aainstitutetext: Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Oxford OX1 3PU, U.K.bbinstitutetext: Mathematical Institute,
University of Oxford, Woodstock Road, Oxford, OX2 6GG, U.K.

# New horizons for inhomogeneous quenches and Floquet CFT

Hanzhi Jiang b and Márk Mezei<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A fruitful avenue in investigating out-of-equilibrium quantum many-body
systems is to abruptly change their Hamiltonian and study the subsequent
evolution of their quantum state. If this is done once, the setup is called a
quench, while if it is done periodically, it is called Floquet driving. We
consider the solvable setup of a two-dimensional CFT driven by Hamiltonians
built out of conformal symmetry generators: in this case, the quantum dynamics
can be understood using two-dimensional geometry. We investigate how the
dynamics is reflected in the holographic dual three-dimensional spacetime and
find new horizons. We argue that bulk operators behind the new horizons are
reconstructable by virtue of modular flow.

##  1 Introduction

Understanding the out-of-equilibrium dynamics in interacting many-body systems
is challenging: there are no conventional small parameters to expand in and
numerical simulations are hampered by the exponentially large Hilbert space
that the system explores. There are a handful of special setups, where
analytical progress can be made: in random, dual unitary, and stabiliser
quantum circuits Fisher:2022qey ; in models of Sachdev-Ye-Kitaev type
Rosenhaus:2018dtp ; Chowdhury:2021qpy ; in systems with a holographic gravity
dual Son:2002sd ; Balasubramanian:2011ur ; Shenker:2013pqa ; and in conformal
field theories (CFTs) driven by unconventional Hamiltonians. This paper
examines the interplay between the latter two, both of which describe quantum
dynamics in terms of geometry: in terms of conformal transformations of the
complex plane and in terms of gravity in three-dimensional asymptotically AdS
spacetimes.

1+1 dimensional CFTs with inhomogeneous deformations have received
considerable attention recently Wen:2018vux ; Wen:2018agb ; Fan:2019upv ;
Wen:2020wee ; Han:2020kwp ; Fan:2020orx ; Wen:2022pyj , as they provide
exactly solvable models that describe quantum critical systems out of
equilibrium. These deformations include the so-called sine square deformation
(SSD) Wen:2018vux , which is a special case of $SL(2)$ deformations of 2d
CFTs. In SSD, one changes the CFT Hamiltonian
$H_{0}=\int_{0}^{l}dx\,T_{00}(x)$ on a circle of circumference $l$ to a new
Hamiltonian with a sine-square enveloping function

$\displaystyle H_{1}=2\int_{0}^{l}dx\ \sin^{2}\Big{(}\frac{\pi
x}{l}\Big{)}T_{00}(x)$ (1.1)

SSD was first introduced in quantum many-body systems to eliminate boundary
effects Gendiar:2008udd ; Hikihara:2011mtb ; gendiar2011suppression . Later
on, 2d CFT with SSD Hamiltonian and its generalization, the Mobius deformed
Hamiltonian was studied in Ishibashi:2015jba ; Ishibashi:2016bey ;
Okunishi:2016zat ; Wen:2016inm , where it was shown Ishibashi:2015jba ;
Ishibashi:2016bey  that the spectrum of SSD CFT is continuous. In Wen:2018vux
explicit expressions for entanglement entropy (EE) after a quantum quench were
found. It was also pointed out in Wen:2018vux  that SSD CFT can be interpreted
as a CFT on curved spacetime.

Floquet CFT Wen:2018agb ; Fan:2019upv ; Wen:2020wee ; Han:2020kwp ;
Fan:2020orx ; Wen:2022pyj  is constructed by periodically driving the CFT with
ordinary and SSD Hamiltonians, $H_{0}$ and $H_{1}$, see Figure 7. The physical
motivation is to provide an exactly solvable toy model that captures some of
the features of the more realisable periodically driven systems including
Floquet topological insulators oka2009photovoltaic ; kitagawa2010topological ;
lindner2011floquet ; Rechtsman:2012qj ; cayssol2013floquet ;
rudner2013anomalous ; titum2015disorder ; titum2016anomalous ;
klinovaja2016topological ; Thakurathi:2016ypz , Floquet symmetry-protected
phases Iadecola:2015tra ; von2016phase ; else2016classification ;
potter2016classification ; po2017radical , as well as Floquet time crystals
else2016floquet ; khemani2016phase ; von2016absolute ; else2017prethermal ;
zhang2017observation ; yao2017discrete . Floquet CFT describes driven quantum
many-body systems at criticality. Floquet CFT was found to have heating and
non-heating phases Wen:2018agb ; Fan:2019upv , corresponding to different
types of $SL(2)$ transformations. In the heating phase, the energy is
localised in peaks that also dominate entanglement entropy. If initially
prepared in a thermal state, the system outside these peaks cools down and
behaves like the vacuum state, leading to the name ”Floquet’s refrigerators”
Wen:2022pyj . In the non-heating phases, the energy and entanglement show
oscillatory behavior without a fixed period.

Given these recent advances in inhomogeneously deformed CFT2, it is then a
natural question to ask what the holographic duals of these deformed CFT2
are.111We consider CFTs with large central charges $c$ and large gaps, i.e.
holographic CFTs. The AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc
; Witten:1998qj  serves as a powerful tool to study the time evolution of
strongly coupled conformal field theories in $d$ spacetime dimensions. For
example, the computation of entanglement entropy in the dual
$(d+1)$-dimensional Lorentzian spacetime boils down to the areas of
codimension-2 extremal surfaces Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt .
Moreover, as we will see shortly, in the holographic dual of SSD CFT, there
are new horizons emerging in the bulk (see Figure 3 and 4), and bulk
excitations falling across these new horizons remain reconstructable. Another
motivation for exploring the holographic dual of SSD and Floquet CFT comes
from holographic complexity. In particular, these holographic constructions
may serve as testing ground for various complexity measures and their possible
relations, including the Bogoliubov-Kubo-Mori information geometry studied in
deBoer:2023lrd  in the Floquet CFT context.

The aim of this paper is to study the gravity duals of SSD and Floquet CFT.
There are already holographic studies of SSD and Floquet CFTs in the
literature. In MacCormack:2018rwq , a $(2+1)$-dimensional bulk metric with SSD
boundary is constructed. Holographic dynamics of SSD quench with thermal
initial states was investigated in Goto:2021sqx , and the time evolution of
mutual information under this framework was further explored in Goto:2023wai .
There are also recent works Bernamonti:2024fgx ; Kudler-Flam:2023ahk
exploring holographic aspects of SSD or Mobius quench in boundary conformal
field theory, whose gravity dual involves end-of-the-world branes. We will
construct the gravity dual spacetime for Floquet CFT at general times, whereas
prior work only considered stroboscopic, i.e. discrete times deBoer:2023lrd ;
Das:2022pez .

###  1.1 Outline and summary

In Section 2 we analyse a special quantum quench from $H_{0}$ to the SSD
Hamiltonian $H_{1}$. In Subsection 2.3 we point out that SSD Hamiltonian leads
to a different foliation of the 2-dimensional spacetime as is shown in Figure
1. Furthermore, there is a triangle-shaped Killing horizon (2.11) with zero
temperature.

Afterwards in Section 3, we investigate holographic aspects of SSD CFT. In
Subsection 3.1, we show that the gravitational dual of SSD CFT prepared in the
ground and thermal states are dual to pure AdS3 and BTZ black hole
Banados:1992gq ; Banados:1992wn  with non-trivial foliations, respectively.
Then Subsection 3.2 and 3.3 demonstrate the existence of new horizons in both
pure AdS3 (3.8) and BTZ (3.13), which are plotted in Figure 3 and 4,
respectively. In Subsection 3.2 we show that the AdS3 new horizon, which is a
Killing horizon, also has zero temperature. In Subsection 3.3 we also compare
our new horizon in BTZ geometry with the BTZ black hole horizon and find that
the new horizon is larger (see Figure 5).

In Subsection 3.4, we comment on the reconstructions of bulk operators in the
presence of the aforementioned new horizons and conclude that bulk operators
remain reconstructable even after falling into these horizons (as long as they
remain outside the BTZ black hole horizon).

In Section 4 we generalise the SSD quench results to quenches governed by
other $SL(2)$ Virasoro subalgebras. In Section 5 we turn our attention to
Floquet CFT whose physics has mostly been analysed at stroboscopic times; here
we spell out what happens for continuous time between the discrete
stroboscopic time steps; see Figure 9 for the relevant plot. Subsequently, in
Section 6 we study the holographic dual of Floquet CFT, and argue that there
are no new horizons in the bulk anymore. We also find that in the thermal
state cases, the BTZ black hole horizon asymptotically approaches the boundary
at two points, which is plotted in Figure 10. This is in agreement with the
results in deBoer:2023lrd  obtained in the slow-driving limit.

Finally, Section 7 is devoted to the study of holographic entanglement entropy
Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt  in the gravity dual of SSD and
Floquet CFT prepared both in the ground and thermal state. We find that due to
the non-trivial foliations of the gravity dual of SSD CFT in the bulk, the
holographic entanglement entropy is computed by spacelike geodesics anchoring
on the boundary that is, in general, at non-equal times, see e.g. Figure 11
for an example. The time evolution of large-radius cutoff surfaces (plotted in
Figure 2) plays a crucial role. In cases where CFTs are initially prepared in
thermal states, we discover that holographic entanglement entropy shows the
entanglement plateau phenomenon Hubeny:2013gta . We find precise agreements
between our holographic results and field-theoretic calculations in all
comparable cases and reproduce holographically the cooling and heating effects
Wen:2022pyj ; Goto:2021sqx  discovered earlier in the field theory context.

Section 8 concludes this work and points out possible future directions. To
ease the flow of the main text, we relegated most technical details to often
very detailed appendices.

##  2 A special quantum quench

###  2.1 Setup

We consider CFT2 on a cylinder $\mathbb{R}\times S^{1}$ of circumference
$l=2\pi$ with periodic boundary conditions,222The references Wen:2018vux ;
Wen:2018agb ; Fan:2019upv ; Wen:2020wee ; Han:2020kwp ; Fan:2020orx ;
Wen:2022pyj  sometimes adopt open boundary conditions. We will not use this
boundary condition in our work but will point out its appearance when
referring to their works. in either the ground state $|0\rangle$ or the
thermal state $\rho=e^{-\beta H}$ at temperature $\beta^{-1}$. We are
interested in the Lonretzian time evolution of driven CFTs, although it is
technically easier to work in Euclidean and then analytically continue to
Lorentzian signature. The coordinate on the Euclidean cylinder is

$\displaystyle w$ $\displaystyle=\tau+ix$ (2.1) $\displaystyle\overline{w}$
$\displaystyle=\tau-ix$

Here, we use $\tau$ and $t$ to denote imaginary and real time, respectively.
The spatial variable $x$ is periodic with periodicity $2\pi$: $x\sim x+2\pi$.
The analytic continuation is given by $\tau\to it$, which makes $w$ and
$\overline{w}$ (real) light cone coordinates.

We will mostly work in the Heisenberg picture: we take a primary operator $O$
of weights $(h,\overline{h})$ on the $t=0$ time slice with $w=ix$,
$\overline{w}=-ix$ and time evolve it using the operator $e^{-\tau H}$ with
$H$ made out of the Virasoro generators. Such a time evolution is geometric,
the operator moves $O$ to new positions $(w_{{\rm
new}}(\tau,x),\overline{w}_{{\rm new}}(\tau,x))$,333Here $w=ix$,
$\overline{w}=-ix$. In later sections we will use the notations
$\frac{\partial w_{{\rm new}}}{\partial w}$ and $\frac{\partial w_{{\rm
new}}}{\partial(ix)}$, $\frac{\partial\overline{w}_{{\rm
new}}}{\partial\overline{w}}$ and $\frac{\partial\overline{w}_{{\rm
new}}}{\partial(-ix)}$ interchangeably.

$\displaystyle O^{(H)}(\tau,x)\equiv e^{\tau H}O(ix,-ix)e^{-\tau
H}=\Big{(}\frac{\partial w_{{\rm new}}}{\partial
w}\Big{)}^{h}\Big{(}\frac{\partial\overline{w}_{{\rm
new}}}{\partial\overline{w}}\Big{)}^{\overline{h}}O(w_{{\rm
new}},\overline{w}_{{\rm new}})\,,$ (2.2)

whose explicit expression depends on the Hamiltonian $H$ that drives the
system. This is a powerful result that applies irrespective of the state of
the system or the boundary conditions (provided they preserve $H$).

For example, if $H$ is the CFT Hamiltonian

$\displaystyle H_{0}=\int_{0}^{2\pi}dx\
T_{00}(x)=L_{0}+\overline{L}_{0}-\frac{c}{12}$ (2.3)

with the last term being the Casimir energy, then after (imaginary) time
$\tau$, $(w_{{\rm new}},\overline{w}_{{\rm new}})$ are simply
$(w,\overline{w})$ in (2.1).

###  2.2 The sine-squared deformed time evolution

The fundamental ingredient of the deformation we would like to study is the
sine-squared deformation (SSD) Wen:2018vux , which deforms the CFT Hamiltonian
(2.3) with the enveloping function

$\displaystyle v_{1}(x)=2\sin^{2}\Big{(}\frac{x}{2}\Big{)}$ (2.4)

The resulting deformed Hamiltonian is

$\begin{split}H_{1}&=2\int_{0}^{2\pi}dx\
\sin^{2}\Big{(}\frac{x}{2}\Big{)}\,T_{00}(x)\\\
&=L_{0}-\frac{L_{-1}+L_{1}}{2}+\overline{L}_{0}-\frac{\overline{L}_{-1}+\overline{L}_{1}}{2}-\frac{c}{12}\end{split}$
(2.5)

Note that the SSD enveloping function $v_{1}(x)$ is inhomogeneous in space, in
particular, it vanishes at $x=0$ (or $2\pi$). From (2.5) one observes
$[H_{0},H_{1}]\neq 0$, because $[L_{0},L_{\pm 1}]=\mp L_{\pm 1}$; however,
$H_{0}$ and $H_{1}$ share the same ground state $|0\rangle$, as
$L_{n}|0\rangle=0$ for $n\geq-1$.

The use of $H_{1}$ as a Hamiltonian is unconventional. However, $H_{1}$ can be
conformally mapped to a dilation Okunishi:2016zat ; Wen:2018vux ; Fan:2019upv
; mapping back to the cylinder, we obtain $(w_{{\rm new}},\overline{w}_{{\rm
new}})$ given by the following logarithms of Mobius transformations
Wen:2018vux ; Goto:2021sqx  (see Appendix A.1 for details)

$\displaystyle w_{{\rm new}}$
$\displaystyle=\log[\frac{\big{(}1+\frac{it}{2}\big{)}e^{ix}-\frac{it}{2}}{\frac{it}{2}e^{ix}+\big{(}1-\frac{it}{2}\big{)}}\Big{]}\,,$
(2.6) $\displaystyle\overline{w}_{{\rm new}}$
$\displaystyle=\log[\frac{\big{(}1+\frac{it}{2}\big{)}e^{-ix}-\frac{it}{2}}{\frac{it}{2}e^{-ix}+\big{(}1-\frac{it}{2}\big{)}}\Big{]}\,,$
(2.7)

where we have analytically continued to real-time $\tau\to it$. The logarithm
is a multi-valued function; we fix the appropriate sheet by demanding
continuity in $t$. The terms inside the logarithm are $SL(2)$ transformations
of the variables $z=e^{ix}$ and $\overline{z}=e^{-ix}$ on the unit
circle.444$z$ and $\overline{z}$ are coordinates on the complex plane, related
to the cylinder via the exponential maps $z=e^{w}$ and
$\overline{z}=e^{\overline{w}}$. The $SL(2)$ transformation has a fixed point,
i.e. a point that stays invariant under the $SL(2)$ transformation at $x=0$
(identified with $x=2\pi$). In analogy with (2.1), we can define Goto:2021sqx

$\displaystyle w_{{\rm new}}$ $\displaystyle=i(t_{{\rm new}}+x_{{\rm
new}})\,,$ (2.8) $\displaystyle\overline{w}_{{\rm new}}$
$\displaystyle=i(t_{{\rm new}}-x_{{\rm new}})\,.$

$t_{{\rm new}}$ and $x_{{\rm new}}$ are real functions of $t$ and $x$.555As
$SL(2)$ transformations preserve unit circles, $w_{{\rm new}}$ and
$\overline{w}_{{\rm new}}$ are completely determined by the argument of the
logarithm function and are, therefore, purely imaginary.  We will see that the
$SL(2)$ transformation (2.6) and (2.7) determines (almost) all the physical
phenomena of interest: in the $(w_{{\rm new}},\overline{w}_{{\rm new}})$
coordinate, the CFT is in equilibrium, and the time-dependent conformal
transformations (2.6) and (2.7) allow us to deduce non-equilibrium phenomena
from known equilibrium physics Wen:2018vux ; moosavi2021inhomogeneous .

There is an alternative way of thinking about the SSD CFT: the driving by the
Hamiltonian (2.5) is equivalent to the natural time evolution of the CFT on
the curved spacetime Wen:2018vux ; MacCormack:2018rwq ; deBoer:2023lrd :

$\begin{split}ds^{2}=-v_{1}(x)^{2}dt^{2}+dx^{2}\,.\end{split}$ (2.9)

For the review of the argument, see Appendix A.1.3. Thus we have three
equivalent ways of thinking about correlation functions

$\begin{split}\left\langle
O^{(H_{1})}(\tau,x)\dots\right\rangle&=\Big{(}\frac{\partial w_{{\rm
new}}}{\partial(ix)}\Big{)}^{h}\Big{(}\frac{\partial\overline{w}_{{\rm
new}}}{\partial(-ix))}\Big{)}^{\overline{h}}\dots\left\langle O(w_{{\rm
new}},\overline{w}_{{\rm new}})\dots\right\rangle_{ds^{2}=dw_{{\rm
new}}d\overline{w}_{{\rm new}}}\\\\[4.0pt] &=\left\langle
O(\tau,x)\dots\right\rangle_{ds^{\prime
2}=v_{1}(x)^{2}d\tau^{2}+dx^{2}}\,.\end{split}$ (2.10)

Quantum quenches with SSD CFT at $t>0$ have been studied for systems prepared
in ground Wen:2018vux  and thermal Goto:2021sqx  states.666Note that in
Wen:2018vux , open boundary conditions has been imposed to set up a quench
problem, which makes the ground state of $H_{0}$ into an excited state of
$H_{1}$. In this work, we study the non-trivial time evolution of $H_{1}$ with
periodic boundary conditions. Although this setup cannot strictly be regarded
as a quench if the initial state is the ground state, the expression for
$w_{{\rm new}}$ (2.6) and $\overline{w}_{{\rm new}}$ (2.7) remain the same.
Some of the most interesting phenomena that have been found are as follows: In
Goto:2021sqx  it was found that the CFT possesses two energy peaks that move
towards the fixed point $x=0$; the entanglement entropy decreases and
approaches the vacuum entanglement entropy if the subregion $A$ does not
include the fixed point $x=0$ (identified with $2\pi$). In the following, we
will provide a geometric perspective on these, predominantly in the context of
holography.

###  2.3 Boundary triangle

We have seen in Section 2.2 that the non-trivial time evolution of SSD CFTs is
completely determined by the coordinate transformations (2.6) and (2.7). The
spacetime coordinates $t_{{\rm new}}(t,x)$ and $x_{{\rm new}}(t,x)$ then
follows from (2.8). In Figure 1, we plot the constant-$t$ lines and
constant-$x$ lines on the $(t_{{\rm new}},x_{{\rm new}})$ plane.777It is more
convenient to plot with the trigonometric form of $t_{{\rm new}}$ (A.19) and
$x_{{\rm new}}$ (A.20).

