Exactly Solvable Floquet Dynamics for Conformal Field Theories in Dimensions
Greater than Two

Diptarka Das1, Sumit R. Das2, Arnab Kundu3,4, Krishnendu Sengupta5,

1 Department of Physics, Indian Institute of Technology, Kanpur, UP 208016,
India.

2 Department of Physics and Astronomy, University of Kentucky, Lexington, KY
40506, U.S.A.

3 Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India.

4 Homi Bhaba National Institute, Training School Complex, Anushaktinagar,
Mumbai 400094, India.

5 School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, India.

## Abstract

We find classes of driven conformal field theories (CFT) in $d+1$ dimensions
with $d>1$, whose quench and floquet dynamics can be computed exactly. The
setup is suitable for studying periodic drives, consisting of square pulse
protocols for which Hamiltonian evolution takes place with different
deformations of the original CFT Hamiltonian in successive time intervals.
These deformations are realized by specific combinations of conformal
generators with a deformation parameter $\beta$; the $\beta<1$ ($\beta>1$)
Hamiltonians can be unitarily related to the standard (Lüscher-Mack) CFT
Hamiltonians. The resulting time evolution can be then calculated by
performing appropriate conformal transformations. For $d\leq 3$ we show that
the transformations can be easily obtained in a quaternion formalism; we use
this formalism to obtain exact expressions for the fidelity, unequal-time
correlator, and the energy density for the driven system for $d=3$. Our
results for a single square pulse drive cycle reveal qualitatively different
behaviors depending on the value of $\beta$, with exponential decays
characteristic of heating for $\beta>1$, oscillations for $\beta<1$ and power
law decays for $\beta=1$. When the Hamiltonians in one cycle involve
generators of a single $SL(2,R)$ subalgebra we find fixed points or fixed
surfaces of the corresponding transformations. Successive cycles lead to
either convergence to one of the fixed points, or oscillations, depending on
the conjugacy class. This indicates that the system can be in different
dynamical phases as we vary the parameters of the drive protocol. Our methods
can be generalized to other discrete and continuous protocols. We also point
out that our results are expected to hold for a broader class of QFTs that
possesses an SL$(2,C)$ symmetry with fields that transform as quasi-primaries
under this. As an example, we briefly comment on celestial CFTs in this
context.

###### Contents

  1. 1 Introduction
  2. 2 The deformed CFTs
    1. 2.1 $\beta<1$ and Möbius quantization
    2. 2.2 $\beta>1$ and Lüscher-Mack Hamiltonians
  3. 3 Dynamics as a Conformal Transformation
    1. 3.1 $d=3$ and quaternions
  4. 4 Floquet Dynamics
    1. 4.1 Fidelity
    2. 4.2 Unequal-time Correlators
    3. 4.3 Local probes of the driven state
    4. 4.4 Multiple cycles and dynamical phases
  5. 5 Discussions
  6. 6 Acknowledgements
  7. A Transformation of the generators
  8. B Conformal Algebra in $d$-dimensions
  9. C SL$(2,C)$, Quasi-primaries, Lorentz symmetry

##  1 Introduction

The study of time evolution of a driven quantum system with a time dependent
Hamiltonian is a valuable tool for gaining insight into its non-equlibrium
properties. Useful drive protocols include quantum quench, ramp, and periodic
or quasi-periodic drive protocols [1, 2, 3, 4]. Recently, properties of
quantum systems driven via periodic protocols have been most intensely
studied; such systems undergo evolution governed by Hamiltonian periodic in
time with a time period $T$. Interestingly, at strobosocopic times $t=nT$,
where $n$ is an integer, the evolution operators of such systems can be
written as $U(nT,0)=\exp[-iH_{F}nT]$; the corresponding dynamics is then
completely controlled by the Floquet Hamiltonian $H_{F}$.

However, most of these studies require numerical work: analytically tractable
models are limited to free field theories or two dimensional conformal field
theories. For the latter class of theories, the underlying infinite
dimensional symmetry algebra provides a powerful tool to calculate physical
quantities of interest. One example involves a sudden quench from a massive
theory to a two dimensional CFT whose IR properties can be well approximated
by a Cardy-Calabrese state [5], and the time evolution can be calculated by
using conformal mapping. A more recent example involves quench and Floquet
dynamics in a two dimensional CFT [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]:
once again the time evolution can be obtained by conformal maps. No such
analytical results are known in higher dimensions, and one has to typically
resort to numerical calculations which are often limited by finite size
effects 111A different class of problems involving fast smooth quenches in
conformal field theories in arbitrary dimension can be addressed using
conformal properties [22, 23]. .

In this paper we demonstrate that a class of time dependent problems can be
exactly studied in conformal field theories in arbitrary number of dimensions
and lead to interesting non-trivial dynamics. Our work is inspired by the work
of [6] who studied a $1+1$ dimensional CFT on a strip which starts from the
usual CFT Hamiltonian and is quenched by a sine squared deformed (SSD) one.
The SSD-CFT2 has been studied previously in [17, 18, 19, 20]. Both these
Hamiltonians belong to the global part of the Virasoro algebra. The time
evolution then becomes a Möbius transformation. The global part of the
Virasoro algebra is (in Euclidean signature) $SO(3,1)$ (generated by
$L_{0},L_{\pm 1}$ and their hermitian conjugates). Floquet generalizations to
periodically driven CFTs by a sequence of the usual CFT Hamiltonian and SSD
Hamiltonian for successive time durations $T_{i}$ were studied in [12, 14,
16]. Generalizations of the SSD to other $SL_{2}$ subgroups of Virasoro were
studied in [15]. Holographic interpretations were investigated in [7, 8, 9,
11] and from a different perspective in [21].

In higher dimensions ($d+1$), the conformal algebra is $SO(d+2,1)$. This
suggests that a natural generalization to higher dimensional systems can be
obtained by considering different generators of the conformal algebra as
Hamiltonians and considering sequences of non-commuting Hamiltonians in
successive time intervals. Similar to the two dimensional case, the dynamics
is expected to be equivalent to conformal transformations.

In this work we consider conformal field theories on $S^{d}\times{\rm time}$.
The class of Hamiltonians are combinations of generators which belong to
$SL(2,R)\sim SU(1,1)$ subrgoups of the conformal group. More specifically we
consider

$H(\beta,\Pi)=2iD+i\beta(K_{\mu}+P_{\mu})\Pi^{\mu}\ ,$ (1)

where the operators $D,K_{\mu},P_{\mu}$ ($\mu,\nu=0,\cdots d$) are generators
of conformal transformations and $\Pi^{\mu}$ is a projector along
$X^{\mu}$-direction. Note that there are $(d+1)$ such different SL$(2,R)$
subalgebras corresponding to $(d+1)$-inequivalent choices of $\Pi^{\mu}$. This
provides a wider class of drive protocols where one can use different members
of the class of Hamiltonians $H(\beta,\Pi)$, for different time intervals.

In terms of the energy momentum tensor, the Hamiltonian may be written as

$H(\beta,\Pi)=2\int d\Omega_{d-1}~{}[1+\frac{1}{2}\beta
Y_{\mu}\Pi^{\mu}]T_{ww}\\\ $ (2)

where $Y^{\mu}$ denote coordinates on a $R^{d+1}$, where the sphere $S^{d}$ is
the surface $Y^{\mu}Y_{\mu}=1$, $d\Omega_{d-1}$ denotes the volume element on
$S^{d}$ and $w$ denotes the Euclidean time.

As will be shown below, when $\beta<1$ a Hamiltonian of the form (1) can be
transformed to the standard CFT Hamiltonian $(2iD)$ (on the plane), while for
$\beta>1$ it can be transformed to the Lüscher-Mack type Hamiltonian
$P_{\mu}+K_{\mu}$.

As is standard in CFT’s, it is convenient to first work in Euclidean signature
and perform a Weyl transformation to a plane $R^{d+1}$

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dw^{2}+d\Omega_{d}^{2}=\frac{1}{r^{2}}\left[\delta_{\mu\nu}dx^{\mu}dx^{\nu}\right]$
(3)

where $x^{\mu}$ are cartesian coordinates on $R^{d+1}$ and

$\displaystyle r^{2}$ $\displaystyle=$
$\displaystyle\delta_{\mu\nu}x^{\mu}x^{\nu}=e^{w}.$ (4)

The operator $(2iD)$ becomes the dilatation operator on the plane, while
$P_{\mu}$ generates translations on the plane. Acting on functions on
$R^{d+1}$ the conformal generators are represented by the following
differential operators

$\displaystyle D$ $\displaystyle=$ $\displaystyle-ix_{\mu}\partial_{\mu},\quad
P_{\mu}=-i\partial_{\mu},\quad
K_{\mu}=-i(2x_{\mu}(x_{\nu}\partial_{\nu})-r^{2}\partial_{\mu}),$
$\displaystyle L_{\mu\nu}$ $\displaystyle=$
$\displaystyle-i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\ .$ (5)

For a fixed $\mu$ the generators $D,P_{\mu},K_{\mu}$ form an SU$(1,1)$
subalgebra of the Euclidean conformal algebra SO$(d+2,1)$. There are $d+1$
such subgroups corresponding to different choice of $\mu$. The ${\rm SU}(1,1)$
nature of these subgroups can be understood by noting that the commutation of
these generators satisfy

$\displaystyle[D,K_{\mu}]$ $\displaystyle=$ $\displaystyle-
iK_{\mu},\quad[D,P_{\mu}]=iP_{\mu},\quad[K_{\mu},P_{\mu}]=2iD\ .$ (6)

Our strategy for calculating the response is analogous to the earlier works in
$1+1$ dimensions. We will perform a Weyl transformation to $R^{d+1}$, where
the expressions for the generators and the resulting conformal transformation
become simple, calculate the time evolution by evaluating the corresponding
conformal transformation, transform back to $S^{d}\times{\rm time}$ and
finally continue to real time. While this is straightforward in principle,
this becomes quickly cumbersome in practice for $d>1$.

The key technical tool which facilitates our calculations is the fact that
when $d\leq 3$, a point on $R^{d+1}$ can be represented by a quaternion and
the action of finite conformal transformations take a simple form,
generalizing Möbius transformations on the complex plane to Möbius
transformations on the field of quaternions, $SL(2,H)$. In general, the
parameters of these transformations are themselves quaternions. However, when
the transformation belongs to an $SL(2,R)$ subgroup of the conformal group one
can judiciously choose the quaternion representation such that these
parameters become real numbers. While this simplifies our calculations
considerably, this necessitates switching quaternionic representations
whenever we switch from one $SL(2,R)$ subgroup to another. In this work, we
provide explicit results for $3+1$ dimensions, though it should be emphasized
that the framework is completely general for any $d\leq 3$.

In the study of Floquet dynamics in $d=1$ a key step was to determine the
fixed points of Möbius transformations on the complex plane and express the
transformation in “normal form”. In this form it is possible to determine the
behavior of a point on the plane after arbitary number $n$ of Floquet cycles.
The behavior for large $n$ depends on the conjugacy class of the Möbius group,
with points converging to one of the fixed points for a hyperbolic class,
oscillating for elliptic class and displaying maginal behavior for the
parabolic class. The conjugacy class of the transformation depends on the
parameters of the period drive protocol and the different behavior of points
on the plane result in different dynamical phases, which are heating (for
hyperbolic) and non-heating (for elliptic).

In higher dimensions $d=2,3$ one needs to look for fixed points or surfaces of
quaternionic Möbius transformations. However, as mentioned above, if the
transformation involves a definite $SL(2,R)$ subgroup, the parameters of these
Möbius transformations can be chosen to be real numbers. We show that in this
case there are fixed points for hyperbolic transformations and fixed surfaces
for elliptic transformations. We then express the corresponding quaternionic
Möbius transformations in a “normal form” and show that, as in $d=1$, points
converge to one of the fixed points for the hyperbolic class and oscillate for
the elliptic class. This would lead to heating or non-heating phases in
Floquet dynamics. We defer the discussion of behavior of physical quantities
for large number of cycles to a future publication.

