spectroscopic data, given the maturity of the field of study.  These
papers exemplify the power of SimBAL analysis.  The forward
modeling semi-empirical approach allows us to directly constrain the
physical parameters of the outflow as a function of velocity in the
most complicated spectra. The relative ease of use allows us to
compile detailed results from moderate-sized samples, which means that
we can not only look for trends, but also place our  analysis on a
statistical footing.

At the same time, it is worth remembering that the quasar phenomenon
is stochastic rather than deterministic.  The stochasticity of the
quasar phenomenon was discussed in the context of Type 2 AGN by
Elitzur & Netzer, 2016 <cit.>.  They argued that the
luminosity cutoff consistent with the presence of a broad line region
depends on the black hole mass, but that the proportionality
coefficient is not a single value but has a range of values due to
factors that are not or cannot be modelled. Thus it is certain that
objects that do not conform to the scenario presented here can be
found.   While the cutoff for different accretion states in
X-ray binaries may be well constrained (e.g., the  $\dot
m_\mathrm{crit}=\dot M/\dot M_\mathrm{Edd}=0.01$–0.02 for the
transition from hard to soft state), it seems reasonable that the
different physics of accretion onto a million or billion times larger
black hole would shift this value.     A significant factor may be the
fact that the accretion disk around a large black hole should emit in
the UV where there is plenty of atomic opacity.  This idea has been
investigated by Jiang et al., 2016 who found that a modeled accretion
disk that included atomic transitions, in particular the iron opacity
that is also important in stellar winds, was more stable than one
without atomic transitions.

Moving forward, we have embarked  on an analysis of higher-redshift
($1.3 < z < 3$) FeLoBALQs; preliminary results have been presented in
Voelker et al., 2021.  Because of the flux-limited nature of SDSS, the
higher-redshift objects are more luminous.  More luminous objects are
known to have faster outflows <cit.>, and
we have already found evidence for a higher incidence of energetic
outflows among these more luminous objects.  An interesting question
is whether we will find many $E1<0$ FeLoBALQs, in particular, the
objects with loitering outflows.  In the sample presented here, we
found that about half of the objects have $E1<0$.  Low accretion rate
objects are predicted to fall out of flux-limited samples at higher
redshifts [Jester et al., 2005], so we expect to find fewer of them.  On
the other hand, the loitering outflow objects have very characteristic
FeLoBAL absorption spectra, and it may be that we can use those
spectra to predict which objects should have low accretion rates.  If
we can find such objects at higher redshifts, we may find very massive
black holes, and possibly be able to place constraints on the scaling
of the critical accretion rate $\dot m_\mathrm{crit}$ with black hole
mass and/or luminosity.

KML acknowledges very useful conversations with Leah Morabito.
Support for SimBAL development and analysis is provided by NSF
Astronomy and Astrophysics Grants No. 1518382, 2006771, and
2007023. This work was performed in part at Aspen Center for Physics,
which is supported by National Science Foundation grant
PHY-1607611. SCG thanks the Natural Science and Engineering Research
Council of Canada.

Long before the University of Oklahoma was established, the land on
which the University now resides was the traditional home of the
“Hasinais” Caddo Nation and “Kirikiris” Wichita & Affiliated
Tribes. This land was also once part of the Muscogee Creek and
Seminole nations.

We acknowledge this territory once also served as a hunting ground,
trade exchange point, and migration route for the Apache, Comanche,
Kiowa and Osage nations. Today, 39 federally-recognized Tribal nations
dwell in what is now the State of Oklahoma as a result of settler
colonial policies designed to assimilate Indigenous peoples.

The University of Oklahoma recognizes the historical connection our
university has with its Indigenous community. We acknowledge, honor
and respect the diverse Indigenous peoples connected to this land. We
fully recognize, support and advocate for the sovereign rights of all
of Oklahoma’s 39 tribal nations.

This acknowledgement is aligned with our university’s core value of
creating a diverse and inclusive community. It is our institutional
responsibility to recognize and acknowledge the people, culture and
history that make up our entire OU Community.

Funding for the SDSS and SDSS-II has been provided by the Alfred
P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the
Max Planck Society, and the Higher Education Funding Council for
England. The SDSS Web Site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the
American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve
University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Joint Institute for Nuclear Astrophysics, the
Kavli Institute for Particle Astrophysics and Cosmology, the Korean
Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington.

Funding for SDSS-III has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science
Foundation, and the U.S. Department of Energy Office of Science. The
SDSS-III web site is http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven
National Laboratory, Carnegie Mellon University, University of
Florida, the French Participation Group, the German Participation
Group, Harvard University, the Instituto de Astrofisica de Canarias,
the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins
University, Lawrence Berkeley National Laboratory, Max Planck
Institute for Astrophysics, Max Planck Institute for Extraterrestrial
Physics, New Mexico State University, New York University, Ohio State
University, Pennsylvania State University, University of Portsmouth,
Princeton University, the Spanish Participation Group, University of
Tokyo, University of Utah, Vanderbilt University, University of
Virginia, University of Washington, and Yale University.

IRTF (SpeX), Sloan, WISE, NEOWISE

Sherpa [Freeman et al., 2001], unTimelyCatalogExplorer [Kiwy, 2022]

§ SYSTEMATIC ERRORS IN THE UNTIMELY CATALOG PHOTOMETRY

The analysis was performed for each of the 30 FeLoBALQs and each of the
132 comparison objects.  For each object, star candidates were
identified from the AllWISE Source Catalog (TAP identifier allwise_p3as_psd) as follows.  The potential star candidates were
objects lying within circular region with a radius of 500 arc seconds
around each quasar
that also had W1$>9$, were less than 0.5 magnitudes dimmer than the
corresponding object, and with color range $-0.2< W1-W2 < 0.2$.  The
time series data was downloaded from the unTimely Catalog using the unTimely_Catalog_explorer [Kiwy, 2022] using a box size of
1000 arc seconds.  All star candidates were required to have both W1 and
W2 light curves.

A comparison of the stellar light curve variance and the
mean catalog error squared for the star candidates for both the W1
and W2 filters; legends apply to both panels.  Most stars do not
vary significantly.  However, we
found that the measured light curve variance (grey) is larger than the
statistical error squared (pink), especially at lower magnitudes.  We
interpreted this as evidence for a systematic underestimation of the
statistical errors.  We defined a modified error as a function of
magnitude based on the observed median (black line) and 16 and 84%
cumulative distribution values (dashed grey lines).