Figure 1: Constant $t$ lines (blue) and constant-$x$ lines (red) in the
$(t_{{\rm new}},x_{{\rm new}})$-plane. At $t<0$ when the driving Hamiltonian
is $H_{0}$, constant-$t$ lines and constant-$x$ lines are uniformly
distributed horizontal and vertical lines, respectively. At $t>0$ when the
driving Hamiltonian is suddenly switched to $H_{1}$, constant-$t$ lines
asymptote a triangle-shaped curve (2.11), which is plotted as a dashed gray
line; constant-$x$ lines converge to the top tip of the triangle, $(\pi,\pi)$.

Figure 1 shows that the action of $H_{1}$ as Hamiltonian changes the foliation
of the $(t_{{\rm new}},x_{{\rm new}})$ spacetime in a non-uniform
manner.888This reflects the inhomogeneous nature of $H_{1}$; in particular,
the fixed point $x=0$ (which maps to $(t_{{\rm new}},x_{{\rm new}})=(0,0)$)
does not time evolve at all.  As time evolves, constant-$t$ lines approach a
triangle-shaped curve

$t_{{\rm new}}=\left\\{\begin{aligned} &x_{{\rm new}},\ \ 0\leq x_{{\rm
new}}<\pi\\\ &2\pi-x_{{\rm new}},\ \ \pi\leq x_{{\rm
new}}<2\pi\end{aligned}\right.$ (2.11)

instead of moving towards $t_{{\rm new}}\to\infty$ homogeneously as is in the
$H_{0}$ evolution case; constant-$x$ lines converge to the top tip of the
triangle, $(\pi,\pi)$. Indeed, from the expression of $t_{{\rm new}}$ (A.19)
and $x_{{\rm new}}$ (A.20), one can verify that

$\displaystyle\lim_{t\to\infty}t_{{\rm new}}=\pi$
$\displaystyle\lim_{t\to\infty}x_{{\rm new}}=\pi$ (2.12)

regardless of $x$ ($x\neq 0,2\pi$). As the triangle (2.11) separates points
that are connected to infinity (outside the triangle) from those that are not
(inside the triangle), it can be regarded as a new horizon in the $(t_{{\rm
new}},x_{{\rm new}})$ plane.

The SSD Hamiltonian (2.5) evolves the CFT along a conformal Killing vector
field, hence the horizon (2.11) is a Killing horizon associated with this
vector field. From the expression (2.5), the conformal Killing vector $k_{1}$
corresponds to $H_{1}$ can be written as

$\displaystyle k_{1}$ $\displaystyle=-z_{{\rm new}}\partial_{z_{{\rm
new}}}-\overline{z}_{{\rm new}}\partial_{\overline{z}_{{\rm
new}}}+\frac{z_{{\rm new}}^{2}+1}{2}\partial_{z_{{\rm
new}}}+\frac{\overline{z}_{{\rm new}}^{2}+1}{2}\partial_{\overline{z}_{{\rm
new}}}$ (2.13) $\displaystyle=(\cosh\tau_{{\rm new}}\ \cos x_{{\rm
new}}-1)\partial_{\tau_{{\rm new}}}+\sinh\tau_{{\rm new}}\ \sin x_{{\rm
new}}\partial_{x_{{\rm new}}}$ (2.14)

where $z_{{\rm new}}=e^{\tau_{{\rm new}}+ix_{{\rm new}}}$ and
$\overline{z}_{{\rm new}}=e^{\tau_{{\rm new}}-ix_{{\rm new}}}$. One can easily
check that $k_{1}$ obeys the conformal Killing equation

$\displaystyle\partial_{(\mu}(k_{1})_{{\nu})}=\delta_{\mu\nu}\partial_{\rho}(k_{1})^{\rho}\,.$
(2.15)

The analytic continuation of the conformal Killing vector (2.14) to Lorentizan
signature is

$\displaystyle k_{1}=(1-\cos t_{{\rm new}}\ \cos x_{{\rm
new}})\partial_{t_{{\rm new}}}+\sin t_{{\rm new}}\ \sin x_{{\rm
new}}\partial_{x_{{\rm new}}}\,,$ (2.16)

where we have removed the overall $i$ so that the vector field implements
Lorentzian time evolution. The conformal Killing horizon is located where the
conformal Killing vector becomes null, so we compute its norm squared:

$\displaystyle|k_{1}|^{2}=(\cos t_{{\rm new}}-\cos x_{{\rm new}})^{2}=0$
$\displaystyle\quad\implies\quad t_{{\rm new}}=\pm x_{{\rm new}}+2n\pi\
(n\in\mathbb{Z})$ (2.17)

As $t_{{\rm new}}\in[0,\pi]$, $t_{{\rm new}}=x_{{\rm new}}$ or $t_{{\rm
new}}=-x_{{\rm new}}+2\pi$, which is exactly the expression of the triangle
(2.11). Since (2.17) has a double zero, this is an extremal conformal Killing
horizon, hence it has zero temperature.999A simple calculation on (2.14)
(after analytic continuation) confirms this reasoning,  $\displaystyle
T=\frac{1}{2\pi}\sqrt{\nabla_{\mu}|k_{1}|\nabla^{\mu}|k_{1}|}\Big{|}_{{\rm
hor.}}=\sqrt{-\sin^{2}t_{{\rm new}}+\sin^{2}x_{{\rm new}}}=0$ (2.18) where
$|k_{1}|$ is given by (2.17).  The physical reason for this is that $H_{0}$
and $H_{1}$ share the _same_ ground state.101010This is in contrast to the
Unruh effect, where the Minkowski vacuum is an excited state of and Rindler
observer and vice versa.

##  3 Gravity dual for the special quantum quench

###  3.1 Bulk geometry

The gravity dual of a generic (global) quench involves the changing of
boundary conditions for the gravity fields, corresponding to uniform injection
of energy into the bulk from the near boundary region. In the simplest
scenario, the spacetime reaches equilibrium by settling to a black hole, the
dual of thermalisation in the boundary QFT.

In the special quench we analyse, the Heisenberg evolution on the cylinder can
be described by the $t$-depending coordinate transformation $(t_{{\rm
new}},x_{{\rm new}})$ (2.8) expressing non-equilibrium physics from
equilibrium results. Correspondingly, the matter stress tensor in the bulk
remains zero and the spacetime does not change. What changes is the cutoff
surface that the field theory lives on. This among other things leads to the
change in energy in the boundary theory.

To construct the holographic dual of SSD CFT, we introduce the emergent bulk
radial coordinate $r_{{\rm new}}\in[0,\infty)$. Holographic CFTs in their
ground and thermal states are dual to pure AdS3 and BTZ black holes
Banados:1992gq ; Banados:1992wn , respectively. As the CFT in $(t_{{\rm
new}},x_{{\rm new}})$ are in equilibrium, the dual bulk metrics in the
$(t_{{\rm new}},r_{{\rm new}},x_{{\rm new}})$ coordinate are given by those of
the pure AdS3 and BTZ black hole metrics,

$\displaystyle|0\rangle:$ $\displaystyle ds^{2}=-(r_{{\rm new}}^{2}+1)dt_{{\rm
new}}^{2}+\frac{dr_{{\rm new}}^{2}}{r_{{\rm new}}^{2}+1}+r_{{\rm
new}}^{2}dx_{{\rm new}}^{2}$ (3.1) $\displaystyle\rho=e^{-\beta H_{1}}:$
$\displaystyle ds^{2}=-(r_{{\rm new}}^{2}-r_{+}^{2})dt_{{\rm
new}}^{2}+\frac{dr_{{\rm new}}^{2}}{r_{{\rm new}}^{2}-r_{+}^{2}}+r_{{\rm
new}}^{2}dx_{{\rm new}}^{2}$ (3.2)

where $r_{+}$ is the radius of the BTZ black hole, related to the inverse
temperature $\beta$ via $r_{+}=\frac{2\pi}{\beta}$.

Besides the movement of the insertion points of operators, there are explicit
factors showing up in formulas (2.2) and (2.10) multiplying equilibrium
correlators. These correspond to taking the holographic cutoff at

$\begin{split}\epsilon={1\over\sqrt{\frac{\partial w_{{\rm new}}}{\partial
w}\frac{\partial\overline{w}_{{\rm new}}}{\partial\overline{w}}}\,R_{{\rm
new}}(t_{\text{new}},x_{\text{new}})}\,,\end{split}$ (3.3)

resulting in a nonuniform cutoff surface in the radial coordinate $R_{{\rm
new}}(t_{\text{new}},x_{\text{new}})$. Note that the formula is applicable for
$t>0$, for $t\leq 0$ we have $R_{{\rm new}}=1/\epsilon$. A plot of the cutoff
surface (3.3) can be found in Figure 2. This formula has appeared in the
literature before in Goto:2021sqx  based on different arguments; in Appendix
B.1 we give a detailed new argument that this nonuniform cutoff produces the
appropriate $\big{(}\frac{\partial w_{{\rm new}}}{\partial w}\big{)}^{h}$
factors in correlators.

Figure 2: Left: Plot of the cutoff surface in $(t_{{\rm new}},r_{{\rm
new}},x_{{\rm new}})$ spacetime. The light blue surface denotes the $t_{{\rm
new}}=t=0$ slice, during which we suddenly switch the Hamiltonian from $H_{0}$
to $H_{1}$. At $t<0$ when the driving Hamiltonian is $H_{0}$, the uniform
cutoff $R_{\text{new}}=\epsilon^{-1}$ is shown in blue. At $t>0$ the driving
Hamiltonian becomes $H_{1}$, and the cutoff surface $R_{{\rm
new}}(t_{\text{new}},x_{\text{new}})$ is no longer uniform according to (3.3).
This significant change leads to the presence of a new horizon, which is
depicted above in light yellow. This new horizons is also plotted in Figure 3
along with its generators. Note that the cutoff surface ends on the horizon
(up to finite cutoff corrections). Right: Cartoon of constant $t$ slices in
$(t_{{\rm new}},r_{{\rm new}},x_{{\rm new}})$ spacetime in global AdS3.

We have seen that in the boundary theory a horizon appears, this in the
holographic dual leads to a new horizon in the bulk. As constant $t$-slices in
the bulk are anchored on the constant $t$-lines on the boundary, this induces
a change in the foliation of the bulk, see Figure 2. (There is a natural
extension of $H_{1}$ into the bulk in the pure AdS case, but not in BTZ.) As
$t\to\infty$, constant-$t$ slices asymptote the bulk null hypersurface
anchored on the boundary triangle (2.11) instead of moving towards $t_{{\rm
new}}\to\infty$. As this hypersurface separates bulk points that are connected
to the boundary spacetime explored by the CFT from those that are not, it is a
new horizon in the bulk.

An alternative approach is provided by the CFT in curved spacetime
perspective: we set the boundary metric to agree with (2.9), and use
Fefferman-Graham coordinates Skenderis:1999nb ; deHaro:2000vlm  to build a
coordinate system adapted to the boundary spacetime. The uniform cutoff
surface in the Fefferman-Graham radial coordinate will coincide with the
cutoff surface found from (3.3). In 3d pure gravity, the Fefferman-Graham
expansion is known to terminate. By solving the Einstein equations order by
order in $r$ we obtain the metric Skenderis:1999nb :

$\begin{split}ds^{2}={dr^{2}\over
r^{2}}+r^{2}\big{(}-v_{1}(x)^{2}dt^{2}+dx^{2}\big{)}+g^{(1)}_{ab}(x^{c})\,dx^{a}dx^{b}+{g^{(2)}_{ab}(x^{c})\,dx^{a}dx^{b}\over
r^{2}}\,,\end{split}$ (3.4)

where $g^{(j)}_{ab}(x^{c})$ is given in Appendix B.2 for the most general
solution. There we also give the explicit form of the metric for the case of
the vacuum and thermal states. Of course, (3.4) is just the reslicing of the
AdS3 (3.1) and BTZ (3.2) spacetimes. (The reslicing of AdS3 was written down
in MacCormack:2018rwq  before studying the vacuum physics of the SSD CFT.) One
advantage of these coordinates is that now the cutoff is at $R=1/\epsilon$
instead of the nonuniform cutoff in $r_{{\rm new}}$ coordinates.

Note that as discussed in Appendix A.1.3, the curved spacetime (2.9) is
obtained by a Weyl transformation from the flat spacetime $ds^{2}=dw_{{\rm
new}}d\overline{w}_{{\rm new}}$. It has been understood from the early days of
AdS/CFT Witten:1998qj ; Henningson:1998gx  that changing the conformal factor
of the boundary metric corresponds to choosing a different cutoff
prescription.

###  3.2 New horizon in AdS3

The isometries of AdS3 form two copies of $SL(2)$ algebras Brown:1986nw . The
Killing vectors $\zeta_{i}$ and $\overline{\zeta_{i}}$ ($i=-1,0,1$) of AdS3
are given in Appendix C. From these Killing vectors, we can find a linear
combination that asymptotes to the conformal Killing vector $k_{1}^{\mu}$
(2.14). This provides the natural extension of the $H_{1}$ evolution into the
bulk:

$\begin{split}K_{1}&=\zeta_{0}+\overline{\zeta_{0}}-\frac{\zeta_{1}+\overline{\zeta_{1}}}{2}-\frac{\zeta_{-1}+\overline{\zeta_{-1}}}{2}\\\
&=\Big{(}1-\frac{r_{{\rm new}}}{\sqrt{r_{{\rm new}}^{2}+1}}\cos t_{{\rm new}}\
\cos x_{{\rm new}}\Big{)}\partial_{t_{{\rm new}}}-\sqrt{r_{{\rm new}}^{2}+1}\
\sin t_{{\rm new}}\ \cos x_{{\rm new}}\partial_{r_{{\rm new}}}\\\
&+\frac{\sqrt{r_{{\rm new}}^{2}+1}}{r_{{\rm new}}}\sin t_{{\rm new}}\ \sin
x_{{\rm new}}\partial_{x_{{\rm new}}}\end{split}$ (3.5)

In the near-boundary $r_{{\rm new}}\to\infty$ limit, the components of
$K_{1}^{\mu}$ (3.5) along the boundary indeed goes to the boundary conformal
Killing vector $k_{1}^{\mu}$ (2.14),

$\displaystyle\lim_{r_{{\rm new}}\to\infty}K_{1}^{\mu}=(1-\cos t_{{\rm new}}\
\cos x_{{\rm new}}),\sin t_{{\rm new}}\ \sin x_{{\rm new}}\big{)}=k_{1}^{\mu}$
(3.6)

The Killing horizon is located where a Killing vector becomes null. From
(3.5), we have for the norm square:

$\displaystyle|K_{1}|^{2}$ $\displaystyle=\big{(}\sqrt{r^{2}+1}\cos t_{{\rm
new}}-r\cos x_{{\rm new}}\big{)}^{2}=0$ (3.7) $\displaystyle\Rightarrow\quad
t_{{\rm new}}(r_{{\rm new}},x_{{\rm new}})=\arccos[\frac{r}{\sqrt{r_{{\rm
new}}^{2}+1}}\cos x_{{\rm new}}\Big{]}\,.$ (3.8)

(3.8) is the new horizon in pure AdS3. Since $|K_{1}|^{2}$ has a double zero,
this is an extremal Killing horizon with zero associated temperature.111111A
direct calculation shows that this is indeed the case:  $\displaystyle
T=\frac{1}{2\pi}\sqrt{\nabla_{\mu}|K_{1}|\nabla^{\mu}|K_{1}|}\Big{|}_{{\rm
hor.}}=\sqrt{r_{{\rm new}}^{2}+1}\cos t_{{\rm new}}-r\cos x_{{\rm new}}=0$
(3.9) where $|K_{1}|$ given by (3.7).  Just like on the CFT side, the
intuition behind this result is that $H_{0}$ and $H_{1}$ share the same vacuum
state. A plot of the Killing horizon (3.8) along with its null generators can
be found in Figure 3.

We can find the horizon (3.8) in an alternative manner, which will be a useful
method in the following. To find the horizon as a null hypersurface, we need
to investigate the null geodesics emanating from a point $(t_{{\rm
new}},x_{{\rm new}})=(y,y)$ on the boundary triangle. The congruence of null
geodesics that maximise $t_{{\rm new}}$ in the bulk then generates the null
hypersurface. Here, for simplicity, we study only half of the triangle, i.e.
$y\in[0,\pi]$; the result of the other half follows from reflection.

We will start with the metric (3.1) in the $(t_{{\rm new}},r_{{\rm
new}},x_{{\rm new}})$ coordinate. The null geodesics anchored on a point
$(t_{{\rm new}},x_{{\rm new}})=(y,y)$ on the boundary triangle (see Appendix
D.1.1 for details) is

$\displaystyle t_{{\rm new}}(r_{{\rm new}},x_{{\rm new}},y)={\rm
arctan}\frac{-r_{{\rm new}}\ \cot(x_{{\rm new}}-y)}{\sqrt{r_{{\rm
new}}^{2}+\csc^{2}(x_{{\rm new}}-y)}}+y-\frac{\pi}{2}$ (3.10)

where $y\in[0,\pi]$. (3.10) increases monotonically with $y$ (see (D.17)).
Hence, the maximal value of $t$ is at $y=\pi$, which is the top tip of the
boundary triangle. After some work detailed in Appendix D.1.1, one arrives at

$\displaystyle t_{{\rm new}}(r_{{\rm new}},x_{{\rm
new}})=\arccos[\frac{r_{{\rm new}}}{\sqrt{r_{{\rm new}}^{2}+1}}\cos x_{{\rm
new}}\Big{]}$ (3.11)

agreeing with (3.8) obtained by the Killing vector method.

Figure 3: A plot of the new Killing horizon (3.11) in global AdS3. The
coloured lines are null geodesics emanating from the top tip of the triangle
$(\pi,\pi)$ that generate the hypersurface (3.11); for visualization, we only
plotted them on half of the horizon. The light blue surface is the quench
surface $t_{{\rm new}}=t=0$.

.

