#  Out-of-time-order correlation and detection of phase structure in Floquet
transverse Ising spin system

Rohit Kumar Shukla<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology (Banaras Hindu University), Varanasi - 221005,
India  Gautam Kamalakar Naik<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi -
221005, India  Sunil Kumar Mishra<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi -
221005, India

###### Abstract

We study the out-of-time-order correlation (OTOC) of the Floquet transverse
Ising model and use it to verify the phase diagram of the system. First, we
present the exact analytical solution of the transverse magnetization OTOC
using the Jorden-Wigner transformation. In order to get the phase structure of
the Floquet transverse Ising system, we use the longitudinal magnetization
OTOC as it is known to serve as an order parameter of the system. We show the
phase structure numerically in the transverse Ising Floquet system by using
the long time average of the longitudinal magnetization OTOC. In both the open
and the closed chain systems, we find distinct phases out of which two are
paramagnetic (0-paramagnetic and $\pi$-paramagnetic), and two are
ferromagnetic (0-ferromagnetic and $\pi$-ferromagnetic) as defined in the
literature.

##  I Introduction

In the last two decades, out-of-time-order correlation (OTOC) has gained a lot
of attention among the researchers of various fields. One field of interest is
the but- terfly effects in quantum chaotic systems Ray  _et al._ (2018); Hosur
_et al._ (2016); Gu and Qi (2016); Ling  _et al._ (2017). Other directions are
quantum information scrambling Bohrdt  _et al._ (2017); Swingle and Yao
(2017); Swingle  _et al._ (2016); Swingle and Chowdhury (2017); Schleier-Smith
(2017); Pappalardi  _et al._ (2018); Klug  _et al._ (2018); Khemani  _et al._
(2018); Hosur  _et al._ (2016); Alavirad and Lavasani (2019) and many-body
localization Maldacena  _et al._ (2016). The nontrivial OTOC as a holographic
tool has been instrumental in determining the interplay of scrambling and
entanglement Shenker and Stanford (2014); Roberts  _et al._ (2015). Many
experiments have been done to measure OTOCs in various systems, e. g.,
trapped-ion quantum magnets Gärttner  _et al._ (2017) and Nuclear Magnetic
resonance quantum simulator Li  _et al._ (2017).

In addition to the above fields of interest, the OTOCs are useful in
determining phases of the quantum critical systems Shen  _et al._ (2017); Sun
_et al._ (2018); Heyl  _et al._ (2018). The phase structures of quantum
critical systems have been studied extensively in the last few decades Shen
_et al._ (2017); Heyl  _et al._ (2013); Pollmann  _et al._ (2010); von
Keyserlingk and Sondhi (2016a); Thakurathi  _et al._ (2013); Turner  _et al._
(2011); von Keyserlingk and Sondhi (2016b); Feng  _et al._ (2007); Bastidas
_et al._ (2012); Jiang  _et al._ (2011); Sun  _et al._ (2018); Fidkowski and
Kitaev (2011); Khemani  _et al._ (2016); Thakurathi  _et al._ (2014). One of
the simplest models to display and analyze the quantum phase transition is one
dimensional transverse Ising model which is given by Hamiltonian
$H=J\sum_{i}\sigma_{i}^{x}\sigma_{i+1}^{x}+h\sum_{i}\sigma_{i}^{z}$. This
system undergoes a phase transition at $J=h$ from the ferromagnetic state
($J>h$) to the paramagnetic phase ($J<h$) Chakrabarti  _et al._ (2008); Heyl
_et al._ (2018); Su  _et al._ (2006); Sun and Chen (2009). Such phase
transitions in time-independent equilibrium systems have been well studied
over the years. In the last few years, the OTOC has emerged as a tool to
detect equilibrium and dynamical quantum phase transitions in the transverse
field Ising (TFI) model and the Lipkin-Meshkov-Glick model (LMG) (Heyl  _et
al._ , 2018).

It has been shown that the OTOC of the ground states and quenched states can
diagnose the QPTs and dynamical phase transition, respectively Heyl  _et al._
(2018). The ferromagnetic ($J>h$)and paramagnetic ($J<h$) phases of the
transverse Ising model can be characterized by nonzero and zero long time
average OTOC, respectivelyHeyl  _et al._ (2018). The periodically driven
interacting quantum system known as the Floquet system, on the other hand,
poses a different problem: one would expect generic Floquet systems to heat to
infinite temperatures. However, specific cases of nonergodic phases with
localization have been observed in Floquet systems Parameswaran  _et al._
(2017); Zhang  _et al._ (2016). In these systems, multiple nonergodic phases
with differing forms of dynamics and ordering have been observed Khemani  _et
al._ (2016).

These multiple phases are characterized by broken symmetries and topological
order. In the case of transverse Ising Floquet systems, Majorana modes are
produced at the ends of the long chain Thakurathi  _et al._ (2013). The zero-
energy Majorana mode corresponds to the long range ferromagnetic order while
the nonzero-energy Majorana mode corresponds to the paramagnetic phase
Thakurathi  _et al._ (2013); von Keyserlingk and Sondhi (2016a). The sharp
phase boundaries of the Floquet Ising system are explained by symmetry
protected Ising order Khemani  _et al._ (2016); von Keyserlingk and Sondhi
(2016b). For binary Floquet drive, two paramagnetic and two ferromagnetic
phases can be seen in the phase diagram. The two paramagnetic/ferromagnetic
phases are distinguished by the combined eigenvalues of the Floquet drives and
the parity operators. On the basis of the combined eigenvalues, the
paramagnetic region is divided into two parts: $0$ and $0\pi$-paramagnetic and
ferromagnetic region is also divided into two parts: 0 and $\pi$-ferromagnetic
[FIG. 1]. In the ferromagnetic region, all of the eigenstates have long range
Ising symmetry broken order. However, in the paramagnetic phase, all of the
eigenstates have long range symmetric order.

Figure 1: Phase structure of the Floquet system with Floquet map given by eq.
1 Khemani  _et al._ (2016); von Keyserlingk and Sondhi (2016a, b). There are
four distinct phases in the $\tau_{0}$-$\tau_{1}$ parameter space. Two of
these phases, the $\pi$ ferromagnetic and the $0\pi$ paramagnetic, and phases
which are unique to Floquet systems.