The stars were chosen from the star candidates by requiring that the
magnitude of the star lay within a range of the magnitude of the
object.  Because there are far more fainter stars than bright ones,
the range depended on the magnitude as $\Delta M=0.7 - 0.125
(W1_{object} - 12.5)$.  In addition, only lightcurves with 10 or more
points were considered.    The variance in the magnitude was computed for
each star candidate.  The mean of the catalog-obtained error squared
was also collected.  Fig. <ref> shows the relationship
between these values for each star.   There were 3293 W1 comparison
stars and 661 W2 comparison stars.  The values are different because
quasars are relatively brighter in W2 compared with W1, and there is a
lower density of brighter stars in the sky. The observed variance was
found to be larger than the mean error squared, with the discrepancy
being larger for the brighter objects.  The origin of this discrepancy
is not known.   For each object, the median of the variance and the
error squared is also plotted.   Because of the outliers with unknown
origin seen in the plot, median was deemed more representative than
the mean. These values are also shown in the plot.

As discussed in <ref>, the principal metric used for
the variability was the excess variance, which is the difference
between the variance of the object and the contribution due to noise
in the data.  We determined the contribution due to the noise in the
data from the distribution of all of the stars shown in
Fig.<ref>.  In bins of magnitude, the median, 16 and
84% values of the distribution were computed.  The results are seen
in Fig. <ref>.  For each object, the
variance due to noise was interpolated from the median relationship
using the mean magnitude of the object.  The uncertainties used in
computing the excess variance in <ref> were taken to
be the 1-sigma bounds shown.

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Restoring the ”new” subscript then gives (7.6) in the main text.

####  D.2.2 BTZ

For BTZ, we have $f(r)=r^{2}-r_{+}^{2}$ in (D.32). Let us consider the cases
in which the spacelike geodesic never enters the black hole. In such a case,
$V(r)\geq 0$ for some $r>r_{+}$ Hubeny:2012ry . To see what this indicates
more concretely, we first note that $V(r)$ peaks at $r=\sqrt{Lr_{+}}$:

$\displaystyle\frac{dV(r)}{dr}=-2r+\frac{2L^{2}r_{+}^{2}}{r^{3}}$
$\displaystyle\frac{dV(r)}{dr}=0\Rightarrow r=\sqrt{Lr_{+}}$ (D.44)

where one needs $\sqrt{Lr_{+}}\geq r_{+}$, i.e. $L\geq r_{+}$. The condition
for the maximal value of $V(r)$, that is $V(\sqrt{Lr_{+}})$, to be non-
negative is

$\displaystyle V(\sqrt{Lr_{+}})=-E^{2}+(L-r_{+})^{2}\geq
0\qquad\Rightarrow\qquad L\geq E+r_{+}$ (D.45)

which is more stringent than $L\geq r_{+}$. Henceforth, the criterion for the
spacelike geodesic not entering the BTZ horizon is $L\geq E+r_{+}$. The
limiting case $L=r_{+}$ when $E=0$ corresponds to the known case that the
spacelike geodesic gets trapped in a circular orbit at the horizon, which is
studied in Hubeny:2012ry . At the minimal radius the spacelike geodesic can
reach, $dr/d\lambda=0$, so the minimal radius is given by the larger positive
zero of $V(r)$

$\displaystyle 0$
$\displaystyle=r^{2}-r_{+}^{2}+E^{2}-L^{2}\frac{r^{2}-r_{+}^{2}}{r^{2}}$
(D.46) $\displaystyle\Rightarrow r_{*}$
$\displaystyle=\sqrt{\frac{1}{2}\Big{[}L^{2}-E^{2}+r_{+}^{2}+\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}\Big{]}}$
(D.47)

One can verify that for $L=E+r_{+}$, $r_{*}=\sqrt{r_{+}(r_{+}+E)}>r_{+}$;
increasing $L$ results in increasing $r_{*}$. Hence $r_{*}>r_{+}$ for $L\geq
E+r_{+}$. By setting $r_{+}^{2}=-1$ in (D.47), we recover (D.37) in pure AdS3.

The regulated geodesic length is then given by integrating (D.34)

$\displaystyle\mathcal{L}_{\rm
reg}=\Big{(}\int_{r_{*}}^{R_{1}}+\int_{r_{*}}^{R_{2}}\Big{)}\frac{dr}{\sqrt{-V(r)}}=-\log\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}+\log
4R_{1}R_{2}$ (D.48)

Note that by setting the BTZ black hole radius $r_{+}^{2}=-1$ in (D.48), we
recover (D.38) in pure AdS3. The temporal and angular distance between the two
geodesic endpoints on the boundary are given by integrating (D.35) and (D.36),
respectively

$\displaystyle\Delta t$
$\displaystyle=2\int_{r_{*}}^{\infty}\frac{E}{f(r)\sqrt{-V(r)}}dr$ (D.49)
$\displaystyle=\frac{1}{r_{+}}\log[\frac{\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}E^{2}}}{(L-E-
r_{+})(L+E+r_{+})}\big{]}$ (D.50) $\displaystyle=\frac{1}{r_{+}}{\rm
arccosh}\Big{[}\frac{L^{2}-E^{2}-r_{+}^{2}}{\sqrt{(L^{2}-E^{2}-r_{+}^{2})^{2}-4r_{+}^{2}E^{2}}}\Big{]}$
(D.51) $\displaystyle\Delta x$
$\displaystyle=2\int_{r_{*}}^{\infty}\frac{L}{r^{2}\sqrt{-V(r)}}dr$ (D.52)
$\displaystyle=\frac{1}{r_{+}}\log[\frac{\sqrt{(E^{2}-L^{2}-r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}}{(L+E-r_{+})(L-E-
r_{+})}\big{]}$ $\displaystyle=\frac{1}{r_{+}}{\rm
arccosh}\Big{[}\frac{L^{2}-E^{2}+r_{+}^{2}}{\sqrt{(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}}}\Big{]}$
(D.53)

which are (7.14) and (7.15) in the main text. Note that
$(L^{2}-E^{2}+r_{+}^{2})^{2}-4r_{+}^{2}L^{2}=(L^{2}-E^{2}-r_{+}^{2})^{2}-4r_{+}^{2}E^{2}$.
One can verify that when $E=0$, (D.53) becomes

$\displaystyle\Delta x=\frac{1}{r_{+}}{\rm
arccosh}\Big{(}\frac{L^{2}+r_{+}^{2}}{L^{2}-r_{+}^{2}}\Big{)}=\frac{2}{r_{+}}{\rm
arctanh}\Big{(}\frac{r_{+}}{L}\Big{)}$ (D.54)

which matches with a known result in section 2.3 of Hubeny:2012ry .