###  3.3 New horizon on BTZ

The $SL(2)$ isometry of AdS3 is broken to $\mathbb{R}\times U(1)$ in BTZ
geometry Keski-Vakkuri:1998gmz . Hence, the new horizon is no longer a Killing
horizon, nor can we associate temperature with it. To find the explicit
expression of the horizon, one needs to investigate the null geodesic equation
using the second method of the previous section. The null geodesics anchored a
point on the boundary triangle $(t_{{\rm new}},x_{{\rm new}})=(y,y)$ (see
Appendix D.1.2 for details) is

$\displaystyle t_{{\rm new}}(r_{{\rm new}},x_{{\rm
new}},y)=\frac{1}{2r_{+}}\log\frac{r_{{\rm
new}}\sqrt{1+r_{+}^{2}\sinh^{2}(x_{{\rm new}}-y)}-r_{+}\sqrt{1+r_{{\rm
new}}^{2}\sinh^{2}(x_{{\rm new}}-y)}}{r_{{\rm
new}}\sqrt{1+r_{+}^{2}\sinh^{2}(x_{{\rm new}}-y)}+r_{+}\sqrt{1+r_{{\rm
new}}^{2}\sinh^{2}(x_{{\rm new}}-y)}}+y$ (3.12)

where $y\in[0,\pi]$. (3.12) increases monotonically with $y$ (see (D.30)).
Hence, the maximal value of $t$ is at $y=\pi$, which is the top tip of the
boundary triangle. After some work detailed in Appendix D.1.2, one arrives at

$\displaystyle t_{{\rm new}}(r,x)=\frac{1}{2r_{+}}\log\frac{r_{{\rm
new}}\sqrt{1+r_{+}^{2}\sinh^{2}(x_{{\rm new}}-\pi)}-r_{+}\sqrt{1+r_{{\rm
new}}^{2}\sinh^{2}(x_{{\rm new}}-\pi)}}{r_{{\rm
new}}\sqrt{1+r_{+}^{2}\sinh^{2}(x_{{\rm new}}-\pi)}+r_{+}\sqrt{1+r_{{\rm
new}}^{2}\sinh^{2}(x_{{\rm new}}-\pi)}}+\pi$ (3.13)

Figure 4: Plot of the new horizon (3.13) in the BTZ geometry. The coloured
lines are null geodesics emanating from the top tip of the triangle
$(\pi,\pi)$ that generate the hypersurface (3.13); for visualization, we only
plotted them on half of the horizon. The light blue surface is the quench
surface $t_{{\rm new}}=t=0$. The BTZ black hole horizon is set to $r_{+}=1$
and plotted in light gray. The new horizon (3.13) is larger than the BTZ black
hole horizon.

.

A plot of the new horizon (3.13) along with its null generators can be found
in Figure 4.

For comparison with ref. Goto:2021sqx , we use the coordinate system they were
working in to plot time slices of both the new horizon (3.13) and the BTZ
horizon.121212The Feffermann-Graham coordinates discussed around (3.4) and
Appendix B.2 generically break down before encountering horizons, as was
recently discussed in Abajian:2023bqv . Hence we cannot use them for
addressing this question. (See Appendix E for details.) From Figure 4 it is
clear that the new horizon will be slightly larger than the BTZ black hole
horizon, and indeed this is what we find in Figure 5. Note also that the new
horizon (3.13) touches the boundary at the fixed point $x=0$.

The horizon (3.13) in Figure 5 develops two peaks that move towards the fixed
point $x=0$, as is observed for the BTZ horizon in Goto:2021sqx . It was also
pointed out in Goto:2021sqx  that the horizon deformation mimics the energy
peaks of the boundary stress tensors. We note that the two peaks of the
horizons (both the BTZ horizon and the new horizon) originate from the Weyl
factor $\frac{\partial w_{{\rm new}}}{\partial
w}\frac{\partial\overline{w}_{{\rm new}}}{\partial\overline{w}}$ in (E.1); in
Goto:2021sqx , the peaks of the boundary stress tensors also originate from
these prefactors $\frac{\partial w_{{\rm new}}}{\partial w}$ and
$\frac{\partial\overline{w}_{{\rm new}}}{\partial\overline{w}}$ in $\langle
T(w)\rangle$ and $\langle\overline{T}(\overline{w})\rangle$, respectively.
This then explains the qualitative131313As the metric (E.1) is not in
Fefferman-Graham gauge Skenderis:1999nb ; deHaro:2000vlm , it is hard to draw
a quantitative connection. correlations between horizon deformations and
energy peaks were first observed in Goto:2021sqx .

Figure 5: Comparison of the new horizon (3.13) and the BTZ black hole horizon
in the coordinates employed by Goto:2021sqx , at $t=0$ (left) and $t=3$
(right), respectively. The blue line denotes the BTZ horizon, whereas the
orange line represents the new horizon (3.13).

###  3.4 Comments on bulk reconstruction

Horizons make information inaccessible. The presence of the horizons (3.11)
and (3.13) in Figure 3 and 4 makes us wonder if an information puzzle can
arise in our setup: e.g. a bulk excitation prepared on the $t=0$ would fall
behind the horizon and become inaccessible leading to tension with unitary
time evolution. Equivalently, one asks how to reconstruct local bulk operators
(dressed to make them diffeomorphism invariant) inside or outside the new
horizons, respectively.

For bulk fields inside the causal wedge, i.e. in the bulk region enclosed by
the null hypersurfaces in Figure 3 or 4 and the quench surface at $t=0$, we
can recover the information using HKLL-like reconstructions Banks:1998dd ;
Hamilton:2006az ; Witten:2023qsv . In our case, the $t_{\text{new}}>0$ part of
this subregion is the triangle. (The $t_{\text{new}}<0$ part is a time strip
of width $\pi/2$ for pure AdS and infinite for BTZ.) The reconstruction
formula is given by

$\displaystyle\phi(X)=\int_{{\rm triangle+strip}}dx\,K(X|x)O(x)$ (3.14)

where $X=(t_{{\rm new}},r_{{\rm new}},x_{{\rm new}})$ and $x=(t_{{\rm
new}},x_{{\rm new}})$ stand for the coordinates on the bulk and boundary,
respectively. $K(X|x)$ is a smearing function adapted to the subregion.

For bulk operators outside the causal wedge, i.e. behind the horizon (3.11)
and (3.13), the aforementioned HKLL-like reconstruction does not apply
anymore. However, information falling through the horizon (3.11) or (3.13)
(but not through the BTZ black hole horizon in the case with (3.13)) remains
reconstructable, by considering modular reconstruction Jafferis:2015del ;
Dong:2016eik . In our case, the modular Hamiltonian is simply $H_{0}$ up to a
coefficient $\frac{\beta}{2\pi}$,141414In the vacuum since the density matrix
is a projector, the modular Hamiltonian is not well defined, so we need to
regulate the vacuum with a small temperature. so the modular evolution is just
$H_{0}$ time evolution with the modular parameter given by $t_{{\rm
new}}=t$.151515Note that when the Hamiltonian is $H_{0}$, we have $t_{{\rm
new}}=t$; $t_{{\rm new}}$ and $t$ are different only when the driving
Hamiltonian is $H_{1}$.  We would like to then consider the modular flow of
local operators $O$ on the boundary, $O(x,t)=U^{-1}(t)O(x,0)U(t)$, where
$U(t)=e^{-i\beta H_{0}t/2\pi}$. Modular reconstruction then states that
modular flows equal bulk modular flows Jafferis:2015del . This conclusion
allows us to recover the information of all bulk operators within the
entanglement wedges, which in our case is the entire spacetime for the ground
state case and all the spacetime regions outside the BTZ black hole horizon
for the thermal state case. In each case, the entanglement wedge is larger
than the casual wedge and, importantly, includes the spacetime region that is
behind the horizon (3.11) or (3.13) (but outside the BTZ horizon in the
thermal state case).

##  4 Quenches governed by other $SL(2)$ Virasoro subalgebras

It is straightforward to generalise the SSD story, by replacing $L_{\pm 1}$ in
the expression of $H_{1}$ in (2.5) with $L_{\pm q}$ with $q>1$. This
corresponds to the envelope function Fan:2019upv

$\displaystyle v_{q}(x)=2\sin^{2}\Big{(}q\cdot\frac{x}{2}\Big{)}$ (4.1)

and the resulting deformed Hamiltonian $H_{q}$ is given by

$\displaystyle H_{q}$ $\displaystyle=2\int_{0}^{2\pi}dx\
\sin^{2}\Big{(}q\cdot\frac{x}{2}\Big{)}\,T_{00}(x)$ (4.2)
$\displaystyle=L_{0}-\frac{L_{-q}+L_{q}}{2}+\overline{L}_{0}-\frac{\overline{L}_{-q}+\overline{L_{q}}}{2}-q\cdot\frac{c}{12}$
(4.3)

In order for $\\{L_{0},L_{q},L_{-q}\\}$ to form the same algebra as
$\\{L_{0},L_{1},L_{-1}\\}$, we have to shift the Virasoro generators
$\\{L_{0},L_{q},L_{-q}\\}$,161616There are subtleties regarding the difference
between $L_{q}$ ($q>1$) and the global conformal charges
$\\{L_{0},L_{1},L_{-1}\\}$, as the former generate transformations that are
not globally well defined and are associated to conformal Killing vectors with
poles. But these poles appear at $0$ or $\infty$ on the complex $z$-plane or
equivalently $\pm\infty$ on the cylinder, where we will not perform
calculations. Therefore, our calculation is unaffected by this subtlety.

$\displaystyle\mathcal{L}_{0}=\frac{1}{q}\big{(}L_{0}-\frac{c}{12}q(q^{2}-1)\big{)}$
$\displaystyle\mathcal{L}_{\pm q}=\frac{1}{q}L_{\pm q}$ (4.4)

Note that such shifts also define a new vacuum $|0\rangle_{q}$ for each
$H_{q}$. $H_{q}$ is now given by

$\displaystyle
H_{q}=q\Big{(}\mathcal{L}_{0}-\frac{\mathcal{L}_{-q}+\mathcal{L}_{q}}{2}+\overline{\mathcal{L}}_{0}-\frac{\overline{\mathcal{L}}_{-q}+\overline{\mathcal{L}}_{q}}{2}-\frac{c}{12}\Big{)}$
(4.5)

which follows the same algebraic commutation rules as that of $H_{1}$. The
relevant coordinate transformations then take the same form as (2.6) and
(2.7), with the circumference $2\pi$ replaced by $2\pi/q$. The system can then
be regarded as $q$ copies of identical theories driven by $H_{1}$, each with a
shrunk length scale $2\pi/q$ living in a subregion on the cylinder (see
Appendix A.1.4 for more details). Hence the constant $t$ lines approach a
$q$-fold triangle, i.e. $q$ triangles, each living in a subregion
$\big{[}(p-1)\frac{2\pi}{q},p\frac{2\pi}{q}\big{]}\ (p=1,2,...,q)$. The
boundary conformal Killing vector is

$\displaystyle k_{q}^{\mu}=\Big{(}\cosh(q\tau_{{\rm new}})\ \cos(qx_{{\rm
new}})-1,\sinh(q\tau_{{\rm new}})\ \sin(qx_{{\rm new}})\Big{)}$ (4.6)

The boundary conformal Killing horizon is then the aforementioned $q$-fold
triangle.

Generalization of the new horizon with $H_{q}\ (q>1)$ as Hamiltonian is
straightforward. From equation (4.6), the boundary horizons when $H_{q}$
drives are $q$-triangles each in a subregion; according to the same
maximisation procedure of null geodesics in (3.10) and (3.12), the new horizon
then consists of $q$-copies of (3.11) or (3.13) when the CFT is in the ground
and thermal state, respectively, each anchoring on one of the $q$ triangles on
the boundary. An example for $q=2$ in the thermal state is plotted in Figure
6. Note that the bulk horizons for $H_{q}\ (q>1)$ in the ground state are no
longer Killing horizons. (The naive generalizations of (3.5) to $q>1$ do not
obey the Killing equation (C.7). This is as anticipated as the bulk does not
have an $SL(2)$ isometry for each $q$.)

Figure 6: A plot of the new horizon in BTZ geometry with $H_{2}$ as driving
Hamiltonian. The light blue surface is the quench surface $t_{{\rm new}}=t=0$.
The BTZ black hole horizon is set to $r_{+}=1$ and plotted in light gray. The
new horizon is larger than the BTZ black hole horizon.

##  5 Floquet CFT

###  5.1 Floquet CFT at stroboscopic times

Floquet CFT is defined by periodically driving the CFT2 with $H_{0}$ and
$H_{1}$, for times $T_{0}$ and $T_{1}$, respectively. See Figure 7.

Figure 7: Floquet CFT. In this example, we have $T_{0}=T_{1}$, and choose
$t=0$ in the middle of a $H_{0}$-driving region.

We will again work in the Heisenberg picture (2.2) and study the coordinate
transformation $w_{{\rm new}}$ and $\overline{w}_{{\rm new}}$. Previous works
Wen:2018agb ; Fan:2019upv ; Wen:2020wee ; Han:2020kwp ; Fan:2020orx ;
Wen:2022pyj  have been focusing on Floquet CFT at stroboscopic time
$t=n(T_{0}+T_{1})$ $(n\in\mathbb{Z}^{+})$, i.e. at the end of each driving
cycle.171717Again, in Wen:2018agb ; Fan:2019upv , open boundary conditions are
adapted for this set up. The time evolution operator in (2.2) (after analytic
continuation) is of the form

$\displaystyle...\underbrace{e^{-iH_{0}T_{0}}e^{-iH_{1}T_{1}}\overbrace{e^{-iH_{0}T_{0}}e^{-iH_{1}T_{1}}}^{{\rm
1\
cycle}}e^{-iH_{0}T_{0}}e^{-iH_{1}T_{1}}...e^{-iH_{0}T_{0}}e^{-iH_{1}T_{1}}}_{n\
{\rm times}}...$ (5.1)

We can start at any time during the driving. For technical simplicity, we
adopt the setup in Lapierre:2019rwj  and start in the middle of a
$H_{0}$-driving region, see Figure 7. Then the system possesses time-reversal
symmetry, i.e. it is invariant under $t\to-t$. The new coordinates $w_{{\rm
new}}$ and $\overline{w}_{{\rm new}}$ on the cylinder at stroboscopic times
are given by Wen:2018agb ; Fan:2019upv ; Wen:2020wee ; Han:2020kwp ;
Fan:2020orx ; Wen:2022pyj ; Lapierre:2019rwj  (see Appendix F for more
details)

$\displaystyle w_{n}$ $\displaystyle=w_{{\rm
new}}|_{t=n(T_{0}+T_{1})}=\log\Big(\frac{A_{n}e^{ix}+B_{n}}{C_{n}e^{ix}+D_{n}}\Big{missing})$
(5.2) $\displaystyle\overline{w}_{n}$ $\displaystyle=\overline{w}_{{\rm
new}}|_{t=n(T_{0}+T_{1})}=\log\Big(\frac{A_{n}e^{-ix}+B_{n}}{C_{n}e^{-ix}+D_{n}}\Big{missing})$
(5.3)

and

$\displaystyle t_{n}=t_{{\rm
new}}|_{t=n(T_{0}+T_{1})}=\frac{1}{2i}(w_{n}+\overline{w}_{n})$ $\displaystyle
x_{n}=x_{{\rm new}}|_{t=n(T_{0}+T_{1})}=\frac{1}{2i}(w_{n}-\overline{w}_{n})$
(5.4)

where

$\displaystyle A_{n}=\gamma_{1}-\eta^{n}\gamma_{2}$ $\displaystyle
B_{n}=-(1-\eta^{n})$ $\displaystyle C_{n}=1-\eta^{n}$ $\displaystyle
D_{n}=-(\gamma_{2}-\eta^{n}\gamma_{1})$ (5.5)

and $\gamma_{1}$, $\gamma_{2}$ and $\eta$ are determined by the two parameters
$T_{0}$, and $T_{1}$ of the system, and are given in Appendix F. The terms
inside the logarithm are again $SL(2)$ transformations of the variables
$z=e^{ix}$ and $\overline{z}=e^{-ix}$ on the unit circle. Here, $\gamma_{1}$
and $\gamma_{2}$ are two fixed point of the $SL(2)$ transformation, i.e. they
are invariant under the $SL(2)$ transformation. In the time-reversal symmetric
case Lapierre:2019rwj , $\gamma_{1}\gamma_{2}=1$. Note that as we are only
considering stroboscopic, i.e. discrete time, we lose the ability to use
continuity to choose the appropriate sheet of the logarithm. We simply use the
principal sheet, which results in $t_{n}$ and $x_{n}$ not containing full
information about the spacetime. We will remedy this situation in Section 5.2.

Depending on the values of $T_{0}$, and $T_{1}$, the system ends up in
different phases, which are known as the heating phase, non-heating phase, and
phase transition, respectively Wen:2018agb ; Fan:2019upv ; Wen:2020wee ;
Han:2020kwp ; Fan:2020orx ; Wen:2022pyj . Their properties are summarised as
follows:

  * •

In the heating phase, $\eta$ is real and $0<\eta<1$. $\gamma_{1}$ and
$\gamma_{2}$ are on the unit circle, $|\gamma_{1}|=|\gamma_{2}|=1$. Together
with $\gamma_{1}\gamma_{2}=1$, $\gamma_{2}$ is then the complex conjugate of
$\gamma_{1}$: $\gamma_{2}^{*}=\gamma_{1}$. They are known as the stable and
unstable fixed point, respectively: for $z\neq\gamma_{2}$, $w_{n}$ flows to
$\log\gamma_{1}$ at late time; whereas for $z=\gamma_{2}$, $w_{n}$ stays at
$\log\gamma_{2}$. The same works for $\overline{w}_{n}$.181818Namely, for
$\overline{z}\neq\gamma_{2}$, $\overline{w}_{n}$ flows to $\log\gamma_{1}$;
while for $\overline{z}=\gamma_{2}$, $\overline{w}_{n}$ stays at
$\log\gamma_{2}$. Since at $t=0$ the coordinates $z$ and $\bar{z}$ are complex
conjugates of each other equal to $e^{ix}$ and $e^{-ix}$ respectively, the
only way the system can end up at an unstable fixed point if
$z=e^{ix}=\gamma_{2}$, but then $\bar{z}=\gamma_{1}$ (the stable fixpoint for
$z$), since $\gamma_{1}$ and $\gamma_{2}$ are conjugate to each other, or vice
versa.

  * •

In the non-heating phase, $\eta$ is a phase, $|\eta|=1$. $\gamma_{1}$ and
$\gamma_{2}$ are both real and, in the time-reversal case, inverses of each
other. They stay on different sides of the unit circle. $w_{n}$ and
$\overline{w}_{n}$ shows oscillatory behavior.

  * •

At the phase transition, (5.5) does not apply anymore, and
$A_{n},B_{n},C_{n},D_{n}$ are instead given by (F.13). In this case, the two
fixed points $\gamma_{1}$ and $\gamma_{2}$ merge into one fixed point
$\gamma$, which is on the unit circle, $|\gamma|=1$; in the time-reversal
case, we also have $\gamma=1$ (see (F.21)). $w_{n}$ flows to $\log\gamma$,
albeit at a slower rate.

In the heating phase, it is shown in Fan:2019upv  that there are energy peaks
forming at the unstable fixed points;191919Again note that results in
Fan:2019upv  are derived with systems prepared in the ground state with open
boundary conditions. moreover, the entanglement entropy of a single interval
$A$ grows linearly if $A$ includes one of the two peaks, i.e. include one
unstable fixed point. These phenomena indicate that at late time, nearly all
degrees of freedom are localised at the two fixed points. At the phase
transition, the two peaks merge into one Fan:2019upv . In the non-heating
phases, energy and entanglement oscillate without a well-defined period
Fan:2019upv . More recently, the work Wen:2022pyj  studied Floquet CFT
prepared in a thermal initial state. They discovered that the system heats up
in the ”heating region” in the small neighborhood of the energy peaks, where
energy and local temperatures grow exponentially and entanglement grows
linearly; the rest of the system, nevertheless cools down, with energy
decreases exponentially and entanglement entropy as well as local temperatures
become that of the ground state.

In this work, we primarily focus on the heating phases mainly because we are
interested in understanding the meaning of the fixed points for continuous
times (beyond the stroboscopic times) and the role they play in the
holographic dual. Also, it was shown in Han:2020kwp  that heating phases
always exist in more generic Floquet CFT setups than we analyse here, whereas
non-heating phases might be absent.