In the following, we calculate fidelities, unequal-time correlation functions
and expectation values of the energy momentum tensor and primary operators in
primary states at the end of a single drive cycle. Each cycle involves
Hamiltonians of the form (1) with various $\Pi^{\mu},\mu=0,1,\cdots 3$ for
successive time intervals of equal value. The value of $\beta$ is either zero,
or some fixed value, i.e. we do not consider different non-zero values of
$\beta$ for different time intervals. These restrictions are for simplicity.
All these diagonistics show that for $\beta<1$ one has a non-heating phase
with oscillatory behavior of these quantities as a function of time, while for
$\beta>1$ one has a heating phase characterized by exponential decays.
$\beta=1$ is a critical value, where the quantities have power law behavior.
These three phases mirror the known dynamical phase structure in $1+1$
dimensions and reflect the three conjugacy classes of $SL(2,R)$. Since the
driving Hamiltonians break rotation invariance on the $S^{d}$ the system
develops inhomogenities. The nature of the inhomogeneties is richer in these
higher dimensional cases, since we have the ability to break different sets of
symmetries by choosing different sequences of $H_{1}\cdots H_{d+1}$. Likewise,
the structure of dynamical phases which can result from Floquet drives will be
richer. It can be easily seen that if we use different nonzero values of
$\beta$ the behavior is oscillatory when all the $\beta$ values are less than
1, power law when all the $\beta$ are equal to one. If any of the $\beta$
exceed 1, the behavior will be exponential.

The plan of the rest of this work is as follows. In Sec. 2 we study the
properties of Hamiltonians of the form (1). This is followed by Sec. 3 which
deals with our strategy for calculating the response to the dynamics for
general $d$ and the formulation in terms of quaternions for $d=3$. Next, in
Sec. 4, we provide results for three quantities under the dynamics. The first
corresponds to the fidelity of an evolving primary state at the end of a drive
cycle; this is discussed in subsection (4.1). The second constitutes behavior
of unequal-time correlation functions of the CFT under such evolution; this is
discussed subsection(4.2). The evolution of the stress tensor, starting from a
primary CFT state, is discussed in subsection (4.3). In section (4.4) we find
the fixed points and fixed surfaces for the special class of quaternionic
Möbius transformations mentioned above and determine the trajectories of
points under successive cycles of a periodic drive. Finally, we discuss our
main results and conclude in Sec. 5. Some details of the calculation are
provided in the appendix.

##  2 The deformed CFTs

In this section we will discuss some properties of deformed CFT’s with a
Hamiltonian of the form (1).

###  2.1 $\beta<1$ and Möbius quantization

When $\beta<1$, for the choice $\Pi^{\mu}=(1,0,0,0)$, the Hamiltonian $H$
(defined in (1)) can be thought of a different quantization of the theory with
the undeformed hamitonian ($\beta=0$) upto a scaling of the new time. For
$d=1$ this has been called “Möbius quantization”. Consider the Hamiltonian

$H^{\prime}=\frac{1}{\sqrt{1-\beta^{2}}}H=\cosh\theta
2iD+i\sinh\theta(K_{0}+P_{0})~{}~{}~{}~{}~{}~{}~{}\beta=\tanh\theta\ .$ (7)

This Hamiltonian is related to the dilatation operator by a similarity
transformation

$\displaystyle U^{-1}H^{\prime}U$ $\displaystyle=$ $\displaystyle(2iD),\quad
U=\exp\left[-\frac{i}{2}\theta(K_{0}-P_{0})\right]\ .$ (8)

This can be verified by explicit calculation. However since the left hand side
of (8) depends only on the commutators of the generators of the $SU(1,1)$
group, the equation must be independent of the specific representation.

To verify (8) it is convenient to use a representation of the $SU(1,1)$ using
Pauli matrices,

$\displaystyle D$ $\displaystyle=$ $\displaystyle i\sigma_{z}/2,\quad
K_{0}=\sigma_{-},\quad
P_{0}=\sigma_{+}\quad\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y}).$ (9)

The transformed special conformal and translation generators can be now
deduced using the identities

$\displaystyle U^{-1}\sigma_{z}U$ $\displaystyle\equiv$
$\displaystyle\tau_{z}=\sigma_{z}\cosh\theta+i\sigma_{x}\sinh\theta$
$\displaystyle U^{-1}\sigma_{x}U$ $\displaystyle\equiv$
$\displaystyle\tau_{x}=\sigma_{x}\cosh\theta-i\sigma_{z}\sinh\theta,\quad\tau_{y}=\sigma_{y}\
.$ (10)

Using the representation (9) this leads to

$\displaystyle D^{\prime}$ $\displaystyle=$ $\displaystyle
D\cosh\theta+\frac{1}{2}(K_{0}+P_{0})\sinh\theta$ $\displaystyle
K^{\prime}_{0}[P^{\prime}_{0}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{0}[P_{0}]-\frac{1}{2}(1-\cosh\theta)P_{0}[K_{0}]-D\sinh\theta\
.$ (11)

To calculate the transformation of the other generators of the conformal
algebra, it is useful to consider the combinations

$A_{\mu}\equiv\frac{1}{2}(K_{\mu}-P_{\mu})~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{\mu}\equiv\frac{1}{2}(K_{\mu}+P_{\mu})\
.$ (12)

The conformal algebra then leads to a closed $SL(2,R)$ subalgebra for the
generators $(A_{0},A_{j},L_{0j})$ for each value of $j=1,2,3$

$[A_{0},L_{0j}]=iA_{j}~{}~{}~{}~{}~{}~{}[A_{0},A_{j}]=iL_{0j}~{}~{}~{}~{}~{}~{}[L_{0j},A_{j}]=iA_{0}\
.$ (13)

Now consider the transformed $A_{j}$ or $L_{0j}$ (these angular momentum
generators are defined in (5)),

$A_{j}^{\prime}=e^{i\theta A_{0}}A_{j}e^{-i\theta
A_{0}}~{}~{}~{}~{}~{}~{}L_{0j}^{\prime}=e^{i\theta A_{0}}L_{0j}e^{-i\theta
A_{0}}$ (14)

Once again, the result is determined entirely in terms of commutators. We can
therefore use any representation of the algebra (13) to perform the
calculation, e.g. the representation in terms of Pauli matrices

$L_{0j}\rightarrow\frac{1}{2}\sigma_{x}~{}~{}~{}~{}A_{j}\rightarrow\frac{1}{2}i\sigma_{z}~{}~{}~{}~{}~{}A_{0}\rightarrow-\frac{1}{2}i\sigma_{y}$
(15)

to obtain

$\displaystyle A_{j}^{\prime}$ $\displaystyle=$ $\displaystyle
A_{j}\cosh\theta-L_{0j}\sinh\theta$ $\displaystyle L_{0j}^{\prime}$
$\displaystyle=$ $\displaystyle L_{0j}\cosh\theta-A_{j}\sinh\theta\ .$ (16)

Furthermore the conformal algebra implies

$[A_{0},K_{j}+P_{j}]=[A_{0},L_{ij}]=0\ .$ (17)

This leads to final form of the deformed generators:

$\displaystyle D^{\prime}$ $\displaystyle=$ $\displaystyle
D\cosh\theta-\frac{1}{2}(K_{0}+P_{0})\sinh\theta$ $\displaystyle
K^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{0}-\frac{1}{2}(1-\cosh\theta)P_{0}-D\sinh\theta$
$\displaystyle P^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)P_{0}-\frac{1}{2}(1-\cosh\theta)K_{0}-D\sinh\theta$
$\displaystyle K^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{j}+\frac{1}{2}(1-\cosh\theta)P_{j}-L_{0j}\sinh\theta$
$\displaystyle P^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)P_{j}+\frac{1}{2}(1-\cosh\theta)K_{j}+L_{0j}\sinh\theta$
$\displaystyle L^{\prime}_{0j}$ $\displaystyle=$
$\displaystyle\cosh\theta\,L_{0j}+\frac{1}{2}\sinh\theta(P_{j}-K_{j}),\quad
L^{\prime}_{ij}=L_{ij}\ .$ (18)

These relations can be alternatively derived by first looking at the
coordinate transformations resulting from the action of $U$ with infinitesimal
$\theta$, and exponentiating them and requiring that the correct commutation
relations are satisfied by the deformed generators. This is detailed in the
appendix.

The results of this subsection imply that for $\beta<1$ the deformed
Hamiltonian is proportional to a standard CFT Hamiltonian (dilatation
operator) which is quantized with a different notion of time. Consequently
aspects of the physics of the deformed Hamiltonian are expected to be
qualitatively similar to the undeformed Hamiltonian.

###  2.2 $\beta>1$ and Lüscher-Mack Hamiltonians

The unitary transformation which relates the deformed theory to the undeformed
theory for $\beta<1$ does not work when $\beta>1$. We will now demonstrate
that for $\beta>1$ the Hamiltonian can be instead deformed to the generator
$K_{0}+P_{0}$. Consider the Hamiltonian

$H^{\prime\prime}=\frac{1}{\sqrt{\beta^{2}-1}}H=2iD\sinh\phi+i(K_{0}+P_{0})\cosh\phi~{}~{}~{}~{}\beta=\coth\phi\
.$ (19)

Using manipulations entirely similar to the previous subsection it is easy to
show that

$U^{-1}H^{\prime\prime}U=i(K_{0}+P_{0}),\quad
U=\exp\left[-\frac{i}{2}\phi(K_{0}-P_{0})\right]\ .$ (20)

The transformations of the other generators can be worked out following a
procedure entirely analogous to the previous subsection. This shows that the
physics of the deformed theory for $\beta>1$ is similar to a theory with a
Lüscher-Mack Hamiltonian $(K_{0}+P_{0})$ rather than the dilatation operator.

As we will see, this difference manifests in dynamical processes like quantum
quench by the appearance of a heating phase for $\beta>1$ and an oscillating
phase for $\beta<1$.

##  3 Dynamics as a Conformal Transformation

As mentioned above, it is convenient to think of the time evolution by
Hamiltonians like (1) by first performing a Weyl transformation to $R^{d+1}$
using Eqs. 3 and 4. Euclidean time evolution with the Hamiltonian
$H(\beta,\Pi)$ is equivalent to a conformal transformation. This
transformation can be obtained by using the Baker-Campbell-Hausdorff formula
[24]

$\displaystyle U_{\mu}\Pi^{\mu}\equiv U(\Pi)$ $\displaystyle=$ $\displaystyle
e^{-(2iwD+iw\beta(P_{\mu}+K_{\mu})\Pi^{\mu})}=e^{\Lambda_{+}K_{\mu}\Pi^{\mu}}e^{\ln\Lambda_{0}iD/2}e^{\Lambda_{-}P_{\mu}\Pi^{\mu}}$
$\displaystyle\Lambda_{0}$ $\displaystyle=$ $\displaystyle\left(\cosh
w\nu_{0}-(w\nu_{0})^{-1}\sinh
w\nu_{0}\right)^{-2},\quad\nu_{0}=\sqrt{1-\beta^{2}},$
$\displaystyle\Lambda_{+}$ $\displaystyle=$
$\displaystyle-\Lambda_{-}=i\beta\nu_{0}^{-1}\sinh(w\nu_{0})\Lambda_{0}^{1/2}$
(21)

To write down the explicit transformations, we separate out the components.
For example, using the above relations, one obtains:

$\displaystyle e^{\alpha_{1}iD}x_{\nu}e^{-\alpha_{1}iD}$ $\displaystyle=$
$\displaystyle e^{\alpha_{1}}x_{\nu},\quad
e^{i\alpha_{2}P_{\mu}}x_{\nu}e^{-i\alpha_{2}P_{\mu}}=x_{\nu},$ $\displaystyle
e^{i\alpha_{3}K_{\mu}}x_{\nu}e^{-i\alpha_{3}K_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{x_{\nu}}{1-2\alpha_{3}x_{\mu}+r^{2}\alpha_{3}^{2}}\ ,$
(22)

where $K_{\mu}$ and $x_{\mu}$ denote the $\mu$-th component of the
corresponding vectors.222To present these formulae in a completely covariant
form, one needs to implement an appropriate projection operator, which we do
not use here. Our notation therefore breaks covariance. Nonetheless, we hope
it is clear from the context whether we are referring to a vector or a
particular component of it. Identifying $\alpha_{1}=(1/2)\ln\Lambda_{0}$,
$\alpha_{2}=-i\Lambda_{-}$, and $\alpha_{3}=-i\Lambda_{+}$, we find, after a
few lines of algebra and for $\nu\neq\mu$

$\displaystyle x^{\prime}_{\nu}=\frac{x_{\nu}}{{\cal{D}}_{\mu}},$ (23)
$\displaystyle\text{where the denominator, }{\cal{D}}_{\mu}=\Big{[}\left(\cosh
w\nu_{0}-(w\nu_{0})^{-1}\sinh
w\nu_{0}\right)^{2}+2x_{\mu}\beta\nu_{0}^{-1}\sinh w\nu_{0}$
$\displaystyle\times\left(\cosh w\nu_{0}-(w\nu_{0})^{-1}\sinh
w\nu_{0}\right)+\beta^{2}r^{2}(w\nu_{0})^{-2}\sinh^{2}\tau\nu_{0}\Big{]}.$

A similar, but more complicated expression can be obtained for $\nu=\mu$ in a
similar manner. One obtains

$\displaystyle x^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle\Big{[}x_{\mu}(2(\cosh^{2}w\nu_{0}-(w\nu_{0})^{-2}\sinh^{2}w\nu_{0})-1)-\beta\nu_{0}^{-1}\sinh
w\nu_{0}(\cosh w\nu_{0}(1-r^{2})$ (24) $\displaystyle-(w\nu_{0})^{-1}\sinh
w\nu_{0}(1+r^{2}))\Big{]}/{\cal{D}}_{\mu}\ .$

These finite conformal transformations can be then used to express time
evolved quantities in terms of the quantities at initial time for any $d$.