From (7.14) and (7.15), one can solve for $E$ and $L$

$\displaystyle E=r_{+}\frac{\sinh(r_{+}\Delta t)}{\cosh(r_{+}\Delta
t)-\cosh(r_{+}\Delta x)}$ $\displaystyle L=r_{+}\frac{\sinh(r_{+}\Delta
x)}{\cosh(r_{+}\Delta t)-\cosh(r_{+}\Delta x)}$ (D.55)

Plugging (D.55) into (D.48), we have,

$\displaystyle\mathcal{L}_{{\rm reg}}$
$\displaystyle=\log[\Big{(}\frac{2}{r_{+}}\Big{)}^{2}\frac{1}{2}\big{|}\cosh(r_{+}\Delta
t)-\cosh(r_{+}\Delta x)\big{|}\Big{]}+\log(R_{{1}}R_{{2}})$ (D.56)
$\displaystyle=\log[\Big{(}\frac{2}{r_{+}}\Big{)}^{2}\Big{|}\sin\Big(r_{+}\frac{w_{1}-w_{2}}{2}\Big{missing})\
\sin\Big(r_{+}\frac{\overline{w}_{1}-\overline{w}_{2}}{2}\Big{missing})\Big{|}\
\Big{]}+\log(R_{{1}}R_{{2}})$ (D.57)

where we have used (7.1) in getting from the second to the third line. This
then gives (7.17) in the main text with the ”new” subscript restored.

##  Appendix E Time evolution of horizons

###  E.1 SSD CFT

In this appendix, we show how the coordinates in Figure 5 are established. In
Goto:2021sqx , a redefinition of the radial coordinate

$\displaystyle r^{\prime}=\sqrt{\frac{\partial w_{{\rm new}}}{\partial
w}\frac{\partial\overline{w}_{{\rm new}}}{\partial\overline{w}}}r_{{\rm
new}}=\frac{1}{\sqrt{4t^{2}\sin^{4}\frac{x}{2}+\Big{(}1-t^{2}\sin^{2}\frac{x}{2}\Big{)}^{2}}}r_{{\rm
new}}$ (E.1)

was introduced, and the BTZ black hole horizon was plotted on constant $t$
slices in the $(t,x,r^{\prime})$ coordinate. In order to compare with their
result, we adopt the same $(t,x,r^{\prime})$ coordinate when plotting Figure
5. Notice that unlike (B.6), the metric in the $(t,x,r^{\prime})$ coordinate
is not in the Fefferman-Graham gauge Skenderis:1999nb ; deHaro:2000vlm , as
$r^{\prime}$ is $t$ and $x$ dependent, leading to non-zero $dr^{\prime}dt$ and
$dr^{\prime}dx$ terms in the metric. Therefore quantitative comparisons with
the boundary stress tensor are in general hard to achieve.

###  E.2 Floquet CFT

As for Floquet CFT, we introduce the analogue of (E.1) at stroboscopic time
$t=n(T_{0}+T_{1})$ to plot Figure 10.

$\displaystyle r^{\prime}$ $\displaystyle=\sqrt{\frac{\partial w_{n}}{\partial
w}\frac{\partial\overline{w}_{n}}{\partial\overline{w}}}r_{n}=\frac{e^{ix}|\left(\eta^{n}-1\right)^{2}-\left(\gamma_{1}\eta^{n}-\gamma_{2}\right)\left(\gamma_{2}\eta^{n}-\gamma_{1}\right)|}{\sqrt{\Xi(x,t)}}r_{n}$
$\displaystyle\Xi(x,t)$
$\displaystyle=\left(e^{ix}\left(\gamma_{1}\eta^{n}-\gamma_{2}\right)-\eta^{n}+1\right)\left(\gamma_{2}+e^{ix}\left(\eta^{n}-1\right)-\gamma_{1}\eta^{n}\right)$
$\displaystyle\left(\gamma_{1}+e^{ix}\left(\eta^{n}-1\right)-\gamma_{2}\eta^{n}\right)\left(e^{ix}\left(\gamma_{2}\eta^{n}-\gamma_{1}\right)-\eta^{n}+1\right)$
(E.2)

where $r_{n}$=$r_{{\rm new}}|_{t=n(T_{0}+T_{1})}$. One can verify that
$\sqrt{\frac{\partial w_{n}}{\partial
w}\frac{\partial\overline{w}_{n}}{\partial\overline{w}}}$ grows as $\eta^{-n}$
at at $z=\gamma_{1}$ and $\overline{z}=\gamma_{1}$, i.e.
$x=\pm\frac{1}{i}\log\gamma_{1}$.

##  Appendix F Review of Floquet CFT

###  F.1 Floquet CFT at stroboscopic time

####  F.1.1 First cycle

In the first cycle, the Floquet Hamiltonian is:

$H(t)=\left\\{\begin{aligned}
H_{1}&=2\int_{0}^{2\pi}dx\,\sin^{2}\Big{(}\frac{x}{2}\Big{)}\,T_{00}(x),\ \
&&0<t<T_{1}\\\ H_{0}&=\int_{0}^{2\pi}dx\,T_{00}(x),\ \
&&T_{1}<t<T_{1}+T_{0}\\\ \end{aligned}\right.$ (F.1)

Each step can be understood separately in Euclidean signature and by working
in the complex $z$-plane. We then analytically continue to Lorentzian time.

