aainstitutetext: Center for High Energy Physics, Indian Institute of Science,
Bengaluru 560012, Indiabbinstitutetext: Ramakrishna Mission Vivekananda
Educational and Research Institute, Belur Math, Howrah-711202, West Bengal,
India

# Notes on heating phase dynamics in Floquet CFTs and Modular quantization

Suchetan Das b Bobby Ezhuthachan b Somnath Porey b Baishali Roy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this article, we explore the connection between the heating phase of
periodically driven CFTs and the Modular Hamiltonian of a subregion in the
vacuum state. We show that the heating phase Hamiltonian corresponds to the
Modular Hamiltonian, with the fixed points mapping to the endpoints of the
subregion. In the bulk dual, we find that these fixed points correspond to the
Ryu-Takayanagi surface of the AdS-Rindler wedge. Consequently, the
entanglement entropy associated to the boundary interval within two fixed
points exactly matches with the Bekenstein-Hawking entropy of AdS-Rindler. We
observe the emergent Virasoro algebra in the boundary quantization of the
Modular Hamiltonian has a striking similarity with the emerging near Horizon
Virasoro algebra. This is a consequence of the fact that while obtaining the
boundary Virasoro algebra, a cut-off with conformal boundary condition around
the fixed point is introduced, which in the bulk is related to a stretched
horizon, with an emergent two-dimensional conformal symmetry. We also argue
that as one tunes the parameter space of Floquet Hamiltonians to transition
from the non-heating to the heating phase the operator algebra type changes
from Von Neumann type $I$ to $III_{1}$ factor, providing a non-equilibrium
analogue of the Hawking-Page transition.

##  1 Introduction

The study of Floquet CFTs has recently garnered considerable attention as
analytically tractable models of physics out of equilibrium. At stroboscopic
times, the evolution of such systems is governed by an effective Hamiltonian,
referred to in the literature as the Floquet Hamiltonian $H_{F}$. One of the
novel features of their dynamics is the existence of three different dynamical
phases, characterized by qualitatively different temporal growths of the
entanglement entropy and energy. In particular, one of the phases, aptly
referred to as the heating phase, is characterized by an exponentially growing
energy and linearly growing entanglement entropy in stroboscopic time.
Moreover, for the special example of a $sl(2)$ Floquet CFT, an OTOC
computation in large-$c$ CFTs in the heating phase also shows a Lyapunov
growth, characteristic of early time chaos Das:2022jrr . In the bulk
description, such a Lyapunov growth in a holographic CFT is understood to be a
consequence of blue-shifting of energy in the presence of a horizon
Shenker:2013pqa . Indeed, as was confirmed in Das:2022pez , there is an
observer-dependent horizon in the dual description of the $sl(2)$ Floquet CFT
in the heating phase 111For other works on the holographic bulk dual of
Floquet CFTs or deformed CFTs see deBoer:2023lrd ; Jiang:2024hgt ; Kudler-
Flam:2023ahk ; MacCormack:2018rwq .. As was shown in that paper, one can
reproduce the Lyapunov exponent from a bulk computation, following the methods
of Shenker:2013pqa .

These features of the heating phase are best explained by noting that $H_{F}$
is also the Modular Hamiltonian of a line interval on the circle in the vacuum
state of the CFT, with the endpoints of the interval being the two fixed
points of the flow under $H_{F}$222This point has been made previously, for
e.g.- seeKhetrapal:2022dzy , as we show in section [2.2]. Thus, the Floquet
CFT in the heating phase captures the dynamics of this sub-system in modular
time. Consequently, in the bulk, it describes the dynamics within the
corresponding Entanglement Wedge (EW). In fact, as we show in section [3.2],
in the bulk, the fixed points of the flow under $H_{F}$ correspond to the RT
surface.

In Das:2022pez , the bulk dual of the heating phase was described in the frame
of an observer moving under the flow generated by the bulk representation of
$H_{F}$, (which we refer to as $H^{b}_{F}$). In the boundary theory, one can
analogously study the heating phase in the CFT directly in a conformal
frame333We call this frame in which time is the modular time, i.e. generated
by $H_{F}$ as $CF_{2}$, while the frame, where the time is the one generated
by the undeformed Hamiltonian is referred to as $CF_{1}$. where the modular
time444This is the same as the stroboscopic time of the driven CFT is the
real-time of the system. Quantization in this frame would provide the boundary
description of the bulk dynamics inside the Entanglement Wedge. As has been
emphasized before in the literature Ohmori:2014eia , to make this quantization
well-defined, one has to impose boundary conditions around the two endpoints
(fixed points of the Floquet Hamiltonian)555after imposing a cut-off
($\epsilon$) around the fixed points. The full Hilbert space of the theory can
formally be then expressed as,

$\mathcal{H}_{CFT}=\bigoplus_{B_{1},B_{2}}\mathcal{H}_{B_{1},B_{2}}^{u}\otimes\mathcal{H}_{B_{2},B_{1}}^{d}.$
(1)

Where $B_{1}$, $B_{2}$ represent the boundary conditions imposed at a cut-off
distance $\epsilon$ away from the two fixed points Ohmori:2014eia ;
Agia:2022srj . Here, $\mathcal{H}^{u(d)}_{B_{1},B_{2}}$, is dual to the region
inside the Entanglement Wedge, while the $\mathcal{H}^{d(u)}_{B_{1},B_{2}}$,
would be dual to the complimentary Wedge (see Fig.1). If we impose conformal
boundary conditions at the cut-off, then the boundary condition will be
preserved by one copy of the Virasoro algebra. Usual AdS/CFT rules imply that
the bulk description must also involve a boundary placed at a small cut-off
away from the RT surface. This is very reminiscent of the stretched Horizon
scenario for black holes tHooft:1984kcu ; Susskind:1993if . Indeed, as we show
in section [3], if one pulls the conformal transformation from $CF_{1}$ to
$CF_{2}$ into the bulk Anand:2017dav , we get the AdS-Rindler metric. In fact,
in these coordinates, the RT surface maps to the Rindler Horizon. Moreover, as
we argue in section [33], the ‘emergent’ boundary Virasoro algebra in the
$CF_{2}$ frame, should be identified with the near horizon Virasoro algebra
that has been discussed previously in the literature Solodukhin:1998tc ;
Carlip:2002be ; Carlip:1999db ; Carlip:2005zn ; Guica:2008mu ; Kang:2004js .

Figure 1:  The illustration depicts a constant time slice, with the blue arc
representing the subregion $u$ between two fixed points denoted by black dots.
Similarly, the yellow arc corresponds to its complement subregion $d$. The
Hilbert spaces associated with each are denoted as
$\mathcal{H}^{u}_{B_{1}B_{2}}$ and $\mathcal{H}^{d}_{B_{1}B_{2}}$. $B_{1}$ and
$B_{2}$ represent the boundary conditions applied at the two ends of $u(d)$ at
a distance $\epsilon$ from the fixed points $z_{f_{1}}$ and $z_{f_{2}}$.