We will explore the OTOCs as a detector of phase transition in the Floquet
Ising spin system. First, we will consider transverse magnetization out-of-
time-order correlation(TMOTOC) and calculate the exact solution using the
Jordan Wigner transformation by mapping the spin operators onto the fermionic
annihilation and creation operators. Next, we will consider the longitudinal
magnetization out-of-time-order correlation(LMOTOC) and explore the various
phases in the Fouquet Ising spin system. Our motivation is to find the phase
structure of the Floquet system using a long time average of LMOTOC.

This manuscript is organized as follows: In section II, we will discuss the
model of the Floquet system. In section III, we will define the longitudinal
and transverse magnetization OTOCs. We will introduce the time average of the
LMOTOC for the detection of phase structures. In section III, we will discuss
the various phases of the Floquet Ising system using a long time averaged
LMOTOC. Subsequently, in section V, we conclude the results.

##  II Model

We consider an integrable Floquet transverse Ising system with binary Floquet
drives. The Floquet map corresponding to this system is

$U=e^{-iH_{xx}\tau_{1}}e^{-iH_{z}\tau_{0}},$ (1)

where $H_{xx}$ is the nearest neighbor Ising interaction given by
$H_{xx}=\sum_{l=1}^{N-1}\sigma_{l}^{x}\sigma_{l+1}^{x}$ for open chain system
and $H_{xx}=\sum_{l=1}^{N}\sigma_{l}^{x}\sigma_{l+1}^{x}$ for closed chain
system with $\sigma_{N+1}^{x}=\sigma_{1}^{x}$.
$H_{z}=\sum_{l=1}^{N}\sigma_{l}^{z}$ is the transverse field in z-direction.
$\tau_{0}$ and $\tau_{1}$ are the time periods. The Hamiltonian corresponding
to the above Floquet operator is:

$\displaystyle
H(t)=H_{xx}+\sum_{n=-\infty}^{\infty}\delta\Big{(}n-\frac{t}{\tau_{1}}\Big{)}\frac{\tau_{0}}{\tau_{1}}H_{z}.$
(2)

##  III  Out-of-time-order Correlation

The out of time order correlation (OTOC) is, in general, defined as
$F(t)=\langle W(t)VW(t)V\rangle$ where $V$ and $W$ are two local Hermitian
operators and $W(t)$ is the Heisenberg evolution of the operator $W$ by time
$t$. If we consider the local operator $V=W=\sigma_{l}^{z}$, the spin operator
perpendicular to the Ising axis (x-axis), we get TMOTOC, which is given by:

$\displaystyle F_{z}(n)$ $\displaystyle=$
$\displaystyle\langle\phi_{0}|\sigma_{l}^{z}(n)\sigma_{l}^{z}\sigma_{l}^{z}(n)\sigma_{l}^{z}|\phi_{0}\rangle,$
(3)

where $l$ can take any value between $1$ to $N$ for closed spin chain. The
time evolution of the spin operator at the position $l$ after $n$ kicks is
defined as $S_{l}^{z}(n)=U^{\dagger n}S_{l}^{z}U^{n}$. In this case we will
start with the initial state

$|\phi_{0}\rangle=|\uparrow\uparrow\uparrow\cdots\uparrow\rangle,$ (4)

where, $\left|\uparrow\right\rangle$ is the eigenstate of $\sigma^{z}$ with
eigenvalue $+1$. Let’s consider the periodic boundary condition and even
number of spins and calculate the Floquet map given by eq. 1 as;

$\displaystyle U$ $\displaystyle=$
$\displaystyle\exp\Big{[}-it_{1}\sum_{l=1}^{N}S_{l}^{x}S_{l+1}^{x}\Big{]}\exp\Big{[}-it_{0}\sum_{l=1}^{N}S_{l}^{z}\Big{]},$
(5)

where, $t_{0}=2\tau_{0}$, $t_{1}=4\tau_{1}$ and $\sigma_{l}^{x}=2S_{l}^{x}$
gives the same unitary operator as defined in eq. 1. We calculate the
analytical expression for the TMOTOC using the Jorden-Wigner transformation
(for detailed calculation refer to Appendix A):

$\displaystyle F_{z}(n)$ $\displaystyle=$ $\displaystyle
1-\Big{(}\frac{2}{N}\Big{)}^{3}\sum_{p,q,r}\Big{(}|\Psi_{r}(n)|^{2}\Phi_{p}(n)^{*}\Phi_{q}(n)-\Psi_{-p}(n)$
(6) $\displaystyle\times$
$\displaystyle\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{q}(n)-\Psi_{q}(n)\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{-r}(n)$
$\displaystyle+$
$\displaystyle\Psi_{q}(n)\Psi_{r}(n)^{*}|\Phi_{p}(n)|^{2}\Big{)},$

where the expansion coefficients $\Phi_{q}(n)$ and $\Psi_{q}(n)$ are defined
as

$\Phi_{q}(n)=|\alpha_{+}(q)|^{2}e^{-it\gamma_{q}}+|\alpha_{-}(q)|^{2}e^{it\gamma_{q}},$
(7)
$\Psi_{q}(n)=\alpha_{+}(q)\beta_{+}(q)e^{-in\gamma_{q}}+\alpha_{-}(q)\beta_{-}(q)e^{in\gamma_{q}}.$
(8)

The phase angle $\gamma_{q}$ and the coefficients $\alpha_{\pm}(q)$ and
$\beta_{\pm}(q)$ are given by

$\cos(\gamma_{q})=\cos(t_{0})\cos(t_{1})-\cos(q)\sin(t_{0})\sin\Big{(}\frac{t_{1}}{2}\Big{)},$
(9)

and

$\alpha_{\pm}(q)^{-1}=\sqrt{1+\Big{(}\frac{\cos(\frac{t_{1}}{2})-cos(\gamma_{q}\pm
t_{0})}{\sin(q)\sin(t_{0})\sin(\frac{t_{1}}{2})}\Big{)}^{2}},$ (10)
$\displaystyle\beta_{\pm}(q)=\alpha_{\pm}(q)\frac{\mp\sin(\gamma_{q})-\cos(t_{0})\cos(q)\sin(\frac{t_{1}}{2})-\sin(t_{0})\cos(\frac{t_{1}}{2})}{\sin(q)\sin(\frac{t_{1}}{2})}e^{-it_{0}}.$
(11)

The allowed value of $p$, $q$ and $r$ are from $\frac{-(L-1)\pi}{L}$ to
$\frac{(L-1)\pi}{L}$ differing by $\frac{2\pi}{L}$ for even number of $N_{F}$
($N_{F}=c_{l}^{\dagger}c_{l}$, Number of Fermion).