  * •

In the first step, $H_{0}$ simply acts as dilation, and

$\left\\{\begin{aligned} z_{{\rm new},0}&=e^{\tau}z\\\ \overline{z}_{{\rm
new},0}&=e^{\tau}\overline{z}\end{aligned}\right.$ (F.2)

  * •

In the second step, $H_{1}$ acts as $SL(2)$ transformation on $z_{{\rm
new},0}$ and $\overline{z}_{{\rm new},0}$ according to (A.12) and (A.13),

$\left\\{\begin{aligned} z_{{\rm
new},1}&=\frac{(1+\frac{\tau}{2})(e^{\tau}z)-\frac{\tau}{2}}{\frac{\tau}{2}(e^{\tau}z)+(1-\frac{\tau}{2})}\\\
\overline{z}_{{\rm
new},1}&=\frac{(1+\frac{\tau}{2})(e^{\tau}\overline{z})-\frac{\tau}{2}}{\frac{\tau}{2}(e^{\tau}\overline{z})+(1-\frac{\tau}{2})}\end{aligned}\right.$
(F.3)

After the completion of one cycle, and analytic continue $\tau=it$, we have
Wen:2018agb ; Fan:2019upv

$\displaystyle z_{1}=f(z)=\frac{az+b}{cz+d}$
$\displaystyle\mathfrak{H}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ (F.4)

where

$\displaystyle a=\Big{(}1+i\frac{T_{1}}{2}\Big{)}e^{\frac{iT_{0}}{2}}$
$\displaystyle b=-i\frac{T_{1}}{2}e^{-i\frac{T_{0}}{2}}$ $\displaystyle
c=i{\frac{T_{1}}{2}e^{i\frac{T_{0}}{2}}}$ $\displaystyle
d=\Big{(}1-i\frac{T_{1}}{2}\Big{)}e^{-i\frac{T_{0}}{2}}$ (F.5)

We further introduce the quantity $\Delta$ through the square of the trace of
$\mathfrak{H}$ Wen:2018agb ,

$\displaystyle({\rm Tr}\mathfrak{H})^{2}$
$\displaystyle=(a+d)^{2}=4(1-\Delta)$ (F.6) $\displaystyle\Delta$
$\displaystyle=\Big{(}1-\Big{(}\frac{T_{1}}{2}\Big{)}^{2}\Big{)}\
\sin^{2}\frac{T_{0}}{2}+\frac{T_{1}}{2}\ \sin T_{0}$ (F.7)

The sign of $\Delta$ determines different types of Mobius transformations.

####  F.1.2 $n$-cycle

Now let us repeat the first cycle for $n$ times, with the time evolution
operator given by (5.1). $n$-cycle driving can be represented via a Mobius
transformation in the normal form Wen:2018agb ; Fan:2019upv ,

$\displaystyle\frac{z_{n}-\gamma_{1}}{z_{n}-\gamma_{2}}=\eta^{n}\
\frac{z-\gamma_{1}}{z-\gamma_{2}}$ (F.8)

where $\gamma_{1}$ and $\gamma_{2}$ are the fixed points of the Mobius
transformation, defined as points that are invariant under the $SL(2)$
transformation (F.4): $f(\gamma_{i})=\gamma_{i},(i=1,2)$; and $\eta$ is given
by $\eta=f^{\prime}(z)|_{z=\gamma_{1}}$. From (F.8) one can solve for $z_{n}$,

$\displaystyle
z_{n}=\frac{(\gamma_{1}-\eta^{n}\gamma_{2})z-(1-\eta^{n})\gamma_{1}\gamma_{2}}{(1-\eta^{n})z-(\gamma_{2}-\eta^{n}\gamma_{1})}=:\frac{A_{n}z+B_{n}}{C_{n}z+D_{n}}$
(F.9)

And the two fixed points $\gamma_{1}$ and $\gamma_{2}$ are determined by

$\displaystyle f(\gamma_{i})=\frac{a\gamma_{i}+b}{c\gamma_{i}+d}=\gamma_{i}\
\Rightarrow\
\gamma_{1,2}=\frac{(a-d)\mp\sqrt{(a-d)^{2}+4bc}}{2c}=\frac{(a-d)\mp
2\sqrt{-\Delta}}{2c}$ (F.10)

where we have used $(a-d)^{2}+4bc=-4\Delta$. $\eta$ can be obtained from

$\displaystyle\eta=f^{\prime}(z)|_{z=\gamma_{1}}=\frac{(a+d)+\sqrt{(a-d)^{2}+4bc}}{(a+d)-\sqrt{(a-d)^{2}+4bc}}=\frac{(a+d)+2\sqrt{-\Delta}}{(a+d)-2\sqrt{-\Delta}}=\frac{c\gamma_{2}+d}{c\gamma_{1}+d}$
(F.11)

At the phase transition, $\gamma_{1}=\gamma_{2}=(a-d)/2c$, and (F.9) does not
apply anymore. Instead we have

$\displaystyle\frac{1}{z_{n}-\gamma}+n=\frac{1}{z-\gamma}+n\cdot\mathfrak{b}$
(F.12)

where $\mathfrak{b}$ is the translation length, $\mathfrak{b}=\frac{a-d}{2c}$
Fan:2019upv . From (F.12), one can solve for $z_{n}$,

$\displaystyle
z_{n}=\frac{(1+n\mathfrak{b}\cdot\gamma)z-n\mathfrak{b}\cdot\gamma^{2}}{n\mathfrak{b}\cdot
z+(1-n\mathfrak{b}\cdot\gamma)}=:\frac{A_{n}z+B_{n}}{C_{n}z+D_{n}}$ (F.13)

Different types of Mobius transformations are characterised by the sign of
$\Delta$ (F.7)

  * •

$\Delta>0$: $\eta$ is a phase. This case corresponds to the non-heating phase.

  * •

$\Delta<0$: $\eta$ is a real number. This case corresponds to the heating
phase. $\eta$ can be either in $(0,1)$ or $(1,\infty)$. From (F.8) we see that
if $0<\eta<1$, then at large $n$, i.e. late stroboscopic time, $z_{n}$
converges to $\gamma_{1}$ if $z\neq\gamma_{2}$ and stays at $\gamma_{2}$ if
$z=\gamma_{2}$; on the other hand, if $\eta>1$, then $z_{n}$ converges to
$\gamma_{2}$ if $z\neq\gamma_{1}$ and stays at $\gamma_{1}$ if $z=\gamma_{1}$.
Therefore, the range of $\eta$ in $(0,1)$ or $(1,\infty)$ is simply a matter
of convention, for we can always take $\eta\to\frac{1}{\eta}$ and swap
$\gamma_{1}\leftrightarrow\gamma_{2}$ to get to the other case. In our
convention, we choose $0<\eta<1$, making $\gamma_{1}$ the stable fixed point
and $\gamma_{2}$ the unstable fixed point.