###  3.1 $d=3$ and quaternions

In  $d\leq 3$ dimensions in general an efficient way to compute the
transformed coordinates under the $SU(1,1)$ subgroup of conformal
transformations is to use the quaternion formulation. In what follows, we
provide details this formulation for $d=3+1$. In Euclidean signature the
coordinates are denoted by $X^{\mu}$ with $x_{0}=\tau$, $x_{1}=x$ and so on.
We shall also define the $2\times 2$ matrices $\tau_{0}=I$ and
$\tau_{j}=-i\sigma_{j}$ where $\sigma_{j}$ are the standard Pauli matrices and
$I$ denotes the identity matrix. The first step of using the quaternion
formulation is then to write the coordinates $x_{\mu}$ using a $2\times 2$
matrix by associating each component of the coordinate to one of the
$\tau_{\mu}$, where $\vec{\tau}=(I,-i\sigma_{j})$, for $j=1,2,3$:

$\displaystyle Q_{\nu}$ $\displaystyle=$ $\displaystyle
Ix_{\nu}-i\sum_{j=1}^{3}\sigma_{j}y_{j}\ .$ (25)

The $y_{j}$ which appears here are the components of $x^{\mu}$ with
$\mu\neq\nu$. The choice of $\nu$ and $\mu$ is arbitrary at this stage. As an
example, we may choose $\nu=1$. The three other coordinates $x^{j}$ which
appear are $y_{1}=x_{0}=\tau,y_{2}=x_{2}=y,y_{3}=x_{3}=z$, leading to

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}x-iz&-i(\tau-iy)\\\
-i(\tau+iy)&x+iz\end{array}\right)$ (28)

Clearly $Q_{\nu}$ is not unique even after we fix $\nu$ and associate
$x_{\nu}$ with the identity matrix since there is freedom of associating the
rest of the coordinates with other Pauli matrices in different ways. As we
shall see, all such choices lead to identical results for final coordinates
under the class of conformal transformations that we discuss.

The action of the SU(1,1) transformations is then obtained by representing the
generators by Pauli matrices. The operator $U(\Pi^{\mu})$ in (21) is then
represented by the $2\times 2$ matrix,

$\displaystyle U(\Pi^{\mu})$ $\displaystyle=$ $\displaystyle
e^{-w(-\sigma_{z}+i\beta\sigma_{x})}=\left(\begin{array}[]{cc}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{array}\right)$ (31) $\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\left(\cosh w\nu_{0}+\nu_{0}^{-1}\sinh w\nu_{0}\right),\quad
a_{4}=\left(\cosh w\nu_{0}-\nu_{0}^{-1}\sinh w\nu_{0}\right)$ $\displaystyle
a_{2}$ $\displaystyle=$ $\displaystyle a_{3}=-i\beta\nu_{0}^{-1}\sinh
w\nu_{0}$ (32)

with $a_{1}a_{4}-a_{2}a_{3}=1$. To find the transformed coordinates, we note
that the quaternion matrix $Q_{\mu}$ transforms, upon action of $U(\Pi^{\mu})$
to $Q^{\prime}_{\mu}$ given by [25, 26, 27, 28]

$\displaystyle Q^{\prime}_{\mu}$ $\displaystyle=$ $\displaystyle
Ix^{\prime}_{\mu}-i\sum_{j\neq\mu}\sigma_{j}x^{\prime}_{j}=(a_{1}Q_{\mu}-ia_{2}I).(ia_{3}Q_{\mu}+a_{4}I)^{-1}$
(35) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\frac{x_{\mu}(2a_{1}a_{4}-1)-i(a_{2}a_{4}-a_{1}a_{3}r^{2})-ix_{\nu_{1}}}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}}&\frac{-i(x_{\nu_{2}}-ix_{\nu_{3}})}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}}\\\
\frac{-i(x_{\nu_{2}}+ix_{\nu_{3}})}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}}&\frac{x_{\mu}(2a_{1}a_{4}-1)-i(a_{2}a_{4}-a_{1}a_{3}r^{2})+ix_{\nu_{1}}}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}}\end{array}\right)$

where $\mu\neq\nu_{1}$, $\nu_{2}$, and $\nu_{3}$. The transformed coordinates
are then obtained from the relation $x^{\prime}_{\alpha}={\rm
Tr}[\tau_{\alpha}^{-1}Q^{\prime}_{\mu}]/2$ and yields

$\displaystyle x^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{x_{\mu}(2a_{1}a_{4}-1)-i(a_{2}a_{4}-a_{1}a_{3}r^{2})}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}},\quad
x^{\prime}_{\nu}=\frac{x_{\nu}}{a_{4}^{2}+2ia_{3}a_{4}x_{\mu}-a_{3}^{2}r^{2}}$
(36)

Substituting Eq. 32 in Eq. 36, one recovers Eqs. 23 and 24. This shows that
the quaternionic approach provides a vastly simpler way to obtain the
transformed coordinates.

In the rest of the paper we will deal with $d=3$. The transformation from
$R\times S^{3}$ to $R^{4}$ is given by

$\displaystyle\tau$ $\displaystyle=$ $\displaystyle e^{w}\cos\theta\
,x=e^{w}\sin\theta\cos\phi\ ,y=e^{w}\sin\theta\sin\phi\cos\psi\
,z=e^{w}\sin\theta\sin\phi\sin\psi\ ,$ (37)

where $w_{\mu}=(w,\theta,\phi,\psi)$ are the coordinates on the cylinder
$R\times S^{3}$, and $w$ denotes time on the cylinder.

While we have illustrated this formalism for $d=3$ the same formalism can be
used for $d=1,2$ by setting some of the coordinates in Eq.(28) to zero.

##  4 Floquet Dynamics

In this section, we shall study Floquet dynamics of $3+1$ dimensional CFTs,
using a square pulse protocol, described below. We will compute various
physical properties at the end of each cycle. The strategy can be generalized
to arbitrary number of cycles. We defer the analysis for multiple cycles to
future work. In what follows, we shall choose the amplitude of the pulses
within a drive period (denoted by $\beta$ below) to be same for simplicity.

The strategy will be to perform the computations on $R^{4}$, (coordinates
$x^{\mu}$) Weyl transforming to $R\times S^{3}$ (coordinates $w^{\mu}$) and
finally analytically continuing to real time $t=iw$. Under the Weyl
transformation, a primary operator with conformal dimension $\Delta$
transforms as

$\displaystyle O_{\Delta}(w_{\mu})$ $\displaystyle=$ $\displaystyle
e^{w\Delta}O_{\Delta}(x_{\mu}).$ (38)

We shall use these relations to transform between the physical coordinates
$w_{\mu}$ on the cylinder and those on the plane ($x_{\mu}$).

As discussed above, unitary time evolution governed by a Hamiltonian of the
form (1) is equivalent to a conformal transformation $x_{\mu}\rightarrow
x_{\mu}^{\prime}$ given by (38). The transformation of the primary operators
of the CFT having conformal dimension $\Delta$ due to such dynamics is given
by [29] :

$\displaystyle O(x_{\mu})$ $\displaystyle\rightarrow$ $\displaystyle
U^{\dagger}O(x_{\mu})U=O(x^{\prime}_{\mu})J_{2}^{\Delta/4},\quad
J_{2}=\left|\frac{\partial x^{\prime}_{\mu}}{\partial x_{\nu}}\right|=|{\rm
Det}[ia_{3}Q+a_{4}I]|^{-4}$ (39)

where the last expression holds if the coordinates ${\bf x}^{\prime}$ and
${\bf x}$ are related by the transformation given by Eqs. 32 and 35.

Figure 1: Schematic representation of the protocol for computing fidelity
$F(t)$. See text for details.

###  4.1 Fidelity

In this section, we shall compute the fidelity $F(T)$ of a primary state
$|\Delta\rangle$ of the CFT at the end of a drive cycle. This is defined, in
Euclidean time, as

$\displaystyle F(T_{0})$ $\displaystyle=$
$\displaystyle\frac{\langle\Delta|U(T_{0},0)|\Delta\rangle}{\langle
h|h\rangle}=\frac{\lim_{x_{2\mu}\to\infty,x_{1\mu}\to 0}\langle 0|\phi({\bf
x}_{2})U(T_{0},0)\phi({\bf
x}_{1})|0\rangle}{\lim_{x_{2\mu}\to\infty,x_{1\mu}\to 0}\langle 0|\phi({\bf
x}_{2})\phi({\bf x}_{1})|0\rangle}$ (40)

where $\phi({\bf x})\equiv\phi(\tau,x,y,z)$ denotes a primary CFT field of
dimension $h$, $T=-iT_{0}$ is the drive period in real time and $|0\rangle$
denotes the CFT vacuum. Throughout this section, we shall work in Euclidean
time and analytically continue to real time whenever necessary.

The protocol we choose for computing $F(T)$ is schematically shown in Fig. 1.
The evolution operator $U(\Pi^{\mu})$ in each cycle is chosen to be

$\displaystyle U(\Pi^{\mu};\tau,0)$ $\displaystyle=$ $\displaystyle
e^{-H(\Pi^{\mu})\tau},\quad H(\Pi^{\mu})=i(2D+\beta(K_{\mu}+P_{\mu})\Pi^{\mu}\
.$ (41)

The total evolution operator at the end of one cycle is given by

$\displaystyle
U=U^{\dagger}(\Pi^{1};0,T_{0}/2)U(\Pi^{3};T_{0}/2,T_{0}/4)U(\Pi^{0};T_{0}/4,0)\
,$ (42) $\displaystyle\Pi^{1}=(0,1,0,0)\ ,\quad\Pi^{3}=(0,0,0,1)\
,\quad\Pi^{0}=(1,0,0,0)\ .$ (43)

Note that we use the different deformed CFT Hamiltonians with same deformation
parameter $\beta$ to generate the evolution operator $U$. The fidelity is
computed at the end of the cycle.