Figure 2: $F_{z}(n)$ for closed chain transverse Ising Floquet system of
system size $N=12$ by using the Numerical calculations (solid line) and using
the formula (point). Here we take $\tau_{0}=\tau_{1}=\epsilon$. Figure 3: (a)
Variation of the real part of (a) TMOTOC and (b) LMOTOC with number of Floquet
periods for a fixed $\tau_{0}=\epsilon$ and $\tau_{1}=\epsilon,$ $2\epsilon$
and $3\epsilon$ in closed chain Floquet systems with system size N=12. In
TMOTOC, we take the initial state as a direct product of the eigenstate of
$\sigma^{z}$ with eigenvalue +1. In LMOTOC, the initial state is the direct
product of the eigenstate of $\sigma^{x}$ with eigenvalue +1. Oscillation of
the mean value of TMOTOC always remains positive value however in LMOTOC have
both positive and zero value which depends on the constant value of $\tau_{0}$
and $\tau_{1}$. In the inset, there is the variation of TMOTOC with the number
of Floquet periods for a fixed $\tau_{1}=\frac{\pi}{24}$ and
$\tau_{0}=\epsilon,$ $2\epsilon$ and $3\epsilon$ in closed chain Floquet
systems with system size N=50.

The values of $F_{z}(n)$ obtained by the analytical expression in eq. 6 match
the values obtained numerically by exact diagonalization. For instance, for
$\tau_{0}=\tau_{1}=\epsilon$, where $\epsilon=\frac{\pi}{28}$, we have shown
the analytical and numerical results in FIG. 2. The advantage of having an
easily computable formula such as eq. (6) is that we can study the TMOTOC as a
function of Floquet periods $\tau_{0}$ and $\tau_{1}$ and see the behavior at
any number of kicks. The analytical expression is of $O(L^{3})$ which has a
significant advantage over exact diagonalization calculations of O($2^{L}$).
Inset of FIG. 3(a) shows the TMOTOC with time (kicks) using eq. 6 for system
size $N=50$ with a periodic boundary condition, where we fix $\tau_{1}=\pi/24$
and $\tau_{0}=\epsilon,2\epsilon$ and $3\epsilon$.

The LMOTOCs have been shown to be useful in detecting the phase transitions
between the paramagnetic and ferromagnetic phases in spin systems Heyl  _et
al._ (2018). However, the same cannot be said about TMOTOC. A comparison of
the behavior of the two quantities with time is shown in FIG. 3. We see from
FIG. 3(a) that TMOTOCs always oscillate about a positive value for all the
pairs of $\tau_{0}$ and $\tau_{1}$, signaling that the long time average of
TMOTOC is always a positive quantity. However, in the case of LMOTOCs, as
shown in FIG. 3(b), we find that the long time average value can be either
zero or a positive quantity. In order to detect the phase structure of the
system, we require the order parameter characterising the distinct phases to
show a sharp contrast between the phases. We see that the LMOTOCs qualify the
criterion to be used as an order parameter, but TMOTOCs fail to do so.

Upon performing a quantum quench from a polarized state, we will use the
saturation value of the LMOTOC as the order parameter to distinguish between
the two phases. It is calculated by numerical methods because it is not
beneficial to calculate it analytically Essler  _et al._ (2012).

Figure 4: Variation of the real part of LMOTOC ($F_{x}(n)$) with number of
Floquet periods for a fixed $\tau_{0}=\epsilon$ and multiple values of
$\tau_{1}$ in (a) closed and (b) open chain Floquet systems with system size
N=12. The initial state is direct product of eigenstate of $\sigma^{x}$ with
eigenvalue $+1$. At $\tau_{1}=0$, $F_{x}(n)$ shows periodic oscillations about
zero. As the value of $\tau_{1}$ is increased, the $F_{x}(n)$ oscillates about
greater non-zero values with lower amplitudes of oscillation and at
$\tau_{1}=\frac{\pi}{4}$, it saturates to the value $1$. As increase constant
value of $\tau_{1}=0$, $F_{x}(n)$ has same and different value as
$\frac{\pi}{2}-\tau_{1}=0$ in closed and open chain respectively.

Considering $V=W=\sigma_{l}^{x}$, the spin operator along the Ising axis, we
get the LMOTOC defined as:

$F_{x}(n)=\langle\psi_{0}|\sigma_{l}^{x}(n)\sigma_{l}^{x}\sigma_{l}^{x}(n)\sigma_{l}^{x}|\psi_{0}\rangle,$
(12)

We study the phase structure of the system [eq. 1] by calculating the LMOTOC
using the fully polarized initial state

$|\psi_{0}\rangle=|\rightarrow\rightarrow\rightarrow\cdots\rightarrow\rangle,$
(13)

where, $\left|\rightarrow\right\rangle$ is the eigenstate of $\sigma^{x}$ with
eigenvalue $+1$. We consider an even number of spins in this analysis. where
can take any values from $l=1$ to $N$ for closed spin chain and
$l=\frac{N}{2}$ for open chain. $\sigma_{l}^{x}(n)=U^{\dagger
n}\sigma_{l}^{x}U^{n}$ is the Heisenberg evolution of $\sigma_{l}^{x}$ after
$n$ Floquet periods. If the LMOTOC denoted by eq. 12 saturates to a particular
value after a long period of time, this value can be determined by taking the
long time average of the LMOTOC. We define the long time-average of the
LMOTOC, $\overline{F}_{x}(T)$ upto $T$ Floquet periods by:

$\overline{F}_{x}=\frac{1}{T}\sum_{n=1}^{T}F_{x}(n).$ (14)

The long time average of the LMOTOC has a direct link with the spectral
properties of the system in consideration de Mello Koch  _et al._ (2019). The
averaged LMOTOC links with the spectral form factor Dyer and Gur-Ari (2017), a
well-known quantity in random matrix theory, which is a quantifier for
discreteness in the spectrum.

##  IV Phase Structure

Figure 5:  Plot of $\overline{F}_{x}(T)$ with $\tau_{1}$ for values of
$\tau_{0}$ varying from $0$ to $\frac{\pi}{4}$ in intervals of $\epsilon$ in
the (a) closed and (b) open chain Floquet systems of system size $N=10$ and
$T=10^{4}$. The variation of $\overline{F}_{x}(T)$ with $\tau_{1}$ for
$\pi/4<\tau_{0}<\pi/2$ is the same as that for $\frac{\pi}{2}-\tau_{0}$. This
plot can be used to find the regions in the $\tau_{0}$ and $\tau_{1}$
parameter space that have $\overline{F}_{x}(T)>0$ and $\overline{F}_{x}(T)=0$
( FIG. 6). Figure 6: Regions with $\overline{F}_{x}(T)>0$ and
$\overline{F}_{x}(T)=0$ in the $\tau_{0}$ and $\tau_{1}$ parameter space with
$T=10^{4}$, for the (a)closed and (b) open chain Floquet systems of system
size N=6(Green), 8(Blue) and 10(Brown). As increasing system size, critical
lines of both (a) and (b) tend towards diagonal critical lines. Hence, this
suggests that the phase structure of the system would contain two
ferromagnetic regions (where $\overline{F}_{x}>0$) and two paramagnetic
regions (where $\overline{F}_{x}=0$).

