon$
(163)

Assuming $h/c\ll 1$ and keeping $h_{w}/c$ fixed, the numerator is again a
4-point function involving two heavy and two light operators. Using the large
central charge results of [10] we can write the correlator as

$C_{4}=\left|\frac{\beta}{\pi z_{\infty}}\sinh\frac{\pi
x_{23}}{\beta}\right|^{-4h}\left|1-z\right|^{4h}G(z,\bar{z})$ (164)

with

$G(z,\bar{z})\simeq\left(\frac{z^{\frac{1-\alpha}{2}}(1-z^{\alpha})\bar{z}^{\frac{1-\alpha}{2}}(1-\bar{z}^{\alpha})}{\alpha^{2}}\right)^{-2h},\qquad\alpha=\sqrt{1-\frac{24h_{\omega}}{c}}\,.$
(165)

In the limit of small $\epsilon/\beta$, the cross-ratios reduce to

$\displaystyle z\simeq 1+\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(\Delta L+\Delta
t)}{\beta}}{\sinh\frac{\pi(L_{1}-t_{-}-t_{\omega})}{\beta}\cosh\frac{\pi(L_{2}-t_{+}-t_{\omega})}{\beta}}$
(166) $\displaystyle\bar{z}\simeq 1-\frac{2\pi
i\epsilon}{\beta}\frac{\cosh\frac{\pi(\Delta L+\Delta
t)}{\beta}}{\sinh\frac{\pi(L_{1}+t_{-}+t_{\omega})}{\beta}\cosh\frac{\pi(L_{2}+t_{+}+t_{\omega})}{\beta}}$
(167)

where $\Delta L=L_{2}-L_{1}$ and $\Delta t=t_{-}-t_{+}$. As in our main text
discussions, the key observation is that the imaginary part of $z$ depends on
the sign of $L_{1}-t_{-}-t_{\omega}$. For $L_{1}>t_{-}+t_{\omega}$, we have
$(z,\bar{z})\sim(1,1)$ and

$C_{4}\simeq\left(\frac{\beta}{\pi
z_{\infty}}\sqrt{\frac{1}{2}\left(\cosh\frac{2\pi\Delta
L}{\beta}+\cosh\frac{2\pi\Delta t}{\beta}\right)}\right)^{-4h}$ (168)

whereas for $L_{1}<t_{-}+t_{\omega}$, $(z,\bar{z})\sim(e^{2\pi i},1)$ and

$C_{4}\simeq\left(\frac{\beta}{\pi
z_{\infty}}\sqrt{\frac{1}{2}\left(\cosh\frac{2\pi\Delta
L}{\beta}+\cosh\frac{2\pi\Delta
t}{\beta}\right)}\right)^{-4h}\left(\frac{\beta}{\pi\epsilon}\frac{\sin(\pi\alpha)}{\alpha}\frac{\sinh\frac{\pi(t_{-}+t_{\omega}-L_{1})}{\beta}\cosh\frac{\pi(L_{2}-t_{+}-t_{\omega})}{\beta}}{\cosh\frac{\pi(\Delta
L+\Delta t)}{\beta}}\right)^{-2h}$ (169)

It can be checked this result precisely matches the gravity computation where
the two-point function is given by the length of a geodesic (153) between two-
boundaries in our back-reacted metric.

One can read off the scrambling time scale from the two-point correlators. For
example, setting $L_{1}=L_{2}=0$ and $t_{-}=t_{+}=0$, the correlation
$\eqref{2ptL}$ for large $t_{\omega}$ is given by

$C_{4}\simeq\left(\frac{\beta}{2\pi
z_{\infty}}\right)^{-4h}\exp\left[-\frac{4\pi
h}{\beta}\left(t_{w}+\frac{\beta}{2\pi}\log\left(\frac{\beta}{\pi\epsilon}\frac{\sin(\pi\alpha)}{\alpha}\right)\right)\right]\,.$
(170)

This reproduces the dependence on the perturbation for the scrambling time
derived in the mutual information analysis.