In Das:2022pez , it was shown that the bulk dual metric of the heating phase
is given by an $AdS_{2}$ black hole patch of $AdS_{3}$. As we show in section
[3], this metric and the AdS Rindler metric are related by a diffeomorphism in
the bulk and represent the same bulk observer666Consequently, they are not
large diffeomorphisms.. Since the Modular Hamiltonian, generates evolution
along the closed lines around the fixed points, (in the Euclidean plane), the
entanglement entropy of the line interval is by definition equal to the
thermal entropy of the associated (Euclidean) cylinder (see Fig[1])777This
should match the AdS Rindler entropy in the bulk. While this entropy would
depend on the boundary condition imposed, the leading contribution (in the
$\epsilon\rightarrow 0$) is independent of it Cardy:2016fqc . In the full
tensor product Hilbert space description given by equation [1], the purified
thermofield double state should be identified with the vacuum state of the CFT
$|0\rangle$:

$|0\rangle\equiv\sum_{B_{1},B_{2}}\sum_{\Delta_{B_{1},B_{2}}}e^{-\frac{\Delta_{B_{1},B_{2}}}{2}}|\Delta_{B_{1},B_{2}}\rangle_{(u)}\otimes|\Delta_{B_{2},B_{1}}\rangle_{(d)}$
(2)

Where $\Delta_{B_{1},B_{2}}$ are the spectrum of $H^{(u)}_{B_{1},B_{2}}$ and
should be identified in the bulk with the microstates of the AdS Rindler
Geometry Czech:2012be 888 Formally, the Modular Hamiltonian of the region $l$,
is given by $K\equiv H^{(u)}_{B_{1},B_{2}}\otimes{\bf
I}^{(d)}_{B_{2},B_{1}}-{\bf I}^{(u)}_{B_{1},B_{2}}\otimes
H^{(d)}_{B_{2},B_{1}}$. The States $|\Delta_{B_{1},B_{2}}\rangle_{(u)}$ are
the eigenstates of $H^{(u)}_{B_{1},B_{2}}$, and in the bulk will be dual to
the fixed area states Dong:2018seb .

The rest of the paper is organized as follows: In the next section, after a
brief review of $sl(2,\mathbb{R})$ driven CFTs in section [2.1], we show the
equivalence of the heating phase Hamiltonian with the Modular Hamiltonian of a
sub-region in the CFT vacuum in section [2.2]. We end this section by
discussing the classification of the dynamical phases in terms of Von Neumann
factors. We explicitly show in section [3.1], that the metric inside the
Entanglement Wedge is given by the AdS Rindler metric, and that the Rindler
Horizon is the RT surface. In section [3.2], we show that in the bulk, the
fixed points of the flow correspond to the Ryu-Takayanagi surface, which ends
on the boundary fixed points. In Section [4.1], we provide a brief review of
heating phase quantization in the conformal frame $CF_{2}$ and the
corresponding Virasoro algebra, as discussed in Tada:2019rls . In Section
[4.2], we begin by reviewing the work of Solodukhin:1998tc , where the near-
horizon Virasoro algebra was constructed. We demonstrate that the structure of
the central extensions in both cases matches exactly, thus providing evidence
for the identification of the emergent Virasoro algebra in the $CF_{2}$ frame
with the near-horizon algebra in the bulk.

In section [5], we explicitly show that the entanglement entropy of the
boundary CFT is indeed the AdS Rindler entropy. We end with a summary of our
results and a discussion of possible future direction in section [6]. The
article has two appendices. In Appendix A, we show that the non-heating phase
Hamiltonian is related to the scaling generators by a unitary transformation,
while in Appendix B, some of the details of section [2.2] is presented.

##  2 Floquet Hamiltonian as a Modular Hamiltonian

To make the article self-contained, we will first briefly review the dynamics
of a driven CFT, where the Hamiltonian of the system is taken to vary
periodically in time, with period T. Depending on the chosen protocol, within
each time period (T), the Hamiltonian may be changing continuously or in
discrete steps. For the sake of simplicity, we focus on the discrete-drive
protocol here.

###  2.1 Brief review of driven Floquet CFTs

We consider a system that is defined on a ring of length $L$. The general form
of the Hamiltonian ($H_{def}^{(i)}$) at the $i^{th}$ discrete step, which
lasts for a time $T_{i}$, is :

$H_{def}^{(i)}=\int^{L}_{0}f_{i}(x)T_{00}(x)dx$ (3)

The functions $f_{i}(x)$ and time interval $T_{i}$ are part of the given
protocol. Restricting to $sl(2,\mathbb{R})$ drives, we choose the functions
such that the Hamiltonians are built out of global generators. With this
restriction, the general form of ‘$f_{i}(x)$’ becomes:

$f_{i}(x)=(a_{0}^{(i)}+b_{0}^{(i)}\cos{\frac{2\pi
x}{L}}+c_{0}^{(i)}\sin{\frac{2\pi x}{L}})$

The corresponding Hamiltonian, can schematically be represented as:

$H=\sum_{n=0}^{\pm 1}p_{n}L_{n}+\sum_{n=0}^{\pm
1}q_{n}\overline{L}_{n}-\frac{c}{12}$

The coefficients $p_{n}$ and $q_{n}$ depends on the specific choice of
$a_{0}^{(i)},b_{0}^{(i)}$ and $c_{0}^{(i)}$. For instance, the choice
$(a_{0}^{(i)}=1,b_{0}^{(i)}=c_{0}^{(i)}=0)$, corresponds to the usual CFT
Hamiltonian $H_{0}=(L_{0}+\bar{L}_{0})$, while
$(a_{0}^{(i)}=1,b_{0}^{(i)}=-1,c_{0}^{(i)}=0)$, corresponds to the so-called
sine-squared deformed Hamiltonian
$H_{1}=\left(L_{0}+\left(\frac{L_{1}+L_{-1}}{2}\right)+c.c.\right)$999$L_{n}$
are the Virasoro generators, defined in the usual way-
$L_{n}=\frac{c}{24}\delta_{n,0}+\frac{L}{2\pi}\int_{0}^{L}\frac{dx}{2\pi}e^{\frac{2\pi
inx}{L}}T(x)$ and
$\bar{L}_{n}=\frac{c}{24}\delta_{n,0}+\frac{L}{2\pi}\int_{0}^{L}\frac{dx}{2\pi}e^{-\frac{2\pi
inx}{L}}\bar{T}(x)$..

As an example of a simple drive protocol, consider evolving the system for
time $T_{0}$ with the Hamiltonian $H_{0}$ and then change the Hamiltonian to
$H_{1}$ and evolve it for $T_{1}$ and keep repeating this process, so that the
time period is $T=T_{0}+T_{1}$. At stroboscopic time (n$T$), the dynamics of
the system are governed by an effective (Floquet) Hamiltonian. For our choices
of the $f_{i}(x)$, the Floquet Hamiltonian of the system will always be some
linear combination of the global Virasoro modes since they form a closed
algebra. The form of the Floquet Hamiltonian can be derived by the requirement
that

$\left(e^{iH_{1}T_{1}}e^{iH_{0}T_{0}}\right)^{n}\mathcal{O}(z_{i},\bar{z}_{i})\left(e^{-iH_{0}T_{0}}e^{-iH_{1}T_{1}}\right)^{n}\equiv
e^{inH_{F}T}\mathcal{O}(z_{i},\bar{z}_{i})e^{-inH_{F}T}$

For any local operator $\mathcal{O}$. For the choices of driving Hamiltonian,
that we are considering here, the most general form of the Floquet Hamiltonian
is,

$H_{F}=(\alpha L_{0}+\beta L_{1}+\gamma L_{-1})+c.c-\frac{c}{12}$ (4)

The above Hamiltonian leads to three distinct dynamics, depending on the
quantity $(\delta=\alpha^{2}-4\beta\gamma)$ defined using the coefficients
$\alpha,\ \beta,\ \gamma$. The three classes of Hamiltonians for $\delta\
(=0,\ <0,\ >0)$ are known respectively as parabolic, hyperbolic, and elliptic
in the literature. It has been shown that two-point and one-point correlation
functions e.g. entanglement entropy and total energy show different dynamical
behaviour when evolved with these three types of Hamiltonians. The
entanglement entropy shows linear, oscillatory, and logarithmic behavior
whereas the total energy shows exponential, oscillatory, and linear behaviour
for $\delta<0,>0$, and $\delta=0$ respectively. Following the authors in
Wen:2018agb ; Wen:2020wee ; Fan:2020orx ; Wen:2022pyj we will be referring to
the three classes as heating phase $(\delta<0)$ and non-heating phase
$(\delta>0)$ separated by a phase boundary $(\delta=0)$.