Figure 7: Heat map of the logarithmic dominant frequencies of
$F(n)-\overline{F}_{x}(T)$ in the discretized $\tau_{0}$-$\tau_{1}$ parameter
space for the (a) closed and (b) open Floquet system of system size $N=10$ and
with $T=10^{4}$. The heat map shows logarithmically small values close to the
transition lines shown in FIG. 6.

The phase structure of the Floquet system given by eq. 1 is known to have four
distinct phases in the two-dimensional parameter space of $\tau_{0}$ and
$\tau_{1}$. The phase diagram is shown in FIG. 1 Khemani  _et al._ (2016); von
Keyserlingk and Sondhi (2016a, b). Paramagnetic and ferromagnetic phases show
behavior similar to their undriven counterparts. The other two of the phases
observed, the $\pi$-ferromagnetic and $0\pi$-paramagnetic are unique to
Floquet systems and are not observed in undriven non-Floquet systems. The
phase transitions between these phases in the $\tau_{0}$ and $\tau_{1}$
parameter space can be detected by calculating LMOTOCs. The LMOTOC has been
shown to saturate to non-zero values in the ferromagnetic region and to zero
in the paramagnetic region, at long times in the undriven systems Heyl  _et
al._ (2018). Hence, the long time averaged LMOTOC serves as a good order
parameter for paramagnetic and ferromagnetic regions in the undriven systems.
In driven Floquet systems, LMOTOCs do not saturate to non-zero and zero values
for all values of $\tau_{0}$ and $\tau_{1}$; we see a continuing oscillating
behaviour about a non-zero or zero mean value (FIG. 4). The time-averaged
LMOTOC ($\overline{F}_{x}(n)$) is seen to saturate at long times in the
thermodynamic limit to non-zero values in the ferromagnetic and $\pi$
ferromagnetic regions, and to zero in the paramagnetic and $0\pi$ paramagnetic
regions of the phase space. FIG. 5 shows the variation of the long tine-
average of the real part of the LMOTOC with $\tau_{1}$, for different values
of $\tau_{0}$ in the closed and open boundary conditions for a system size
$N=10$.

The critical points, where the phase transition occurs, are identified along
the constant $\tau_{0}$ line at the points where LMOTOC goes from zero to non-
zero. These critical points, when mapped in the $\tau_{0}$ and $\tau_{1}$
parameter space for $N=6,8$ and $10$ give plots as shown in FIG. 6. There
exists a symmetry along $\tau_{1}=\frac{\pi}{4}$, in the closed boundary
condition but this symmetry is absent in the open boundary condition because
in close chain Floquet system behavior of LMOTOC is same for Floquet period
$\tau_{1}$ and $\frac{\pi}{2}-\tau_{1}$, but in case of open chain both are
not the same for long time. However, there is a symmetry about
$\tau_{0}=\frac{\pi}{4}$ in both the open and the closed chain [FIG. 4]. The
tips of the regions with $\overline{F}_{x}=0$ can be seen to be moving closer
to each other with increasing the system size. In the open boundary condition,
the tips which start out in the upper half of the parameter space, also move
downwards towards the point $(\frac{\pi}{4},\frac{\pi}{4})$ with increasing
system size. For system size $N\rightarrow\infty$, we can expect the tips to
meet at the centre, giving the critical lines as shown in FIG. 1, which is the
expected phase diagram for this system. Hence, the time-averaged LMOTOC
[$\overline{F}_{x}(T)$] for large $T$ and $N\rightarrow\infty$ can be used as
an order parameter to distinguish the phases of a driven transverse field
Floquet Ising model. It must be noted that the time-averaged LMOTOC does not
distinguish between the ferromagnetic and the $\pi$ ferromagnetic phase or the
paramagnetic and the $0\pi$ paramagnetic phase. However, these distinct phases
can be identified by observing the combined eigenvalues of the unitary
operator which is defined in eq. 1 and the parity operator
($P=\prod_{l}\sigma_{l}^{z}$) von Keyserlingk and Sondhi (2016a, b).
Considering the operators $U$ and $P$ have eigenvalues $u$ and $p$
respectively, the different phases can be distinguished by observing the
eigenvalues along the outer edges of the phase diagram. The eigenvalues have
protected multiplets of the form: $(u,p)$ in the paramagnetic,
$[(u,p),(u,-p),(-u,p),(-u,-p)]$ in the $0\pi$ paramagnetic, $[(u,p),(u,-p)]$
in the ferromagnetic and $[(u,p),(-u,-p)]$ in the $\pi$ ferromagnetic regions.

The frequencies of oscillations of the LMOTOC also provide a signature of the
phase transitions. Here, the dominant frequencies have been numerically
determined by taking the argument maxima of the discrete Fourier transforms of
the deviation of the LMOTOC from its mean value
($F_{x}(n)-\overline{F}_{x}(T)$). Mathematically, it can be defined as:

$\mathcal{F}(\nu)=\frac{1}{T}\sum_{n=1}^{T}[F_{x}(n)-\overline{F}_{x}(T)]e^{-i\frac{2\pi}{T}\nu
n}$ (15)

A heatmap of the dominant frequencies of $F(\nu)$ in the logarithmic scale is
shown in FIG. 7 for $N=10$ in the closed and open boundary conditions. These
plots show that the dominant frequencies logarithmically drop close to the
critical lines and at the edges. Comparing FIG. 7 and FIG. 6, we observe that
heatmap displays indications of the phase transition in the Floquet Ising
system.