###  5.2 Floquet CFT at general time

The field theoretic study of Floquet CFT Wen:2018agb ; Fan:2019upv ;
Wen:2020wee ; Han:2020kwp ; Fan:2020orx ; Wen:2022pyj  focuses primarily on
stroboscopic time $t=n(T_{0}+T_{1})$. To build the dual bulk spacetime,
however, one needs to study the time evolution of $t_{{{\rm new}}}$ and
$x_{{{\rm new}}}$ at general $t>0$. It will be more convenient to do so in the
complex $z$-plane, related to the $w$-cylinder (2.6), (2.7) via the
logarithmic map $w=\log z$ and $\overline{w}=\log\overline{z}$. On the
$z$-plane, the effect of $H_{1}$ is an $SL(2)$ transformation of the $z$ and
$\overline{z}$ coordinates; the explicit expressions of $z_{n}$ and
$\overline{z}_{n}$ are given by (F.19) and (F.20), respectively.

To obtain the expression of $z_{{{\rm new}}}$ at general time, one can start
with (F.19) at a stroboscopic time step, and then study how $z_{n}$ time
evolves in the $(n+1)$-th cycle, i.e. $n(T_{0}+T_{1})\leq
t\leq(n+1)(T_{0}+T_{1})$. The full time evolution then follows inductively. In
each cycle, the time evolution of $z_{{{\rm new}}}$ depends on whether the
Hamiltonian is $H_{0}$ or $H_{1}$. As the effect of $H_{0}$ and $H_{1}$ are
ordinary time translations and $SL(2)$ transformations according to (2.6) (see
also (A.12)), respectively, the expression of $z_{{{\rm new}}}$ in the
$(n+1)$-th cycle is:

$z_{{{\rm new}}}=\left\\{\begin{aligned} &e^{i\mathfrak{t}_{n+1}}z_{n},\ \
&&0\leq\mathfrak{t}_{n+1}\leq\frac{T_{0}}{2}\\\
&\frac{\Big{(}1+\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}\Big{)}\left(e^{\frac{iT_{0}}{2}}z_{n}\right)-\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}}{\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}\left(e^{\frac{iT_{0}}{2}}z_{n}\right)+\Big{(}1-\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}\Big{)}},\
\ &&\frac{T_{0}}{2}<\mathfrak{t}_{n+1}\leq\frac{T_{0}}{2}+T_{1}\\\
&e^{i(\mathfrak{t}_{n+1}-(T_{0}+T_{1}))}z_{n+1},\ \
&&\frac{T_{0}}{2}+T_{1}\leq\mathfrak{t}_{n+1}\leq T_{0}+T_{1}\\\
\end{aligned}\right.$ (5.6)

where $\mathfrak{t}_{n+1}\equiv t-n(T_{0}+T_{1})\in[0,T_{0}+T_{1}]$ is the
time in the $(n+1)$-th cycle, and $z_{n}$ is replaced by $z=e^{ix}$ in the
first cycle, i.e. when $n=0$. For $\overline{z}_{{{\rm new}}}$ we simply have
$z_{n}\to\overline{z}_{n}$ in (5.6), where $\overline{z}_{n}$ (F.20) is given
by $z\to\overline{z}=e^{-ix}$ in (F.19), see (G.7) for the full expression. A
proof of the formula (5.6) using mathematical induction can be found in
Appendix G.1.

The rather complicated equation (5.6) simplifies greatly in the heating phase
due to properties of the fixed points. As $0<\eta<1$, at late stroboscopic
time (F.19) becomes

$\displaystyle\lim_{n\to\infty}z_{n}=\lim_{n\to\infty}\frac{(\gamma_{1}-\eta^{n}\gamma_{2})z-(1-\eta^{n})}{(1-\eta^{n})z-(\gamma_{2}-\eta^{n}\gamma_{1})}=\frac{\gamma_{1}z-1}{z-\frac{1}{\gamma_{1}}}=\gamma_{1}$
$\displaystyle(z\neq\gamma_{2})$ (5.7)

where we have used $\gamma_{1}\gamma_{2}=1$. For $z=\gamma_{2}$,
$z_{n}=\gamma_{2}$ by the definition of fixed point. Henceforth, we have

$\lim_{n\to\infty}z_{n}=\left\\{\begin{aligned} &\gamma_{1},\ \
z\neq\gamma_{2}\\\ &\gamma_{2},\ \ z=\gamma_{2}\end{aligned}\right.$ (5.8)

From the middle equation of (5.6), $H_{1}$ sends $e^{\frac{iT_{0}}{2}}z_{n}$
to $e^{-\frac{iT_{0}}{2}}z_{n+1}$ (see Appendix (G.3) for details), and the
Floquet dynamics can be summarised as the following iteration

$\displaystyle...\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}z_{n}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}z_{n}\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}z_{n+1}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}z_{n+1}\overset{H_{1}}{\longrightarrow}...$
(5.9)

As $z_{n}$ either approaches the stable fixed points $\gamma_{1}$ or stays at
the unstable fixed point $\gamma_{2}$ at late time (5.8), the recursion (5.9)
eventually becomes the repetition of the two fixed points

$\displaystyle...\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}\gamma_{1}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}\gamma_{1}\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}\gamma_{1}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}\gamma_{1}\overset{H_{1}}{\longrightarrow}...$
(5.10)
$\displaystyle...\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}\gamma_{2}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}\gamma_{2}\overset{H_{1}}{\longrightarrow}e^{-\frac{iT_{0}}{2}}\gamma_{2}\overset{H_{0}}{\longrightarrow}e^{\frac{iT_{0}}{2}}\gamma_{2}\overset{H_{1}}{\longrightarrow}...$
(5.11)

where we have used the definition of fixed point as points that leave (F.19)
invariant. The analysis (5.7)-(5.11) also work for the anti-holomorphic
coordinate with $z\to\overline{z}$ and $z_{n}\to\overline{z}_{n}$.

From (5.6), (5.10) (5.11), we see that the term ”fixed points” is meaningful
only in the stroboscopic sense; in general time, they undergo repetitive
motions on the unit circle in the complex plane and return to the same
location after completing each cycle. Thus, after the transient dies out, the
late-time behavior of Floquet CFT boils down to that of the two fixed points
$\gamma_{1}$ and $\gamma_{2}$ (or even one point, considering that
$\gamma_{2}$ is the complex conjugate of $\gamma_{1}$ in the time-reversal
case).

There is a second sense that the ”fixed points” are not really fixed: so far
we have used a principal sheet prescription for the logarithm, but continuity
in $t$ leads to additions of multiples of $\pi$ to $t_{\text{new}}$ compared
to $t_{n}$. In preparation for the investigation of the holographic dual, we
need to study the continuous time evolution of $t_{{\rm new}}$ and $x_{{\rm
new}}$. Using (5.6) and (G.7), we plot $t_{{{\rm new}}}$ and $x_{{{\rm new}}}$
as functions of $t$ for different values of $x$ in Figure 8, and further plot
constant-$t$ lines and constant-$x$ lines on the $(t_{{\rm new}},x_{{\rm
new}})$ plane in the first cycle in Figure 9. To better understand $t_{{{\rm
new}}}$ and $x_{{{\rm new}}}$, it is instructive to first plot
$\frac{1}{i}w_{{{\rm new}}}=\frac{1}{i}\log z_{{{\rm new}}}=t_{{{\rm
new}}}+x_{{{\rm new}}}$ and $\frac{1}{i}\overline{w}_{{{\rm
new}}}=\frac{1}{i}\log\overline{z}_{{{\rm new}}}=t_{{{\rm new}}}-x_{{{\rm
new}}}$, which we include in Appendix G.2 with discussions.

Figure 8: Plot of $t_{{{\rm new}}}$ and $x_{{{\rm new}}}$ at general $t$ in
the heating phase. Here we set $T_{0}=4$ and $T_{1}=8$. We plotted for
$0,\frac{\pi}{2},\frac{1}{i}\log\gamma_{1}\approx
1.76076,\frac{2\pi}{3},\frac{3\pi}{4}$ for the first 5 cycles of $t_{{{\rm
new}}}$ (as $t_{{{\rm new}}}$ is symmetric under $x\to-x$) and for
$x=-\frac{3\pi}{4},-\frac{2\pi}{3},\frac{1}{i}\log\gamma_{2}\approx-1.76076,-\frac{\pi}{6},0,\frac{\pi}{2},\frac{1}{i}\log\gamma_{1}\approx
1.76076,\frac{2\pi}{3},\frac{3\pi}{4}$ for the first two cycles of $x_{{{\rm
new}}}$.

From Figure 8 and Figure 9, it is evident that $t_{{{\rm new}}}$ and $x_{{{\rm
new}}}$ indeed converge to different curves as $t$ evolves. In particular,

  * •

At $|x|<\frac{1}{i}\log\gamma_{1}$, $x_{{{\rm new}}}$ converges to $x_{{{\rm
new}}}=0$, and $t_{{{\rm new}}}$ converges to the curve whose stroboscopic
value is $\frac{1}{i}\log\gamma_{1}$. This is because both the
$\frac{1}{i}w_{{{\rm new}}}$ and the $\frac{1}{i}\overline{w}_{{{\rm new}}}$
curve converge to that of the stable fixed point $\gamma_{1}$.

  * •

At $|x|=\frac{1}{i}\log\gamma_{1}$, $x_{{{\rm new}}}$ follows repetitive
motions with stroboscopic values $x_{{{\rm
new}}}=\pm\frac{1}{i}\log\gamma_{1}$; $t_{{{\rm new}}}$ follows a curve whose
stroboscopic value is $0$. This is because either the $\frac{1}{i}w_{{{\rm
new}}}$ or the $\frac{1}{i}\overline{w}_{{{\rm new}}}$ curve corresponds to
the unstable fixed point $\gamma_{2}$.202020$z_{n}$ and $\overline{z}_{n}$
cannot both be at the unstable fixed point $\gamma_{2}$, since intially they
are complex conjugates.

  * •

At $|x|>\frac{1}{i}\log\gamma_{1}$, $x_{{{\rm new}}}$ converges to $x_{{{\rm
new}}}=\pm\pi$ (these are the same point); $t_{{{\rm new}}}$ again converges
to the curve whose stroboscopic value is $\frac{1}{i}\log\gamma_{1}$, but is
shifted down by $\pi$ compared to the first case above. This is because either
the $\frac{1}{i}w_{{{\rm new}}}$ or the $\frac{1}{i}\overline{w}_{{{\rm
new}}}$ curve is on another Riemann sheet and is therefore $2\pi$ lower, see
Appendix G.2 for details. These subtle $\pi$ shifts were not spelled out in
previous work that focused on the stroboscopic dynamics Wen:2018agb ;
Fan:2019upv ; Wen:2020wee ; Han:2020kwp ; Fan:2020orx ; Wen:2022pyj .

Figure 9: Constant $t$ lines (blue) and constant-$x$ lines (red) in the
$(t_{{\rm new}},x_{{\rm new}})$-plane. Here we set $T_{0}=4$ and $T_{1}=8$,
and plot for the first cycle. At $0<t<2$ and $10<t<12$, the driving
Hamiltonian is $H_{0}$, constant-$t$ lines and constant-$x$ lines are
uniformly distributed straight lines. At $2<t<10$ the driving Hamiltonian is
$H_{1}$, constant-$t$ lines asymptote two triangles, whose tips are at
$x_{{\rm new}}=0$ and $x_{{\rm new}}=\pm\pi$, respectively (notice that
$x_{{\rm new}}$ is periodic: $x_{{\rm new}}\sim x_{{\rm new}}+2\pi$);
constant-$x$ lines converge to these two tips, depending on the range of $x$.
The dashed orange lines are the trajectories of the two fixed points.

##  6 The gravity dual of Floquet CFT

###  6.1 Gravity dual of Floquet CFT in heating phases

The dual gravitational picture can be established similarly to that of the SSD
quench case. The gravity dual for the ground and thermal state is pure AdS3
and BTZ black hole, respectively, whose metrics are given by (3.1) and (3.2),
with $t_{{\rm new}}$ and $x_{{\rm new}}$ obtained from (5.6) and (G.7). The
cutoff surface is also nonuniform in the radial coordinate, like in (3.3) in
the SSD quench case.

From Figure 8, the gravity dual of Floquet CFT in the $(t_{{\rm new}},x_{{\rm
new}},r_{{\rm new}})$ coordinate can thus be deduced as follows: as $t$
evolves, constant $x$ lines on the $(t_{{\rm new}},x_{{\rm new}})$ cylinder
converge to $x_{{\rm new}}=0$ or $x_{{\rm new}}=\pi$, or stay at
$\pm\frac{1}{i}\log\gamma_{1}$ as evolving forward in $t_{{\rm new}}$,
depending on whether $|x|$ are smaller than, greater than or equal to
$\frac{1}{i}\log\gamma_{1}$. Since $t_{{\rm new}}\to\infty$ as $t\to\infty$
for any $x\in[-\pi,\pi]$, there are no new horizons in the bulk anymore.

Next, we study the evolution of the BTZ horizon $r_{{\rm new}}=r_{+}$ in the
$(t,x,r)$ coordinates. Using the foliation in Appendix E.2, the stroboscopic
dynamics of the horizon are shown in Figure 10.

Figure 10: Time evolution of the BTZ black hole horizon. We chose $r_{+}=3$,
$T_{0}=2$ and $T_{1}=8$. We plotted the cutoff surface at $n=0,1,2$. We used
the ${\rm arctan}$ function to make the plot, so the boundary is at
$r=\frac{\pi}{2}$, which is plotted in a dashed gray line.

We observe that the horizon approaches the boundary at the two fixed points,
i.e. $x=\pm\frac{1}{i}\log\gamma_{1}$, developing two peaks. These peaks
originate from the prefactor $\sqrt{\frac{\partial w_{n}}{\partial
w}\frac{\partial\overline{w}_{n}}{\partial\overline{w}}}$ in (E.2), which
grows exponentially at the two heating points, see (E.2). In deBoer:2023lrd ,
the evolution of BTZ horizon in the slow-driving limit (i.e. when
$v(x)=1+O(\epsilon)$ in (A.22)) for gravity dual of generally deformed CFT was
studied,212121Assuming that these driven CFTs obey the slow-driving condition
at least within a range of parameters. and it was concluded that the horizon
approaches the conformal boundary at the heating point(s). Here, our Floquet
setup is not in the slow-driving limit. This provides evidence that the
phenomenon of BTZ horizons evolving toward the boundary at the fixed points is
likely generic in heating phases.

For CFT prepared in the thermal states on a circle, there are energy peaks
forming at the unstable fixed points. To see this, note first that in a
holographic CFT the thermal stress tensor can be approximated by that of a
high energy pure state with dimension $h$ as in the thermal SSD quench case
Goto:2021sqx , which is $\langle T(z_{n})\rangle=\frac{h}{z_{n}^{2}}$ on the
complex $z$-plane. Transforming back to the $w$-cylinder,

$\displaystyle\langle T(w)\rangle=\Big{(}\frac{\partial z}{\partial
w}\Big{)}^{2}\Big{(}\frac{\partial z_{n}}{\partial
z}\Big{)}^{2}\frac{h}{z_{n}^{2}}-\frac{c}{24}$ (6.1)

where the last term comes from ${\rm Sch}(z,w)$. The remaining analysis
parallels that in Fan:2019upv : the Jacobian factor $\big{(}\frac{\partial
z_{n}}{\partial z}\big{)}^{2}$ determines the chiral energy peak. The anti-
chiral peak is specified by
$\big{(}\frac{\partial\overline{z}_{n}}{\partial\overline{z}}\big{)}^{2}$ in
$\langle\overline{T}(\overline{w})\rangle$. The two energy peaks locate at
$x=\pm\frac{1}{i}\log\gamma_{1}$, which are positions at which the black hole
horizon approaches the boundary in Figure 10. This is because both the horizon
peaks and energy peaks originate from $\frac{\partial z_{n}}{\partial z}$ or
equivalently $\frac{\partial w_{n}}{\partial w}$ as well as their anti-
holomorphic counterparts in the $SL(2)$ transformations. They are big at the
unstable fixed point, since in their neighborhood there is a strong
sensitivity to the initial data. This is in contrast with the vicinity of the
stable fixed point that cools down because it is insensitive to small changes
in the initial data.

###  6.2 Non-heating phases and phase transition

At the phase transition, the two fixed points $\gamma_{1}$ and $\gamma_{2}$
merge into one, which is $\gamma=1$ in the time-reversal symmetric case. In
the non-heating phase, $z_{{\rm new}}$ shows oscillatory behavior instead of
approaching the fixed points (which are no longer on the unit circle
Fan:2019upv ). Therefore, one would expect $t_{{\rm new}}$ to show stronger
$x$-dependence. In both cases, $t_{{\rm new}}$ increases for different
$x\in[0,\pi]$ just as in the heating phases: when $H_{0}$ acts as Hamiltonian,
$t_{{\rm new}}$ increases linearly; when $H_{1}$ drives, it is also
straightforward to show from (2.8) that $t_{{\rm new}}$ is monotonically
increasing. Therefore, $t_{{\rm new}}\to\infty$ at any initial $x\in[0,\pi]$
as $t\to\infty$. Consequently, there are no new horizons in the bulk in non-
heating phases or at the phase transition either.

##  7 Dynamics of entanglement entropy

In this section, we study the holographic entanglement entropy of a single
interval $A=[x_{1},x_{2}]$ at a constant $t$-slice. To this end, we again work
in the $(t_{{\rm new}},x_{{\rm new}},r_{{\rm new}})$ geometry, and then
perform a coordinate transformation back to $(t,x,r)$. At time $t$, the
endpoints of $A=[x_{1},x_{2}]$ are obtained from (2.6), (2.7), (2.8),

$\displaystyle w_{{{\rm new}},j}=\frac{1}{2i}(t_{{{\rm new}},j}+x_{{{\rm
new}},j})$ $\displaystyle\overline{w}_{{{\rm new}},j}=\frac{1}{2i}(t_{{{\rm
new}},j}-x_{{{\rm new}},j})$ $\displaystyle(j=1,2)$ (7.1)

The holographic entanglement entropy of a single interval $A$ is given by the
length of the spacelike geodesic anchored on $\partial A=\\{(t_{{{\rm
new}},1},x_{{{\rm new}},1}),(t_{{{\rm new}},2},x_{{{\rm new}},2})\\}$ on the
boundary Ryu:2006bv ; Ryu:2006ef :

$\displaystyle S_{A}=\frac{\mathcal{L}}{4G_{N}}=\frac{c}{6}\mathcal{L}$ (7.2)

where we have used the Brown-Henneaux central charge, $c=3/2G_{N}$
Brown:1986nw . Therefore, to calculate the holographic entanglement entropy,
one needs to obtain the geodesic length $\mathcal{L}$, using the temporal and
angular separation $\Delta t_{{{\rm new}}}=t_{{{\rm new}},2}-t_{{{\rm
new}},1}$ and $\Delta x_{{{\rm new}}}=x_{{{\rm new}},2}-x_{{{\rm new}},1}$ as
inputs, and then transform back to $(t,x,r)$. Note that in general $\Delta
t_{{{\rm new}}}\neq 0$, so our calculation is not of equal time (”time” as
$t_{{\rm new}}$). See Figure 11 for a demonstration.

Figure 11: A demonstration of non-equal-$t_{{\rm new}}$ spacelike geodesics,
plotted as green curves in the bulk anchoring on the two endpoints of the
subregion $A$ on the boundary constant-$t$ lines. These spacelike geodesics
compute the holographic entanglement entropy.