In the next section, we will show that for the case $\delta<0$, $H_{F}$ can be
expressed as the Modular Hamiltonian of a subregion.

###  2.2 Heating phase as Modular Hamiltonian

The Modular Hamiltonian101010By which we mean extended Modular hamiltonian for
a sub-region ($R_{1}$, $R_{2}$) in the vacuum of a CFT, defined on a ring of
length L is Cardy:2016fqc ,

$K=\frac{L}{\pi}\int_{0}^{L}\frac{\sin{\frac{\pi(x-R_{1})}{L}}\sin{\frac{\pi(R_{2}-x)}{L}}}{\sin{\frac{\pi(R_{2}-R_{1})}{L}}}T_{00}(x)dx$
(5)

If we compare (4) and (5) to find out the subregion for which (4) corresponds
to a Modular Hamiltonian, we get

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\cot{\frac{\pi(R_{1}-R_{2})}{L}}$ (6)
$\displaystyle(\beta+\gamma)$ $\displaystyle=$
$\displaystyle-\frac{\cos{\frac{\pi(R_{1}+R_{2})}{L}}}{\sin{\frac{\pi(R_{1}-R_{2})}{L}}}$
(7) $\displaystyle i(\beta-\gamma)$ $\displaystyle=$
$\displaystyle-\frac{\sin{\frac{\pi(R_{1}+R_{2})}{L}}}{\sin{\frac{\pi(R_{1}-R_{2})}{L}}}$
(8)

Therefore, the quadratic Casimir is,

$\delta=\alpha^{2}-4\beta\gamma=-1$

Since the quadratic Casimir is negative, we can conclude that the Hamiltonian
in the heating phase corresponds to the extended Modular Hamiltonian of an
interval in the vacuum state of a CFT, defined on a ring of length L. In
Tada:2019rls , the authors considered a simpler choice where $\beta=\gamma$.
In that case,

$\displaystyle\alpha=-\cot{\frac{2\pi R}{L}}$ (9)
$\displaystyle\beta=\frac{1}{2}\csc\frac{2\pi R}{L}$

In the above case also we can come to the same conclusion since the Casimir is
again

$\delta=\alpha^{2}-4\beta^{2}=-1$

As described in Tada:2019rls , when we quantize (4) for $(\delta<0)$, the
constant time $t$ curve will end at the fixed point of the quantization which
is located at
$z=\mathcal{Z}_{f_{1,2}}=\frac{-\alpha\pm\sqrt{\alpha^{2}-4\beta^{2}}}{2\beta}$.
Using (9), we can see that $\mathcal{Z}_{f_{1,2}}=e^{\pm\frac{2\pi iR}{L}}$.
Therefore, these two fixed points correspond to two endpoints of the interval
$(R,-R)$. The more general case, when ($\beta\neq\gamma$), is discussed in
detail in appendix B.

Thus, the Floquet dynamics in the heating phase describes the evolution of
operators in modular time111111now identified with the stroboscopic time,
inside the corresponding Entanglement Wedge. This, Modular dynamics, can be
better studied by going to a conformal frame ($CF_{2}$), where the modular
time is identified as time. The conformal map to this new frame is given by:

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\tau+ix=\frac{\sqrt{|\delta|}}{2\beta}\tan\left[\frac{\sqrt{|\delta|}z}{2}\right]$
(10) $\displaystyle\bar{\omega}$ $\displaystyle=$ $\displaystyle\tau-
ix=\frac{\sqrt{|\delta|}}{2\beta}\tan\left[\frac{\sqrt{|\delta|}\bar{z}}{2}\right]$

The holomorphic coordinates ($\omega$) describe a finite size cylinder of
length
$l=2\log\left[\frac{L\sin\frac{\pi(R_{2}-R_{1})}{L}}{\pi\epsilon}\right]$ as
shown in Fig. 2.

To describe a consistent theory on the sub-region, one needs to impose
boundary conditions on the fixed points, after cutting off a small region
($\epsilon$) around itOhmori:2014eia . Imposing conformal boundary conditions,
one ends up with a BCFT on the cylinder, with the Modular Hamiltonian
generating translations along the Euclidean modular time circle 121212since
the Modular Hamiltonian generates rotations around the fixed points
Cardy:2016fqc )..

By definition, the BCFT on the Euclidean cylinder represents a thermal state,
whose thermal entropy is the same as the entanglement entropy of the interval.
This construction has been used to compute the entanglement entropy in 2D CFTs
Holzhey:1994we . The spectrum of the Modular Hamiltonian would then be the
thermal spectrum of a BCFT of length
$l=2\log\left[\frac{L\sin\frac{\pi(R_{2}-R_{1})}{L}}{\pi\epsilon}\right]$. In
the limit $\epsilon\rightarrow 0$, this becomes effectively an infinite length
system and therefore one expects that this spectrum to become continuous in
this limit 131313This provides us with an effective regulator for 2D CFTs on
an infinite line Tada:2019rls . See SDasToAppear , for a detailed analysis of
the spectrum in this Quantization..

Figure 2: The reduced density matrix for a single interval subregion with
endpoints at $R_{1}$ and $R_{2}$ can be calculated as a path integral on a
Euclidean spacetime $z$, which can be conformally mapped via $w=f(z)$ to a
cylinder of fixed width $l=f(R_{2}-\epsilon)-f(R_{1}+\epsilon)$ after removing
circular regions of radius $\epsilon$ from the two ends of the subregion. The
corresponding modular Hamiltonian $K$, which creates flow along the entangling
points at the ends of this subregion, is defined on a horizontal line of this
cylinder, generating flow along the circular direction.

This should be contrasted with the non-heating phase, where the spectrum is
discrete Ishibashi:2016bey . Moreover, as we show in appendix A, the
Hamiltonian in the non-heating phase is unitarily equivalent to the generator
of scaling transformations. Therefore, the non-heating phase dynamics are
equivalent to radial quantization. This difference in the spectrum has a
direct consequence on the behavior of unequal time two-point functions in the
two phases. As was shown in Wen:2022pyj , in the non-heating phase, the
unequal two-point function shows an oscillatory behavior, while in the heating
phase, it shows an exponential decay. Thus in the heating phase, operators
cluster in time and so in the formulation of Furuya:2023fei , are ‘mixing’
operators.

Dynamical phase classification via Von-Neumann Factors

We end this section by noting that the identification of the heating phase
Hamiltonian with a Modular Hamiltonian of a subregion, provides an interesting
non-equilibrium analogue of the Hawking-Page transition between the non-
heating and heating phases in these systems 141414For a recent study see
Miyata:2024gvr .. In Das:2022pez , such a transition was also observed in the
bulk dual in the first derivative of the free energy of a probe Brane, as one
moved between the heating and non-heating phases. Here we point out an
analogous transition directly in the CFT, in the language of Von-Neumann
factors.

On the one hand, the fact that the operators in the CFT are mixing operators
under the heating phase Hamiltonian as well as the clustering property of
(connected) two-point unequal time correlators under the flow generated by
heating phase Hamiltonian in the vacuum state 151515The vacuum is a cyclic and
separating state which serves the purpose of a TFD state in our context as we
discussed in the introduction and hence satisfy KMS condition with respect to
Heating phase Hamiltonian or the modular Hamiltonian.Das:2022pez , implies
that one can associate a type $III_{1}$ algebra with the KMS state associated
with the heating phase HamiltonianFuruya:2023fei .

On the other hand, the unitary equivalence of the non-heating phase
Hamiltonian with the usual CFT Hamiltonian on a ring implies that at finite
$c$, the thermal entropy is finite and so, one can associate a type-$I$
algebra wrt the KMS state (TFD state) associated with this Hamiltonian.