To obtain $F(T)$, we first note that the two point correlation function of a
primary operator with dimension $\Delta$ can be written in the quaternion
formalism as [27]

$\displaystyle C_{0}$ $\displaystyle=$ $\displaystyle\langle 0|\phi({\bf
x}_{2})\phi({\bf x}_{1})|0\rangle=\frac{1}{({\rm Det}[Q({\bf x}_{2})-Q({\bf
x}_{1})])^{\Delta}}$ (44)

In the above equation we have not specified a subscript for the quaternions
since this particular result is independent of which $Q_{(\mu)}$ we use. For
the unequal-time correlation function under a transformation by
$U_{\mu}(T,0)$, this leads to

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle\langle
0|U_{\mu}^{\dagger}(T,0)\Pi^{\mu}\phi({\bf
x}_{2})U_{\mu}(T,0)\Pi^{\mu}\phi({\bf x}_{1})|0\rangle$ (45) $\displaystyle=$
$\displaystyle\frac{1}{{\rm Det}[Q^{\prime}_{\mu}({\bf
x^{\prime}}_{2})-Q_{\mu}({\bf x}_{1})]^{\Delta}}{\rm Det}[(ia_{3}Q_{\mu}({\bf
x}_{2})+a_{4}I)^{-1}]^{\Delta}$ $\displaystyle=$ $\displaystyle\frac{1}{{\rm
Det}[(a_{1}Q_{\mu}({\bf x}_{2})-ia_{2}I)-Q_{\mu}({\bf
x}_{1})(ia_{3}Q_{\mu}({\bf x}_{2})+a_{4}I)]^{\Delta}}.$

Note that the second term in the denominator vanishes when $x_{1\mu}\to 0$.
Further when $x_{2\mu}\to\infty$, we find $C_{1}\to 1/{\rm
Det}[a_{1}Q_{\mu_{1}}({\bf x}_{2})]^{\Delta}$. Furthermore similar analysis
shows for multiple subsequent transformations given by

$\displaystyle U_{\mu_{i}}\Pi^{\mu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}a_{i}&b_{i}\\\
c_{i}&d_{i}\end{array}\right),\quad a_{i}d_{i}-b_{i}c_{i}=1$ (48)

$C_{1}\to 1/{\rm Det}[(\prod_{i}a_{i})Q_{\mu_{i}}({\bf x}_{2})]^{\Delta}$.
This allows us to write the final expression for $F(T_{0})$

$\displaystyle F(T_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{(a_{0}(T_{0})a_{3}(T_{0})a_{1}(T_{0}))^{\Delta}}$ (49)

Figure 2: Plot of $|F(T)|$ as a function of $\beta$ and $T$ with $\Delta=1$.
For $\beta<1$, $|F(T)|$ oscillates with $T$ characterizing the non-heating
phase and reaches $\simeq 1$ for $\nu=4n\pi$ where $n$ is an integer. For
$\beta>1$, $|F(T)|$ decays exponentially with $T$ which is a signature of the
heating phase.

To compute the $a_{0}$, $a_{1}$ and $a_{3}$, we use the $2\times 2$ matrix
representation of the operators $D$, $K_{\mu}$ and $P_{\mu}$ given in Eq. 9.
Using this one can write

$\displaystyle U_{\mu}(\tau)\Pi^{\mu}$ $\displaystyle=$ $\displaystyle
e^{-\tau\sqrt{1-\beta}^{2}(-n_{z}\sigma_{z}+n_{x}\sigma_{x})},\quad
n_{z}=\frac{1}{\sqrt{1-\beta^{2}}},\quad
n_{x}=\frac{i\beta}{\sqrt{1-\beta^{2}}}$ (50)

Defining $\nu=\sqrt{1-\beta^{2}}T/4$, we find, after analytically continuing
to real time $T=-iT_{0}$,

$\displaystyle a_{0}(T)$ $\displaystyle=$ $\displaystyle
a_{3}(T)=(\cos\nu+iT(4\nu)^{-1}\sin\nu),\quad a_{1}(T)=(\cos 2\nu-
iT(2\nu)^{-1}\sin 2\nu)$ $\displaystyle F(T)$ $\displaystyle=$
$\displaystyle\frac{1}{[(\cos\nu+iT(4\nu)^{-1}\sin\nu)^{2}(\cos 2\nu-
iT(2\nu)^{-1}\sin 2\nu)]^{\Delta}}\quad{\rm for}\,\,\beta\neq 1$ (51)
$\displaystyle=$
$\displaystyle\left(\frac{1}{(1+iT/4)^{2}(1-iT/2)}\right)^{\Delta}\quad{\rm
for}\,\,\beta=1$

The behavior of each of the $a_{i}$ and hence $F(T)$ depends crucially on the
value of the parameter $\beta$. For $\beta<1$ these are oscillatory functions
of the time $T$, while for $\beta>1$ they decay exponentially for large $T$.
These two behaviors are higher dimensional versions of the non-heating and
heating phases. In between these phases there is a critical point $\beta=1$
where we have a power law decay in time, $|F(T)|\sim
1/[(1+T^{2}/16)^{2}(1+T^{2}/4)]^{\Delta/2}$. This behavior is charted out in
Fig 2 which shows the behavior $|F(T)|$ for $\Delta=1$. In between, at
$\beta=1$, we find a line which represents a critical line separating the two
phases.

Our analysis of $F(T)$ indicates special frequencies at which $|F(T)|=1$
indicating perfect overlap of the driven state with the initial states. These
frequencies, which exist only in the non-heating phase can be read off from
Eq. 51 and are given by $T=T_{n}^{\ast}(\beta)$ where

$\displaystyle T_{n}^{\ast}(\beta)$ $\displaystyle=$ $\displaystyle
4n\pi/\sqrt{1-\beta^{2}},$ (52)

and $n$ is an integer. The existence of these frequencies can also be seen in
Fig. 2.

Thus we find that under a drive schematically represented by Fig. 1, $F$
exhibits perfect revival at special frequencies in the non-heating phase and
an exponential (power-law) decay with $T$ in the heating phase (on the
critical line).

The different behaviors for different values of $\beta$ reflect the properties
of the Hamiltonians used for time evolutions which have been discussed in the
previous section. Each of these Hamiltonians represent a different $SU(1,1)$
subgroup of the conformal group. We chose the same $\beta$ for each of the
three Hamiltonians, so the conjugacy classes of each of these $SU(1,1)$ are
the same. The results can be trivially extended to the cases where the three
$\beta$’s are different.

###  4.2 Unequal-time Correlators

In this section, we compute the unequal-time two-point correlation function of
the primary fields with conformal dimension $\Delta$ in the presence of a
drive. We consider the fields on a cylinder and map them onto the plane using
Eq. 38. The initial coordinates on the plane corresponds to

$\displaystyle{\bf x_{2}}$ $\displaystyle=$
$\displaystyle(\tau_{2},x_{2},y_{2},z_{2})=(\cos\theta_{2},\sin\theta_{2}\cos\phi_{2},\sin\theta_{2}\sin\phi_{2}\cos\psi_{2},\sin\theta_{2}\sin\phi_{2}\sin\psi_{2}),$
$\displaystyle{\bf x}_{1}$ $\displaystyle=$
$\displaystyle(\tau_{1},x_{1},y_{1},z_{1})=(1,0,0,0)$ (53)

where we have taken initial time on the cylinder $w=0$ without loss of
generality. We have also chosen the coordinates $\theta_{1}=0$ for simplicity.
In what follows we shall consider the correlation function

$\displaystyle C_{1}(T)$ $\displaystyle=$ $\displaystyle\langle
0|U^{\dagger}(T,0)O({\bf x}_{2})U(T,0)O({\bf
x}_{1})|0\rangle=\frac{J_{2}^{\Delta/4}}{({\rm Det}[Q({\bf
x}^{\prime}_{2})-Q({\bf x}_{1})])^{\Delta}}$ (54)

where ${\bf x}^{\prime}_{2}$ represents the transformed coordinate and $J_{2}$
denotes the Jacobian of the coordinate transformation.

Figure 3: Plot of $|C_{1}(T)/C_{1}(0)|$ as a function of $\theta_{2}$ and $T$
for $\beta=0.2$ (left panel) and $\beta_{1}=1.2$ (right panel) corresponding
to a square pulse protocol. See text for details.

To study the dynamics, we first consider a square pulse protocol given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{(-)}=2iD~{}~{}~{}~{}~{}~{}{\rm for}~{}~{}t\leq T/2$ $\displaystyle H$
$\displaystyle=$ $\displaystyle
H_{(+)}=2iD+i\beta\left(K_{0}+P_{0}\right)~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}t>T/2.$ (55)

We will calculate the correlator at time $T$.

The evolution operator for this is

$\displaystyle U(T,0)$
$\displaystyle=U_{0}(T,0)=e^{-T_{0}\left(2iD+i\beta\left(K_{0}+P_{0}\right)\right)/2}e^{-iDT_{0}}=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)$ (58) $\displaystyle a$
$\displaystyle=d^{\ast}=\left(\cosh\eta T/2+\frac{i}{\eta}\sinh\eta
T/2\right)e^{iT/2},\quad b=c^{\ast}=e^{-iT/2}\frac{\beta}{\eta}\sinh\eta T/2,$
(59) $\displaystyle Q_{0}({\bf x}^{\prime}_{2})$ $\displaystyle=(aQ_{0}({\bf
x_{2}})-ibI).(icQ_{0}({\bf x}_{2})+dI)^{-1},\,\,J_{2}=|({\rm Det}[icQ_{0}({\bf
x}_{2})+dI])^{\Delta}|^{-4}$ (60)

where $\eta=\sqrt{\beta^{2}-1}$ and $I$ is the $2\times 2$ identity matrix.
Here we have used Eqs. 32 and 36, and performed a Wick rotation $T=-iT_{0}$.
Substituting Eqs. 60 in Eq. 54, we find after transforming back to cylinder
coordinates 333The Weyl factors cancel in the following expressions

$\displaystyle C_{1}(T)$ $\displaystyle=$ $\displaystyle
C_{1}(0)\left[\frac{2(1-\cos\theta_{2})}{(a-ic)^{2}-(b-id)^{2}-2i(a-ic)(b-id)\cos\theta_{2}}\right]^{\Delta},$
$\displaystyle C_{1}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2(1-\cos\theta_{2})^{\Delta}}$ (61)

The dependence of $C_{1}(T)$ on both $T$ and $\beta$ can be obtained by
substituting the expressions of $a$, $b$, $c$ and $d$ from Eq. 60. We note
that for $\beta>1$, the system is in the heating phase and the correlation
decays exponentially; in contrast, for $\beta<1$, the dynamics is oscillatory.
Also, we find that $C(T)$ depends only on $\theta_{2}$ and is independent of
other coordinates. This is a consequence of the choice of the drive protocol
which involves only $K_{0}$ and $P_{0}$. These features of the correlation
function are shown in Fig. 3 where $|C_{1}(T)/C_{1}(0)|$ is plotted for
$\Delta=1$ as a function of $\theta_{2}$ and $T$ for $\beta=0.2$ (left panel)
and $\beta=1.2$ (right panel).

We next consider a drive protocol involving two different sets of generators
given by

$\displaystyle H_{(-)}$ $\displaystyle=2iD+i\beta(K_{(0)}+P_{(0)}),\quad
H_{(+)}=2iD+i\beta(K_{(3)}+P_{(3)})$ $\displaystyle U(T_{0},0)$
$\displaystyle=U_{(+)}(T_{0},T_{0}/2)U_{(-)}(T_{0}/2,0),\quad
U_{(\pm)}(\tau,0)=e^{-\tau
H_{(\pm)}/\hbar}=\left(\begin{array}[]{cc}a_{\pm}&b_{\pm}\\\
c_{\pm}&d_{\pm}\end{array}\right)$ (64)

To obtain the correlator corresponding to this protocol, we consider the
action of $U_{-}$ on $Q({\bf x}_{2})\equiv Q_{2}$ which is given by

$\displaystyle Q_{2}^{(1)}$ $\displaystyle=$
$\displaystyle(a_{-}Q_{2}-ib_{-}I).(ic_{-}Q_{2}+d_{-}I)^{-1}=\left(\begin{array}[]{cc}\tau^{\prime}-iz^{\prime}&-i(x^{\prime}-iy^{\prime})\\\
-i(x^{\prime}+iy^{\prime})&\tau^{\prime}+iz^{\prime}\end{array}\right)$ (67)
$\displaystyle\tau^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\tau(2a_{-}d_{-}-1)-i(a_{-}c_{-}-b_{-}d_{-}r^{2})}{d_{-}^{2}+2ic_{-}d_{-}\tau-
c_{-}^{2}r^{2}},\quad x^{\prime}_{j}=\frac{x_{j}}{d_{-}^{2}+2ic_{-}d_{-}\tau-
c_{-}^{2}r^{2}}$ (68)

where $x_{j}=(x,y,z)$ for $j=(1,2,3)$. Next, we rewrite $Q_{2}^{(1)}$ as
$Q_{2}^{{}^{\prime}(1)}=z^{\prime}I--
i\sigma_{z}\tau^{\prime}-i\sigma_{x}x^{\prime}-i\sigma_{y}y^{\prime}$ and
perform the second transformation

$\displaystyle Q_{2}^{(2)}$ $\displaystyle=$
$\displaystyle(a_{+}Q_{2}^{{}^{\prime}(1)}-ib_{+}I).(ic_{+}Q_{2}^{{}^{\prime}(1)}+d_{+}I)^{-1}=\left(\begin{array}[]{cc}z^{\prime\prime}-i\tau^{\prime\prime}&-i(x^{\prime\prime}-iy^{\prime\prime})\\\
-i(x^{\prime\prime}+iy^{\prime\prime})&z^{\prime\prime}+i\tau^{\prime\prime}\end{array}\right)$
(71) $\displaystyle z^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{z^{\prime}(2a_{+}d_{+}-1)-i(a_{+}c_{+}-b_{+}d_{+}r^{\prime
2})}{d_{+}^{2}+2ic_{+}d_{+}z^{\prime}-c_{+}^{2}r^{\prime 2}},\quad
x^{\prime\prime}_{j}=\frac{x^{\prime}_{j}}{d_{+}^{2}+2ic_{+}d_{+}z^{\prime}-c_{+}^{2}r^{\prime
2}}$ (72)

where
$x_{j}^{\prime\prime}=(\tau^{\prime\prime},x^{\prime\prime},y^{\prime\prime})$
for $j=(1,2,3)$.