The length of the spacelike geodesic anchoring on the boundary is infinite,
therefore, we need to regulate it by imposing a cutoff at some large radius
$r_{{\rm new}}=R_{{\rm new}}$, and adopt the regulated length
$\mathcal{L}_{{\rm reg}}$. As is discussed in Subsection 3.1, the cutoff
surface is non-uniform in the radial direction, with $R_{{\rm
new}}(t_{\text{new}},x_{\text{new}})$ given by (3.3). See Figure 2 for a plot
of the cutoff surface.

###  7.1 Gravity dual of SSD: ground state

####  7.1.1 Holographic computation

The metric (3.1) leads to two conserved quantities for geodesics, $E=(r_{{\rm
new}}^{2}+1)\frac{dt_{{\rm new}}}{d\lambda}$ and $L=r_{{\rm
new}}^{2}\frac{dx_{{\rm new}}}{d\lambda}$, where $\lambda$ is the affine
parameter along the geodesic. In Hubeny:2012ry , it was found that in pure
AdS3, the temporal and angular distances are given in terms of the two
conserved quantities by

$\displaystyle\cos(\Delta t_{{\rm new}})$
$\displaystyle=\frac{L^{2}-E^{2}+1}{\sqrt{(L^{2}-E^{2}+1)^{2}+4E^{2}}}$ (7.3)
$\displaystyle\cos(\Delta x_{{\rm new}})$
$\displaystyle=\frac{L^{2}-E^{2}-1}{\sqrt{(L^{2}-E^{2}-1)^{2}+4L^{2}}}$ (7.4)

From which one can solve for $E$ and $L$. Using these $E$ and $L$, we compute
the regulated length of the spacelike geodesic Hubeny:2012ry  in Appendix
D.2.1

$\displaystyle\mathcal{L}_{{\rm reg}}$
$\displaystyle=-\log\sqrt{(E^{2}-L^{2}+1)^{2}+4L^{2}}+\log(4R_{{\rm
new,1}}R_{{\rm new,2}})$ (7.5)
$\displaystyle=\log\left[4\Big{|}\sinh\big{(}\frac{1}{2}(w_{{{\rm
new}},1}-w_{{{\rm new}},2})\big{)}\
\sinh\big{(}\frac{1}{2}(\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2})\big{)}\Big{|}\ \right]+\log(R_{{\rm new,1}}R_{{\rm new,2}})\,,$
(7.6)

where $\log(4R_{{\rm new,1}}R_{{\rm new,2}})$ is the universal diverging
piece. From (3.3), we have for the large-radius cutoff

$\displaystyle R_{{\rm new},i}=\frac{R_{i}}{\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}}}$ (7.7)

where $i=1,2$ labels the two large-radius cutoffs near the two endpoints of
the geodesic. With (7.7), the holographic entanglement entropy is:222222Note
that as $w_{{\rm new}}$ is purely imaginary, the terms inside the logarithm
are in fact $\sin$ functions of $t_{{\rm new}}$ and $x_{{\rm new}}$.

$\displaystyle S_{A}=\frac{\mathcal{L}_{{\rm
reg}}}{4G_{N}}=\frac{c}{6}\log\left[4\frac{\Big{|}\sinh\big{(}\frac{1}{2}(w_{{{\rm
new}},1}-w_{{{\rm new}},2})\big{)}\
\sinh\big{(}\frac{1}{2}(\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2})\big{)}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]$ (7.8)

where we have introduced the regularization parameter
$\epsilon_{i}=R_{i}^{-1}$. One can easily verify that $S_{A}=S_{A^{c}}$, which
is a property of pure states. An example of the spacelike geodesic computing
the holographic entanglement entropy is in Figure 11.

####  7.1.2 Match with field theory result

In 1+1 dimensional CFT, the entanglement entropy of a single interval
$A=[x_{1},x_{2}]$ can be evaluated from the two-point function of twist
operators Calabrese:2004eu ,

$\displaystyle S_{A}=-\lim_{n\to 1}\frac{\partial}{\partial n}{\rm
Tr}\rho_{A}^{n}$ $\displaystyle{\rm Tr}\rho_{A}^{n}$
$\displaystyle\propto\langle\mathcal{T}_{n}(w_{1},\overline{w}_{1})\mathcal{T}_{-n}(w_{2},\overline{w}_{2})\rangle$
(7.9)

with dimension $h=\overline{h}=\frac{c}{24}(n-\frac{1}{n})$. The time
evolution of the two-point function is captured by the $SL(2)$ transformation
$(w,\overline{w})\to(w_{{\rm new}},\overline{w}_{{\rm new}})$:

$\displaystyle\langle\mathcal{T}_{n}(w_{1},\overline{w}_{1})\mathcal{T}_{-n}(w_{2},\overline{w}_{2})\rangle=\prod_{i=1}^{2}\Big{(}\frac{\partial
w_{{\rm new},i}}{\partial w}\Big{)}^{h}\Big{(}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}}\Big{)}^{h}\langle\mathcal{T}_{n}(w_{{{\rm
new}},1},\overline{w}_{{{\rm new}},1})\mathcal{T}_{-n}(w_{{{\rm
new}},2},\overline{w}_{{{\rm new}},2})\rangle$ (7.10)

The two-point function $\langle\mathcal{T}_{n}(w_{1,{{\rm
new}}},\overline{w}_{1,{{\rm new}}})\mathcal{T}_{-n}(w_{2,{{\rm
new}}},\overline{w}_{2,{{\rm new}}})\rangle$ is evaluated following the
methods in Calabrese:2004eu  but paying close attention to the difference
between holomorphic and antiholomorphic parts (see Appendix H for details),

$\displaystyle\langle\mathcal{T}_{n}(w_{{{\rm new}},1},\overline{w}_{{{\rm
new}},1})\mathcal{T}_{-n}(w_{{{\rm new}},2},\overline{w}_{{{\rm
new}},2})\rangle$ $\displaystyle=$
$\displaystyle\Big{[}4\sinh\Big{(}\frac{1}{2}(w_{{{\rm new}},1}-w_{{{\rm
new}},2})\Big{)}\ \sinh\Big{(}\frac{1}{2}(\overline{w}_{{{\rm
new}},1}-\overline{w}_{{{\rm new}},2})\Big{)}\Big{]}^{-2h}\,,$ (7.11)

where

$\displaystyle w_{{\rm new}}=\tau_{{\rm new}}+ix_{{\rm new}}$
$\displaystyle\overline{w}_{{\rm new}}=\tau_{{\rm new}}-ix_{{\rm new}}\,.$
(7.12)

Substituting (7.11) into (7.10) and then into (7.9) and performing the
analytic continuation $\tau_{{\rm new}}\to it_{{\rm new}}$, the CFT
entanglement entropy is then

$\displaystyle S_{A}$
$\displaystyle=\frac{c}{6}\log\left[\frac{4\Big{|}\sinh\big{(}\frac{1}{2}(w_{{{\rm
new}},1}-w_{{{\rm new}},2})\big{)}\
\sinh\big{(}\frac{1}{2}(\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2})\big{)}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]$ (7.13)

where we have introduced the UV cutoff $\epsilon_{i}$. The terms multiplying
$\epsilon_{i}$ in the denominator in (7.13) come from the Jacobian factors in
(7.10). The field theory calculation (7.13) precisely matches our earlier
holographic result (7.8).232323Note, however, that the field theory result is
valid for generic CFTs, while our holographic result is limited to
holographic, i.e. large $c$, large gap CFTs only.

###  7.2 Gravity dual of SSD: thermal state

####  7.2.1 Holographic computations

The metric (3.2) has two conserved quantities, $E=(r_{{\rm
new}}^{2}-r_{+}^{2})\frac{dt_{{\rm new}}}{d\lambda}$ and $L=r_{{\rm
new}}^{2}\frac{dx_{{\rm new}}}{d\lambda}$, where $\lambda$ is the affine
parameter along the geodesic. We will focus on the region outside the BTZ
horizon, $r>r_{+}$. In Appendix D.2.2, we calculated the temporal and angular
separation of the two endpoints of the spacelike geodesic

$\displaystyle\cosh(r_{+}\Delta t_{{\rm new}})$
$\displaystyle=\frac{L^{2}-E^{2}-r_{+}^{2}}{\sqrt{(L^{2}-E^{2}-r_{+}^{2})^{2}-4r_{+}^{2}E^{2}}}$
(7.14) $\displaystyle\cosh(r_{+}\Delta x_{{\rm new}})$
$\displaystyle=\frac{L^{2}-E^{2}+r_{+}^{2}}{\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}}$
(7.15)

where $L\geq E+r_{+}$ so that the spacelike geodesic does not enter the BTZ
horizon (see Appendix D.2.2 for details).242424Note that if one sets the BTZ
black hole radius $r_{+}^{2}=-1$ and hence replaces $\cosh$ with $\cos$ in
(7.14) and (7.15), one recovers (7.3) and (7.4) in pure AdS3, respectively.
Therefore one can solve for $E$ and $L$. Using these $E$ and $L$, we computed
the regulated length of the spacelike geodesic in Appendix D.2.2.

$\displaystyle\mathcal{L}_{{\rm reg}}$
$\displaystyle=-\log\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}+\log(4R_{{\rm
new,1}}R_{{\rm new,2}})$ (7.16)
$\displaystyle=\log\left[\Big{(}\frac{2}{r_{+}}\Big{)}^{2}\Big{|}\sin\Big(r_{+}\frac{w_{{{\rm
new}},1}-w_{{{\rm new}},2}}{2}\Big{missing})\
\sin\Big(r_{+}\frac{\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2}}{2}\Big{missing})\Big{|}\ \right]+\log(R_{{\rm new,1}}R_{{\rm
new,2}})$ (7.17)

For the BTZ black hole, the radius and inverse temperature are related via
$r_{+}={2\pi/\beta}$. The large-radius cutoff is the same as (7.7). The
holographic entanglement entropy is:252525Note that as $w_{{\rm new}}$ is
purely imaginary, the terms inside the logarithm are in fact $\sinh$ functions
of $t_{{\rm new}}$ and $x_{{\rm new}}$.

$\displaystyle S_{A}^{(1)}=\frac{\mathcal{L}_{{\rm
reg}}}{4G_{N}}=\frac{c}{6}\log\left[\frac{\big{(}\frac{\beta}{\pi}\big{)}^{2}\Big{|}\sin\big(\frac{\pi}{\beta}(w_{{{\rm
new}},1}-w_{{{\rm new}},2})\big{missing})\
\sin\big(\frac{\pi}{\beta}(\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2})\big{missing})\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]$ (7.18)

Where the meaning of the superscript $(1)$ will be clear in later subsections.
$S_{A}^{(1)}$ and $S_{A^{c}}^{(1)}$ are not equal, as the state is no longer
pure but thermal. An example of the spacelike geodesic computing the
holographic entanglement entropy is in the left plot of Figure 13.

####  7.2.2 Non-equal-time entanglement plateau

When the state of the total system is a thermal density matrix, $S_{A}$ and
$S_{A^{c}}$ are not equal, as the minimal surfaces $M_{A}$ and $M_{A^{c}}$
computing them encircle complementary parts of the black hole horizon
Ryu:2006bv ; Ryu:2006ef  (see Figure 12). Their difference $\Delta S_{A}$ is
bounded from above by the Araki-Lieb inequality Araki:1970ba ,

$\displaystyle|\Delta S_{A}|=|S_{A}-S_{A^{c}}|\leq S_{A\cup A^{c}}=S_{\rho}$
(7.19)

where $S_{\rho}$ is the entropy of the total density matrix, which in our case
is the entropy of the BTZ black hole $S_{\rho}=\frac{c}{3}\pi
r_{+}=\frac{2\pi^{2}c}{3\beta}$. If we used the geodesics on Figure 12 to
determine the entanglement entropy of the CFT at finite temperature using the
static BTZ geometry, for large $A$ we would get a violation of (7.19). The way
out is to consider another minimal surface besides $M_{A}$ that also satisfies
the homology constraint Headrick:2007km , $M_{A^{c}}\cup\mathbb{S}_{\rm hor}$
with $\mathbb{S}_{\rm hor}$ the bifurcation surface of the black hole. Hence

$\begin{split}S_{A}=\frac{1}{4G_{N}}{\rm
min}\\{\mathcal{L}(M_{A}),\mathcal{L}(M_{A^{c}})+\mathcal{L}(\mathbb{S}_{\rm
hor})\\}\,,\end{split}$ (7.20)

where $\mathcal{L}(\mathbb{S}_{\rm hor})=2\pi r_{+}$. When
$M_{A^{c}}\cup\mathbb{S}_{\rm hor}$ is minimal in (7.20), the Araki-Lieb
inequality is saturated, which ref. Hubeny:2013gta  called the plateau
phenomenon.

Figure 12: A demonstration of the minimal surface for a subregion $A$ and its
complementary $A^{c}$.

To study whether our non-equal time holographic entanglement entropy (7.18)
shows the plateau phenomenon, let us check if (7.18) obeys the Araki-Lieb
inequality (7.19),

$\displaystyle\Delta
S_{A}^{(1)}=|S_{A}^{(1)}-S_{A^{c}}^{(1)}|=\frac{c}{6}\log\left[\sinh^{2}\Big{(}\frac{\pi}{\beta}\Delta
t_{{\rm new}}\Big{)}\ {\rm csch}\Big{(}\frac{\pi}{\beta}(\Delta t_{{\rm
new}}+2\pi)\Big{)}\ {\rm csch}\Big{(}\frac{\pi}{\beta}(\Delta t_{{\rm
new}}-2\pi)\Big{)}\right]$ (7.21)

Recall that in (2.12), as $t\to\infty$, the two endpoints $(t_{{\rm
new},1},x_{{\rm new},1})$ and $(t_{{\rm new},2},x_{{\rm new},2})$ flow to the
top tip of the triangle, $(t_{{\rm new}},x_{{\rm new}})=(\pi,\pi)$. Therefore,
$\Delta t_{{\rm new}}\to 0$ in (7.21) as $t$ evolves. In this limit $\Delta
S_{A}^{(1)}$ diverges, which means that (7.21) violates the Araki-Lieb
inequality (7.19). Consequently, our non-equal time holographic entanglement
entropy (7.18) also possesses the plateau phenomenon. The non-equal-time
generalization of Hubeny:2013gta  is straightforward: in addition to the
connected minimal surface (see the red curve in the right plot of Figure 13)
that computes (7.18), there is also a disconnected minimal surface, given by
the union of the minimal surface for $A^{c}$ (i.e. $\Delta t_{{\rm new}}$
stays the same while $\Delta x_{{\rm new}}$ becomes the complementary value
$2\pi-\Delta x_{{\rm new}}$, see the green curve in the right plot of Figure
13) and the bifurcation surface of the event horizon.262626The bifurcation
surface is minimal in all static black hole geometries Hubeny:2013gta . The
holographic entanglement entropy is then computed by,

$\displaystyle S_{A}={\rm min}\\{S_{A}^{(1)},S_{A}^{(2)}\\}$ (7.22)

where $S_{A}^{(1)}$ is given by (7.18), and

$\displaystyle S_{A}^{(2)}$
$\displaystyle=\frac{2\pi^{2}c}{3\beta}+\frac{c}{6}\log\left[\Big{(}\frac{\beta}{\pi}\Big{)}\frac{\big{|}\sinh\big{[}\frac{\pi}{\beta}((t_{{{\rm
new}},1}-t_{{{\rm new}},2})+(2\pi-(x_{{{\rm new}},1}-x_{{{\rm
new}},2}))\big{]}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\epsilon_{i}}}\right]$
$\displaystyle+\frac{c}{6}\log\left[\Big{(}\frac{\beta}{\pi}\Big{)}\frac{\big{|}\sinh\big{[}\frac{\pi}{\beta}((t_{{{\rm
new}},1}-t_{{{\rm new}},2})-(2\pi-(x_{{{\rm new}},1}-x_{{{\rm
new}},2}))\big{]}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}\epsilon_{i}}}\right]\,.$ (7.23)

When $S_{A}^{(2)}$ is minimal, the Araki-Lieb inequality (7.19) is saturated
automatically.

The plateau phenomenon in thermal SSD quench was first proposed in
Goto:2021sqx . We remark that while our results are qualitatively similar, our
holographic construction is quantitatively different, as the holographic
entanglement entropy is computed by spacelike geodesics anchoring on the
boundary at non-equal times.

####  7.2.3 Holographic cooling/heating effect

To further understand the time-evolution of holographic entanglement entropy,
let us first note that the endpoints of the interval $t_{{\rm new},i}$ and
$x_{{\rm new},i}$ $(i=1,2)$ can flow to the top tip of the triangle (2.12)
from the same or different side, depending on whether $A$ includes the fixed
point $x=0$ or not.

  * •

If $A$ does not include $x=0$, $(t_{{\rm new},1},x_{{\rm new},1})$ and
$(t_{{\rm new},2},x_{{\rm new},2})$ flow to the top tip of the triangle from
the same side, and $|x_{{{\rm new}},1}-x_{{{\rm new}},2}|\to 0$ in (7.18). See
the left plot of Figure 13. Therefore, $S_{A}^{(1)}$ is always smaller than
$S_{A}^{(2)}$, so we have $S_{A}=S_{A}^{(1)}$ at all $t>0$. At late-time, one
can show Goto:2021sqx  that $S_{A}$ is given by the vacuum entanglement
entropy,

$\displaystyle
S_{A}=S_{A}^{(1)}\to\frac{c}{3}\log[2\Big{(}\sin\frac{1}{2}(x_{2}-x_{1})\Big{)}\Big{]}$
(7.24)

  * •

If $A$ includes $x=0$, $(t_{{\rm new},1},x_{{\rm new},1})$ and $(t_{{\rm
new},2},x_{{\rm new},2})$ flow to the top tip of the triangle from different
sides, making $|x_{{{\rm new}},1}-x_{{{\rm new}},2}|\to 2\pi$ in (7.18). At
some point, therefore, the entanglement entropy switches from $S_{A}^{(1)}$ to
$S_{A}^{(2)}$. See the right plot of Figure 13. At late time, the second and
third terms in (7.23) become much smaller than the first term,272727For this
the time has to compensate the smallness of the cutoff $\epsilon_{i}$. so that
$S_{A}$ is dominated by the thermal entanglement entropy,

$\displaystyle S_{A}=S_{A}^{(2)}\to S_{\rho}=\frac{2\pi^{2}c}{3\beta}\,.$
(7.25)

Figure 13: Non-equal time spacelike geodesics in BTZ geometry. The two
geodesic endpoints flow towards the top tip of the triangle $(\pi,\pi)$ as $t$
evolves, from either the same (left) or different (right) side. In the latter
case, at late time, the minimal surfaces computing holographic entanglement
entropy switches from the red surface homologous to $A$ to the union of the
green surface homologous to $A^{c}$ and the bifurcation surface of the black
hole horizon (also coloured in green).

.

We conclude that whether the entanglement entropy cools down to the vacuum
value or heats up and eventually reaches thermal value has a simple geometric
interpretation in the bulk, which is whether the spacelike geodesic computing
the holographic entanglement entropy winds around the BTZ black hole or not at
late time. Again, (7.24) and (7.25) match the results in Goto:2021sqx
qualitatively, albeit the setup is quantitatively different.