##  V Conclusion

We calculate the exact analytical expression for TMOTOC as a function of
$\tau_{0}$ and $\tau_{1}$. Further, we study the phase structure of the
traverse field Floquet Ising system given by eq. 1 using numerical calculation
of LMOTOC. We use LMOTOC defined in equation eq. 12 to distinguish between the
paramagnetic and ferromagnetic phases of the chosen Floquet system.
ferromagnetic and the $\pi$ ferromag-netic phase or the paramagnetic and the
$0\pi$ paramagnetic phase are distinguished by the combined eigenvalues of $U$
and parity operator $P$. We numerically find the time averaged LMOTOC
[$\overline{F}_{x}(T)$] for the system sizes up to $N=10$ and plot the regions
of the parameter space that have $\overline{F}_{x}(T)=0$ and
$\overline{F}_{x}(T)>0$ for $T=10^{4}$. We observe that the plot showing the
critical lines of phase transition for $N\rightarrow\infty$ tends to the
expected plot FIG. 1.

In the limit $N\rightarrow\infty$, the regions with $\overline{F}_{x}(T)>0$
for large $T$ are ferromagnetic and those with $\overline{F}_{x}(T)=0$ for
large $T$ are paramagnetic. We also see that the dominant frequency of
oscillations in [$F_{x}(n)-\overline{F}_{x}(T)$] logarithmically decreases
close to the critical line and the boundary of phase space for the system
sizes considered.

OTOCs can be experimentally calculated Li  _et al._ (2017), and Floquet
systems can be experimentally realized Allen (2007); Chitsazi  _et al._
(2017). Our study outlines how LMOTOC can be a useful tool to distinguish the
phases of a Floquet system.

##  VI Acknowledge

We would like to thank V. Subrahmanyam for his fruitful suggestion in the
analytical calculation of TMOTOC.

##  Appendix A Calculation of transverse magnetization OTOC

We transform the spin variables to fermionic creation $c_{l}^{\dagger}$ and
annihilation $c_{l}$ operators at site $l$ by using the Jordan-Wigner
transformation Jordan and Wigner (1928)

$S_{l}^{x}=-\frac{1}{2}\prod_{j=1}^{l-1}(2c_{j}^{\dagger}c_{j}-1)(c_{l}^{\dagger}+c_{l})\quad{\rm
and}\quad S_{l}^{z}=c_{l}^{\dagger}c_{l}-\frac{1}{2}.$ (16)

The operators $c_{l}$ and $c_{l}^{\dagger}$ obey the the usual fermion
anticommutation rules. The unitary operator for the closed chain is given as

$\displaystyle U$ $\displaystyle=$
$\displaystyle\exp\Big{[}\frac{-it_{1}}{4}\Big{(}\sum_{l=1}^{N-1}(c_{l}^{\dagger}-c_{l})(c_{l+1}^{\dagger}-c_{l+1})$
(17) $\displaystyle-$
$\displaystyle(-1)^{N_{F}}(c_{N}^{\dagger}-c_{N})(c_{l+1}^{\dagger}-c_{N+1})\Big{)}\Big{]}$
$\displaystyle\times$
$\displaystyle\exp\Big{[}-it_{0}\sum_{l=1}^{N}\big{(}c_{l}^{\dagger}c_{l}-\frac{1}{2}\big{)}\Big{]},$

where $N_{F}=\sum_{l=1}^{L}c_{l}^{\dagger}c_{l}$ is the total number of
fermions. We move in the momentum space using the Fourier transform of $c_{l}$
which is defined as

$c_{q}=\frac{exp(i\frac{\pi}{4})}{\sqrt{N}}\sum_{l=1}^{N}e^{-iql}c_{l}.$ (18)

Hence U can be written as Lakshminarayan and Subrahmanyam (2005)

$U=e^{(-it_{0}\frac{N}{2})}\prod_{q>0}\mathcal{V}_{q}.$ (19)

The operator $\mathcal{V}_{q}$ in the above expression has the form

$\displaystyle\mathcal{V}_{q}$ $\displaystyle=$
$\displaystyle\exp\Big{(}-i\frac{t_{1}}{2}[\cos(q)(c_{q}^{\dagger}c_{q}+c_{-q}^{\dagger}c_{-q})+\sin(q)(c_{q}c_{-q}$
(20) $\displaystyle+$ $\displaystyle
c_{-q}^{\dagger}c_{q}^{\dagger})]\Big{)}\exp\Big{(}-2it_{0}(c_{q}^{\dagger}c_{q}+c_{-q}^{\dagger}c_{-q})\Big{)}.$

For $\mathcal{V}_{q}$, the four basis state are $|0\rangle$ , $|\pm
q\rangle=c_{\pm q}^{\dagger}|0\rangle$,
$|-qq\rangle=c_{-q}^{\dagger}c_{q}^{\dagger}|0\rangle$

The eigenstates of $\mathcal{V}_{q}$ are given by

$\mathcal{V}_{q}|\pm
q\rangle=e^{\big{(}-\frac{t_{1}}{2}cos(q)-it_{0}\big{)}}|\pm q\rangle,$
$\mathcal{V}_{q}|\pm\rangle=e^{\big{(}-\frac{t_{1}}{2}cos(q)-it_{0}}\big{)})e^{\pm
i\gamma_{q}}|\pm\rangle,$ (21)

where

$|\pm\rangle=\alpha_{\pm q}|0\rangle+\beta_{\pm q}|-qq\rangle.$ (22)

In the above equation $\alpha_{\pm q}$ and $\beta_{\pm q}$ are given by eq. 10
and eq. (11) respectively. The initial unentangled state is
$|\psi_{L}(0)\rangle=|0\rangle^{\otimes L}$. In Fock space, it is treated as
vacuum. Time evolution operator of the fermionic annihilation operatoer in the
momentum space is given as

$c_{q}(n)=\mathcal{V}_{q}^{\dagger
n}c_{q}\mathcal{V}_{q}^{n}=\Phi_{q}(n)^{*}c_{q}-\Psi_{q}(n)c_{-q}^{\dagger}.$
(23)

The expansion coefficients $\Phi_{q}(n)$ and $\Psi_{q}(n)$ are defined in eq.
7 and eq. 8, respectively, and phase angle ($\gamma_{q}$) is defined in eq. 9.
Let us apply the first spin operator on the initial state, we get

$S_{l}^{z}(0)|0\rangle=\Big{(}c_{l}^{\dagger}c_{l}-\frac{1}{2}\Big{)}|0\rangle=-\frac{1}{2}|0\rangle.$
(24)

Since, $c_{l}^{\dagger}c_{l}$ is the number operator. The operation of the
number operator on the vaccum gives zero eigenvalue. Time evolution of the
spin operator at position $l$ is