The rest of the calculation is cumbersome but straightforward. A somewhat
lengthy algebra yields

$\displaystyle C_{2}(T)$
$\displaystyle=C_{1}(0)\left(\frac{2(1-\cos\theta_{2})}{\alpha+\beta_{1}\cos\theta_{2}+\beta_{2}\sin\theta_{2}+\beta_{3}\cos
2\theta_{2}+\gamma(\theta_{2})\sin\theta_{2}\sin\phi_{2}\sin\psi_{2}}\right)^{\Delta}.$
(73)

Here $\alpha$, $\beta_{i}$ ($i=1,2,3$), and $\gamma$ are functions of the
drive parameters through functions $a$, $b$, $c$ and $d$ given by

$\displaystyle a$ $\displaystyle=$ $\displaystyle d^{\ast}=\left(\cosh\eta
T/2+\frac{i}{\eta}\sinh\eta T/2\right),\quad b=c=\frac{\beta}{\eta}\sinh\eta
T/2,$ (74) $\displaystyle\gamma(\theta_{2})$ $\displaystyle=$
$\displaystyle-4ib(a-d)(b^{2}-d^{2}-2ibd\cos\theta_{2}),\quad\beta_{2}=b^{2}-a^{2}$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}8a^{3}d(1-ad)+a^{2}(-1+8(b+2id)d^{2})-8iabd(d+b(2bd-i))$
$\displaystyle+b^{2}(1+8d(ib-d(b^{2}-d^{2})))\Big{]}$ $\displaystyle\beta_{1}$
$\displaystyle=$ $\displaystyle
4i\Big{[}2a^{4}bd-c^{3}b(1+2d^{2})+a(b^{3}+ib^{2}d(1+4b^{2})+2b^{3}d^{2}-id^{3})-b^{3}d(1+4d^{2})$
$\displaystyle+a^{2}d(b-2b^{3}-4ib^{2}c4+2id^{3})+b(2b^{4}d+ibd^{2}+2d^{5}-ib^{3}(1+2d^{2}))\Big{]}$

where we have analytically continued to real time $T=-iT_{0}$. The function
$\alpha$ is cumbersome but can be expressed in terms of $a$, $b$, $c$ and $d$.

Eq. 73 indicates a clear rotational symmetry breaking in the correlation
function. This is a consequence of application of two different sets of
generators $(K_{0},P_{0})$ and $(K_{3},P_{3})$ for constructing $U(T,0)$. This
dependence is shown for a fixed $T=1$, $\theta_{2}=\pi/4$ and $\beta=0.2$ in
Fig. 4 where $|C_{2}(T)/C_{2}(0)|$ is plotted as a function of $\phi_{2}$ and
$\psi_{2}$.

Figure 4: Plot of $|C_{2}(T)/C_{2}(0)|$ as a function of $\phi_{2}$ and
$\psi_{2}$ for $\beta=0.2$, $T=5$ and $\theta=\pi/4$. See text for details.

###  4.3 Local probes of the driven state

In this section we will probe the time-evolving state with local operators,
i.e., we are interested in computing :

$\displaystyle\langle\Delta|U^{\dagger}_{\mu}(T,0)\Pi^{\mu}A(w^{\alpha})U_{\mu}(T,0)\Pi^{\mu}|\Delta\rangle.$
(75)

We will find here as in the $d=1$ case [30] that there is localization in the
local observables.

#### Energy density

Consider first the energy density, i.e. when the operator $A(w^{\alpha})$ is
chosen to be $T_{ww}(w^{\alpha})$. Therefore we will need the transformation
of the stress tensor. For $d=3$, the conformal dimension of the energy-
momentum tensor is $4$, so that the Weyl factor is simply $e^{4w}$. Along with
the factors coming from the transformation of a spin-2 tensor, the energy
density translates to

$\displaystyle T_{ww}(w^{\mu})$ $\displaystyle=e^{4w}\frac{\partial
x^{\rho}}{\partial w}\frac{\partial x^{\sigma}}{\partial
w}T_{\rho\sigma}(x)+\frac{3a}{8\pi^{2}},$ (76)

where the last term comes from the Weyl anomaly. Conjugation of the $R^{d+1}$
stress tensor $T_{\rho\sigma}(x)$ gets with the deformed evolution operators
$U_{\mu}(T,0)\Pi^{\mu}$ leads to a conformal transformation. In $D$ flat
spacetime for a spin-2 conformal primary like the stress-tensor the
transformation rule is:

$\displaystyle U^{\dagger}T_{\mu\nu}(x)U$
$\displaystyle=T^{\prime}_{\mu\nu}(x)=J^{\frac{D-2}{D}}\frac{\partial
x^{\prime\rho}}{\partial x^{\mu}}\frac{\partial x^{\prime\sigma}}{\partial
x^{\nu}}T_{\rho\sigma}(x^{\prime}),$ (77)

where $J$ denotes the Jacobian of the conformal transformation. Finally we
shall be left to compute a plane three point function involving a stress-
tensor and two primaries. This is given by [31]:

$\displaystyle\langle O(x_{1})O(x_{2})T^{\mu\nu}(x_{3})\rangle$
$\displaystyle=-\frac{4\Delta}{6\pi^{2}}\frac{H^{\mu\nu}(x_{1},x_{2},x_{3})}{|x_{12}|^{2\Delta-2}|x_{13}|^{2}|x_{23}|^{2}},$
(78) $\displaystyle\text{where, }H^{\mu\nu}$
$\displaystyle=V^{\mu}V^{\nu}-\frac{1}{4}V^{2}\delta^{\mu\nu},\,\,\,\text{with,
}V^{\mu}=\frac{x_{13}^{\mu}}{x_{13}^{2}}-\frac{x_{23}^{\mu}}{x_{23}^{2}}.$

Putting everything together, the time-dependent piece (i.e. modulo the
anomalous piece) in the energy density of the driven state is:

$\displaystyle\langle\Delta|U^{\dagger}T_{ww}(w^{\mu})U|\Delta\rangle_{T}$
$\displaystyle=-2\frac{\Delta}{3\pi^{2}}\frac{e^{4w}J_{2}^{-1/2}}{|x^{\prime}(w^{\mu})|^{2}}\frac{\partial
x^{\rho}}{\partial w}\frac{\partial x^{\sigma}}{\partial w}\frac{\partial
x^{\prime\alpha}}{\partial x^{\rho}}\frac{\partial x^{\prime\beta}}{\partial
x^{\sigma}}H_{\alpha\beta}(x^{\prime}(w^{\mu})),$ (79)
$\displaystyle\text{where, }H_{\rho\sigma}(x^{\prime})$
$\displaystyle=\frac{x^{\prime}_{\rho}x^{\prime}_{\sigma}-\frac{x^{\prime
2}}{4}\delta_{\rho\sigma}}{(x^{\prime})^{4}}.$

We consider a square pulse drive given by:

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{(-)}=2iD~{}~{}~{}~{}~{}~{}{\rm for}~{}~{}t\leq T/2$ $\displaystyle H$
$\displaystyle=$ $\displaystyle
H_{(+)}=2iD+i\beta\left(K_{(1)}+P_{(1)}\right)~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}t>T/2.$ (80)

For this protocol the evolution operator is parametrized as :
$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ with $a,b,c,d$ as in
Eq. 59. Below in Fig. 5 and Fig. 6 we plot the absolute value of the
normalized energy density

$\displaystyle
E(\theta,\phi,\psi,T)=\frac{\langle\Delta|U^{\dagger}T_{ww}(w^{\mu})U|\Delta\rangle_{T}}{\langle\Delta|U^{\dagger}T_{ww}(w^{\mu})U|\Delta\rangle_{T=0}}$
(81)

in various regimes. Note that $\beta$ denotes the amplitude of the deformation
and can be used to enter and leave the heating regime. When $\beta>1$ and we
are in the heating regime we find clear signatures of localization of the
energy density in the angular directions, whereas the non-heating regime is
characterized by oscillations. Furthermore the localization occurs in both the
angular directions $\theta$ and $\phi$ as is clear from the density plot of
Fig. 7, and is independent of the $\psi$ direction. Due to the latter feature
we omit $\psi$ from $E(\theta,\phi,\psi,T)$.

Figure 5: Density plots of normalized density $|E(\theta,\phi,T)|$ in the
$\phi$, $T$ plane for $\theta=\pi/2$. Left Panel: In the heating phase with
$\beta=1.2$ we find progressive localization. Right panel: In the non-heating
phase with $\beta=0.09$ we find oscillations.

Figure 6: Plots of normalized density $|E(\theta,\phi,T)|$ as a function of
$\phi$ for $\theta=\pi/2$ in the heating phase, $\beta=1.2$, and for several
representative values of $T$ : 1.5 (Green), 2 (Orange), 7 (Blue), 9 (Brown)
and 12 (Black).

Figure 7: Density plots of normalized density $|E(\theta,\phi,T)|$ in the
$\phi$, $\theta$ plane for $T=6$, in the heating phase, $\beta=1.1$.

The explicit formula for the energy density as obtained from the analytic
expression Eq. 79 turns out to be too complicated. Thus we turn our attention
to a simpler observable, namely, that of a local primary probe in the driven
state, which allows for an explanation of the angular localization.

#### Primaries

Here we choose the probe to be another primary operator in place of
$A(w^{\alpha})$ in Eq. 75. The transformation rule of a primary is already
given as in Eq. 39, using which along with the Weyl factor, Eq. 38, and the
universal formula for three point primary correlator, we obtain:

$\displaystyle\langle\Delta|U^{\dagger}_{\mu}\Pi^{\mu}(T,0)O_{\Delta_{1}}(w^{\alpha})U_{\mu}(T,0)\Pi^{\mu}|\Delta\rangle$
$\displaystyle=e^{w\Delta_{1}}J_{2}^{\Delta_{1}/4}\frac{C_{\Delta\Delta\Delta_{1}}}{|x^{\prime}|^{\Delta_{1}}},$
(82)

where $C_{\Delta\Delta\Delta_{1}}$ is the operator product expansion
coefficient which is part of the CFT data. The above expression in the
cylindrical coordinates, with initial time $w=0$, for a generic $SU(1,1)$
drive in the $x$ direction, takes the form :

$\displaystyle\langle\Delta|O_{\Delta_{1}}(T)|\Delta\rangle$
$\displaystyle=C_{\Delta\Delta\Delta_{1}}\frac{[((1+a_{2}a_{3})^{2}-a_{2}^{2}a_{4}^{2})^{2}+4a_{2}^{2}a_{4}^{2}(1+a_{2}a_{3})^{2}\cos^{2}\phi\sin^{2}\theta]^{-\Delta_{1}/4}}{a_{4}^{-4\Delta_{1}}[a_{4}^{4}(a_{3}^{2}-a_{4}^{2})^{2}+4a_{3}^{2}a_{4}^{6}\sin^{2}\theta\cos^{2}\phi]^{3\Delta_{1}/4}}$
$\displaystyle=C_{\Delta\Delta\Delta_{1}}O(\theta,\phi,T)_{\Delta}.$ (83)

Once again there is independence in $\psi$. Notice that when either $\theta=0$
or $\phi=\pi/2$ the above becomes space independent. Once again, we consider
the square-pulse protocol involving deformation in the $x$ direction, as in
Eq. 80. The parameter $\beta$ denotes the amplitude of the drive and can be
used to enter and leave the heating regime. In Fig.8 on the $\beta,T$ plane we
obtain a density plot very similar to Fig.2. In particular we notice, that
oscillations die into exponential fall-offs as the $\beta=1$ line is crossed
from below.