Thus at finite $c$, one sees that as one changes the sign of $d$, one makes a
transition from a type $I$ algebra to a type $III$ algebra. This is in
contrast to the Hawking Page transition which happens strictly in the infinite
$c$ limit, whereas one changes $\beta$, above a particular value of
$\beta=\beta_{HP}$, the algebra associated with the TFD state transitions from
type $I$ to type $III_{1}$ Leutheusser:2022bgi . In the non-equilibrium case,
the role of $\beta$ is played by $\delta$\- the parameter that interpolates
between the two phases.

##  3 A view from the bulk

In this section, we will shift focus to the bulk description of these driven
CFTs. In Das:2022pez , the holographic dual-bulk geometries corresponding to
the three distinct dynamics of the Floquet Hamiltonian $H_{F}$ in the boundary
were discussed. The idea was to write down the bulk counterparts of the
boundary Floquet Hamiltonians, by replacing the global Virasoro generators
$L_{0},L_{1},L_{-1}$ with their bulk representations and then solving the
tangent curve generated by this bulk Hamiltonian.

The three phases of the $sl(2,\mathbb{R})$ drive correspond to the three
distinct geometries, i.e.- $AdS_{3}$ metrics foliated by different $AdS_{2}$
slices. In particular, $AdS_{2}$ black hole, global $AdS$, and Poincaré patch
correspond to the heating, non-heating, and phase transition respectively.

The curve equations in the bulk corresponding to the heating phase are given
in Das:2022pez 161616Here, we have changed the conventions of $x$ and $\tau$
compared to Das:2022pez because we define the complex coordinate on the
boundary as $\omega=\tau+ix$ and $\bar{\omega}=\tau-ix$. Also, we have changed
the definition of $\delta$ to be $\delta=4\beta\gamma-\alpha^{2}$.

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{\sqrt{|\delta|}}{4\beta}\left(\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}+i\theta)\right]+\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}-i\theta)\right]\right)$
(11) $\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{\sqrt{|\delta|}}{4\beta
i}\cos{\phi}\left(\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}+i\theta)\right]-\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}-i\theta)\right]\right)$
$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{\sqrt{|\delta|}}{4\beta
i}\sin{\phi}\left(\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}+i\theta)\right]-\tan\left[\frac{\sqrt{|\delta|}}{2}(\sigma^{0}-i\theta)\right]\right)$

Putting these equations of the curves into the Poincaré metric yields a
$AdS_{2}$ black hole foliation of $AdS_{3}$. Notice that if one takes the
boundary limit, $\phi=0$, in (11), one recovers the conformal transformation
in the CFT, between the conformal frames $CF_{1}$ and $CF_{2}$, discussed in
equation (10), as one should.

###  3.1 AdS Rindler description of the heating phase

Given this boundary conformal transformation (10), there is a well-known way
to lift them to bulk diffeomorphisms, thus obtaining a bulk metric. The bulk
diffeomorphisms corresponding to boundary conformal transformations are given
by Anand:2017dav :

$\displaystyle\omega$ $\displaystyle\rightarrow$ $\displaystyle
f(z)-\frac{2y^{2}\left(f^{\prime}(z)\right)^{2}\bar{f}^{\prime\prime}(\bar{z})}{4f^{\prime}(z)\bar{f}^{\prime}(\bar{z})+y^{2}f^{\prime\prime}(z)\bar{f}^{\prime\prime}(\bar{z})}$
(12) $\displaystyle\bar{\omega}$ $\displaystyle\rightarrow$
$\displaystyle\bar{f}(\bar{z})-\frac{2y^{2}\left(\bar{f}^{\prime}(\bar{z})\right)^{2}f^{\prime\prime}(z)}{4f^{\prime}(z)\bar{f}^{\prime}(\bar{z})+y^{2}f^{\prime\prime}(z)\bar{f}^{\prime\prime}(\bar{z})}$
$\displaystyle u$ $\displaystyle\rightarrow$ $\displaystyle
y\frac{4\left(f^{\prime}(z)\bar{f}^{\prime}(\bar{z})\right)^{\frac{3}{2}}}{4f^{\prime}(z)\bar{f}^{\prime}(\bar{z})+y^{2}f^{\prime\prime}(z)\bar{f}^{\prime\prime}(\bar{z})}$

And the transformed metric is:

$ds^{2}=\frac{dy^{2}+dzd\bar{z}}{y^{2}}-\frac{1}{2}S\left(f,z\right)dz^{2}-\frac{1}{2}S\left(\bar{f},\bar{z}\right)d\bar{z}^{2}+y^{2}\frac{S\left(f,z\right)S\left(\bar{f},\bar{z}\right)}{4}dzd\bar{z}$
(13)

where
$S\left(f,z\right)\equiv\frac{f^{\prime\prime\prime}(z)f^{\prime}(z)-\frac{3}{2}\left(f^{\prime\prime}(z)\right)^{2}}{\left(f^{\prime}(z)\right)^{2}}$.

For the conformal transformations given in (10), the Schwarzians are:

$\displaystyle S\left(f,z\right)$ $\displaystyle=$
$\displaystyle\frac{|\delta|}{2}$ (14) $\displaystyle
S\left(\bar{f},\bar{z}\right)$ $\displaystyle=$
$\displaystyle\frac{|\delta|}{2}$

Therefore, the metric in this case is:

$ds^{2}=\left(\frac{dy}{4}-\frac{1}{y}\right)^{2}(d\sigma^{0})^{2}+\left(\frac{dy}{4}+\frac{1}{y}\right)^{2}d\theta^{2}+\frac{dy^{2}}{y^{2}}$
(15)

If we transform the coordinate $y=\frac{2\left(\sqrt{r^{2}-|\delta|}\ +\
r\right)}{|\delta|}$ and analytically continue $\sigma^{0}\rightarrow
i\sigma^{0}$, the metric becomes:

$ds^{2}=-(r^{2}-|\delta|)(d\sigma^{0})^{2}+\frac{1}{(r^{2}-|\delta|)}dr^{2}+r^{2}d\theta^{2}$
(16)

This is precisely the three-dimensional AdS-Rindler metric. This is
’equivalent’ to the bulk geometry corresponding to the heating phase,
discussed in Das:2022pez , which was an $AdS_{2}$ black hole foliation of
$AdS_{3}$, in the sense that both describe the same bulk observers (since the
time coordinate in both descriptions is the same) and are related by local
diffeomorphisms.

One can repeat this procedure for the case of the non-heating phase, the
boundary conformal transformations for which are given by:

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\tau+ix=-\frac{\sqrt{\delta}}{2\beta}\coth\left[\frac{\sqrt{\delta}z}{2}\right]$
(17) $\displaystyle\bar{\omega}$ $\displaystyle=$ $\displaystyle\tau-
ix=-\frac{\sqrt{\delta}}{2\beta}\coth\left[\frac{\sqrt{\delta}\bar{z}}{2}\right]$

Hence, in this case, the Schwarzians are
$S\left(f,z\right)=S\left(\bar{f},\bar{z}\right)=-\frac{\delta}{2}$, and the
metric is:

$ds^{2}=\left(\frac{dy}{4}+\frac{1}{y}\right)^{2}(d\sigma^{0})^{2}+\left(\frac{dy}{4}-\frac{1}{y}\right)^{2}d\theta^{2}+\frac{dy^{2}}{y^{2}}$
(18)

Following the coordinate redefinition
$y=\frac{2\left(\sqrt{\rho^{2}+\delta}+\rho\right)}{\delta}$ and analytical
continuation of $\sigma^{0}\rightarrow i\sigma^{0}$, the metric can be written
as:

$ds^{2}=-(\rho^{2}+\delta)(d\sigma^{0})^{2}+\frac{1}{(\rho^{2}+\delta)}d\rho^{2}+\rho^{2}d\theta^{2}$
(19)

As we can notice, the above metric (19) corresponds to the global $AdS_{3}$
geometry. On the phase boundary, the boundary conformal transformations are
simply:

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle\tau+ix=-\frac{1}{\beta
z}$ (20) $\displaystyle\bar{\omega}$ $\displaystyle=$ $\displaystyle\tau-
ix=-\frac{1}{\beta\bar{z}}$

Therefore, in this case, the Schwarzians of the transformations are $0$. As a
result, after the analytic continuation of $\sigma^{0}$, the metric is:

$ds^{2}=\frac{-(d\sigma^{0})^{2}+d\theta^{2}+dy^{2}}{y^{2}}$ (21)

The above metric describes the Poincaré patch of three-dimensional AdS
spacetime.