  * •

$\Delta=0$: $\eta=1$. This case corresponds to the phase transition.

###  F.2 Time-reversal symmetric case

We can start one cycle at the middle of a $H_{0}$-driving region as in
Lapierre:2019rwj , see Figure 7 in the main text. The Hamiltonian in this
setup can be written as

$H(t)=\left\\{\begin{aligned} &H_{0},\ \ &&0<t\ {\rm
mod}(T_{0}+T_{1})<\frac{T_{0}}{2}\\\ &H_{1},\ \ &&\frac{T_{0}}{2}<t\ {\rm
mod}(T_{0}+T_{1})<\frac{T_{0}}{2}+T_{1}\\\ &H_{0},\ \
&&\frac{T_{0}}{2}+T_{1}<t\ {\rm mod}(T_{0}+T_{1})<T_{1}+T_{0}\\\
\end{aligned}\right.$ (F.14)

With the Hamiltonian (F.14), the system is invariant under time-reversal,
$t\to-t$.

####  F.2.1 First cycle

Repeating the procedure in Appendix F.1.1 of for the Hamiltonian (F.14) in the
first cycle, we have Lapierre:2019rwj

$\displaystyle z_{1}=f(z)=\frac{az+b}{cz+d}$ (F.15)

where

$\displaystyle a=\Big{(}1+i\frac{T_{1}}{2}\Big{)}e^{\frac{iT_{0}}{2}}$
$\displaystyle b=-i\frac{T_{1}}{2}$ $\displaystyle c=i\frac{T_{1}}{2}$
$\displaystyle d=\Big{(}1-i\frac{T_{1}}{2}\Big{)}e^{-i\frac{T_{0}}{2}}$ (F.16)

####  F.2.2 $n$-cycle

The $n$-cycle case also parallels that in Appendix F.1.2, with the
coefficients $a,b,c,d$ given by (F.16). By plugging (F.16) into the general
formula (F.10), it is straightforward to verify that Lapierre:2019rwj

$\displaystyle\gamma_{1}\gamma_{2}=1$ $\displaystyle\Rightarrow$
$\displaystyle\gamma_{2}=\gamma_{1}^{*}\ ({\rm heating\ phase})$ (F.17)

where we have used the fact that in the heating phase
$|\gamma_{1}|=|\gamma_{2}|=1$. $\gamma_{2}$ is thus the complex conjugate of
$\gamma_{1}$ in the heating phase. This then simplifies (F.9) to a
particularly nice form Lapierre:2019rwj ,

$\displaystyle A_{n}=\gamma_{1}-\eta^{n}\gamma_{2}$ $\displaystyle
B_{n}=-(1-\eta^{n})$ $\displaystyle C_{n}=(1-\eta^{n})$ $\displaystyle
D_{n}=-(\gamma_{2}-\eta^{n}\gamma_{1})$ (F.18)

which then yields

$\displaystyle z_{n}$
$\displaystyle=\frac{A_{n}z+B_{n}}{C_{n}z+D_{n}}=\frac{(\gamma_{1}-\eta^{n}\gamma_{2})z-(1-\eta^{n})}{(1-\eta^{n})z-(\gamma_{2}-\eta^{n}\gamma_{1})}$
(F.19) $\displaystyle\overline{z}_{n}$
$\displaystyle=\frac{A_{n}\overline{z}+B_{n}}{C_{n}\overline{z}+D_{n}}=\frac{(\gamma_{1}-\eta^{n}\gamma_{2})\overline{z}-(1-\eta^{n})}{(1-\eta^{n})\overline{z}-(\gamma_{2}-\eta^{n}\gamma_{1})}$
(F.20)

where we also listed the expressions of $\overline{z}_{n}$ for the anti-
holomorphic coordinate $\overline{z}$.

In the phase transition case, we have, from (F.10) and (F.16),

$\displaystyle\Delta\propto(a-d)^{2}+4bc=0$
$\displaystyle\gamma=\frac{a-d}{2c}=1$ (F.21)

i.e. the two fixed points that are conjugate to each other merge to $z=1$ on
the real axis symmetrically.

###  F.3 General cases

In Section 5, we have introduced the fundamentals of Floquet CFT built upon
SSD CFT with Hamiltonian $H_{1}$. We remark, however, that Floquet setups for
CFT2 could be more general. Floquet CFT based on $H_{0}$ and $H_{\theta}$ were
studied in Wen:2020wee . Floquet CFT with $q\geq 2$ Hamiltonian were
systematically investigated in Han:2020kwp , where it was shown that heating
phases always exist, whereas non-heating phases might or might not be present
subject to certain criteria Han:2020kwp . More generally, one can construct
Floquet CFT with deformed Hamiltonian involving the full Virasoro algebra (not
just the $SL(2)$ sub-algebra) Fan:2020orx ; Lapierre:2020ftq . Both works
indicate that much of the conclusions for Floquet CFT with $H_{0}$ and $H_{1}$
generalise to these cases at least qualitatively: the presence/absence of the
fixed points (or higher-periodic points) in the stroboscopic evolution of CFT
operators characterises the heating/non-heating phases; in the heating phases,
energy accumulates at the unstable fixed points rapidly and forms up peaks,
while entanglement entropy grows linearly in time if the subregion $A$
includes at least one (but not all) of unstable fixed points; in the non-
heating phases, energy and entanglement show quasi-periodic oscillatory
behavior.

##  Appendix G Details on Floquet CFT at general time

###  G.1 Proof of $z_{{{\rm new}}}$ and $\overline{z}_{{{\rm new}}}$ at
general $t$

In this appendix, we prove that the formula $\eqref{znewGeneralTime}$
correctly returns to $\eqref{zn}$ at stroboscopic time $t=n(T_{0}+T_{1})$ for
general $n\in\mathbb{Z}^{+}$ using mathematical induction.