####  7.2.4 Comparison with field theory result on a line

Computing the entanglement entropy of 1+1 dimensional CFT on a finite interval
at finite temperature $\beta$ is difficult. However, one can compare the
holographic result (7.18) with that of the thermal state of CFT2 on a line.
This CFT result can be obtained by Wick rotating $w_{{\rm new}}\to iw_{{\rm
new}}$ and rescaling $2\pi\to\beta$ the cylinder that computes (7.13) (see
Appendix H for details),

$\displaystyle S_{A}=\frac{\mathcal{L}_{{\rm
reg}}}{4G_{N}}=\frac{c}{6}\log\left[\frac{\big{(}\frac{\beta}{\pi}\big{)}^{2}\Big{|}\sin\big(\frac{\pi}{\beta}(w_{{{\rm
new}},1}-w_{{{\rm new}},2})\big{missing})\
\sin\big(\frac{\pi}{\beta}(\overline{w}_{{{\rm new}},1}-\overline{w}_{{{\rm
new}},2})\big{missing})\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial w_{{\rm
new},i}}{\partial w_{i}}\frac{\partial\overline{w}_{{\rm
new},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]\,.$ (7.26)

This result matches with the holographic result $S_{A}^{(1)}$ before the
switch in the dominant extremal surface (7.23). In Asplund:2014coa , it is
explained that the coincidence of (7.18) and (7.26) is due to properties of
large-$c$ Virasoro identity blocks arising from a CFT calculation obtaining
entanglement entropy from four point functions with two heavy and two light
operators of the cyclic orbifold CFT${}^{n}/\mathbb{Z}_{n}$. In particular,
the identity block in the dominant channel corresponds to the spacelike
geodesic computing (7.18).

###  7.3 Gravity dual of Floquet CFT

We have seen from Figure 9 that points with
$x\neq\pm\frac{1}{i}\log\gamma_{1}$ quickly converges to $x_{{\rm new}}=0$ or
$\pm\pi$ on the $(t_{{\rm new}},x_{{\rm new}})$ cylinder while evolving
forward in $t_{{\rm new}}$. Therefore, to study the time evolution of
holographic entanglement entropy, it is convenient to work in the stroboscopic
time, and the general-time results follow. Adopting the same methods in
Subsection 7.1.1 and 7.2.1, we find the holographic entanglement entropy at
$t=n(T_{0}+T_{1})$ for ground state

$\displaystyle
S_{A}=\frac{c}{6}\log\left[\frac{4\Big{|}\sinh\big{(}\frac{1}{2}(w_{{{n}},1}-w_{{{n}},2})\big{)}\
\sinh\big{(}\frac{1}{2}(\overline{w}_{{{n}},1}-\overline{w}_{{{n}},2})\big{)}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial
w_{{n},i}}{\partial
w_{i}}\frac{\partial\overline{w}_{{n},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]$
(7.27)

and thermal state

$\displaystyle
S_{A}^{(1)}=\frac{c}{6}\log\left[\frac{\big{(}\frac{\beta}{\pi}\big{)}^{2}\Big{|}\sin\big(\frac{\pi}{\beta}(w_{{{n}},1}-w_{{{n}},2})\big{missing})\
\sinh\big{(}\frac{\pi}{\beta}(\overline{w}_{{{n}},1}-\overline{w}_{{{n}},2})\big{)}\Big{|}}{\prod_{i=1}^{2}\sqrt{\frac{\partial
w_{{n},i}}{\partial
w_{i}}\frac{\partial\overline{w}_{{n},i}}{\partial\overline{w}_{i}}}\epsilon_{i}}\right]$
(7.28)

Using the approach in Subsection 7.1.2, one can find agreements between the
ground state entanglement entropy (7.27) and the corresponding CFT result. The
thermal state entanglement entropy (7.28) matches with that of field theory on
a line at finite temperature $\beta^{-1}$ in Wen:2022pyj  exactly, for reasons
explained in Asplund:2014coa  and re-iterated in Subsection 7.2.4. Therefore,
the cooling/heating effects in Wen:2022pyj  can be reproduced using
holography. In what follows, we discuss the geometric understanding of the
cooling/heating effects like in Subsection 7.2.3.

####  7.3.1 Holographic cooling/heating effect

For a given subregion $A=[x_{1},x_{2}]$, the two endpoints are mapped to two
distinct points $(w_{n,1},\overline{w}_{n,1})$ and
$(w_{n,2},\overline{w}_{n,2})$ or equivalently $(t_{n,1},x_{n,1})$ and
$(t_{n,2},x_{n,2})$ on the $(t_{{\rm new}},x_{{\rm new}})$ cylinder at
stroboscopic time, and the holographic entanglement entropy is computed by the
non-equal-time spacelike geodesic anchoring at these two points, resulting in
(7.28). The time evolution of holographic entanglement entropy then boils down
to those of $(t_{n,1},x_{n,1})$ and $(t_{n,2},x_{n,2})$, which was discussed
in Section 6.1: for $|x|\neq\frac{1}{i}\log\gamma_{1}$, $(t_{n},x_{n})$ flows
to either $\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ or
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$, depending on whether
$|x|<\frac{1}{i}\log\gamma_{1}$ or $\frac{1}{i}\log\gamma_{1}<|x|<\pi$. This
is already evident in the first cycle, see Figure 9. At general time, point
$\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ and
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$ evolves forward in $t_{{\rm
new}}$ along the lines $x_{{\rm new}}=0$ or $x_{{\rm new}}=\pi$. There are
three different possible cases:

  * •

$A=[x_{1},x_{2}]$ includes none of the two fixed points
$x=\frac{1}{i}\log\gamma_{1}$ or $x=\frac{1}{i}\log\gamma_{2}$. In that case,
$x_{n,1}$ and $x_{n,2}$ will both flow to
$\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ or
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$, depending on whether
$|x_{j}|<\frac{1}{i}\log\gamma_{1}$ or $|x_{j}|>\frac{1}{i}\log\gamma_{1}$
$(j=1,2)$. The spacelike geodesic computing the holographic entanglement
entropy then shrinks to the neighborhood of
$\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ or
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$ at stroboscopic time,
resulting in $\Delta w_{n}\to 0$ and $\Delta\overline{w}_{n}\to 0$. Plugging
in (7.28), it can be shown that the entanglement entropy equals that of the
vacuum entanglement entropy Wen:2022pyj , and the system is said to be in the
cooling region.

  * •

$A=[x_{1},x_{2}]$ includes one of the two fixed points
$x=\frac{1}{i}\log\gamma_{1}$ or $x=\frac{1}{i}\log\gamma_{2}$. In that case,
one of the two $x_{n,j}$ will flow to the point
$\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$, while the other instead to the
point $\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$. The holographic
entanglement entropy is then computed by the non-equal-time spacelike geodesic
ending at point $\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ and
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$, thus $\Delta t_{n}=\pi$,
and $\Delta x_{n}=\pm\pi$, leading to

$\displaystyle\frac{1}{i}\Delta w_{n}=\Delta t_{n}+\Delta x_{n}=2\pi$
$\displaystyle\frac{1}{i}\Delta\overline{w}_{n}=\Delta t_{n}-\Delta x_{n}=0$
(7.29)

or

$\displaystyle\frac{1}{i}\Delta w_{n}=\Delta t_{n}+\Delta x_{n}=0$
$\displaystyle\frac{1}{i}\Delta\overline{w}_{n}=\Delta t_{n}-\Delta x_{n}=\pi$
(7.30)

Substituting in (7.28), one can show that the entanglement entropy grows
linearly with slope $\frac{c}{6}\log\frac{1}{\eta}$ Wen:2022pyj , and the
system is said to be in the heating region.

Note that it is impossible to have $\frac{1}{i}\Delta
w_{n}=\frac{1}{i}\Delta\overline{w}_{n}=\pi$, as the $-\pi$ in
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,0\big{)}$ originates from either
$\frac{1}{i}w_{{\rm new}}$ or $\frac{1}{i}\overline{w}_{{\rm new}}$ being
$2\pi$ lower. There does not exist an $x$ where both would be $2\pi$ lower
(see Appendix G.2). Equivalently speaking, $x_{1},x_{2}$ cannot be at the two
different sides of both $\frac{1}{i}\log\gamma_{1}$ and
$\frac{1}{i}\log\gamma_{2}$. Thus, the heating effect is induced by either the
chiral or the anti-chiral component of the CFT; these two parts cannot both
heat.

Another interesting feature of the heating case is that as $\Delta x_{n}$
enclose half of the $(t_{{\rm new}},x_{{\rm new}})$ cylinder, the extremal
geodesic computing holographic entanglement entropy will not switch to the
disconnected one, and $S_{A}$ will always be computed by $S_{A}^{(1)}$. This
is in contrast to the thermal SSD quench case in Subsection 7.2.3, where the
spacelike geodesic computing holographic entanglement entropy winds the black
hole at late time.

  * •

$A=[x_{1},x_{2}]$ includes both fixed points. In that case, again, $x_{n,1}$
and $x_{n,2}$ will both flow to $\big{(}\frac{1}{i}\log\gamma_{1},0\big{)}$ or
$\big{(}\frac{1}{i}\log\gamma_{1}-\pi,\pi\big{)}$ depending on $|x_{j}|$, and
the system is again in the cooling region.

We conclude that whether the entanglement entropy cools down to the vacuum
value or heats up and grows linearly has a simple geometric interpretation in
the bulk, which is whether the spacelike geodesic computing the holographic
entanglement entropy encloses half of the BTZ black hole or not at late time.
The difference between heating in Floquet CFT and in SSD quench 7.2.3 is
because in the latter case, one injects energy into the CFT only once at
$t=0$, and then allows it to equilibrate. That the extremal surface winds the
black hole at late time and the entanglement entropy is computed by the
thermal entropy is a signature of this equilibration; this could perhaps be
made more precise. In Floquet CFT, however, one keeps adding energy to the CFT
at each cycle by switching Hamiltonian from $H_{0}$ to $H_{1}$ (therefore the
energy increases exponentially Wen:2022pyj ), so that the CFT will never reach
equilibrium, and entanglement keeps growing.

##  8 Conclusions and outlook

In this work, we studied the gravitational dual of SSD CFT and Floquet CFT
prepared in both vacuum and thermal states with periodic boundary conditions.
We found that the holographic dual of the non-trivial time evolution of SSD
CFT in the ground and thermal states can be understood as pure AdS3 and BTZ
black hole spacetime with novel foliations, respectively. We further
discovered that new horizons are presented in the bulk in both cases, which
are given by equations (3.8) and (3.13), and plotted in Figure 3 and 4,
respectively. We then argued that bulk operators outside the BTZ horizon can
be reconstructed regardless of whether they are inside or outside the new
horizons. Subsequently, we studied the gravity dual of Floquet CFT after
working out Floquet dynamics in general time, and concluded that there are no
new horizons in the bulk anymore. Lastly, we computed the holographic
entanglement entropy in the gravity dual of SSD and Floquet CFT, respectively.
We found agreement with CFT results in all cases that are comparable and
reproduced the cooling/heating effects holographically.

There are two important technical results of the paper from which most of the
results follow. First, the expressions $w_{{\rm new}}$ (2.6) and
$\overline{w}_{{\rm new}}$ (2.7) lead to the causal structure of spacetime,
such as the horizon (3.11) and (3.13), and the Floquet dual gravitational
picture. Second, the Jacobians $\frac{\partial{\rm new}}{\partial{\rm old}}$,
$\frac{\partial\overline{{\rm new}}}{\partial\overline{{\rm old}}}$ determine
the curved spacetime CFT that the driving is equivalent to and lead to the new
slicing of the bulk in holography; they lead to the energy peaks in SSD and
Floquet CFT, hence they result in the exponential growth of energy Fan:2019upv
; Wen:2022pyj ; they determine how the twist operator two-point function
transform under the change of coordinates (7.10), hence they result in the
linear growth of entanglement entropy Fan:2019upv ; Wen:2022pyj .

Throughout our work, we focused only on one of the simplest $SL(2)$
deformations: the sine-square deformation as well as the Floquet CFT
constructed from it, both with periodic boundary conditions. There are several
straightforward generalizations. First of all, it would be interesting to
explore holographic aspects of more general $SL(2)$ deformed CFTs such as
Mobius deformations (A.27), using the approaches in this work. Time evolution
of entanglement entropy after a Mobius quench was studied in Wen:2018vux ,
Floquet CFT with ordinary and Mobius deformed CFT were constructed in
Wen:2020wee , and a version of holographic dynamics of Mobius quench with
thermal initial states was investigated in Goto:2021sqx , where it was pointed
out that entanglement entropy for subsystems after a Mobius quench shows
periodic oscillations. It would also be interesting to go beyond the subgroup
$SL(2)$ and explore holographic aspects of deformations involving the full
Virasoro algebra moosavi2021inhomogeneous ; Fan:2020orx ; Lapierre:2020ftq ;
Erdmenger:2021wzc ; deBoer:2023lrd . Another interesting direction is to study
holographic constructions of deformed CFT with open (instead of periodic)
boundary conditions. While $H_{0}$, $H_{1}$ and the Mobius deformed
Hamiltonian $H_{\theta}$ (A.27) share the same ground states with periodic
boundary conditions and hence lead to a very simple quench dynamics, imposing
open boundary conditions can set up a non-trivial quantum quench problem for
CFTs prepared in the ground state by suddenly switching the CFT Hamiltonian
from $H_{0}$ to $H_{1}$ or $H_{\theta}$ at $t=0$. This is indeed the
construction adopted in Wen:2018vux ; Wen:2018agb ; Fan:2019upv ;
Bernamonti:2024fgx . Holographic studies of SSD or Mobius deformed CFT with
open boundary condition are carried out in Bernamonti:2024fgx ; Kudler-
Flam:2023ahk . It would be interesting to generalise our holographic
constructions in section 3.1 to deformed CFTs with open boundary conditions,
and make comparisons with their results.

One of the most interesting discoveries in our work is the presence of new
horizons (3.11) and (3.13). While we have confirmed the entanglement wedge
reconstructability of bulk operators behind them, it would be interesting to
explore whether any trace of the easy (HKLL) and hard (entanglement wedge)
reconstruction dichotomy persists away from the large $c$ limit, in systems
more relevant for condensed matter.

The recently proposed BKM metric deBoer:2023lrd  provides an interesting
information-theoretic viewpoint of driven CFTs, which links SSD and Floquet
CFT with circuit complexity. Therefore, our holographic construction offers a
possible testing ground for holographic complexity. In addition to the BKM
information metric, there are also other complexity measures of relevance. For
example, it was shown that Krylov complexity for states (a.k.a. spread
complexity) Balasubramanian:2022tpr  can be defined for the Hamiltonian

$\displaystyle H=\alpha(L_{-1}+L_{1})+\gamma L_{0}+\delta\mathbb{I}\,,$ (8.1)

which includes (the holomorphic part of) $H_{1}$ as a special case with
$\alpha=-1/2$, $\gamma=1$ and $\delta=0$. Another possible theory of relevance
is the path integral optimization formalism Caputa:2017urj ; Caputa:2017yrh ,
where foliations of (Euclidean) AdS3 can be obtained from optimizing the
Liouville action Polyakov:1981rd . The path integral optimization formalism
was extended to Lorentzian time in Boruch:2021hqs . It would be interesting to
explore the possibility of reproducing our results in Figure 2 from path
integral optimization.

Another interesting direction is to extend the formalism we developed to
”quantum scars” based on the Virasoro algebras Caputa:2022zsr ; Liska:2022vrd
. In these works, generalised coherent states Perelomov:1974yw  of the form

$\displaystyle|\Psi_{n}(\xi)\rangle=e^{\xi
L_{-n}-\overline{\xi}L_{n}}|h\rangle$ (8.2)

are interpreted as scarred states, where $|h\rangle$ is the highest weight
state in 2d CFT and $\xi$ is a complex parameter. In Liska:2022vrd , the
effects of displacement operators corresponding to scarred states (8.2) are
also elucidated as inducing coordinate transformations on the complex plane,
similar to that of $H_{1}$ in our work. Therefore, it would be interesting to
explore the bulk causal structures of the gravity dual of scarred states of
the form (8.2).

## Acknowledgments

We thank Yiming Chen, Ruihua Fan, Zohar Komargodski, Conghuan Luo, Matthew
Roberts, Kotaro Tamaoka, Mao Tian Tan, Apoorv Tiwari, Jie-qiang Wu, Jingxiang
Wu, Zhou Yang, and especially Xueda Wen for discussions and/or correspondence.
HJ is supported in part by Lady Margaret Hall, University of Oxford. MM is
supported in part by the STFC grant ST/X000761/1.

##  Appendix A Review of SSD CFT

In the main text, we introduced the cylinder $\mathbb{R}\times S^{1}$ with
coordinate (2.1). To study CFT2, we map the cylinder to the complex plane
$\mathbb{C}$ via the exponential map,

$\left\\{\begin{aligned} z&=e^{w}=e^{\tau+ix}\\\
\overline{z}&=e^{\overline{w}}=e^{\tau-ix}\end{aligned}\right.$ (A.1)

$H_{0}$ is dilation on the $z$-plane. Analytically continuing to real time
$\tau\to it$, we have

$\left\\{\begin{aligned} z&=\lambda e^{w}=e^{i(t+x)}\\\ \overline{z}&=\lambda
e^{\overline{w}}=e^{i(t-x)}\end{aligned}\right.$ (A.2)

where $\lambda=e^{\tau}$ in Euclidean signature. In Lorentzian signature,
$z_{{\rm new}}$ and $\overline{z}_{{\rm new}}$ are on the unit circle.

###  A.1 SSD CFT

####  A.1.1 The action of $H_{1}$

Under the exponential map (A.1), the SSD Hamiltonian (2.5) becomes

$\displaystyle H_{1}$ $\displaystyle=2\int_{0}^{2\pi}dx\
\sin^{2}\Big{(}\frac{x}{2}\Big{)}T_{00}(x)$ (A.3)
$\displaystyle=\oint\frac{dz}{2\pi
i}\Big{(}-\frac{1}{2}+z-\frac{z^{2}}{2}\Big{)}T(z)-(z\to\bar{z})-\frac{c}{12}$
(A.4)
$\displaystyle=L_{0}-\frac{L_{-1}+L_{1}}{2}+\bar{L}_{0}-\frac{\bar{L}_{-1}+\bar{L_{1}}}{2}-\frac{c}{12}$
(A.5)

While $H_{0}$ acts as dilation, the action of $H_{1}$ is a more general
$SL(2)$ transformation. To see this, we perform a coordinate transformation
$(z,\overline{z})\to(\chi,\overline{\chi})$:

$\begin{split}H_{1}&=\oint\frac{dz}{2\pi i}\
\Big{(}-\frac{1}{2}(z-1)^{2}\Big{)}T(x)+h.c\\\ &=\oint\frac{d\chi}{2\pi i}\
\frac{dz}{d\chi}\Big{(}-\frac{1}{2}(z-1)^{2}\Big{)}\Big{[}\Big{(}\frac{d\chi}{dz}\Big{)}^{2}T(\chi)+\frac{c}{12}{\rm
Sch}(\chi,z)\Big{]}+h.c\,,\end{split}$ (A.6)

where ${\rm Sch}(\chi,z)$ is the Schwarzian. By requiring that $H_{1}$ is
dilation on the $\chi$-plane Fan:2019upv , i.e. that the prefactor of
$T(\chi)$ is equal to $\chi$, we obtain a differential equation with the
solution:282828The most general solutions take the form
$\chi(z)=c\,e^{\frac{z+1}{z-1}}$ and
$\overline{z}=\overline{c}\,e^{\frac{\overline{z}+1}{\overline{z}-1}}$; the
constants $c$ and $\overline{c}$ have no effect on the coordinate
transformations.