###  3.2 Fixed points as Ryu-Takayanagi surface in the bulk

In this section, we discuss the bulk meaning of the fixed points under the
$sl(2,\mathbb{R})$ valued Floquet Hamiltonian flows in the boundary CFT. For
the Hamiltonian, that we considered in (4), the fixed points are at

$\mathcal{Z}_{f_{1,2}}=\frac{\alpha\pm i\sqrt{|\delta|}}{2\beta}$ (22)

In the bulk, the fixed points correspond to the set of points that do not flow
in time. From the tangent equations of the curve generated by the Hamiltonian
in the bulk ($u\neq 0$), we get 171717See appendix B of Das:2022pez for more
details.

$\displaystyle\frac{du(s)}{ds}$ $\displaystyle=$ $\displaystyle 2\beta
u\left(\tau-\frac{\alpha}{2\beta}\right)$ (23) $\displaystyle\frac{dx(s)}{ds}$
$\displaystyle=$ $\displaystyle 2\beta
x\left(\tau-\frac{\alpha}{2\beta}\right)$ (24)
$\displaystyle\frac{d\tau(s)}{ds}$ $\displaystyle=$
$\displaystyle\beta\left(\left(\tau-\frac{\alpha}{2\beta}\right)^{2}-x^{2}-u^{2}-|\delta|\right)$
(25)

Then, by definition, on the fixed points we will have
$\frac{dz}{ds}=\frac{dx}{ds}=\frac{d\tau}{ds}=0$. Using these conditions, if
we solve the above equations, we get the equations of the fixed points to be

$\displaystyle\tau=\frac{\alpha}{2\beta}$ (26) $\displaystyle
u^{2}+x^{2}=-\frac{|\delta|}{4\beta^{2}}$ (27)

There are a few important points to mention here. Firstly, while solving the
equations we have assumed that, $z\neq 0$ which makes sense since we are
solving the equations in bulk. Also, as we can see, (27) has solutions only
when $\delta\leq 0$, i.e. in the heating phase and on the phase boundary. This
simply means that there are no bulk-fixed points in the non-heating phase, and
therefore a bulk observer can access the full spacetime and not just a sub-
region like the entanglement wedge.

The above equation corresponds to the equation of the RT surface between two
boundary fixed points. This can be verified very easily from the above
equations. In the heating phase, when $\delta<0$, from (27) we get
$x=\pm\frac{\sqrt{|\delta|}}{2\beta}$ on the boundary ($u=0$) and from the
(27) we get $\tau=\frac{\alpha}{2\beta}$. Therefore, the corresponding
coordinates on the plane are, $z\equiv\tau+ix=\frac{\alpha\pm
i\sqrt{|\delta|}}{2\beta}$ which are precisely the fixed points (22).

##  4 Virasoro algebra from the boundary and the bulk

In section 2.2, we argued that the heating-phase Hamiltonian acts as a modular
Hamiltonian for a subregion between two fixed points. To study quantization
under the heating-phase Hamiltonian, it is necessary to introduce an
infinitesimally small ($\epsilon$) cut-off around the fixed points. This
results in a discrete Virasoro algebra with a distinct central extension term.
The algebra becomes continuous in the $\epsilon\rightarrow 0$ limit. A cut-off
in the boundary theory with conformal boundary conditions corresponds to a
cut-off surface near the RT surface in the bulk. On the other hand, there has
been a lot of discussion in the literature of the existence of Virasoro
algebra in the near horizon limit of any $D\geq 3$ dimensional black holes
Solodukhin:1998tc ; Carlip:2002be ; Carlip:1999db ; Carlip:2005zn ;
Guica:2008mu ; Kang:2004js . In this section, we will first review the heating
phase quantization and the corresponding Virasoro algebra following the work
of Tada:2019rls . Then we will go through the general strategy as discussed in
Solodukhin:1998tc , where the near horizon Virasoro algebra was constructed.
Notably, we observe that the central extensions in both these cases match
exactly.

###  4.1 A brief review of the Heating-Phase Quantization

To make this section self-consistent, we will go through the basic steps to
the heating phase quantization discussed in 181818See Arzano:2024ogp for a
recent work on similar ideas. Tada:2019rls . Usually in 2D CFTs, one deals
with the un-deformed Hamiltonian $H_{CFT}=L_{0}+\bar{L}_{0}-\frac{c}{12}$
which is the dilatation generator and generates time translation along the
radial direction on the z-plane. This is known as the “radial quantization"
where the Hilbert space of a CFT defined on a constant (Euclidean) time slice
on the cylinder $S^{1}\times R^{1}$ gets mapped to the Hilbert space on a
constant radius surface on the plane $R^{2}$.

In the Floquet CFT setup, one considers a more general effective Hamiltonian
consisting of a linear combination of $L_{0},L_{1},L_{-1}$ and the
corresponding conjugates.

Given a heating phase Hamiltonian (4) with $\delta<0$, one can use the
prescription by Tada et al. Ishibashi:2016bey ; Tada:2019rls to define the
conserved charges corresponding to new time translation generated by (4), as
the following

$\mathcal{L}_{n}=\frac{1}{2\pi i}\int_{C}g(z)f_{n}(z)T(z)dz\quad\
and\quad\mathcal{L}_{0}=\frac{1}{2\pi i}\int_{C}g(z)T(z)dz$ (28)

along with their respective conjugates, where $T(z)(\bar{T}(z))$ is the chiral
(anti-chiral) component of the energy-momentum tensor) and the function $g(z)$
is obtained by rewriting the heating phase Hamiltonian in terms of the
corresponding classical generators $l_{0},l_{1},l_{-1}$ and their differential
forms $z\partial_{z},z^{2}\partial_{z},\partial_{z}$. In terms of parameters
of $H_{F}$, $g(z)=\beta z^{2}+\alpha z+\gamma$ with
$\delta=\alpha^{2}-4\beta\gamma<0$. To be specific,
$H_{F}=\mathcal{L}_{0}+\bar{\mathcal{L}_{0}}$, whereas $\mathcal{L}_{n}$s are
its eigenmodes with eigenvalue $n$. One must note that just like radial
quantization, these conserved charges (28) are defined on a contour $C$ where
the time generated by the floquet Hamiltonian $H_{F}$ in the heating phase
remains constant. In radial quantization, one defines the usual Virasoro
generators on the concentric circles of different radii at different times. In
particular, $t=-\infty$ corresponds to the center, effectively giving rise to
the well-known notion of state-operator correspondence. On the other hand, in
this case of heating phase quantization, the time slices are arcs connecting
two fixed points, and the constant spatial slices encircle the two fixed
points with varying radii as well as varying centers (See Fig.3 for
reference).