Figure 8: Plot of $|O(\theta=0,\phi,T)|_{\Delta}$ as a function of $\beta$ and
$T$ with $\Delta_{1}=1$. For $\beta<1$, we find oscillations with $T$
characterizing the non-heating phase and for $\beta>1$, the one-point function
decays exponentially with $T$ which is a signature of the heating phase.

At $\beta=1$ the functional dependence of the absolute value squared of the
one point function in the driven state is :

$\displaystyle|O(\theta,\phi,T)|_{\Delta}^{2}=\bigg{(}(8+4T^{2}+T^{4}+T^{2}(T^{2}-4)\cos
2T+4T^{3}\sin 2T)^{2}-4T^{2}\cos^{2}\phi(T^{2}(4+T^{2})^{2}$
$\displaystyle+(8+4T^{2}+T^{4})((T^{2}-4)\cos 2T+4T\sin
2T))\sin^{2}\theta+4T^{4}(4+T^{2})^{2}\cos^{4}\phi\sin^{4}\theta\bigg{)}^{-\Delta_{1}}.$
(84)

At large $T$, dropping the rapidly oscillatory pieces, we find:

$\displaystyle|O(\theta,\phi,T)|^{2}_{\Delta}$
$\displaystyle\sim\frac{1}{T^{8\Delta_{1}}(1-2\cos^{2}\phi\sin^{2}\theta)^{2\Delta_{1}}}+{\cal
O}(T^{-10\Delta_{1}}).$ (85)

It turns out that the higher order terms in the $1/T$ series (with rapidly
oscillating terms dropped) contains higher order singularities of
$1-2\cos^{2}\phi\sin^{2}\theta$ :

$\displaystyle|O(\theta,\phi,T)|^{2}_{\Delta}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{f_{n,\Delta_{1}}(\cos^{2}\phi\sin^{2}\theta)}{T^{\Delta_{1}(6+2n)}(1-2\cos^{2}\phi\sin^{2}\theta)^{\tfrac{1}{2}(2n+1-(-1)^{n})\Delta_{1}}}.$
(86)

These singularities give rise to localization in the angular directions, and
as we numerically investigate next, also persists in the heating, $\beta>1$
regime.

Below in Fig.9 we plot the normalized one point function amplitude in various
regimes. We find clear signatures of localization of the amplitude in the
angular directions, where as the non-heating regime is characterized by
oscillations.

Figure 9: Density plots of normalized $|O(\theta,\phi,T)|_{\Delta}$ in the
$\theta$, $T$ plane for $\phi=0$. Left Panel: In the heating phase with
$\beta=1.1$ we find progressive localization. Right panel: In the non-heating
phase with $\beta=0.1$ we find oscillations. The conformal dimension is taken
to be $\Delta_{1}=0.4$.

Localization occurs in both the $\phi$ as well as $\theta$ directions as is
clear from the left panel of Fig.10. In the right panel of Fig.10 we have
plotted the contours of the function $\cos^{2}\phi\sin^{2}\theta$ since at
least in the $\beta=1$ line we expect from Eq. (86) localization along the
$\tfrac{1}{2}$ contour. The plots are indicative of the fact that similar
singularities extend into the heating regime as well. It is very natural that
a similar effect is responsible for the localization in the energy density
when the primary gets replaced by the stress-tensor.

Figure 10: Left Panel : Localization from the density plots of normalized
$|O(\theta,\phi,T)|_{\Delta}$ in the $\phi$, $\theta$ plane for $T=3.3$, in
the heating phase, $\beta=1.1$. The conformal dimension is set to
$\Delta_{1}=0.4$. Right Panel : Contour lines of $\cos^{2}\phi\sin^{2}\theta$.
See text for details.

###  4.4 Multiple cycles and dynamical phases

In the sections above we have presented the results for a single cycle of a
periodic protocol. However the formalism developed above can be recursively
applied to multiple cycles, and Floquet dynamics as a function of the
stroboscopic time can be studied. By varying the time intervals we expect to
observe dynamical phases and transitions between them.

When a cycle contains Hamiltonians belonging to one $SL(2,R)$ algebra, we can
judiciously choose quaternionic representation of the coordinates on $R^{4}$,
such that the parameters of the quaternionic Möbius transformation $SL(2,H)$
are real, and the discussion of multiple cycles becomes simple.

Consider a general conformal transformation (we omit the index $\mu$ for the
quaternion $Q_{\mu}$ below)

$Q^{\prime}=(aQ+b)\cdot(cQ+d)^{-1}$ (87)

where $a,b,c,d$ are themselves quaternions, satisfying

${\rm Det}|ac^{-1}dc-bc|^{2}=1$ (88)

For transformations which involve the subalgebra formed by $D,K_{0},P_{0}$,
one can choose

$Q=x_{0}I-i\sum_{i=1}^{3}\sigma_{i}x_{i}$ (89)

Then for such transformations $a,b,c,d$ are real numbers satisfying

$ad-bc=1$ (90)

A fixed point ${\bar{Q}}$ satisfies

${\bar{Q}}(c{\bar{Q}}+d)=a{\bar{Q}}+b$ (91)

Using (91) and standard properties of the Pauli matrices it is then easy to
see that there are three kinds of fixed points or surfaces.

  1. 1.

${\bar{Q}}^{\pm}={\bar{x}}_{0}^{\pm}I$, i.e. ${\bar{x}}_{i}=0$. Here

${\bar{x}}_{0}^{\pm}=\frac{1}{2c}\left[(a-d)\pm\sqrt{(a+d)^{2}-4}\right]$ (92)

In deriving (92) we have used (90). These are two fixed points and requires

$(a+d)^{2}>4$ (93)

The corresponding matrix

$\begin{bmatrix}a&b\\\ c&d\\\ \end{bmatrix}\quad$ (94)

is in the hyperbolic conjugacy class.

  2. 2.

When $(a+d)^{2}<4$ there is a fixed two dimensional hypersurface defined by

${\bar{x}}_{0}=\frac{a-d}{2c}~{}~{}~{}~{}~{}\sum_{i=1}^{3}({\bar{x}}_{i})^{2}=\frac{1}{4c^{2}}\left(4-(a+d)^{2}\right)$
(95)

and now the matrix in (94) is in the elliptic conjugacy class.

  3. 3.

The case $(a+d)^{2}=4$ is marginal and corresponds to the parabolic conjugacy
class.

It is now straightforward to see that the transformation (87) can be re-
expressed in the following form for hyperbolic and elliptic transformations:

$(Q^{\prime}-{\bar{Q}}^{+})(Q^{\prime}-{\bar{Q}}^{-})^{-1}=(c{\bar{Q}}^{-}+d)(Q-{\bar{Q}}^{+})(Q-{\bar{Q}}^{-})^{-1}(c{\bar{Q}}^{+}+d)^{-1}$
(96)

For the elliptic conjugacy class, the ${\bar{Q}}^{\pm}$ in this equation refer
to two antipodal points on the fixed 2-surface defined in (95).

To prove (96) we substitute (87) in the left hand side of this equation so
that the left hand side becomes

$\left[(a-c{\bar{Q}}^{+})Q+(b-d{\bar{Q}}^{+})\right]\left[(a-c{\bar{Q}}^{-})Q+(b-d{\bar{Q}}^{-})\right]^{-1}$
(97)

while the right hand side of (96) becomes

$\left[(c{\bar{Q}}^{-}+d)Q-(c{\bar{Q}}^{-}+d){\bar{Q}}^{+}\right]\left[(c{\bar{Q}}^{+}+d)Q-(c{\bar{Q}}^{+}+d){\bar{Q}}^{-}\right]^{-1}$
(98)

For the hyperbolic case, ${\bar{Q}}^{\pm}$ are both proportional to the
identity matrix. For the elliptic case, we we can take the antipodal points on
the fixed 2-surface to be along the $3$ axis without any loss of generality.
For both these cases one can then explicitly check that

${\bar{Q}}^{+}+{\bar{Q}}^{-}=\frac{a-d}{c}\cdot
I~{}~{}~{}~{}~{}{\bar{Q}}^{+}{\bar{Q}}^{-}={\bar{Q}}^{-}{\bar{Q}}^{+}=-\frac{b}{c}\cdot
I$ (99)

Using these relations and the relation (90) it may be checked that (97) and
(98) are equal term by term.

The relationship (87) can be iterated to yield

$\displaystyle Q^{\prime(n)}$
$\displaystyle=\bigg{(}{I}-(c\,Q_{-}+d)^{n}\alpha_{\pm}(Q)(c\,Q_{+}+d)^{-n}\bigg{)}^{-1}\bigg{(}Q_{+}-\left((c\,Q_{-}+d)^{n}\alpha_{\pm}(Q)(c\,Q_{+}+d)^{-n}\right)Q_{-}\bigg{)},$
(100)

where

$\alpha_{\pm}(Q)=\left[(Q-Q_{+})(Q-Q_{-})^{-1}\right].$

which can be used to determine the trajectory of a point under successive
transformations.

The Euclidean distance between two points represented by quaternions $Q_{1}$
and $Q_{2}$ are given by $\sqrt{|{\rm Det}(Q_{1}-Q_{2})|}$. Equation (96)
implies

$\frac{{\rm Det}(Q^{\prime}-{\bar{Q}}^{+})}{{\rm
Det}(Q^{\prime}-{\bar{Q}}^{-})}=R^{2}\frac{{\rm Det}(Q-{\bar{Q}}^{+})}{{\rm
Det}(Q-{\bar{Q}}^{-})}$ (101)

where

$\displaystyle R^{2}$ $\displaystyle=$ $\displaystyle\frac{{\rm
Det}(cQ^{-}+d)}{{\rm
Det}(cQ^{+}+d)}=\left(\frac{a+d-\sqrt{(a+d)^{2}-4}}{a+d+\sqrt{(a+d)^{2}-4}}\right)^{2}\qquad({\rm{Hyperbolic}})$
$\displaystyle R^{2}$ $\displaystyle=$ $\displaystyle
1\qquad\qquad\qquad\qquad\qquad({\rm{Elliptic}})$ (102)

The equation can be recursively applied to yield

$\frac{{\rm Det}(Q^{(n)}-{\bar{Q}}^{+})}{{\rm
Det}(Q^{(n)}-{\bar{Q}}^{+})}=R^{2n}\frac{{\rm Det}(Q-{\bar{Q}}^{+})}{{\rm
Det}(Q-{\bar{Q}}^{+})}$ (103)

where $Q^{n}$ denotes the transformed point after $n$ such transformations.
The expressions for $R^{2}$ then immediately implies that for hyperbolic
transformations a point converges to one of the fixed points for large $n$,
whereas for ellipic transformations, the point keeps moving on a circle of
fixed radius around a point on the fixed surface.

Using the relation (100) we can calculate various physical quantities under a
Floquet drive of this type, after $n$ cycles. The result depends on the
conjugacy class of the transformation at the end of a cycle. However the
general features will be similar to the $d=1$ case, i.e. hyperbolic classes
lead to a heating phase while elliptic classes lead to oscillatory phase [12,
14, 16].

When a cycle involves two different $SL(2,R)$ subgroups of the conformal
group, e.g. two different $\Pi_{\mu}$’s, one can no longer represent the net
transformation by a $SL(2,H)$ transformation with parameters proportional to
the identity. The analysis of fixed points (surfaces) etc. needs to be re-
done, and the result of such Floquet drives will be quite different.

Explicit results for physical quantities under Floquet dynamics will appear in
a future publication.

##  5 Discussions

In this paper we initiated a program of studying periodic quantum dynamics of
driven deformed conformal field theories in arbitrary number of dimensions,
inspired by recent work in $1+1$ dimensions. More concretely, we studied the
properties of driven $3+1$ dimensional CFTs using square protocols which
involve evolution by non-commuting Hamiltonians in successive time intervals
after a single drive cycle.

There are several points worth emphasizing regarding this approach. First, the
method of exactly computing the dynamics works in arbitrary number of
dimensions. We performed explicit calculations in $3+1$ dimensions in this
work. But our use of quaternion formalism provides a simple way to perform
these calculations all $d\leq 3$. Secondly, our method can be
straightforwardly generalized to other drive protocols including continuous
ones; we leave this as a subject of future work. For a class of drive
protocols which uses same $\mu$ but different $\beta$ for each pulse within a
drive cycle, we have shown how to obtain the necessary transformations for
arbitrary number of cycles. This is similar to $d=1$, and we can have dynamic
control over the transition from the heating to the non-heating phase similar
to that found in $d=1$ driven CFTs [12, 14, 16]. Explicit results for multiple
cycles of this type will appear in a future paper. For pulses with different
$\mu$, the situation is less clear and this issue is left as a subject of
future study.