$\displaystyle\chi(z)=e^{\frac{z+1}{z-1}}\,,$
$\displaystyle\overline{\chi}(\overline{z})=e^{\frac{\overline{z}+1}{\overline{z}-1}}\,,$
(A.7)

Plugging in this map into (A.6), we get

$\displaystyle H_{1}$ $\displaystyle=\oint\frac{d\chi}{2\pi i}\Big{(}\chi
T_{00}(\chi)-\frac{c}{24}\frac{1}{\chi}\Big{)}+h.c\,,$ (A.8)

and evaluating the contour integral (together with an identical contribution
coming from the anti-holomorphic term) gives the Casimir energy $-c/12$.

To our knowledge, the map (A.7) has not appeared in the literature, but we
show around (A.27) that it can be obtained as a limit of maps known from prior
work. Note that the maps contain essential singularities at $z=1$ and
$\overline{z}=1$, respectively, which means that we need to exclude them in
defining (A.7). These are the fixed points of the $SL(2)$ transformations
(A.12) and (A.13), so we simply have $z_{{\rm new}}=z=1$ and
$\overline{z}_{{\rm new}}=\overline{z}=1$ in these two excluded points. On the
$\chi$ plane, we then have the Heisenberg evolution

$\displaystyle e^{\tau H_{1}}O(\chi,\bar{\chi})e^{-\tau
H_{1}}=O(\lambda\chi,\lambda\bar{\chi})$ (A.9)

where $\lambda=e^{\tau}$. Back to the $z$-plane, the effect of $H_{1}$ is to
shift the operator $O$ from $z$ to $z_{{\rm new}}$, where $z_{{\rm new}}$ is
related to $z$ via Wen:2018vux

$\displaystyle\chi(z_{{\rm new}})$ $\displaystyle=\lambda\chi(z)$ (A.10)

from which one can solve for $z_{{\rm new}}$:

$\displaystyle e^{\frac{z_{{\rm new}}+1}{z_{{\rm new}}-1}}$
$\displaystyle=e^{\tau}\cdot e^{\frac{z+1}{z-1}}$ (A.11)
$\displaystyle\Rightarrow z_{{\rm new}}$
$\displaystyle=\frac{(1+\frac{\tau}{2})z-\frac{\tau}{2}}{\frac{\tau}{2}z+(1-\frac{\tau}{2})}$
(A.12)

For the anti-holomorphic part, we repeat the above calculation with
$z\to\overline{z}$, arriving at

$\displaystyle\overline{z}_{{\rm new}}$
$\displaystyle=\frac{(1+\frac{\tau}{2})\overline{z}-\frac{\tau}{2}}{\frac{\tau}{2}\overline{z}+(1-\frac{\tau}{2})}$
(A.13)

(A.12) and (A.13) are $SL(2)$ transformations on the complex plane.
Transforming back to the cylinder via the inverse of (A.1), and performing
analytic continuation $\tau=it$ then yields (2.6) and (2.7) in the main text.

The relation among the cylinder (left), complex plane (middle), and the
$SL(2)$ transformed plane (right) are summarised as follows:

$\displaystyle\begin{CD}w,\overline{w}@>{z=e^{w}}>{\overline{z}=e^{\overline{w}}}>z,\overline{z}@>{\chi=e^{\frac{z+1}{z-1}}}>{\overline{\chi}=e^{\frac{\overline{z}+1}{\overline{z}-1}}}>\chi,\overline{\chi}\\\
@V{H_{1}}V{}V@V{H_{1}}V{}V@V{H_{1}}V{\rm Dilations}V\\\ w_{{\rm
new}},\overline{w}_{{\rm new}}@>{z_{{\rm new}}=e^{w_{{\rm
new}}}}>{\overline{z}_{{\rm new}}=e^{\overline{w}_{{\rm new}}}}>z_{{\rm
new}},\overline{z}_{{\rm new}}@>{\lambda\chi=e^{\frac{z_{{\rm new}}+1}{z_{{\rm
new}}-1}}}>{\lambda\overline{\chi}=e^{\frac{\overline{z}_{{\rm
new}}+1}{\overline{z}_{{\rm
new}}-1}}}>\lambda\chi,\lambda\overline{\chi}\end{CD}$ (A.14)

####  A.1.2 Trigonometric forms of the coordinate transformations

In Lorentzian signature, $|z_{{\rm new}}|=|\overline{z}_{{\rm new}}|=1$ as
Mobius transformations preserve the unit circles (A.2) $z_{{\rm new}}$ and
$\overline{z}_{{\rm new}}$ lives on. For technical conveniences in making
Figure 1 and 2, we follow Goto:2021sqx  and write $z_{{\rm new}}$ and
$\overline{z}_{{\rm new}}$ in trigonometric forms: (A.12) and (2.6) can be
written as

$\displaystyle z_{{\rm new}}=\cos\ 2\varphi+i\ \sin\ 2\varphi$ $\displaystyle
w_{{\rm new}}=2i\varphi\,,$ (A.15)

where

$\displaystyle\sin\varphi=\frac{\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1-t\
\sin x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}$
$\displaystyle\cos\varphi=\frac{\cos\big(\frac{x}{2}\big{missing})-t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1-t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\,.$ (A.16)

We then have $\varphi\in[\frac{x}{2},\pi]\in[0,\pi]$ Goto:2021sqx  as $t$
ranges from $[0,\infty)$. Similarly, (A.13) and (2.7) can be written as

$\displaystyle\overline{z}_{{\rm new}}=\cos\ 2\overline{\varphi}+i\ \sin\
2\overline{\varphi}$ $\displaystyle\overline{w}_{{\rm
new}}=2i\overline{\varphi}\,,$ (A.17)

where

$\displaystyle\sin\overline{\varphi}=\frac{-\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1+t\
\sin x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}$
$\displaystyle\cos\overline{\varphi}=\frac{\cos\big(\frac{x}{2}\big{missing})+t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1+t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\,.$ (A.18)

We then have $\overline{\varphi}\in[-\frac{x}{2},0]\in[-\pi,0]$ Goto:2021sqx
as $t$ ranges from $[0,\infty)$. Note that $\varphi$ and $\overline{\varphi}$
are in general not complex conjugates.

The explicit expressions of $t_{{\rm new}}(t,x)$ and $x_{{\rm new}}(t,x)$ can
be found from (2.6), (2.7), (2.8), (A.16) and (A.18)

$\displaystyle t_{{\rm new}}(t,x)$
$\displaystyle=\arccos[\frac{\cos\big(\frac{x}{2}\big{missing})-t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1-t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\Big{]}-\arccos[\frac{\cos\big(\frac{x}{2}\big{missing})+t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1+t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\Big{]}$ (A.19) $\displaystyle
x_{{\rm new}}(t,x)$
$\displaystyle=\arccos[\frac{\cos\big(\frac{x}{2}\big{missing})-t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1-t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\Big{]}+\arccos[\frac{\cos\big(\frac{x}{2}\big{missing})+t\
\sin\big(\frac{x}{2}\big{missing})}{\sqrt{1+t\ \sin
x+t^{2}\sin^{2}\big{(}\frac{x}{2}\big{)}}}\Big{]}\,.$ (A.20)

####  A.1.3 SSD CFT as a CFT on curved spacetime

Under Weyl transformations, primary correlators transform as

$\begin{split}\left\langle O(z,\bar{z})\dots\right\rangle_{ds^{\prime
2}=\Omega\bar{\Omega}\,ds^{2}}=\Omega(z)^{-h}\,\bar{\Omega}(\bar{z})^{-\bar{h}}\dots\left\langle
O(z,\bar{z})\dots\right\rangle_{ds^{2}}\end{split}$ (A.21)

Now let us choose $\Omega=\big{(}\frac{\partial w_{{\rm
new}}}{\partial(ix)}\big{)}^{-1}$ and similarly for $\bar{\Omega}$. Then the
second quantity in (2.10) becomes the RHS of the above equation. It only
remains to compute metric $ds^{\prime 2}$ in $\tau,x$ coordinates. To this
end, we write:

$\begin{split}ds^{\prime 2}&=\Omega\bar{\Omega}\ dw_{{\rm
new}}d\overline{w}_{{\rm new}}\\\ &=\Big{(}\frac{\partial w_{{\rm
new}}}{\partial(ix)}\Big{)}^{-1}\Big{(}\frac{\partial\overline{w}_{{\rm
new}}}{\partial(-ix))}\Big{)}^{-1}\Big{(}\frac{\partial w_{{\rm
new}}}{\partial\tau}d\tau+\frac{\partial w_{{\rm new}}}{\partial
x}dx\Big{)}\Big{(}\frac{\partial\overline{w}_{{\rm
new}}}{\partial\tau}d\tau+\frac{\partial\overline{w}_{{\rm new}}}{\partial
x}dx\Big{)}\\\ &=\Big{(}{\partial_{\tau}w_{{\rm new}}\over\partial_{x}w_{{\rm
new}}}\,d\tau+dx\Big{)}\Big{(}{\partial_{\tau}\overline{w}_{{\rm
new}}\over\partial_{x}\overline{w}_{{\rm new}}}\,d\tau+dx\Big{)}\\\
&=\Big{(}-iv_{1}(x)\,d\tau+dx\Big{)}\Big{(}iv_{1}(x)\,d\tau+dx\Big{)}\\\
&=v_{1}(x)^{2}d\tau^{2}+dx^{2}\,,\end{split}$ (A.22)

which is the curved space metric (2.9). Thus the Weyl transformation indeed
directly gives the third quantity in (2.10).

####  A.1.4 $H_{q}$ with $q>1$

As for $H_{q}$ with $q>1$, the conformal map $z=e^{qw}$ takes the cylinder to
a $q$-sheet Riemann surface, where $H_{q}$ takes the same form as $H_{1}$

$\displaystyle H_{q}$
$\displaystyle=2\int_{0}^{2\pi}dx\,\sin^{2}\Big{(}q\cdot\frac{x}{2}\Big{)}T_{00}(x)$
(A.23) $\displaystyle=q\oint\frac{dz}{2\pi
i}\Big{(}-\frac{1}{2}+z-\frac{z^{2}}{2}\Big{)}T(z)-(z\to\bar{z})-q\frac{c}{12}$
(A.24)

The calculation then follows the same as above, with $2\pi\to 2\pi/q$. Namely,

$\displaystyle z_{{\rm new}}$
$\displaystyle=\frac{(1+q\frac{\tau}{2})z-q\frac{\tau}{2}}{q\frac{\tau}{2}z+(1-q\frac{\tau}{2})}$
(A.25) $\displaystyle ds^{2}$ $\displaystyle=q^{2}\frac{e^{2q\tau}}{(1-\cos\
qx)^{2}}\Big{[}(1-\cos\ qx)^{2}d\tau^{2}+dx^{2}\Big{]}$ (A.26)

Note that now $z$ and $\overline{z}$ are coordinates on the $q$-sheeted
Riemann surface and there is a branch cut in the $z$-plane. The fact that the
system is $q$ copies of identical theories transcribes to the crossing of
sheets on the $z$-plane.

###  A.2 General Cases

Wen:2018vux ; Goto:2021sqx  also studied quantum quench with the one-parameter
generalization of the SSD Hamiltonian $H_{1}$, the Mobius Hamiltonian, The
Mobius deformed Hamiltonian $H_{\theta}$ is given by

$\displaystyle H_{\theta}=L_{0}-{\rm tanh}\ 2\theta\
\frac{L_{-1}+L_{1}}{2}+\overline{L}_{0}-{\rm tanh}\ 2\theta\
\frac{\overline{L}_{-1}+\overline{L}_{1}}{2}-\frac{c}{12}$ (A.27)

The $\theta\to 0$ and $\theta\to\infty$ limit of $H_{\theta}$ are $H_{0}$ and
$H_{1}$, respectively. This family of Hamiltonians act as dilatations on the
$\xi$ plane as
$H_{\theta}^{(\xi)}={1\over\cosh(2\theta)}\left(L^{(\xi)}_{0}+\bar{L}_{0}^{(\xi)}\right)-{c\over
12}$ with

$\begin{split}\xi=-{\cosh\theta\,z-\sinh\theta\over\sinh\theta\,z-\cosh\theta}\,.\end{split}$
(A.28)

By raising this map to the power $\cosh(2\theta)$, we obtain a properly
normalised dilatation operator:292929Through the exponential map we can see
that raising the map to some power corresponds to rescaling time on the
cylinder.

$\begin{split}\chi_{\theta}=\left(-\xi\right)^{\cosh(2\theta)}=\left({\cosh\theta\,z-\sinh\theta\over\sinh\theta\,z-\cosh\theta}\right)^{\cosh(2\theta)}\,,\end{split}$
(A.29)

which has the the limit

$\begin{split}\chi=\lim_{\theta\to\infty}\chi_{\theta}=e^{\frac{z+1}{z-1}}\,.\end{split}$
(A.30)

This agrees with (A.7).

The methodology of describing the non-trivial time evolution of inhomogeneous
non-equilibrium CFT as coordinate transformations reviewed in Section 2.2 is
valid beyond SSD (where only $SL(2)$ deformations are considered) and extends
to deformations involving the full Virasoro algebras. In these cases, it is
shown moosavi2021inhomogeneous  that the projective unitary representation of
the universal cover of the orientation-preserving diffeomorphisms of the
circle $\widetilde{{\rm Diff}}_{+}(S^{1})$ allows one to flatten out the
deformation $v(x)$ and map the expectations to ordinary equilibrium CFT under
coordinate transformations. Like in the SSD case, these coordinate
transformations capture the full information of the time evolution of the
deformed CFT. In deBoer:2023lrd , it is further demonstrated that just as in
SSD CFT, these generally deformed CFTs can be understood as CFTs on curved
spacetime as well.

##  Appendix B Changing the cutoff in AdS/CFT

###  B.1 AdS/CFT with a nonuniform cutoff

We will show on the example of a scalar field how the extrapolate dictionary
Banks:1998dd  of AdS/CFT changes under changing the cutoff prescription. In
the absence of a source of a scalar field, we have

$\begin{split}\phi(\rho,y^{\mu})=(2\Delta-d)\rho^{-\Delta}\,O(y^{\mu})+\text{(subleading)}\,,\end{split}$
(B.1)

where $\rho$ is the bulk radial direction and $O(y^{\mu})$ is the boundary CFT
primary field. It is clear that if we choose two different bulk radial
coordinates related by $r=\Omega(y^{\mu})r_{\text{new}}$ then the same bulk
physics will give rise to boundary correlators that differ by some factors

$\begin{split}\left\langle O(y^{\mu})\dots\right\rangle_{\text{CFT,
flat}}&=\lim_{r_{\text{new}}\to\infty}{r_{\text{new}}^{\Delta}\over
2\Delta-d}\dots\left\langle\phi(r_{\text{new}},y^{\mu})\dots\right\rangle_{\text{AdS}}\\\\[4.0pt]
&=\Omega(y^{\mu})^{-\Delta}\dots\left\langle
O(y^{\mu})\dots\right\rangle_{\text{wiggly bdy}}\,,\end{split}$ (B.2)

where the last object is the boundary correlation function that is computed
from the nonuniform cutoff prescription on the gravity side

$\begin{split}\epsilon={1\over
R}={1\over\Omega(y^{\mu})R_{\text{new}}}\,,\end{split}$ (B.3)

where in our application the choice $\Omega=\sqrt{\frac{\partial w_{{\rm
new}}}{\partial w}\frac{\partial\overline{w}_{{\rm
new}}}{\partial\overline{w}}}$ is made. $\left\langle
O(y^{\mu})\dots\right\rangle_{\text{wiggly bdy}}$ is hence equal to the second
object in (2.10), as stated in the main text.

It is also equal to the curved space CFT correlation function, as we show on
the CFT side from applying a Weyl transformation and on the gravity side by
adapting Fefferman-Graham coordinates to the nonuniform cutoff surface.

###  B.2 The Fefferman-Graham expansion with a curved boundary

Let us change the spatial coordinate as

$\begin{split}x=2\arccot X\,,\end{split}$ (B.4)

then the boundary metric (2.9) takes the conformally flat form:

$\begin{split}ds_{\text{bdy}}^{2}={4\over(1+X^{2})^{2}}\left(-dt^{2}+dX^{2}\right)\,.\end{split}$
(B.5)

The Fefferman-Graham expansion with this boundary metric takes the form:

$\begin{split}ds^{2}=&{dr^{2}\over
r^{2}}-{(4r^{2}+X^{2}-(1+X^{2})^{2}\,L(t+X))(4r^{2}+X^{2}-(1+X^{2})^{2}\,R(t-X))\over
4r^{2}(1+X^{2})^{2}}\,dt^{2}\\\
&+{(4r^{2}-1+(1+X^{2})^{2}\,L(t+X))(4r^{2}-1+(1+X^{2})^{2}\,R(t-X))\over
4r^{2}(1+X^{2})^{2}}\,dX^{2}\\\ &+{(8r^{2}-1+X^{2})(L(t+X)-R(t+X))\over
4r^{2}}\,dt\,dX\,,\end{split}$ (B.6)

where $L(t+X)$ and $R(t-X)$ are arbitrary left and right moving functions
corresponding to the holomorphic and anti-holomorphic stress tensor. The
geometry (B.6) first considered in Skenderis:1999nb  is the generalisation of
the well-known Banados geometries Banados:1998gg  appropriate for a flat
boundary, and reduce to them in the limit of small $X$. We obtained these
expressions by solving Einstein’s equations in a $1/r$ series expansion, a
method reviewed in Skenderis:1999nb ; deHaro:2000vlm . We can easily convert
the metric to $x$ coordinates using the relation (B.4).

If we set $t=0$ and convert $X$ back to $x$, the resulting metric on the $t=0$
slice should match with the metric on the $t_{\text{new}}=0$ slice of the
empty AdS3 or BTZ metrics. Converting them to Fefferman-Graham form allows us
to read off the functions $L$ and $R$. For the AdS3 case, we get that $L=R=0$.
In this case, a redefinition of $r$ reproduces the metric presented in
MacCormack:2018rwq . For BTZ we find:

$\begin{split}L(y)=R(y)={1+r_{+}^{2}\over(1+y^{2})^{2}}\,.\end{split}$ (B.7)

##  Appendix C Asymptotic symmetries of AdS3

In this appendix we present the Killing vectors of global AdS3 in Hartman :

$\displaystyle\zeta_{-1}$
$\displaystyle=\frac{1}{2}e^{-i(t+x)}\Big{(}\frac{r}{\sqrt{r^{2}+1}}\partial_{t}+\frac{\sqrt{r^{2}+1}}{r}\partial_{x}+i\sqrt{r^{2}+1}\partial_{r}\Big{)}$
(C.1) $\displaystyle\zeta_{0}$
$\displaystyle=\frac{1}{2}(\partial_{t}+\partial_{x})$ (C.2)
$\displaystyle\zeta_{1}$
$\displaystyle=\frac{1}{2}e^{i(t+x)}\Big{(}\frac{r}{\sqrt{r^{2}+1}}\partial_{t}+\frac{\sqrt{r^{2}+1}}{r}\partial_{x}-i\sqrt{r^{2}+1}\partial_{r}\Big{)}$
(C.3) $\displaystyle\overline{\zeta_{-1}}$
$\displaystyle=\frac{1}{2}e^{-i(t-x)}\Big{(}\frac{r}{\sqrt{r^{2}+1}}\partial_{t}-\frac{\sqrt{r^{2}+1}}{r}\partial_{x}+i\sqrt{r^{2}+1}\partial_{r}\Big{)}$
(C.4) $\displaystyle\overline{\zeta_{0}}$
$\displaystyle=\frac{1}{2}(\partial_{t}-\partial_{x})$ (C.5)
$\displaystyle\overline{\zeta_{1}}$
$\displaystyle=\frac{1}{2}e^{i(t-x)}\Big{(}\frac{r}{\sqrt{r^{2}+1}}\partial_{t}-\frac{\sqrt{r^{2}+1}}{r}\partial_{x}-i\sqrt{r^{2}+1}\partial_{r}\Big{)}$
(C.6)

where we dropped the ”new” scripts for notation simplicity. These vectors are
indeed obey the Killing equations

$\displaystyle\nabla_{(\mu}(\zeta_{i})_{\nu)}=0$
$\displaystyle\nabla_{(\mu}(\overline{\zeta_{i}})_{\nu)}=0$
$\displaystyle(i=0,\pm 1)\,.$ (C.7)

The Killing vectors obey the $SL(2,\mathbb{R})_{L}\times SL(2,\mathbb{R})_{R}$
algebra:

$\displaystyle i\\{\zeta_{1},\zeta_{-1}\\}=2\zeta_{0}$ $\displaystyle
i\\{\zeta_{1},\zeta_{0}\\}=\zeta_{1}$ $\displaystyle
i\\{\zeta_{-1},\zeta_{0}\\}=-\zeta_{-1}$ (C.8) $\displaystyle
i\\{\overline{\zeta_{1}},\overline{\zeta_{-1}}\\}=2\overline{\zeta_{0}}$
$\displaystyle
i\\{\overline{\zeta_{1}},\overline{\zeta_{0}}\\}=\overline{\zeta_{1}}$
$\displaystyle
i\\{\overline{\zeta_{-1}},\overline{\zeta_{0}}\\}=-\overline{\zeta_{-1}}$
(C.9)

where $\\{.\,,.\\}$ here denotes the Lie bracket and hence we have extra $i$’s
on the LHS compared to when writing commutators.