$\displaystyle
S_{l}^{z}(n)=\frac{1}{N}\sum_{a,b}e^{i(a-b)l}c_{a}^{\dagger}(n)c_{b}(n)-\frac{1}{2},$

where $a$ and $b$ are indices in momentum space. By using eq. (23), we can
write

$\displaystyle S_{l}^{z}(n)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{a,b}e^{i(a-b)l}\Big{[}\Phi_{a}(n)c_{a}^{\dagger}-\Psi_{a}(n)^{*}c_{-a}\Big{]}$
$\displaystyle\times$
$\displaystyle\Big{[}\Phi_{b}(n)^{*}c_{b}-\Psi_{b}(n)c_{-b}^{\dagger}\Big{]}-\frac{1}{2},$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{a,b}e^{i(a-b)l}\Big{[}\Phi_{a}(n)\Phi_{b}(n)^{*}c_{a}^{\dagger}c_{b}-\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}$
$\displaystyle-$
$\displaystyle\Psi_{a}(n)\Phi_{b}(n)^{*}c_{-a}c_{b}+\Psi_{a}(n)^{*}\Psi_{b}(n)c_{-a}c_{-b}^{\dagger}\Big{]}-\frac{1}{2},$

Application of time evolved spin operator on the vacuum gives

$\displaystyle S_{l}^{z}(n)|0\rangle$ $\displaystyle=$
$\displaystyle\Big{[}-\frac{1}{N}\sum_{a,b}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}$
(26) $\displaystyle+$
$\displaystyle\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{]}|0\rangle,$

and the Hermitian conjugate of the above equation is

$\displaystyle\langle 0|S_{l}^{z}(n)$ $\displaystyle=$ $\displaystyle\langle
0|\Big{[}-\frac{1}{N}\sum_{p,r}e^{-i(p-r)l}\Phi_{p}(n)^{*}\Psi_{r}(n)^{*}c_{-r}c_{p}$
(27) $\displaystyle+$
$\displaystyle\frac{1}{N}\sum_{p}|\Psi_{p}(n)|^{2}-\frac{1}{2}\Big{]},$

where $p$ and $r$ are indices in the momentum space. We can calculate
$S_{l}^{z}(n)S_{l}^{z}(0)|0\rangle$ as

$\displaystyle S_{l}^{z}(n)S_{l}^{z}(0)|0\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{[}-\frac{1}{N}\sum_{a,b}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}$
(28) $\displaystyle+$
$\displaystyle\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{]}|0\rangle.$

Applying the third spin operator $S_{l}^{z}(0)$ on the state
$S_{l}^{z}(n)S_{l}^{z}(0)|0\rangle$ we get

$\displaystyle S_{l}^{z}(0)S_{l}^{z}(n)S_{l}^{z}(0)|0\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{[}\frac{1}{N}\sum_{x,y}e^{i(x-y)l}c_{x}^{\dagger}c_{y}-\frac{1}{2}\Big{]}\Big{[}-\frac{1}{N}\sum_{a,b}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}+\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{]}|0\rangle,$
(29) $\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{[}-\frac{1}{N^{2}}\sum_{x,y,a,b}e^{i(x-y)l}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)\Big{(}c_{x}^{\dagger}c_{-b}^{\dagger}\delta(a,y)-c_{x}^{\dagger}c_{a}^{\dagger}\delta(-b,y)\Big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2N}\sum_{a,b}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}-\frac{1}{2}\Big{(}\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{)}\Big{]}|0\rangle,$

where $x$ and $y$ are the indices in the momentum space. Now we take the
scalar product of the states given by eq. (27) and eq. (29) and get TMOTOC as

$\displaystyle F_{z}(n)$ $\displaystyle=$ $\displaystyle 2^{4}\langle
0|S_{l}^{z}(n)S_{l}^{z}(0)S_{l}^{z}(n)S_{l}^{z}(0)|0\rangle,$ (30)
$\displaystyle=$ $\displaystyle-2^{3}\langle
0|\Big{[}-\frac{1}{N}\sum_{p,r}e^{-i(p-r)l}\Phi_{p}(n)^{*}\Psi_{r}(n)^{*}c_{-r}c_{p}+\frac{1}{N}\sum_{p}|\Psi_{p}(n)|^{2}-\frac{1}{2}\Big{]}\Big{[}-\frac{1}{N^{2}}\sum_{x,y,a,b}e^{i(x-y)l}e^{i(a-b)l}\Phi_{a}(n)$
$\displaystyle\times$
$\displaystyle\Psi_{b}(n)\Big{(}c_{x}^{\dagger}c_{-b}^{\dagger}\delta(a,y)-c_{x}^{\dagger}c_{a}^{\dagger}\delta(-b,y)\Big{)}+\frac{1}{2N}\sum_{a,b}e^{i(a-b)l}\Phi_{a}(n)\Psi_{b}(n)c_{a}^{\dagger}c_{-b}^{\dagger}-\frac{1}{2}\Big{(}\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{)}\Big{]}|0\rangle,$
$\displaystyle=$
$\displaystyle-2^{3}\Big{[}\frac{1}{N^{3}}\Big{(}\sum_{p,a,r}|\Psi_{r}(n)|^{2}\Phi_{p}^{*}(n)\Phi_{a}(n)-\Psi_{-p}(n)\Psi_{r}(n)^{*}\Phi_{p}^{*}(n)\Phi_{a}(n)-\Psi_{a}(n)\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{-r}(n)+\Psi_{a}(n)$
$\displaystyle\times$
$\displaystyle\Psi_{r}(n)^{*}|\Phi_{p}(n)|^{2}\Big{)}-\frac{1}{2N^{2}}\sum_{p,r}\Big{(}|\Psi_{p}(n)|^{2}|\Phi_{r}(n)|^{2}-\Psi_{-p}(n)\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{-r}(n)\Big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\Big{(}\frac{1}{N}\sum_{p}|\Psi_{p}(n)|^{2}-\frac{1}{2}\Big{)}\Big{(}\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{)}.$

Since, the term

$\displaystyle\frac{1}{2N^{2}}\sum_{p,r}\Big{(}|\Psi_{p}(n)|^{2}|\Phi_{r}(n)|^{2}-\Psi_{-p}(n)\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{-r}(n)\Big{)}+\frac{1}{2}\Big{(}\frac{1}{N}\sum_{p}|\Psi_{p}(n)|^{2}-\frac{1}{2}\Big{)}\Big{(}\frac{1}{N}\sum_{a}|\Psi_{a}(n)|^{2}-\frac{1}{2}\Big{)}$

is constant for all number of kicks $(n)$ and system size $(N)$ which comes
out to be $\frac{1}{2^{3}}$. Since $a$ is a dummy index, we replace it by $q$.
Hence, the final formula of TMOTOC is