For $n=1$, a direct computation shows that

$\displaystyle\begin{pmatrix}A_{1}&B_{1}\\\
C_{1}&D_{1}\end{pmatrix}=\begin{pmatrix}\gamma_{1}-\eta\gamma_{2}&-(1-\eta)\\\
1-\eta&-(\gamma_{2}-\eta\gamma_{1})\end{pmatrix}=\frac{2}{c}\frac{\sqrt{-4\Delta}}{\sqrt{-4\Delta}-(a+d)}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ (G.1)

where $A_{1},B_{1},C_{1},D_{1}$ are given by (F.18) with $n=1$, and $a,b,c,d$
are given by (F.16). Therefore $\eqref{znewGeneralTime}$ holds after the
completion of the first cycle. In other words, the expression $z_{1}$ (F.16)
is consistent with $z_{n}$ (F.19) when $n=1$,

$\displaystyle z_{1}=\frac{az+b}{cz+d}=\frac{A_{1}z+B_{1}}{C_{1}z+D_{1}}$
(G.2)

Assume that $\eqref{znewGeneralTime}$ reproduces $\eqref{zn}$ at stroboscopic
time for $n$ ($n\geq 1$), and consider the $(n+1)$-th cycle. We can then take
$z_{{\rm new}}$ at the beginning of the $(n+1)$-th cycle to be $z_{n}$. The
time evolution follows the same logic as that in Appendix F.1.1. After the
completion of the $H_{1}$-driving period, i.e. at time
$t-n(T_{0}+T_{1})=\frac{T_{0}}{2}+T_{1}$, $\eqref{znewGeneralTime}$ becomes

$\displaystyle z_{{\rm new}}$
$\displaystyle=\frac{\Big{(}1+\frac{iT_{1}}{2}\Big{)}(e^{\frac{iT_{0}}{2}}z_{n})-\frac{iT_{1}}{2}}{\frac{iT_{1}}{2}(e^{\frac{iT_{0}}{2}}z_{n})+\Big{(}1-\frac{iT_{1}}{2}\Big{)}}=e^{-\frac{iT_{0}}{2}}\frac{\Big{(}1+\frac{iT_{1}}{2}\Big{)}e^{\frac{iT_{0}}{2}}\cdot
z_{n}-\frac{iT_{1}}{2}}{\frac{iT_{1}}{2}\cdot
z_{n}+\Big{(}1-\frac{iT_{1}}{2}\Big{)}e^{-\frac{iT_{0}}{2}}}=e^{-\frac{iT_{0}}{2}}\frac{az_{n}+b}{cz_{n}+d}$
(G.3)

where $a,b,c,d$ are given by $\eqref{abcdTR}$. In other words, $H_{1}$ sends
$e^{\frac{iT_{0}}{2}}z_{n}$ to $e^{-\frac{iT_{0}}{2}}z_{n+1}$ after driving
for $T_{1}$. The remaining $H_{0}$-driving period that lasts for
$\frac{T_{0}}{2}$ then eliminates the $e^{-\frac{iT_{0}}{2}}$ prefactor in
(G.3). So, after the completion of the $(n+1)$-th cycle,
$\eqref{znewGeneralTime}$ is

$\displaystyle
z_{n+1}=\frac{az_{n}+b}{cz_{n}+d}=\frac{(aA_{n}+bC_{n})z+aB_{n}+bD_{n}}{(cA_{n}+dC_{n})z+cB_{n}+dD_{n}}$
(G.4)

where $A_{n},B_{n},C_{n},D_{n}$ are given by (F.18). A direct calculation
shows that

$\displaystyle\begin{pmatrix}aA_{n}+bC_{n}&aB_{n}+bD_{n}\\\
cA_{n}+dC_{n}&cB_{n}+dD_{n}\end{pmatrix}=\frac{1}{2}(a+d-\sqrt{-4\Delta})\begin{pmatrix}A_{n+1}&B_{n+1}\\\
C_{n+1}&D_{n+1}\end{pmatrix}$ (G.5)

with $A_{n+1},B_{n+1},C_{n+1},D_{n+1}$ in (F.18) with $n\to n+1$. So $z_{{\rm
new}}$ after the completion of the $(n+1)$-th cycle, i.e. at
$t=(n+1)(T_{0}+T_{1})$ correctly returns to the form (F.19),

$\displaystyle z_{n+1}=\frac{A_{n+1}z+B_{n+1}}{C_{n+1}z+D_{n+1}}$ (G.6)

For $\overline{z}_{{\rm new}}$ (G.7), the proof parallels the above with
$z\to\overline{z}$ and therefore $z_{n}\to\overline{z}_{n}$.

###  G.2 Plots of $\frac{1}{i}w_{{\rm new}}$ and
$\frac{1}{i}\overline{w}_{{\rm new}}$

In Figure 14, we plot $\frac{1}{i}w_{{{\rm new}}}=\frac{1}{i}\log z_{{{\rm
new}}}=\frac{1}{2}(t_{{{\rm new}}}+x_{{{\rm new}}})$ and
$\frac{1}{i}\overline{w}_{{{\rm new}}}=\frac{1}{i}\log\overline{z}_{{{\rm
new}}}=\frac{1}{2}(t_{{{\rm new}}}-x_{{{\rm new}}})$. To avoid confusions, we
list the analogue of (5.6) for the anti-holomorphic coordinate
$\overline{z}=e^{\overline{w}}=e^{-ix}$,

$\overline{z}_{{{\rm new}}}=\left\\{\begin{aligned}
&e^{i\mathfrak{t}_{n+1}}\overline{z}_{n},\ \
&&0\leq\mathfrak{t}_{n+1}\leq\frac{T_{0}}{2}\\\
&\frac{\Big{(}1+\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}\Big{)}(e^{\frac{iT_{0}}{2}}\overline{z}_{n})-\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}}{\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}(e^{\frac{iT_{0}}{2}}\overline{z}_{n})+\Big{(}1-\frac{i(\mathfrak{t}_{n+1}-T_{0}/2)}{2}\Big{)}},\
\ &&\frac{T_{0}}{2}<\mathfrak{t}_{n+1}\leq\frac{T_{0}}{2}+T_{1}\\\
&e^{i(\mathfrak{t}_{n+1}-(T_{0}+T_{1}))}\overline{z}_{n+1},\ \
&&\frac{T_{0}}{2}+T_{1}\leq\mathfrak{t}_{n+1}\leq T_{0}+T_{1}\\\
\end{aligned}\right.$ (G.7)

In the first cycle, $\overline{z}_{n}$ is $\overline{z}=e^{-ix}$.