While the power of conformal symmetry allows us to calculate observables and
the details associated with the phase structure, perhaps it is useful to
ponder over a simpler physical intuition of the underlying physics. In this
regard, it is useful to consider e.g. free theories in $(1+1)$-dimensions. For
free Bosons, the stress-tensor $T(w)=(-1/2):\partial\phi(w)\partial\phi(w):$
and for free Fermions it is given by $T(w)=(1/2):\psi(w)\partial\psi(w):$ and
subsequently the Virasoro modes are extracted from the stress-tensor:
$L_{n}=(1/(2\pi i))\oint w^{n+1}dwT(w)$. The standard radial quantization in
CFT chooses a Hamiltonian $H\sim L_{0}$, which corresponds to a Hamiltonian-
density of the form: $w\partial\phi(w)\partial\phi(w)$ for Bosons and
$w\psi(w)\partial\psi(w)$ for Fermions. The $w$-factor in front of the kinetic
terms can be interpreted to be yielding a red-shift physics for the system.
Now, choosing a different quantization basically entails distinct choices for
the Hamiltonian, involving $\\{L_{m},L_{-m}\\}$-modes. Such choices will
generate a non-trivial red-shift compared to the standard quantization and a
thermal-phase may appear as a result of this.444Qualitatively, this is similar
to the Rindler-physics, although it appears more subtle and layered. Note,
however, that this is too crude to pass as an argument since we have not
factored in the inequivalent conjugacy classes and we have also been imprecise
about where the “red-shift” matters, i.e. near the origin of the complex plane
or near infinity. Nonetheless, it raises an an intriguing possibility of a
more precise physical picture along these lines and we hope to address this in
future.

For holographic CFT’s, it would be insightful to get a dual gravitational
description of the dynamics. This can be done from two different perspectives:
One that is based on the explicit time-dependent Hamiltonian and the other
that is based on the Floquet Hamiltonian. While the former yields a dynamical
geometry, the latter is capable of describing static “patches” of the geometry
which correspond to the different phases (heating and non-heating) and the
phase boundary. For the $1+1$ dimensional case, the dynamical scenario has
been explored in  [7, 8, 9], while the Floquet-Hamiltonian based approach is
discussed in [21]. In [7], a non-trivial dynamical horizon structure has been
observed, whereas the different geometric patches of [21] were obtained by
solving integral curves that are generated by the (appropriate combination of)
bulk Killing vector that is dual to the CFT Floquet Hamiltonian.
Interestingly, the integral curve approach is also explored in a completely
different context of “bulk reconstruction” in which a local bulk observer in
an AdS-geometry is described in terms of the CFT Hamiltonian, see e.g. [32,
33], .

The higher dimensional generalization of the bulk geometric description is
potentially very interesting. First, it is rather non-trivial to obtain
horizons with structures in higher dimensional black holes. Secondly, it is
also technically involved to obtain dynamical horizons, outside Vaidya-type
geometries. Moreover, from the integral curve perspective, a local bulk
observer seems readily describable in terms of the CFT Floquet Hamiltonian
using the bulk Killing vectors that describe the SL$(2,C)$ sub-algebra of the
conformal algebra, see for example appendix B for explicit expressions of the
corresponding generators. We are currently exploring this aspect in detail.

Let us now offer some comments that are not necessarily limited to CFTs. It is
evident that the crucial ingredients of our results are the following: (i) The
existence of an SL$(2,C)$ algebra as a sub-algebra of the symmetry group of
the system and (ii) a “quasi-primary” representation of the fields under this
SL$(2,C)$.555A similar statement holds for an SL$(2,R)$ algebra as well.
Interestingly, scattering matrix elements in $(3+1)$ and $(2+1)$-dimensions
can be recast into correlation functions of quasi-primary operators at null
infinity known as the celestial sphere, see e.g. [35, 36, 37] for explicit
details on this. These quasi-primaries form the continuous principal series,
$\Delta=\frac{d-1}{2}+is$, in $(d+1)$-dimensions and $s$ is real-valued; we
have collected some relevant and explicit formulae in appendix C for a more
direct comparison. Even though the “CFT-spectrum” is continuous and complex-
valued, the corresponding correlators are real-valued since it involves both
$\Delta$ and $\bar{\Delta}$. Thus, the presence of a heating and a non-heating
phase along with a phase boundary is expected in this case as well. Note,
however, that this entails a rather non-trivial quantization of the system:
the Minkowski coordinates, $X^{\mu}$, are first mapped to a stereographic
coordinates on the null sphere, $w$; subsequently, the $w$-plane is mapped to
the cylinder by $\zeta={\rm exp}(2\pi w/L)$. Identifying $\zeta=T+ix$ and
analytically continuing $T\to iT_{L}$ yields the Lorentzian system (with
Lorentzian time $T_{L}$) for which the phases are expected to exist for
different conjugacy classes of the SL$(2,C)$-transformations. It will be very
interesting to explore this aspect as well as its connection with scattering
matrix physics in detail, which we leave for future.

##  6 Acknowledgements

We would like to thank Bobby Ezhuthachan, Akavoor Manu, Masahiro Nozaki and
Koushik Ray for discussions. The work of S.R.D. is supported by a National
Science Foundation grant NSF-PHY/211673 and Jack and Linda Gill Chair
Professorship. S.R.D. would like to thank Yukawa Institute of Theoretical
Physics and Isaac Newton Institute for Mathematical Physics for hospitality
during the completion of this work. KS thanks DST for support through SERB
project JCB/2021/000030. AK is partially supported by CEFIPRA $6304-3$, DAE-
BRNS $58/14/12/2021$-BRNS and CRG$/2021/004539$ of Govt. of India. D.D.
acknowledges support by the Max Planck Partner Group grant MAXPLA/PHY/2018577
and from MATRICS grant SERB/PHY/2020334.

##  Appendix A Transformation of the generators

In this section, we use infinitesimal coordinate transformations to obtain
expressions of the generators in the deformed frame as eluded to in the main
text. To this end, we recollect from the main text that the transformation of
the generators $D$, $K_{0}$ and $P_{0}$ can be easily obtained as follows

$\displaystyle U^{-1}\sigma_{z}U$ $\displaystyle=$
$\displaystyle\tau_{z}=\cosh\theta\sigma_{z}+\sinh\theta i\sigma_{x}$
$\displaystyle U^{-1}\sigma_{x}U$ $\displaystyle=$
$\displaystyle\tau_{x}=\cosh\theta\sigma_{x}-i\sigma_{z}\sinh\theta,\quad\tau_{y}=\sigma_{y}$
(104)

where $U=\exp[-i\theta(K_{0}-P_{0})/2]$ as defined in the main text and the
last relation follows from the fact that $U$ commutes with $\sigma_{y}$. Eq.
104 therefore species the relations between the generators $D$, $K_{t}$ and
$P_{t}$ and their deformed versions $D^{\prime}$, $K^{\prime}_{0}$ and
$P^{\prime}_{0}$. These are given by

$\displaystyle D^{\prime}$ $\displaystyle=$ $\displaystyle
D\cosh\theta-\frac{1}{2}(K_{0}+P_{0})\sinh\theta$ $\displaystyle
K^{\prime}_{0}[P^{\prime}_{0}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{0}[P_{0}]-\frac{1}{2}(1-\cosh\theta)P_{0}[K_{0}]-D\sinh\theta$
(105)

where we have used Eq. 9 to obtain the generators from the Pauli matrices.

To construct rest of the transformed generators, we need to use their
coordinate representation. For this, it is convenient to first consider an
infinitesimal transformation $U_{\rm inf}=1-i\theta(K_{0}-P_{0})/2$. The
transformation of coordinates $(\tau,x,y,z)\equiv(\tau,x_{j})$ under the
action of $U_{\rm inf}$ can be computed and yields

$\displaystyle\tau^{\prime}$ $\displaystyle=$ $\displaystyle
U\tau=\tau-\frac{\theta}{2}(1+r^{2}-2\tau^{2}),\quad
x^{\prime}_{j}=Ux_{j}=x_{j}(1+\theta\tau)$ (106)

The reverse transformation expressing $(\tau,x_{j})$ in terms of
$(\tau^{\prime},x^{\prime}_{j})$ can be obtained from Eq. 106 and yields

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\tau^{\prime}+\frac{\theta}{2}(1+(r^{\prime})^{2}-2\tau^{\prime
2}),\quad x_{j}=x^{\prime}_{j}(1-\theta\tau^{\prime})$ (107)

We now use the transformed coordinates to write the expressions of the
generators. For example, one finds

$\displaystyle P^{\prime}_{0}$ $\displaystyle=$
$\displaystyle-i\partial_{\tau^{\prime}}=-i\partial_{\tau}+i\theta(\tau\partial_{\tau}+\sum_{j}x_{j}\partial_{j})=P_{0}-\theta
D$ (108)

where we have used Eq. 107. Note that this coincides with the infinitesimal
version of Eq. 105 as expected; this can also be explicitly checked for $D$
and $K_{0}$.

Next we find the other generators where the use of the transformed coordinates
provides us the infinitesimal version of the generators in the deformed frame.
For example, one finds

$\displaystyle P^{\prime}_{j}$ $\displaystyle=$
$\displaystyle-i\partial_{x^{\prime}_{j}}=-i(\partial_{x_{j}}+\theta(x_{j}\partial_{\tau}-\tau\partial_{x_{j}}))=P_{j}+\theta
L_{0j}$ (109) $\displaystyle K^{\prime}_{j}$ $\displaystyle=$
$\displaystyle-i(x^{\prime}_{j}(\tau^{\prime}\partial_{\tau^{\prime}}+\sum_{j}x^{\prime}_{j}\partial_{x^{\prime}_{j}})-(r^{\prime})^{2}\partial_{x^{\prime}_{j}})$
(110) $\displaystyle=$
$\displaystyle-i(x_{j}(\tau\partial_{\tau}+\sum_{j}x_{j}\partial_{x_{j}})-r^{2}\partial_{x_{j}})-i\theta(\tau\partial_{x_{j}}-x_{j}\partial_{\tau})=K_{j}-\theta
L_{0j}$

where $L_{\mu\nu}=i(x_{\mu}\partial_{x_{\nu}}-x_{\nu}\partial_{x_{\mu}})$ are
the angular momentum generators. A simlar analysis yields

$\displaystyle L^{\prime}_{0j}$ $\displaystyle=$ $\displaystyle
L_{0j}+\frac{\theta}{2}(P_{j}-K_{j}),\quad L^{\prime}_{ij}=L_{ij}.$ (111)

It can be easily checked that the generators obtained in Eqs. 109, 110, and
111, together with the infinitesimal version of $D$, $K_{0}$ and $P_{0}$
obtained from Eq. 105, satisfy the conformal algebra to ${\rm O}(\theta)$.