$\displaystyle F_{z}(n)$ $\displaystyle=$ $\displaystyle
1-\Big{(}\frac{2}{N}\Big{)}^{3}\sum_{p,q,r}\Big{(}|\Psi_{r}(n)|^{2}\Phi_{p}^{*}(n)\Phi_{q}(n)-\Psi_{-p}(n)\Psi_{r}(n)^{*}\Phi_{p}^{*}(n)\Phi_{q}(n)$
(32) $\displaystyle-$
$\displaystyle\Psi_{q}(n)\Psi_{r}(n)^{*}\Phi_{p}(n)^{*}\Phi_{-r}(n)+\Psi_{q}(n)\Psi_{r}(n)^{*}|\Phi_{p}(n)|^{2}\Big{)}.$

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group imputation strategy than the population one. In blue, the opposite is
true. The crossed area describes settings satisfying Theorem 4.3, i.e. when
both strategies result in larger reconstruction errors for group $g$ but
population mean imputation reduces the fairness gap in comparison to its group
imputation alternative. The dotted area presents the generalisation of the
Theorem when population mean imputation reduces the absolute fairness gap.

##  Appendix B Experiments

This section provides additional details on the experimental design.

###  B.1 Simulation study

#### Data Generation.

The proposed synthetic population consists of 100,000 points for the majority
group and 1,000 for the marginalised group resulting in a sample size of
$N=101,000$ with a ratio of 100:1. Each individual is represented in this
dataset as a pair of covariates, i.e., $X\in\mathbb{R}^{2}$. For each group,
2/3 presents the condition, i.e., $\mathbb{P}(Y_{i}=1)=0.66$. Negatives are
drawn from the normal distribution $\mathcal{N}((0,0),0.25)$. The condition
characterisation, i.e., the boundary between positive and negatives, differs
between groups with positive from the majority (resp. the marginalised group)
sampled from $\mathcal{N}((1,0),0.25)$ (resp. $\mathcal{N}((0,1),0.25)$).
Figure 9 shows the density distribution of the generated population.

Figure 9: Density distributions of the generated population.

#### Missingness.

In this synthetic population, 50% of the dimension $X_{2}$ is removed in a
given subgroup to enforce the three clinical presence scenarios. We enforce
the following clinical missingness:

  * •

Limited access to quality care (S1): $O_{2}\mid[G=1]\sim\text{Bernoulli}(0.5)$

  * •

(Mis)-informed collection (S2):
$O_{2}\mid[X_{1}>0.5]\sim\text{Bernoulli}(0.5)$

  * •

Confirmation bias (S3): $O_{2}\mid[X_{2}>0.5]\sim\text{Bernoulli}(0.5)$

With $O_{2}$ the observation indicator associated with $X_{2}$ and $G$, the
group membership ($G=1$ indicates a member of the marginalised group).

#### Modelling.

We generate 100 datasets and enforce the different missingness patterns before
running a logistic regression with an l2 penalty ($\lambda=1$). Results are
computed on the 20% test set and averaged over the 100 iterations with 95%
confidence bounds reported.

#### Tabular results.

Table 4 presents the reconstruction error differentiated by groups, scenarios
and imputation strategies averaged over 100 simulations presented in Section
5.3. Similarly, Table 5 presents the AUC averaged over the different
imputation strategies as presented in Section 5.4.

Table 4: Group-specific reconstruction errors and error gap $\Delta$ mean (std) across scenarios on 100 synthetic experiments. If $\Delta<0$, the marginalised group has a smaller reconstruction error than the majority. Lower reconstruction error is better. |  | Imputation strategy
---|---|---
Scenario | Group | Population Mean | Group Mean
(S1) | Overall | 0.493 (0.016) | 0.062 (0.004)
Majority | 0.000 (0.000) | 0.000 (0.000)
Marginalised | 0.493 (0.016) | 0.062 (0.004)
2-4 | $\Delta_{Marginalised}^{\mathcal{I}}$ | 0.493 (0.016) |  0.062 (0.004)
(S2) | Overall | 0.332 (0.007) | 0.236 (0.006)
Majority | 0.286 (0.007) | 0.286 (0.007)
Marginalised | 0.490 (0.019) | 0.062 (0.005)
2-4 | $\Delta_{Marginalised}^{\mathcal{I}}$ |  0.204 (0.021) | -0.224 (0.009)
(S3) | Overall | 0.333 (0.002) | 0.325 (0.002)
Majority | 0.333 (0.002) | 0.325 (0.002)
Marginalised | 0.020 (0.010) | 0.370 (0.035)
2-4 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.313 (0.010) |  0.045 (0.035)

#### Missingness Indicators.

In Section 5.4, we only considered the addition of the missingness indicators
to the Group MICE. In this Appendix, we present how this technique impacts
each imputation strategy performance. Table 6 presents the AUC for each
pipeline relying on the imputed data resulting from the different imputation
strategies and their concatenation with the missingness indicator (as
described in Section 5.2). These results echo the same insights. :

  * •

Different imputation strategies may result in equal downstream performance at
the population level while having different group performance gaps

  * •

No strategy consistently outperforms the others across clinical presence
scenarios

  * •

Current recommendations for group-specific imputation can increase the
performance gap and yield a worse performance for the marginalised group.

When comparing Table 5 and Table 6, these experiments underline that
predictive performances are often improved by the addition of the missingness
indicators. However, this gain is not equally distributed across groups and
may increase the fairness gaps.