##  Appendix D Geodesic equations

In this appendix, we study null and spacelike geodesic equations, on pure AdS3
and BTZ black holes, respectively. Our method is largely based on
Hubeny:2012ry . We will be working in geometries in the $(t_{{\rm
new}},x_{{\rm new}},r_{{\rm new}})$ coordinate throughout this appendix, so we
will drop the ”new” subscript for notation simplicity.

We consider Lorentzian (2+1)-dimensional metric:

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}dx^{2}$ (D.1)

where $\lim_{r\to\infty}f(r)=r^{2}$ so that the spacetime is asymptotically
AdS3. As the metric is independent of $t$ and $x$, there are two conserved
quantities:

$\displaystyle E=f(r)\frac{dt}{d\lambda}$ $\displaystyle
L=r^{2}\frac{dx}{d\lambda}$ (D.2)

which correspond to the energy and angular momentum, respectively. The norm of
the tangent vector is:

$\displaystyle\kappa$
$\displaystyle=g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=-\frac{1}{f(r)}E^{2}+\frac{1}{f(r)}\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}+\frac{1}{r^{2}}L^{2}$
(D.3)

where $\kappa=+1,0,-1$ for spacelike, null, and timelike geodesics,
respectively. We will focus on null and spacelike geodesics. Equation (D.3)
can be put into the general form

$\displaystyle\Big{(}\frac{dt}{d\lambda}\Big{)}^{2}+V(r)=0$ $\displaystyle
V(r)=-E^{2}-\kappa f(r)+L^{2}\frac{f(r)}{r^{2}}$ (D.4)

where $V(r)$ can be thought of as an effective potential. Geodesic motion
requires $\Big{(}\frac{dt}{d\lambda}\Big{)}^{2}\geq 0$. Consequently, the
condition for a geodesic to reach the boundary is $\lim_{r\to\infty}V(r)\leq
0$; for a geodesic emanating from a point on the boundary, the deepest
distance, i.e. the smallest $r$ it can reach $r_{*}$ is given by the largest
positive zero of the effective potential $V(r)$ Hubeny:2012ry .

###  D.1 Null geodesics

For null geodesic, we have $\kappa=0$ in (D.4), and the two conserved
quantities can be combined to a single parameter $\ell=L/E$ by rescaling the
affine parameter $\lambda$ Hubeny:2012ry . (D.4) then becomes

$\displaystyle\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}+\ell^{2}\frac{f(r)}{r^{2}}-1=0\,.$
(D.5)

From (D.5), it is clear that in order for the null geodesic to reach the
boundary, one must have $\ell^{2}\leq 1$; $\ell^{2}=1$ corresponds to the case
that the null geodesic stays on the boundary Hubeny:2012ry .

####  D.1.1 Pure AdS3

For pure AdS3, we have $f(r)=r^{2}+1$ in (D.5), which integrates to

$\displaystyle r(\lambda)=\sqrt{(1-\ell^{2})\
\lambda^{2}+\frac{\ell^{2}}{1-\ell^{2}}}$
$\displaystyle\lambda\in\big{[}0,\infty\big{)}$ (D.6)

The null geodesic reaches the boundary, i.e. $r\to\infty$ when
$\lambda\to\infty$.

Substituting (D.6) into the two conserved quantities (D.2) with
$f(r)=r^{2}+1$, one finds

$\displaystyle E$
$\displaystyle=(r^{2}+1)\frac{dt}{d\lambda}=\frac{(1-\ell^{2})^{2}\
\lambda^{2}+1}{1-\ell^{2}}\frac{dt}{d\lambda}=1$ (D.7) $\displaystyle L$
$\displaystyle=r^{2}\frac{dx}{d\lambda}=\frac{(1-\ell^{2})^{2}\
\lambda^{2}+\ell^{2}}{1-\ell^{2}}\frac{dx}{d\lambda}=\ell$ (D.8)

which integrate to

$\displaystyle t(\lambda)$ $\displaystyle={\rm
arctan}[(1-\ell^{2})\lambda]+C_{t}$ (D.9) $\displaystyle x(\lambda)$
$\displaystyle={\rm
arctan}\Big{[}\frac{1}{\ell}(1-\ell^{2})\lambda\Big{]}+C_{x}$ (D.10)

When the null geodesic approaches the boundary

$\displaystyle\lim_{\lambda\to\infty}t(\lambda)=C_{t}+\frac{\pi}{2}=y$
$\displaystyle\lim_{\lambda\to\infty}x(\lambda)=C_{x}+\frac{\pi}{2}=y$ (D.11)

The other half of the triangle can be studied by reflection, i.e. by
performing a change of variable $x^{\prime}=2\pi-x\in[0,\pi]$. The general
expressions of null geodesics equations are therefore

$\left\\{\begin{aligned} r(\lambda)&=\sqrt{(1-\ell^{2})\
\lambda^{2}+\frac{\ell^{2}}{1-\ell^{2}}}\\\ t(\lambda)&={\rm
arctan}[(1-\ell^{2})\lambda]+y-\frac{\pi}{2}\\\ x(\lambda)&={\rm
arctan}\Big{[}\frac{1}{\ell}(1-\ell^{2})\lambda\Big{]}+y-\frac{\pi}{2}\end{aligned}\right.$
(D.12)

From the expression of $x(\lambda)$, we can solve for $\ell$

$\displaystyle-\cot(x-y)$ $\displaystyle=\frac{1}{\ell}(1-\ell^{2})\lambda$
(D.13) $\displaystyle\ell$
$\displaystyle=\frac{r}{\sqrt{1+r^{2}+\cot^{2}(x-y)}}\leq 1$ (D.14)

Plugging (D.13), (D.14) and $r(\lambda)$ into $t(\lambda)$

$\displaystyle t(r,x,y)$ $\displaystyle={\rm
arctan}\Big{[}\ell\cdot\frac{1}{\ell}(1-\ell^{2})\lambda\Big{]}+y-\frac{\pi}{2}$
(D.15) $\displaystyle={\rm
arctan}\Big{[}\frac{-r\cot(x-y)}{\sqrt{r^{2}+\csc^{2}(x-y)}}\Big{]}+y-\frac{\pi}{2}$
(D.16)

which is repeated in the main text in (3.10). It is easy to see that $t$
increases monotonically with respect to $y$,

$\displaystyle\frac{dt}{dy}=1-\underbrace{\frac{r}{\sqrt{r^{2}+\csc^{2}(x-y)}}}_{\leq
1}\geq 0$ (D.17)

So $t$ maximises at $y=\pi$.

$\displaystyle t(r,x,y=\pi)=-{\rm arctan}\Big{[}\frac{r\cot
x}{\sqrt{r^{2}+\csc^{2}x}}\Big{]}+\frac{\pi}{2}$ (D.18)

Further simplifications depend on the range of $x$:

  * •

For $\cot x>0$, i.e. $0<x<\pi/2$, we have

$\displaystyle t(r,x)={\rm arctan}\Big{[}\frac{\sqrt{r^{2}+\csc^{2}x}}{r\cot
x}\Big{]}=\arccos[\frac{r}{\sqrt{r^{2}+1}}\cos x\Big{]}$ (D.19)

  * •

For $\cot x<0$, i.e. $\pi/2<x<\pi$, we have

$\displaystyle t(r,x)$ $\displaystyle=-{\rm arctan}\Big{[}\frac{r\cot
x}{\sqrt{r^{2}+\csc^{2}x}}\Big{]}-\frac{\pi}{2}+\pi={\rm
arctan}\Big{[}\frac{\sqrt{r^{2}+\csc^{2}x}}{r\cot x}\Big{]}+\pi$
$\displaystyle=\arccos[\frac{r}{\sqrt{r^{2}+1}}\cos x\Big{]}$ (D.20)

both of which are (3.11) in the main text.

####  D.1.2 BTZ

For BTZ, we have $f(r)=r^{2}-r_{+}^{2}$ in (D.5), which integrates to

$\displaystyle r(\lambda)=\sqrt{(1-\ell^{2})\
\lambda^{2}-\frac{\ell^{2}r_{+}^{2}}{1-\ell^{2}}}$
$\displaystyle\lambda\in\Big{[}\frac{\ell r_{+}}{1-\ell^{2}},\infty\big{)}$
(D.21)

The null geodesic reaches the boundary, i.e. $r\to\infty$ when
$\lambda\to\infty$.

Substituting (D.21) into the two conserved quantities (D.2) with
$f(r)=r^{2}-r_{+}^{2}$, we find

$\displaystyle E$
$\displaystyle=(r^{2}-r_{+}^{2})\frac{dt}{d\lambda}=\frac{(1-\ell^{2})^{2}\
\lambda^{2}-r_{+}^{2}}{1-\ell^{2}}\frac{dt}{d\lambda}=1$ (D.22) $\displaystyle
L$ $\displaystyle=r^{2}\frac{dx}{d\lambda}=\frac{(1-\ell^{2})^{2}\
\lambda^{2}-\ell^{2}r_{+}^{2}}{1-\ell^{2}}\frac{dx}{d\lambda}=\ell$ (D.23)

which outside the horizon, $r\geq r_{+}$ for integrate to:

$\displaystyle t(\lambda)$
$\displaystyle=\frac{1}{2r_{+}}\log\frac{(1-\ell^{2})\ \lambda-
r_{+}}{(1-\ell^{2})\ \lambda+r_{+}}+C_{t}$ (D.24) $\displaystyle x(\lambda)$
$\displaystyle=\frac{1}{2r_{+}}\log\frac{(1-\ell^{2})\ \lambda-\ell
r_{+}}{(1-\ell^{2})\ \lambda+\ell r_{+}}+C_{x}$ (D.25)

When the null geodesic approaches the boundary

$\displaystyle\lim_{\lambda\to\infty}t(\lambda)=y=C_{t}$
$\displaystyle\lim_{\lambda\to\infty}x(\lambda)=y=C_{x}$ (D.26)

The general expressions of null geodesics equations outside the horizon are
therefore

$\left\\{\begin{aligned} r(\lambda)&=\sqrt{(1-\ell^{2})\
\lambda^{2}-\frac{\ell^{2}r_{+}^{2}}{1-\ell^{2}}}\\\
t(\lambda)&=\frac{1}{2r_{+}}\log\frac{(1-\ell^{2})\ \lambda-
r_{+}}{(1-\ell^{2})\ \lambda+r_{+}}+y\\\
x(\lambda)&=\frac{1}{2r_{+}}\log\frac{(1-\ell^{2})\ \lambda-\ell
r_{+}}{(1-\ell^{2})\ \lambda+\ell r_{+}}+y\end{aligned}\right.$ (D.27)

From the expression of $x(\lambda)$, we can solve for $\ell$

$\displaystyle\ell=\frac{(-1+e^{2r_{+}(x-y)})r}{\sqrt{(-1+e^{2r_{+}(x-y)})^{2}r^{2}+4r_{+}^{2}e^{2r_{+}(x-y)}}}\leq
1$ (D.28)

Plugging (D.28) and $r(\lambda)$ into $t(\lambda)$ gives

$\displaystyle
t(r,x,y)=\frac{1}{2r_{+}}\log\frac{r\sqrt{1+r_{+}^{2}\sinh^{2}(x-y)}-r_{+}\sqrt{1+r^{2}\sinh^{2}(x-y)}}{r\sqrt{1+r_{+}^{2}\sinh^{2}(x-y)}+r_{+}\sqrt{1+r^{2}\sinh^{2}(x-y)}}+y$
(D.29)

which is (D.1.2) in the main text. It is easy to see that $t$ increases
monotonically with respect to $y$ outside the BTZ horizon

$\displaystyle\frac{dt}{dy}\propto 4r(r^{2}-r^{2}_{+})\ \sinh(x-y)\geq 0$
(D.30)

So $t$ is maximised at $y=\pi$.

$\displaystyle
t(r,x,y=\pi)=\frac{1}{2r_{+}}\log\frac{r\sqrt{1+r_{+}^{2}\sinh^{2}(x-\pi)}-r_{+}\sqrt{1+r^{2}\sinh^{2}(x-\pi)}}{r\sqrt{1+r_{+}^{2}\sinh^{2}(x-\pi)}+r_{+}\sqrt{1+r^{2}\sinh^{2}(x-\pi)}}+\pi$
(D.31)

which is (3.12) in the main text.

###  D.2 Spacelike geodesics

For spacelike geodesic, we have $\kappa=1$ in (D.3)

$\displaystyle\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}+V(r)=0$ $\displaystyle
V(r)=-f(r)-E^{2}+L^{2}\frac{f(r)}{r^{2}}=0$ (D.32)

To compute the holographic entanglement entropy, one needs to compute the
length of the spacelike geodesic obeying the appropriate homology constraints.
In our case, the spacelike geodesic equation is for non-equal time. The length
of the spacelike geodesic is given in terms of the temporal and angular
separation of the two points on the boundary on which our geodesic anchors.

The length of the spacelike geodesic is given by

$\displaystyle\mathcal{L}=2\int_{r_{*}}^{\infty}d\lambda$ (D.33)

with the factor of two coming from symmetry. But this is of course infinite;
to get a finite result, we impose the large-radius cutoffs $R_{1}$, $R_{2}$ at
the two geodesic endpoints as in Hubeny:2012ry . Note that from (D.4), one can
change integration variable $d\lambda=\frac{dr}{\sqrt{-V(r)}}$, which is
appropriate since $V(r)<0$ in $r\in(r_{*},\infty)$. Thus, we have

$\displaystyle\mathcal{L}_{{\rm
reg}}=\Big{(}\int_{r_{*}}^{R_{1}}+\int_{r_{*}}^{R_{2}}\Big{)}\frac{dr}{\sqrt{-V(r)}}$
(D.34)

and the universal diverging piece is simply $\log 4R_{1}R_{2}$, since
$V(r)\propto-r^{2}$ for spacelike geodesics as $r$ gets large in (D.32).

The temporal and angular separation are then given by Hubeny:2012ry

$\displaystyle\Delta t$
$\displaystyle=2\int_{t_{*}}^{\infty}dt=2\int_{\lambda_{*}}^{\infty}\frac{dt}{d\lambda}d\lambda=2\int_{\lambda_{*}}^{\infty}\frac{E}{f(r)}d\lambda=2\int_{r_{*}}^{\infty}\frac{E}{f(r)\sqrt{-V(r)}}dr$
(D.35) $\displaystyle\Delta x$
$\displaystyle=2\int_{x_{*}}^{\infty}dx=2\int_{\lambda_{*}}^{\infty}\frac{dx}{d\lambda}d\lambda=2\int_{\lambda_{*}}^{\infty}\frac{L}{r^{2}}d\lambda=2\int_{r_{*}}^{\infty}\frac{L}{r^{2}\sqrt{-V(r)}}dr$
(D.36)

where the conserved quantities (D.2) have been substituted in.

####  D.2.1 Pure AdS3

For pure AdS3, we have $f(r)=r^{2}+1$ in (D.32). The relevant quantities
needed in our calculation have been worked out in Hubeny:2012ry . The minimal
radius the spacelike geodesic can reach is

$\displaystyle
r_{*}=\sqrt{\frac{1}{2}\Big{[}-(E^{2}-L^{2}+1)+\sqrt{(E^{2}-L^{2}+1)^{2}+4L^{2}}\Big{]}}$
(D.37)

The regulated length of the spacelike geodesic is

$\displaystyle\mathcal{L}_{\rm
reg}=\Big{(}\int_{r_{*}}^{R_{1}}+\int_{r_{*}}^{R_{2}}\Big{)}\frac{dr}{\sqrt{-V(r)}}=-\log\sqrt{(E^{2}-L^{2}+1)^{2}+4L^{2}}+\log(4R_{1}R_{2})$
(D.38)

where we added the universal diverging term $\log(4R_{1}R_{2})$ that
Hubeny:2012ry  subtracted off, as it will play a role in our analysis later
on. The temporal and angular distance between the two geodesic endpoints on
the boundary are

$\displaystyle\Delta t$
$\displaystyle=2\int_{r_{*}}^{\infty}\frac{E}{f(r)\sqrt{-V(r)}}dr=\frac{\pi}{2}+\arcsin[\frac{E^{2}-L^{2}-1}{\sqrt{(E^{2}-L^{2}-1)^{2}+4E^{2}}}\Big{]}$
(D.39) $\displaystyle\Delta x$
$\displaystyle=2\int_{r_{*}}^{\infty}\frac{L}{r^{2}\sqrt{-V(r)}}dr=\frac{\pi}{2}+\arcsin[\frac{E^{2}-L^{2}+1}{\sqrt{(E^{2}-L^{2}+1)^{2}+4L^{2}}}\Big{]}$
(D.40)

which are (7.3) and (7.4) in the main text. Note that
$(E^{2}-L^{2}-1)^{2}+4E^{2}=(E^{2}-L^{2}+1)^{2}+4L^{2}$.

From (7.3) and (7.4), we can solve for $E$ and $L$:

$\displaystyle E=\frac{\sin\Delta t}{\cos\Delta t-\cos\Delta x}$
$\displaystyle L=\frac{\sin\Delta x}{\cos\Delta t-\cos\Delta x}$ (D.41)

Plugging (D.41) into (D.38), we have,

$\displaystyle\mathcal{L}_{{\rm reg}}$
$\displaystyle=\log[\frac{1}{2}\big{|}\cos\Delta t-\cos\Delta
x\big{|}\Big{]}+\log(4R_{{1}}R_{{2}})$ (D.42)
$\displaystyle=\log[4\Big{|}\sinh\Big{(}\frac{w_{1}-w_{2}}{2}\Big{)}\
\sinh\Big{(}\frac{\overline{w}_{1}-\overline{w}_{2}}{2}\Big{)}\Big{|}\
\Big{]}+\log(R_{{1}}R_{{2}})$ (D.43)

where we have used (7.1) in getting from the second to the third line.