Figure 3: The figures illustrate two different quantization methods. The left-
hand side figure represents radial quantization, while the right-hand side
picture represents the Heating-phase quantization. In these figures, the red
and blue colours indicate constant time and spatial slices, respectively, and
the arrow represents the direction of time flow. In the second figure, the
points $z_{f_{1}}$ and $z_{f_{2}}$ represents the two fixed points of the
underlying dynamics

The $f_{n}(z)$ which appeared in the definition of $\mathcal{L}_{n}$ in (28),
is defined as

$f_{n}(z)=e^{n\int g(z)dz}$ (29)

However, it is important to note that the function $f_{n}(z)$ depends
crucially on the choice of $g(z)$. Therefore, unlike the usual Virasoro
generators the label $n$ can take any value (continuous or discrete) depending
on the final expression of $f_{n}(z)$ (for a given $g(z)$) and its single
valued-ness on the complex plane. For the heating phase, Tada:2019rls showed
that the $f_{n}(z)$ has divergences at the fixed points of the dynamics. This
makes the modes $\mathcal{L}_{n}$ to be non-analytic at the fixed points. In
particular, to obtain the algebra of $\mathcal{L}_{n}$s, one needs to take
care of these divergences. Therefore, one needs to introduce a cut-off
$\epsilon$ circling the fixed points. Thus, introduction of cut-off will
modify the constant-time contour $C\rightarrow C_{\epsilon}$ and the
eigenmodes $\mathcal{L}_{n}\rightarrow\mathcal{L}_{n}^{\epsilon}$. Once we
demand the $\mathcal{L}_{n}^{\epsilon}$s satisfy Virasoro algebra with finite
central extension term 191919The corresponding classical space-time generators
will satisfy the Witt algebra Ishibashi:2016bey , the generator labels $n$ in
(28) are restricted to take discrete values, rendering the spectrum discrete
Tada:2019rls 202020See SDasToAppear for a review and the final form of the
modified Virasoro algebra.. In particular, $n$ takes an integer or half-
integer value multiplied by a $\epsilon$ dependent fractional term. After a
proper rescaling Tada:2019rls , one would end up with the following form of
the Virasoro algebra, where $n\in\mathbb{Z}$:

$[\mathcal{L}_{m}^{\epsilon},\mathcal{L}_{n}^{\epsilon}]=(m-n)\
\mathcal{L}_{n+m}^{\epsilon}+\frac{c}{12}\left(m^{3}+\rho^{2}|\delta|m\right)\
\delta_{n+m,0}$ (30)

Here, $\rho$ is a divergent quantity that depends on the system parameters:
the distance between the two fixed points $L_{\delta}$, the Hamiltonian
parameter $\beta$ and the cut-off $\epsilon$, and is given by
$\rho=\frac{\log(\frac{L_{\delta}}{\epsilon})}{\beta\pi L_{\delta}}$ and
$L_{\delta}=\frac{\sqrt{|\delta|}}{\beta}$.

It is noteworthy to mention that, due to the boundary condition placed at the
cut-off, the whole problem is getting mapped to a BCFT on a finite cylinder as
we mentioned earlier. As argued in SDasToAppear , the boundary conditions on
those two cut-offs can be fixed by conformal boundary conditions. Hence, from
the usual BCFT point of view, we will end up with a single copy of Virasoro
algebra where the holomorphic and anti-holomorphic Virasoro generators would
not be independent anymore. This makes (30) to be a unique and distinct
Virasoro algebra from the usual one.

###  4.2 Bulk realization of the Virasoro algebra

The author argued in Solodukhin:1998tc that the presence of the horizon leads
to the breaking of one of the conformal symmetries, resulting in one copy of
the Virasoro algebra near the horizon. To construct the conformal generators
and the corresponding Virasoro algebra, one begins with the Einstein-Hilbert
action and computes the action for the metric under consideration. In the case
where $D\geq 3$, it was demonstrated that the action can be reformulated as
the action of a 2D Liouville scalar field theory by integrating out the
remaining compact directions. It was shown that in the near horizon limit,
this scalar field theory becomes a 2D conformal field theory in flat space-
time, and the generator of the conformal transformation can be expressed using
the components of the energy-momentum tensor of the scalar field. For a metric
like (16), one introduces a coordinate

$z=\int\frac{dr}{r^{2}-|\delta|}=\frac{1}{2\sqrt{|\delta|}}\log{\left(\frac{r-\sqrt{|\delta|}}{r+\sqrt{|\delta|}}\right)}$

in which the location of the horizon($r=\sqrt{|\delta|}$) is replaced by
$z=-\infty$ and the boundary location ($r=\infty$) is replaced by $z=0$. In
terms of this coordinate, the near-horizon conformal generators are given by

$T[\zeta]=\int_{-\tilde{l}}^{0}dzT_{++}(z)\zeta(z)$ (31)

where $T_{++}(z)$ denotes the right-moving energy-momentum tensor of the
scalar field in the z coordinate, while $\zeta(z)$ represents the vector
fields that generate diffeomorphisms preserving the horizon symmetry. Here we
have introduced a cut-off $\tilde{l}$ in the $z$ coordinate, which we will
take to infinity at the end of the computations. To be precise, $l$ is a near-
horizon cut-off. In the usual AdS-Rindler coordinate (16), this amounts to
take a cut-off at $r=\sqrt{|\delta|}+\zeta$. In terms of $\zeta$, the form of
$l$ would be

$\displaystyle\tilde{l}=\frac{2}{\sqrt{|\delta|}}\log\left(\frac{2\sqrt{|\delta|}}{\zeta}\right).$
(32)

Following this, we expand the vector fields into Fourier modes and introduce
the bulk Virasoro generators.

$L_{n}^{b}=\frac{l}{2\pi}\int_{-\tilde{l}}^{0}dzT_{++}(z)e^{i\frac{2\pi}{\tilde{l}}nz}$
(33)

As a result, the Poisson algebra of the charges (33) reproduces the Virasoro
algebra as

$i\\{L_{k}^{b},L_{n}^{b}\\}_{PB}=(k-n)\
L_{n+k}^{b}+\frac{c}{12}k\left(k^{2}+|\delta|\frac{\tilde{l}^{2}}{4\pi^{2}}\right)\
\delta_{n+k,0}$ (34)

In the above equation, the central charge depends on the classical value of
the field on the horizon. One can immediately observe the resemblance between
(34) and (30) if $\tilde{l}=2\pi\rho$. Therefore,

$\log\left({\frac{\sqrt{|\delta|}}{\beta\epsilon}}\right)=\frac{\sqrt{|\delta|}}{2}\tilde{l}$
(35)

In other words, from (32) we get $\zeta=2\beta\epsilon$. Hence, the boundary
cut-off $\epsilon$ is related to near-horizon cut-off.

In the next section, we will see that the same relation of the form (35)
between the boundary cut-off and the bulk IR (near boundary) cut-off would
appear if one tries to match the entanglement entropy of the region between
two fixed points with the entropy of the AdS-Rindler. This would suggest the
near boundary bulk IR cut-off is proportionally related to the near horizon
bulk UV cut-off. This fact could be well-explained if we study the same in the
equivalent bulk picture in coordinates ($\phi,\sigma^{0},\theta$) described in
Das:2022pez . In that case, the compactified $\phi$ direction can be
integrated out to carry out the similar analysis in the AdS2 black hole patch
in $(\sigma^{0},\theta)$ coordinates. In the boundary limit $(\phi\rightarrow
0,\pi)$, the horizon of the AdS2 black hole exactly coincides with the fixed
points, as we observed previously. Thus, the cut-off in $\theta$ is the same
cut-off near the horizon of the 2D boundary black hole. The bulk description
of the metric can be thought of as the pulling out of the boundary black hole
in the bulk along the $\phi$ direction 212121At each constant $\phi$ slice,
there is a 2D AdS black hole.. The equivalence of this metric with the AdS-
Rindler in the bulk would explain the equivalence of near-horizon and boundary
cut-off as we observed.

##  5 Entanglement Entropy as BH Entropy

In this section, we will highlight that the entanglement entropy between the
two fixed points is equal to the Bekenstein Hawking entropy corresponding to
the AdS-Rindler metric. More specifically, we will compute the holographic
Entanglement entropy by computing the length of the RT surface and comparing
it with the Rindler entropy.