Figure 14: Plot of $\frac{1}{i}w_{{{\rm new}}}$ and
$\frac{1}{i}\overline{w}_{{{\rm new}}}$ at general $t$ in the heating phase.
Here we set $T_{0}=4$ and $T_{1}=8$, and plotted the first 5 cycles for
$x=-\frac{3\pi}{4},-\frac{2\pi}{3},\frac{1}{i}\log\gamma_{2}\approx-1.76076,-\frac{\pi}{6},0,\frac{\pi}{2},\frac{1}{i}\log\gamma_{1}\approx
1.76076,\frac{2\pi}{3},\frac{3\pi}{4}$.

To understand Figure 14, we first notice that $\frac{1}{i}w_{{{\rm
new}}}=\frac{1}{i}\log z_{{{\rm new}}}$ is monotonically increasing with
respect to $x$ when the driving Hamiltonian is $H_{1}$. In other words,
$\frac{1}{i}w_{{{\rm new}}}$ initially with smaller $x\in[-\pi,\pi]$ cannot
catch up with those with larger $x$, and $\frac{1}{i}w_{{{\rm new}}}$ curves
with different $x$’s therefore do not intersect at general $t>0$. This can be
proved by simply taking the $x$ derivative of (2.6). A similar analysis on
(2.7) shows that $\frac{1}{i}\overline{w}_{{{\rm
new}}}=\frac{1}{i}\log\overline{z}_{{{\rm new}}}$ is monotonically decreasing
with respect to $x$.

It is observed that both $\frac{1}{i}w_{{{\rm new}}}$ and
$\frac{1}{i}\overline{w}_{{{\rm new}}}$ in Figure 14 asymptote three curves.
To understand this phenomenon, first note that at stroboscopic time,
$z_{n}\to\gamma_{1}$ for $z\neq\gamma_{2}$ and $z_{n}=\gamma_{2}$ for
$z=\gamma_{2}$, see, (5.7), (5.8). Thus, for initial
$x\neq\frac{1}{i}\log\gamma_{2}$, the $\frac{1}{i}w_{{{\rm new}}}$ curve
converges to that of $x=\frac{1}{i}\log\gamma_{1}$, whose stroboscopic values
are $w_{n}=\frac{1}{i}\log\gamma_{1}$; while for
$x=\frac{1}{i}\log\gamma_{2}$, $\frac{1}{i}w_{{{\rm new}}}$ follows a
different curve that is slightly lower (coloured red in Figure 14), with
stroboscopic values $w_{n}=\frac{1}{i}\log\gamma_{2}$. There is, however, a
subtlety, as the stroboscopic $\frac{1}{i}w_{n}=\frac{1}{i}\log z_{n}$ only
captures the principal value of $w_{{{\rm new}}}$. Due to the monotonicity of
$\frac{1}{i}w_{{{\rm new}}}$ with respect to $x$, $\frac{1}{i}w_{{{\rm new}}}$
curve with $-\pi<x<\frac{1}{i}\log\gamma_{2}$ should be lower than that with
$x=\frac{1}{i}\log\gamma_{2}$ (i.e. the red curve in Figure 14). These curves
thus converge to the $\frac{1}{i}\log\gamma_{1}$ curve that is on another
Riemann sheet, which results in the curve being $2\pi$ lower than those with
$x>\frac{1}{i}\log\gamma_{2}$. The same reasoning works for
$\frac{1}{i}\overline{w}_{{{\rm new}}}$. At stroboscopic time,
$\overline{z}_{n}\to\gamma_{1}$ for $\overline{z}\neq\gamma_{2}$ and
$\overline{z}_{n}=\gamma_{2}$ for $\overline{z}=\gamma_{2}$. As
$\overline{z}=e^{-ix}$ initially, the unstable fixed point
$\overline{z}=\gamma_{2}$ is instead at
$x=-\frac{1}{i}\log\gamma_{2}=\frac{1}{i}\log\gamma_{1}$. For initial
$x\neq\frac{1}{i}\log\gamma_{1}$, the $\frac{1}{i}\overline{w}_{{{\rm new}}}$
curve converges to $x=\frac{1}{i}\log\gamma_{2}$ (i.e.
$\overline{z}=\gamma_{1}$), whose stroboscopic values are
$\overline{w}_{n}=\frac{1}{i}\log\gamma_{1}$; while for
$x=\frac{1}{i}\log\gamma_{1}$ (i.e. $\overline{z}=\gamma_{2}$),
$\frac{1}{i}\overline{w}_{{{\rm new}}}$ follows a different curve that is
slightly lower (coloured blue in Figure 14), with stroboscopic values
$\overline{w}_{n}=\frac{1}{i}\log\gamma_{2}$. Since
$\frac{1}{i}\overline{w}_{{{\rm new}}}$ monotonically decreases with $x$,
curves with $\frac{1}{i}\log\gamma_{1}<x<\pi$ are $2\pi$ lower.

Averaging $\frac{1}{i}w_{{{\rm new}}}$ and $\frac{1}{i}\overline{w}_{{{\rm
new}}}$, the $t_{{{\rm new}}}$ curves are therefore $\pi$ lower for
$\frac{1}{i}\log\gamma_{1}<|x|<\pi$ than for $0<|x|<\frac{1}{i}\log\gamma_{1}$
as in Figure 8; subtracting $\frac{1}{i}w_{{{\rm new}}}$ and
$\frac{1}{i}\overline{w}_{{{\rm new}}}$, the $x_{{{\rm new}}}$ curves thus
converge to 0 for $0<x<\frac{1}{i}\log\gamma_{1}$ and $\pm\pi$ for
$\frac{1}{i}\log\gamma_{1}<|x|<\pi$ as in Figure 8. Note that the reason for
$t_{{{\rm new}}}$ to be $\pi$ lower and $x_{{{\rm new}}}$ to converge at $\pi$
(instead of 0) is due to either $\frac{1}{i}w_{{{\rm new}}}$ or
$\frac{1}{i}\overline{w}_{{{\rm new}}}$ being $2\pi$ lower.
$\frac{1}{i}w_{{{\rm new}}}$ and $\frac{1}{i}\overline{w}_{{{\rm new}}}$
cannot both be $2\pi$ lower, as the $x$ regions for them to be $2\pi$ lower
are $-\pi<x<\frac{1}{i}\log\gamma_{2}$ and $\frac{1}{i}\log\gamma_{1}<x<\pi$,
respectively, which do not intersect.