Next, we look for the transformed generators for finite conformal
transformation. Instead of a direct calculation via coordinate transformations
which is cumbersome, we use the results obtained for infinitesimal
transformations and the criteria that these generators must satisfy the
conformal algebra. It turns out that these two criteria can uniquely specify
the deformed generators. To see this, we first consider $P_{j}$. Eq. 109
dictates the infinitesimal version of $P^{\prime}_{j}$; taking cue from this
and Eq. 105, we write

$\displaystyle P^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)P_{j}+\frac{1}{2}(1-\cosh\theta)K_{j}+L_{0j}\sinh\theta$
(112)

The first check to ensure that this is the correct form is to verify the
commutator $[D^{\prime},P_{j}]=iP^{\prime}_{j}$. Written explicitly, using
standard commutation relation of conformal generators, this yields

$\displaystyle[D^{\prime},P^{\prime}_{j}]$ $\displaystyle=$ $\displaystyle
i\Big{(}\frac{\cosh\theta(1+\cosh\theta)}{2}P_{j}-\frac{\cosh\theta(1-\cosh\theta)}{2}K_{j}-\frac{1}{2}\sinh^{2}\theta(P_{j}+K_{j})$
(113) $\displaystyle+\sinh\theta L_{0j}\Big{)}=iP^{\prime}_{j}$

Next, we need to check the commutation relation between $P^{\prime}_{j}$ and
$K^{\prime}_{j}$. For this one needs to know $K^{\prime}_{j}$. Using Eq. 110
and taking cue from the form of $P^{\prime}_{j}$, we write

$\displaystyle K^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{j}+\frac{1}{2}(1-\cosh\theta)P_{j}-L_{0j}\sinh\theta$
(114)

This form ensures $[D^{\prime},K^{\prime}_{j}]=-iK^{\prime}_{j}$ and this can
be checked in a similar manner. The commutation of $K^{\prime}_{j}$ and
$P^{\prime}_{j}$ can be computed to be

$\displaystyle[K^{\prime}_{j},P^{\prime}_{j}]$ $\displaystyle=$ $\displaystyle
2i\left(\frac{(1+\cosh\theta)^{2}-(1-\cosh\theta)^{2}}{4}D-\sinh\theta(K_{0}+P_{0})\right)=2iD^{\prime}$
(115)

The commutation of $P^{\prime}_{0}$ and $K^{\prime}_{j}$ can also be checked
and yields

$\displaystyle[K^{\prime}_{j},P^{\prime}_{0}]$ $\displaystyle=$ $\displaystyle
2i\left(\frac{(1+\cosh\theta)^{2}-(1-\cosh\theta)^{2}}{4}L_{0j}+\frac{1}{2}\sinh\theta
D(P_{j}-K_{j})\right)$ (116) $\displaystyle=$ $\displaystyle
2i\left(\cosh\theta\,L_{0j}+\sinh\theta(P_{j}-K_{j})/2\right)=2iL^{\prime}_{0j}$

Note that the form of
$L^{\prime}_{0j}=\cosh\theta\,L_{0j}+\sinh\theta(P_{j}-K_{j})/2$ so obtained
is consistent with Eq. 111. Further it can be checked that
$[D^{\prime},L^{\prime}_{0j}]=0$. The other commutation relations involving
$L^{\prime}_{ij}$ and $L^{\prime}_{0j}$ also holds provided we set
$L^{\prime}_{ij}=L_{ij}$. Thus the final expressions of the deformed
generators, used in the main text, are given by

$\displaystyle D^{\prime}$ $\displaystyle=$ $\displaystyle
D\cosh\theta-\frac{1}{2}(K_{0}+P_{0})\sinh\theta$ $\displaystyle
K^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{0}-\frac{1}{2}(1-\cosh\theta)P_{0}-D\sinh\theta$
$\displaystyle P^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)P_{0}-\frac{1}{2}(1-\cosh\theta)K_{0}-D\sinh\theta$
$\displaystyle K^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)K_{j}+\frac{1}{2}(1-\cosh\theta)P_{j}-L_{0j}\sinh\theta$
$\displaystyle P^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cosh\theta)P_{j}+\frac{1}{2}(1-\cosh\theta)K_{j}+L_{0j}\sinh\theta$
$\displaystyle L^{\prime}_{0j}$ $\displaystyle=$
$\displaystyle\cosh\theta\,L_{0j}+\frac{1}{2}\sinh\theta(P_{j}-K_{j}),\quad
L^{\prime}_{ij}=L_{ij}$ (117)

Thus Eq. 117 provides a complete description of the deformed CFT in terms of
the new generators.

##  Appendix B Conformal Algebra in $d$-dimensions

In this appendix, for completeness, we briefly collect the well-known
conformal algebra and the corresponding generators. We will closely follow the
notations of [34]. The conformal algebra in $R^{d}$ is given by the
SO$(d+1,1)$ algebra, with the following commutation relations:

$\displaystyle[D,P_{\mu}]=iP_{\mu}\ ,[D,K_{\mu}]=-iK_{\mu}\
,[K_{\mu},P_{\nu}]=2i\left(\delta_{\mu\nu}D-M_{\mu\nu}\right)\ ,$ (118)
$\displaystyle[M_{\mu\nu},P_{\alpha}]=i(\delta_{\nu\alpha}P_{\mu}-\delta_{\mu\alpha}P_{\nu})\
,[M_{\mu\nu},K_{\alpha}]=i(\delta_{\nu\alpha}K_{\mu}-\delta_{\mu\alpha}K_{\nu})\
,$ (119)
$\displaystyle[M_{\alpha\beta},M_{\mu\nu}]=i(\delta_{\alpha\mu}M_{\beta\nu}+\delta_{\beta\nu}M_{\alpha\mu}-\delta_{\beta\mu}M_{\alpha\nu}-\delta_{\alpha\nu}M_{\beta\mu})\
.$ (120)

The differential operator representation of this algebra is given by

$\displaystyle P_{\mu}=-i\partial_{\mu}\ ,D=-ix^{\mu}\partial_{\mu}\
,M_{\mu\nu}=-i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\ ,$ (121)
$\displaystyle K_{\mu}=-2ix_{\mu}x^{\nu}\partial_{\nu}+ix^{2}\partial_{\mu}\
.$ (122)

The corresponding Euclidean AdS spacetime can be described by the hyperboloid
in $R^{d+1,1}$:

$\displaystyle-(X^{0})^{2}+(X^{1})^{2}+\ldots(X^{d+1})^{2}=-L^{2}\ .$ (123)

In the global patch, the AdS metric is given by

$\displaystyle
ds^{2}=L^{2}\left(\cosh^{2}\rho\,\,d\tau^{2}+d\rho^{2}+\sinh^{2}\rho\,\,dY_{d-1}^{2}\right)\
,\quad Y\cdot Y=1\ .$ (124)

The SO$(d+1,1)$ algebra is generated by the following Killing vectors in
global coordinate:

$\displaystyle D$
$\displaystyle=-i\partial_{\tau},\,\,\,M_{\mu\nu}=-i\bigg{(}Y_{\mu}\tfrac{\partial}{\partial
Y^{\nu}}-Y_{\nu}\tfrac{\partial}{\partial Y^{\mu}}\bigg{)},$ $\displaystyle
P_{\mu}$
$\displaystyle=-ie^{-\tau}\bigg{[}Y_{\mu}(\partial_{\rho}+\tanh\rho\,\,\partial_{\tau})+\frac{1}{\tanh\rho}\nabla_{\mu}\bigg{]},\,\,\,\nabla_{\mu}=\tfrac{\partial}{\partial
Y^{\mu}}-Y_{\mu}Y^{\nu}\tfrac{\partial}{\partial Y^{\nu}},$ $\displaystyle
K_{\mu}$
$\displaystyle=-ie^{\tau}\bigg{[}Y_{\mu}(-\partial_{\rho}+\tanh\rho\,\,\partial_{\tau})-\frac{1}{\tanh\rho}\nabla_{\mu}\bigg{]}.$
(125)

One can check that on the boundary $\rho=\infty$ these Killing vectors
generate via Eq.(2) the Lüscher-Mack term $K_{\mu}+P_{\mu}$.

##  Appendix C SL$(2,C)$, Quasi-primaries, Lorentz symmetry

In this appendix we collect some relevant formulae and basic statements
regarding the SL$(2,C)$ action of the Lorentz group in $(3+1)$-dimensions. We
will closely follow [35, 36, 37, 38]. It is well-known that given a four-
vector $X^{\mu}$, we can associate a hermitian matrix:

$\displaystyle X=\begin{bmatrix}X^{0}-X^{3}&X^{1}+iX^{2}\\\
X^{1}-iX^{2}&X^{0}+X^{3}\\\ \end{bmatrix}\ ,\quad{\rm det}(X)=-X^{\mu}X_{\mu}\
.$ (126)

Now, an SL$(2,C)$ matrix $\Theta$ acts on the hermitian matrix $X$ as:

$\displaystyle X^{\prime}=\Theta X\Theta^{\dagger}\
,\quad\Theta=\begin{bmatrix}a&b\\\ c&d\\\ \end{bmatrix}\ ,\quad{\rm with}\quad
ad-bc=1\ ,$ (127)

such that ${\rm det}(X^{\prime})={\rm det}(X)$.666In $(2+1)$-dimensions a
similar description holds for a real-symmetric matrix[38]:  $\displaystyle
X=\begin{bmatrix}X^{0}-X^{2}&X^{1}\\\ X^{1}&X^{0}+X^{2}\\\ \end{bmatrix}\
,\quad X^{\prime}=\Lambda X\Lambda^{T}\ ,\quad\Lambda\in SL(2,R)\ .$ (128)

A more detailed action can be obtained by noting that the Lorentz group in
$(3+1)$-dimensions acts as the global conformal group on the celestial sphere
at infinity. More explicitly, given the Minkowski coordinates $X^{\mu}$,
$\mu=0,\ldots 3$, the celestial sphere is defined by
$\eta_{\mu\nu}X^{\mu}X^{\mu}=0$, on which we can define:
$w=(X^{1}+iX^{2})/(X^{0}+iX^{3})$. Under a Lorentz transformation:
$X^{\prime\mu}=\Lambda_{\nu}^{\mu}X^{\nu}$, we get:
$w^{\prime}=(aw+b)/(cw+d)$, where $ad-bc=1$ and $a,b,c,d$ are complex-valued.
Subsequently, the $(3+1)$-dimensional scattering amplitudes can be expressed
as celestial CFT correlators.

For an explicit realization, [35, 36, 37] has defined quasi-primary fields
under this SL$(2,C)$ in the principal series, with a conformal dimension
$\Delta=1+is$, where $s$ is real-valued.777In $(d+1)$-dimensions, the
corresponding conformal dimensions are $\Delta=\frac{d-1}{2}+is$.  Moreover,
there is a precise state-operator correspondence as well. The states are
described by $\left|h,\bar{h},w,\bar{w}\right\rangle$ where $\\{w,\bar{w}\\}$
are the stereographic coordinates of the celestial sphere. Lorentz
transformation, denoted by $U(\Lambda)$ acts on these states as follows:

$\displaystyle
U(\Lambda)\left|h,\bar{h},w,\bar{w}\right\rangle=\frac{1}{(cw+d)^{h}}\frac{1}{(\bar{c}\bar{w}+\bar{d})^{\bar{h}}}\left|h,\bar{h},\frac{aw+b}{cw+d},\frac{\bar{a}\bar{w}+\bar{b}}{\bar{c}\bar{w}+\bar{d}}\right\rangle\
,$ (129) $\displaystyle\Lambda\in SL(2,C)\ ,\quad h=\frac{1+is-\sigma}{2}\
,\quad\bar{h}=\frac{1+is+\sigma}{2}\ ,$ (130)

where $\sigma$ denotes the helicity of the massless particle.

The corresponding basis quasi-primary states are given by

$\displaystyle\left|s,\sigma,w=0=\bar{w}\right\rangle=\frac{1}{\sqrt{8\pi^{4}}}\int_{0}^{\infty}dEE^{is}\left|E,0,0,E,\sigma\right\rangle\
,$ (131)

such that the states are normalized as follows:

$\displaystyle\left\langle
s_{1},\sigma_{1},w_{1},\bar{w}_{1}|s_{2},\sigma_{2},w_{2},\bar{w}_{2}\right\rangle=\delta\left(s_{1}-s_{2}\right)\delta^{(2)}\left(w_{1}-w_{2}\right)\delta_{\sigma_{1}\sigma_{2}}\
,$ (132)

where

$\displaystyle\left|s,\sigma,w,\bar{w}\right\rangle=\left(\frac{1}{1+w\bar{w}}\right)^{1+is}U\left(R(w,\bar{w})\right)\left|s,\sigma,0,0\right\rangle\
,$ (133)

in which $U\left(R(w,\bar{w})\right)$ are the unitary rotation operators. The
action of the Lorentz group on these states then yields equation (129), which
essentially is comprised of SL$(2,C)$ matrix multiplications. The
corresponding operators, denoted by ${\cal O}_{h,\bar{h}}$ will evolve in the
Heisenberg picture according to:

$\displaystyle U(\Lambda){\cal
O}_{h,\bar{h}}\left(w,\bar{w}\right)U(\Lambda)^{\dagger}=\left(\frac{\partial\Lambda
w}{\partial
w}\right)^{h}\left(\frac{\partial\Lambda\bar{w}}{\partial\bar{w}}\right)^{\bar{h}}{\cal
O}_{h,\bar{h}}\left(\Lambda w,\Lambda\bar{w}\right)\ .$ (134)

A very similar description exists in $(2+1)$-dimensions as well, we refer the
interested Reader to e.g. [38] for more details on this.

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