## References

  * [1] M. Wolf, F. Verstraete, M. Hastings, and J.I. Cirac. "Area laws in quantum systems: mutual information and correlations," Physical Review Letters 100, no. 7 (2008): 070502.
  * [2] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131].
  * [3] J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 0304 (2003) 021 [hep-th/0106112].
  * [4] P. Hayden and J. Preskill, “Black holes as mirrors: Quantum information in random subsystems,” JHEP 0709 (2007) 120 [arXiv:0708.4025 [hep-th]].
  * [5] Y. Sekino and L. Susskind, “Fast Scramblers,” JHEP 0810 (2008) 065 [arXiv:0808.2096 [hep-th]].
  * [6] S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP 1403 (2014) 067 [arXiv:1306.0622 [hep-th]].
  * [7] S. H. Shenker and D. Stanford, “Multiple Shocks,” JHEP 1412 (2014) 046 [arXiv:1312.3296 [hep-th]].
  * [8] S. H. Shenker and D. Stanford, “Stringy effects in scrambling,” arXiv:1412.6087 [hep-th].
  * [9] M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099].
  * [10] A. L. Fitzpatrick, J. Kaplan and M. T. Walters, “Universality of Long-Distance AdS Physics from the CFT Bootstrap,” JHEP 1408 (2014) 145 [arXiv:1403.6829 [hep-th]].
  * [11] A. L. Fitzpatrick, J. Kaplan and M. T. Walters, “Virasoro Conformal Blocks and Thermality from Classical Background Fields,” arXiv:1501.05315 [hep-th].
  * [12] T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955 [hep-th].
  * [13] E. Hijano, P. Kraus and R. Snively, “Worldline approach to semi-classical conformal blocks,” arXiv:1501.02260 [hep-th].
  * [14] P. Calabrese and J. L. Cardy, “Entanglement and correlation functions following a local quench: a conformal field theory approach,” J. Stat. Mech. 0710 P10004, arXiv:0708.3750.
  * [15] M. Nozaki, T. Numasawa and T. Takayanagi, “Quantum Entanglement of Local Operators in Conformal Field Theories,” Phys. Rev. Lett. 112 (2014) 111602 [arXiv:1401.0539 [hep-th]].
  * [16] S. He, T. Numasawa, T. Takayanagi and K. Watanabe, “Quantum dimension as entanglement entropy in two dimensional conformal field theories,” Phys. Rev. D 90 (2014) 4, 041701 [arXiv:1403.0702 [hep-th]].
  * [17] M. Nozaki, “Notes on Quantum Entanglement of Local Operators,” JHEP 1410 (2014) 147 [arXiv:1405.5875 [hep-th]].
  * [18] M. Nozaki, T. Numasawa and T. Takayanagi, “Holographic Local Quenches and Entanglement Density,” JHEP 1305 (2013) 080 [arXiv:1302.5703 [hep-th]].
  * [19] P. Caputa, M. Nozaki and T. Takayanagi, “Entanglement of local operators in large-N conformal field theories,” PTEP 2014, no. 9, 093B06 (2014) [arXiv:1405.5946 [hep-th]].
  * [20] C. T. Asplund, A. Bernamonti, F. Galli and T. Hartman, “Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches,” arXiv:1410.1392 [hep-th].
  * [21] S. Giusto and R. Russo, “Entanglement Entropy and D1-D5 geometries,” Phys. Rev. D 90 (2014) 6, 066004 [arXiv:1405.6185 [hep-th]].
  * [22] J. de Boer, A. Castro, E. Hijano, J. I. Jottar and P. Kraus, “Higher Spin Entanglement and WN Conformal Blocks,” arXiv:1412.7520 [hep-th].
  * [23] W. Z. Guo and S. He, “Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs,” arXiv:1501.00757 [hep-th].
  * [24] P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, “Quantum Entanglement of Localized Excited States at Finite Temperature,” JHEP 1501 (2015) 102 [arXiv:1410.2287 [hep-th]].
  * [25] T. Ugajin, “Two dimensional quantum quenches and holography,” arXiv:1311.2562 [hep-th].
  * [26] C. T. Asplund and A. Bernamonti, “Mutual information after a local quench in conformal field theory,” Phys. Rev. D 89 (2014) 6, 066015 [arXiv:1311.4173 [hep-th]].
  * [27] D. A. Roberts, D. Stanford and L. Susskind, “Localized shocks,” arXiv:1409.8180 [hep-th].
  * [28] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602
  * [29] V. E. Hubeny, M. Rangamani and T. Takayanagi, “A Covariant holographic entanglement entropy proposal,” JHEP 0707, 062 (2007) [arXiv:0705.0016 [hep-th]].
  * [30] D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect,” arXiv:1412.5123 [hep-th].
  * [31] J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” arXiv:1503.01409 [hep-th].
  * [32] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech. 0406, P06002 (2004) [hep-th/0405152].
  * [33] J. L. Cardy, O. A. Castro-Alvaredo and B. Doyon, “Form factors of branch-point twist fields in quantum integrable models and entanglement entropy,” J. Statist. Phys. 130, 129 (2008) [arXiv:0706.3384 [hep-th]].
  * [34] P. Calabrese and J. L. Cardy, “Evolution of Entanglement Entropy in One-Dimensional Systems,” J. Stat. Mech. 04 (2005) P04010, cond-mat/0503393.
  * [35] I. A. Morrison and M. M. Roberts, “Mutual information between thermo-field doubles and disconnected holographic boundaries,” arXiv:1211.2887 [hep-th].
  * [36] T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors,” JHEP 1305 (2013) 014 [arXiv:1303.1080 [hep-th]].
  * [37] P. Caputa, G. Mandal and R. Sinha, “Dynamical entanglement entropy with angular momentum and U(1) charge,” JHEP 1311, 052 (2013) [arXiv:1306.4974 [hep-th]].
  * [38] P. Calabrese, J. Cardy and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal field theory II,” J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482 [hep-th]].
  * [39] E. Perlmutter, “Comments on Renyi entropy in AdS3/CFT2,” JHEP 1405 (2014) 052 [arXiv:1312.5740 [hep-th]].
  * [40] G. T. Horowitz and N. Itzhaki, “Black holes, shock waves, and causality in the AdS / CFT correspondence,” JHEP 9902, 010 (1999) [hep-th/9901012].