##  Appendix H Two-point functions of twist operators

In this appendix, we first evaluate the two-point function
$\langle\mathcal{T}_{n}(w_{1,{{\rm new}}},\overline{w}_{1,{{\rm
new}}})\mathcal{T}_{-n}(w_{2,{{\rm new}}},\overline{w}_{2,{{\rm
new}}})\rangle$ for a CFT2 on a compact space of length $2\pi$, and then
evaluate the thermal two-point function $\langle\mathcal{T}_{n}(w_{1,{{\rm
new}}},\overline{w}_{1,{{\rm new}}})\mathcal{T}_{-n}(w_{2,{{\rm
new}}},\overline{w}_{2,{{\rm new}}})\rangle_{\beta}$ for a CFT2 at temperature
$T=\beta^{-1}$ on a non-compact space. Throughout this appendix, we will be
working exclusively in $z_{{\rm new}}$ and $w_{{\rm new}}$ coordinate, so we
drop the ”new” subscript for notation simplicity. Our method is largely based
on Calabrese:2004eu . Note, however, that we are working in the coordinates

$\displaystyle w=\tau+ix$ $\displaystyle\overline{w}=\tau-ix$ (H.1)

on the Euclidean cylinder as is mentioned in the main text. This results in a
different behavior of the holomorphic and anti-holomorphic parts of the two-
point function, in contrast to Calabrese:2004eu . We can use the conformal
transformation (A.1) that maps the Euclidean cylinder to the complex plane,
where the two-point function of twist operator is simply

$\displaystyle\langle\mathcal{T}_{n}(z_{1},\overline{z}_{1})\mathcal{T}_{-n}(z_{2},\overline{z}_{2})\rangle=\frac{1}{(z_{1}-z_{2})^{2h}}\cdot\frac{1}{(\overline{z}_{1}-\overline{z}_{2})^{2h}}$
(H.2)

For the holomorphic part, under a conformal transformation $z\to w$,

$\displaystyle\langle\mathcal{T}_{n}(z_{1})\mathcal{T}_{-n}(z_{2})\rangle=\frac{1}{(z_{1}-z_{2})^{2h}}=\Big{(}\frac{\partial
w_{1}}{\partial z_{1}}\Big{)}^{h}\Big{(}\frac{\partial w_{2}}{\partial
z_{2}}\Big{)}^{h}\langle\mathcal{T}_{n}(w_{1})\mathcal{T}_{-n}(w_{2})\rangle$
(H.3)

Substituting (A.1),
$\langle\mathcal{T}_{n}(w_{1})\mathcal{T}_{-n}(w_{2})\rangle$ on the Euclidean
cylinder is then

$\displaystyle\langle\mathcal{T}_{n}(w_{1})\mathcal{T}_{-n}(w_{2})\rangle=\Big{[}2\sinh\Big{(}\frac{1}{2}(w_{1}-w_{2})\Big{)}\Big{]}^{-2h}=\Big{[}2\sinh\Big{(}\frac{1}{2}((\tau_{1}-\tau_{2})+i(x_{1}-x_{2}))\Big{)}\Big{]}^{-2h}$
(H.4)

The anti-holomorphic part is simply given by $w_{i}\to\overline{w}_{i}$,

$\displaystyle\langle\mathcal{T}_{n}(\overline{w}_{1})\mathcal{T}_{-n}(\overline{w}_{2})\rangle=\Big{[}2\sinh\Big{(}\frac{1}{2}(\overline{w}_{1}-\overline{w}_{2})\Big{)}\Big{]}^{-2h}=\Big{[}2\sinh\Big{(}\frac{1}{2}((\tau_{1}-\tau_{2})-i(x_{1}-x_{2}))\Big{)}\Big{]}^{-2h}$
(H.5)

Note that (H.4) and (H.5) are not equal due to (H.1) in our setup. 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 then given by the product of (H.4) and (H.5) and restoring
the ”new” subscript, which is (7.11) in the main text.

To calculate the thermal two-point function,
$\langle\mathcal{T}_{n}(w_{1,{{\rm new}}},\overline{w}_{1,{{\rm
new}}})\mathcal{T}_{-n}(w_{2,{{\rm new}}},\overline{w}_{2,{{\rm
new}}})\rangle_{\beta}$ on non-compact space, we simply perform a Wick
rotation of the exponential map, and rescale the circumference of the cylinder
from $2\pi$ to $\beta$ Calabrese:2004eu ,

$\displaystyle z=e^{\frac{2\pi}{\beta}iw}$
$\displaystyle\overline{z}=e^{\frac{2\pi}{\beta}i\overline{w}}$ (H.6)

this amounts to $w\to iw$ and $2\pi\to\beta$ in (H.4) and (H.5),

$\displaystyle\langle\mathcal{T}_{n}(w_{1})\mathcal{T}_{-n}(w_{2})\rangle_{\beta}$
$\displaystyle=\Big{[}\frac{\beta}{\pi}\sin\Big(\frac{\pi}{\beta}(w_{1}-w_{2})\Big{missing})\Big{]}^{-2h}=\Big{[}\frac{\beta}{\pi}\sinh\Big{(}\frac{\pi}{\beta}((\tau_{1}-\tau_{2})+i(x_{1}-x_{2}))\Big{)}\Big{]}^{-2h}$
(H.7)
$\displaystyle\langle\mathcal{T}_{n}(\overline{w}_{1})\mathcal{T}_{-n}(\overline{w}_{2})\rangle_{\beta}$
$\displaystyle=\Big{[}\frac{{}_{\beta}}{\pi}\sin\Big(\frac{\pi}{\beta}(\overline{w}_{1}-\overline{w}_{2})\Big{missing})\Big{]}^{-2h}=\Big{[}\frac{{}_{\beta}}{\pi}\sinh\Big{(}\frac{\pi}{\beta}((\tau_{1}-\tau_{2})-i(x_{1}-x_{2}))\Big{)}\Big{]}^{-2h}$
(H.8)

The thermal two-point function $\langle\mathcal{T}_{n}(w_{1,{{\rm
new}}},\overline{w}_{1,{{\rm new}}})\mathcal{T}_{-n}(w_{2,{{\rm
new}}},\overline{w}_{2,{{\rm new}}})\rangle_{\beta}$ is then given by the
product of the two, and restoring the ”new” subscript.

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