Entanglement Entropy
To compute the entanglement entropy using the Ryu-Takayanagi formula, one
needs to find the length of the RT surface. For this, one computes the
geodesic length between two arbitrary points ($u_{1},t_{1},x_{1}$) and
($u_{2},t_{2},x_{2}$), which is given by the following relation in AdS-
Poincaré coordinates,

$D=\cosh^{-1}\left(\frac{(x_{1}-x_{2})^{2}-(t_{1}-t_{2})^{2}+u_{1}^{2}+u_{2}^{2}}{2u_{1}u_{2}}\right)$
(36)

where, $x,t$ are the boundary coordinates while $u$ extends into the bulk. To
avoid the divergence, one usually sets $u_{1}=u_{2}=\epsilon$. In our setup,
$t_{1}=t_{2}=\frac{\alpha}{2\beta}$, $x_{1}=\frac{\sqrt{|\delta|}}{2\beta}$
and $x_{2}=-\frac{\sqrt{|\delta|}}{2\beta}$. Then the RT formula tells us that
the entanglement entropy is given by

$S_{ent}=\frac{D}{4G_{\text{N}}^{(3)}}=\frac{1}{2G_{\text{N}}^{(3)}}\log\left(\frac{\sqrt{|\delta|}}{\beta\epsilon}\right)=\frac{c}{3}\log\left(\frac{\sqrt{|\delta|}}{\beta\epsilon}\right)$
(37)

Note that, in the last equation, we have used
$G_{\text{N}}^{(3)}=\frac{3}{2c}$.

Entropy of AdS-Rindler
The entropy of AdS-Rindler is defined as Compere:2018aar ,

$S_{\text{H}}=\frac{1}{4G_{\text{N}}^{(3)}}\int_{-\frac{1}{\epsilon^{\prime}}}^{\frac{1}{\epsilon^{\prime}}}d\theta\sqrt{g_{\text{ind}}}$
(38)

where $\sqrt{g_{\text{ind}}}$ is the determinant of the induced metric on the
horizon. At a constant time
$ds_{\text{ind}}^{2}=ds^{2}[r=\sqrt{|\delta|},t=constant]=\delta(d\theta)^{2}$,
so

$S_{\text{H}}=\frac{\sqrt{|\delta}|}{2G_{\text{N}}\epsilon^{\prime}}$ (39)

Therefore, (39) and (39) matches if we identify $\epsilon^{\prime}$ and
$\epsilon$ as

$\frac{\sqrt{|\delta|}}{\epsilon^{\prime}}=\log\left(\frac{\sqrt{|\delta|}}{\beta\epsilon}\right)$
(40)

This can be understood from the relation between the coordinate u and
$\theta$, which we know using (10) in (12) to be

$u=\left(\frac{\sqrt{|\delta|}}{2\beta}\right)\frac{(\cosh{(\sqrt{|\delta|}\sigma^{0})}+\cosh{(\sqrt{|\delta|}\theta)})}{(\cosh{(\sqrt{|\delta|}\sigma^{0})}+\cosh{(\sqrt{|\delta|}\theta)})\cosh{(\sqrt{|\delta|}\theta})}$

In the large $\theta$ limit this reduces to,

$u=\frac{\sqrt{|\delta|}}{\beta}e^{-\sqrt{|\delta|}\theta}$ (41)

If we put $u=\epsilon$ and $\theta=\frac{1}{\epsilon^{\prime}}$, it precisely
reproduces (40).

##  6 Discussion

In this article, we have explored some consequences that follow from the
identification of the heating phase Hamiltonian with the Modular Hamiltonian
of a sub-region, in the vacuum state. From this perspective, the heating phase
quantization describes the evolution of operators inside the causal diamond of
the subregion, between two fixed points of the $sl(2,\mathbb{R})$ flow, in
modular/stroboscopic time. To study the modular quantization of this
subregion, one needs to transform to a conformal frame, wherein the time
coordinate is identified with the modular time. One then imposes a conformal
boundary condition at an $\epsilon$ cut-off away from the fixed points of the
Floquet Hamiltonian flows. This reduces the system of study to that of a BCFT
on a strip. We also showed that holographically, this BCFT describes the
dynamics inside the RT Entanglement Wedge, described by an AdS Rindler metric,
where the Rindler Horizon is identified with the RT surface, and that the
boundary cut-off, near the fixed points, translates to a cut-off, analogous to
a stretched horizon222222See SDasToAppear and references therein where the
connection to the stretched horizon has been explored in details., near the RT
surface or the Rindler horizon. We further argued that the single copy of the
conformal algebra preserved by the boundary conditions of the BCFT should be
dual to the single copy of the Virasoro symmetry at the stretched Horizon.
From the perspective of driven CFTs, we also argued that the different
dynamical phases of these $CFT$s can be classified by the type of Von Neumann
factors. In particular as one moves from the non-heating to the heating phase,
by tuning the external control parameters like the frequency or time period of
the drive, the factor changes from type $I$ to type $III$. This, again, is a
direct consequence of the identification of the heating phase dynamics with
Modular quantization.

Since in the bulk description, the qualitative differences in the three
dynamical phases correspond to differences in observations made by three
different classes of observers Das:2022pez , it follows that the Von-Neumann
algebra of observables accessible to the observer, dual to the heating phase
would be type $III$, while the corresponding Von-Neumann algebra of
observables accessible to the observer dual to the non-heating phase should be
Type $I$. It would be interesting to understand this better along the lines of
recent investigations Witten:2023qsv ; Penington:2023dql ; Strohmaier:2023opz
; Chen:2024rpx .

Given that the boundary theory in our work describes the AdS Rindler Black
hole, it should be possible to explicitly construct the microstates of the
black hole directly from the heating phase of the driven CFT. In fact, these
geometries would be the so-called fixed area states, since these would be
eigenstates of the $H^{(u)}_{B_{1}B_{2}}$, which in the large $c$ is
identified with the area operatorDong:2018seb . It would be nice to explicitly
construct these states directly in the CFT232323Partial progress along this
line has been done in SDasToAppear ..

It would also be interesting to understand better the connection between the
‘emergent’ horizon Virasoro algebra and the ‘emergent’ boundary Virasoro
algebra in Modular quantization. In fact, as shown in SDasToAppear , the
microcanonical counting of states of this Hamiltonian which also describes the
horizon CFT, would exactly reproduce the Bekenstein-Hawking entropy of BTZ. In
the context of horizon CFT, this implies that these states live in the
(vicinity of) horizon rather than at an asymptotic boundary. Also, a similar
structure of near horizon Virasoro algebra exists in any higher dimensional
asymptotic AdS or flat spacetime, as argued in Solodukhin:1998tc . Hence, we
suspect the appearance of this conformal field theory and the emergence of
horizon Virasoro symmetry have far-reaching consequences, beyond AdS/CFT. We
hope to explore some of these issues in the immediate future.

## Acknowledgments

The authors are thankful to Souvik Banerjee, Parthajit Biswas, Diptarka Das,
Alok Laddha, and Partha Paul for several useful discussions. The authors thank
Arnab Kundu and Krishnendu Sengupta for their helpful discussions on related
topics. BR would like to thank Aninda Sinha for the three-week visit to IISc,
where, some final works of this project were completed. BE acknowledges the
support provided by the SERB grant CRG/2021/004539. The research work of SD is
supported by a DST Inspire Faculty Fellowship. The work of SP and BR is
supported by the Senior Research Fellowship(SRF) funded by the University
Grant Commission(UGC) under the CSIR-UGC NET Fellowship.

##  Appendix A Relation among different Non-heating phase Hamiltonians

As discussed earlier in section 2.2, it was stated that all the Hamiltonians
in the non-heating phase are unitarily equivalent. This section aims to
furnish the necessary mathematical details to substantiate this claim. Note
that, in what follows, we will only discuss using the holomorphic part of the
theory, which remains consistent for the anti-holomorphic counterparts. The
first key point to keep in mind is that the subalgebra of the Virasoro
algebra,

$[J_{0},J_{\pm}]=\pm J_{\pm},[J_{-},J_{+}]=2J_{0}$ (42)

obtained by redefining $L_{\pm n},L_{0}$ as

$J_{\pm}=\frac{L_{\mp n}}{n}\,\ \
J_{0}=\frac{L_{0}}{n}+\frac{c}{24n}(n^{2}-1)$

enables to show that the following condition is true

$e^{\alpha J_{0}+\beta J_{+}+\gamma J_{-}}=e^{bJ_{+}}e^{aJ_{0}}e^{cJ_{-}}$
(43)

where the coefficients $\alpha,\beta,\gamma$ and $a,b,c$ are related by the
following relation as discussed in Liska:2022vrd ; Matone:2015wxa ,

$\displaystyle
a=Exp\bigg{[}\frac{1}{(\cosh\mathcal{P}-\frac{\alpha}{2\mathcal{P}}\sinh\mathcal{P})^{2}}\bigg{]},$
$\displaystyle
b=\frac{\frac{\beta}{\mathcal{P}}\sinh\mathcal{P}}{\cosh\mathcal{P}-\frac{\alpha}{2\mathcal{P}}\sinh\mathcal{P}},\
c=\frac{\frac{\gamma}{\mathcal{P}}\sinh\mathcal{P}}{\cosh\mathcal{P}-\frac{\alpha}{2\mathcal{P}}\sinh\mathcal{P}}$
(44)

where,

$\mathcal{P}=\frac{\sqrt{\alpha^{2}-4\beta\gamma}}{2}$ (45)

The condition in (43) then allows writing the following similarity
transformation ($\mathcal{S}=e^{(\alpha J_{0}+\beta J_{+}+\gamma J_{-})}$)
acting on $J_{0}$ as,

$\mathcal{S}J_{0}\mathcal{S}^{-1}=e^{bJ_{+}}e^{aJ_{0}}e^{cJ_{-}}J_{0}e^{-cJ_{-}}e^{-aJ_{0}}e^{-bJ_{+}}$
(46)

This crucial step makes it all easy and enables us to utilize the Baker-
Campbell-Hausdorff formula and exploit the sub-algebra to carry out the
computation step by step to find,

$e^{cJ_{-}}J_{0}e^{-cJ_{-}}=1+c[J_{-},J_{0}]+\frac{c^{2}}{2!}[j_{-},[J_{-},[J_{-},J_{0}]]+\dots=J_{0}+cJ_{-}$
(47)

Similarly ,

$e^{aJ_{0}}e^{cJ_{-}}J_{0}e^{-cJ_{-}}e^{-aJ_{0}}=e^{aJ_{0}}(J_{0}+cJ_{-})e^{-aJ_{0}}=J_{0}+(ce^{-a})J_{-}$
(48)

and

$e^{bJ_{+}}e^{aJ_{0}}e^{cJ_{-}}J_{0}e^{-cJ_{-}}e^{-aJ_{0}}e^{-bJ_{+}}=(1-2bce^{-a})J_{0}+(cb^{2}e^{-a}-b)J_{+}+(ce^{-a})J_{-}$
(49)

At this stage, one should check that while this similarity transformation
takes $J_{0}\rightarrow(pJ_{0}+qJ_{+}+rJ_{-})$, the Casimir $p^{2}-4qr$
remains the same, specifically, $p^{2}-4qr=1$ for the above case. From this
one can conclude that the usual CFT hamiltonian in radial quantization
$L_{0}+\bar{L}_{0}$, which is an example of a non-heating phase hamiltonian
gets similarity transformed to another non-heating phase hamiltonian by
retaining the Casimir invariant.

Using a very similar approach, one can also show that a unitary transformation
of $L_{0}$ with the floquet generator $U_{F}=e^{-iH_{F}s}$, where,
$H_{F}=\alpha L_{0}+\beta L_{+}+\gamma L_{-}$ takes $L_{0}$ to some other
linear combination of $L_{0}$ retaining the casimir invariant. Note that, here

$L_{+}=\frac{L_{1}+L_{-1}}{2},L_{-}=\frac{L_{1}-L_{-1}}{2i}$

and $\alpha,\beta,\gamma$ are real. Following the previous discussion, one
begins with the equivalence,

$e^{iH_{F}s}L_{0}e^{-iH_{F}s}=(e^{ikL_{-}}e^{igL_{+}}e^{ifL_{0}})L_{0}(e^{-ifL_{0}}e^{-igL_{+}}e^{-ikL_{-}})$
(50)

where a relation similar to that of (A) exists among the coefficients
$(\alpha,\beta,\gamma)$ and $(f,g,k)$. Using (50) and exploiting the BCH
formula and

$[L_{\pm},L_{0}]=\pm iL_{\mp},\ \ \ \ [L_{+},L_{-}]=iL_{0}$ (51)

one can find that,

$\displaystyle e^{igL_{\pm}}L_{0}e^{-igL_{\pm}}$ $\displaystyle=$
$\displaystyle(\cosh g)L_{0}\mp(\sinh g)L_{\mp}$ (52)

and therefore, the RHS of (50) to be $(\cosh g\cosh k)L_{0}+(\cosh g\cosh
k)L_{+}-\sinh g$. Note that the Casimir ($\alpha^{2}-\beta^{2}-\gamma^{2}$)
remains invariant ($=1$) for this too.

##  Appendix B Fixed points as the end points of subregion

In this section, we want to show explicitly that the Hamiltonian in the
heating phase corresponds to the modular Hamiltonian of a sub-region, with the
endpoints being the fixed points of the conformal transformation generated by
the Hamiltonian. The Hamiltonian in consideration is,

$H_{F}=(\alpha L_{0}+\beta L_{1}+\gamma L_{-1})+c.c-\frac{c}{12}$ (54)

As the above Hamiltonian is made out of only the global generators, it
generates Möbius transformation ($z^{\prime}=\frac{az+b}{cz+d}$) in the
complex plane. In the heating phase, when $(\alpha^{2}-4\beta\gamma)<0$, the
parameters of the transformation generated by (54) are

$\displaystyle a$
$\displaystyle=-\cos{\frac{|\delta|\pi\tau}{L}}-\frac{\alpha}{|\delta|}\sin{\frac{|\delta|\pi\tau}{L}}$
$\displaystyle b$
$\displaystyle=-\frac{\beta+i\gamma}{|\delta|}\sin{\frac{|\delta|\pi\tau}{L}}$
$\displaystyle c$
$\displaystyle=-\frac{\beta-i\gamma}{|\delta|}\sin{\frac{|\delta|\pi\tau}{L}}$
$\displaystyle d$
$\displaystyle=-\cos{\frac{|\delta|\pi\tau}{L}}+\frac{\alpha}{|\delta|}\sin{\frac{|\delta|\pi\tau}{L}}$
(55)

Now, for a conformal transformation $z\rightarrow
z^{\prime}=\frac{az+b}{cz+d}$, the fixed points are

$\mathcal{Z}_{f_{1,2}}=\frac{a-d\pm\sqrt{(a-d)^{2}+4bc}}{2c}$ (56)

We put (7), (8), (8) in (B) to write the parameters of the conformal
transformation in terms of $R_{1}$ and $R_{2}$ and then we use (56) to get

$\displaystyle\mathcal{Z}_{f_{1}}$ $\displaystyle=e^{\frac{2\pi iR_{2}}{L}}$
$\displaystyle\mathcal{Z}_{f_{2}}$ $\displaystyle=e^{\frac{2\pi iR_{1}}{L}}$
(57)

Therefore, we can conclude that two endpoints (on the cylinder) of the region
corresponding to the Modular Hamiltonian correspond to the two fixed points.

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