Table 5: Group-specific AUCs and gaps $\Delta$ mean performance (std) across scenarios on 100 synthetic experiments. If $\Delta<0$, the marginalised group has lower AUC than the majority. Higher AUC is better. |  | Imputation strategy
---|---|---
Scenario | Group | Population Mean | Group Mean | MICE | Group MICE
(S1) | Overall | 0.995 (0.000) | 0.995 (0.000) | 0.996 (0.000) | 0.995 (0.000)
Majority | 0.997 (0.000) | 0.997 (0.000) | 0.997 (0.000) | 0.997 (0.000)
Marginalised | 0.679 (0.037) | 0.872 (0.026) | 0.651 (0.039) | 0.795 (0.033)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.318 (0.037) |  -0.125 (0.026) | -0.347 (0.040) | -0.201 (0.033)
(S2) | Overall | 0.994 (0.000) | 0.993 (0.000) | 0.995 (0.000) | 0.994 (0.000)
Majority | 0.996 (0.000) | 0.995 (0.000) | 0.997 (0.000) | 0.996 (0.000)
Marginalised | 0.785 (0.031) | 0.753 (0.040) | 0.815 (0.028) | 0.804 (0.032)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.212 (0.031) | -0.242 (0.040) |  -0.182 (0.028) | -0.192 (0.032)
(S3) | Overall | 0.983 (0.001) | 0.984 (0.001) | 0.967 (0.001) | 0.968 (0.001)
Majority | 0.987 (0.001) | 0.987 (0.001) | 0.970 (0.001) | 0.972 (0.001)
Marginalised | 0.641 (0.037) | 0.650 (0.037) | 0.652 (0.037) | 0.630 (0.038)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.346 (0.037) | -0.337 (0.037) | -0.318 (0.037) | -0.341 (0.038)
Table 6: Group-specific AUCs and gaps $\Delta$ mean performance (std) across scenarios on 100 synthetic experiments when using missingness indicators as a predictor. If $\Delta<0$, the marginalised group has lower AUC than the majority. Higher AUC is better. |  | Imputation strategy with missingness indicator
---|---|---
Scenario | Group | Population Mean | Group Mean | MICE | Group MICE
(S1) | Overall | 0.996 (0.000) | 0.995 (0.000) | 0.996 (0.000) | 0.996 (0.000)
Majority | 0.997 (0.000) | 0.997 (0.000) | 0.997 (0.000) | 0.997 (0.000)
Marginalised | 0.725 (0.034) | 0.872 (0.026) | 0.684 (0.037) | 0.694 (0.036)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.272 (0.034) |  -0.125 (0.026) | -0.313 (0.037) | -0.304 (0.036)
(S2) | Overall | 0.995 (0.000) | 0.993 (0.000) | 0.995 (0.000) | 0.994 (0.000)
Majority | 0.997 (0.000) | 0.995 (0.000) | 0.997 (0.000) | 0.996 (0.000)
Marginalised | 0.835 (0.027) | 0.753 (0.040) | 0.831 (0.027) | 0.804 (0.032)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ |  -0.161 (0.027) | -0.242 (0.040) | -0.166 (0.027) | -0.192 (0.032)
(S3) | Overall | 0.991 (0.001) | 0.984 (0.001) | 0.990 (0.001) | 0.990 (0.001)
Majority | 0.993 (0.001) | 0.987 (0.001) | 0.993 (0.001) | 0.993 (0.001)
Marginalised | 0.773 (0.032) | 0.650 (0.037) | 0.776 (0.032) | 0.774 (0.032)
2-6 | $\Delta_{Marginalised}^{\mathcal{I}}$ | -0.220 (0.032) | -0.337 (0.037) |  -0.216 (0.032) | -0.219 (0.032)

#### Sensitivity to distribution.

The proposed simulations have focused on a given underlying distribution of
data in which the marginalised group presents the condition differently than
the majority. In this section, we propose to study when the condition
manifestation is the same across groups but the condition _prevalence_
differs. As discussed in Section 3, this group difference may result in
different group-specific missingness processes.

Consider a population of $N=101,000$ patients with a ratio of 100:1 for the
majority. Patients without the condition are drawn from the normal
distribution $\mathcal{N}((0,0),0.25)$ and positives are sampled from
$\mathcal{N}((1,1),0.25)$. Contrarily to the previous simulations, the
marginalised group has a prevalence of 50% while the rest of the population,
10%. We then enforce the three previously described missingness processes.
Figure 10 illustrates how the same three proposed missingness processes would
be expressed in this population. Importantly, due to differences in the
prevalence, the missingness processes still differentially affect the two
groups. For instance, the proposed (S2) affects the positive cases of both
groups, representing 10% resp. 50% of these groups.

Figure 10: Graphical summary of clinical missingness in the simulation
experiments with identical condition manifestation but different prevalence
across groups. Missingness is enforced on $X_{2}$, affecting 50% of the shaded
regions for the indicated group.

Figures 11 and 12 present the associated reconstruction error and AUC
performance differentiated by scenario and imputation strategies. This set of
experiments shows that group-specific condition prevalence would lead to
similar discrepancies in missingness process, reconstruction errors and
performance gaps. Importantly, no imputation strategy would consistently lead
to a better or fairer performance in the studied settings. This result stands
despite no difference in condition manifestation, highlighting the need for a
thorough evaluation of different imputation strategies.

Figure 11: Reconstruction error gap (on the left) and group-specific
reconstruction errors (on the right) across scenarios on 100 synthetic
experiments with the same condition manifestation across groups. If
$\Delta<0$, the marginalised group has a smaller reconstruction error than the
majority. Lower reconstruction error is better. Figure 12: AUC performance
gaps $\Delta$ (on the left) and group-specific AUCs (on the right) across
scenarios on 100 synthetic experiments with the same condition manifestation
across groups. If $\Delta<0$, the marginalised group has lower AUC than the
majority. Higher AUC is better.

#### Correlated covariates.

The introduced distributions do not present correlations between the
covariates at the group level. While the missingness process may introduce
informative correlation that MICE imputation may leverage for improved
imputation, this setting may not present the full strength of MICE imputation
strategies. In this final set of experiments, we enforce correlation between
$X_{1}$ and $X_{2}$ using the same simulations than in Section 5.1 except that
the value of $X_{1}$ is added to $X_{2}$. This results in the distribution
with correlated covariates as schematised in Figure 13, in which we enforce
the same three missing scenarios.

Figure 13: Graphical summary of clinical missingness in the simulation
experiments with correlated covariates. Missingness is enforced on $X_{2}$,
affecting 50% of the shaded regions for the indicated group.

Figures 11 and 12 present the associated reconstruction error and AUC
performance differentiated by scenario and imputation strategies. These
figures echo the results presented in Section 5. Importantly, MICE and its
group alternative do not consistently present superior performance. For
instance, Group Mean presents the smallest reconstruction gap in (S3) and
Population Mean reduced predictive difference in both (S2) and (S3).

Figure 14: Reconstruction error gap (on the left) and group-specific
reconstruction errors (on the right) across scenarios on 100 synthetic
experiments with correlated covariates. If $\Delta<0$, the marginalised group
has a smaller reconstruction error than the majority. Lower reconstruction
error is better. Figure 15: AUC performance gaps $\Delta$ (on the left) and
group-specific AUCs (on the right) across scenarios on 100 synthetic
experiments with correlated covariates. If $\Delta<0$, the marginalised group
has lower AUC than the majority. Higher AUC is better.

###  B.2 MIMIC III

#### Dataset.

After preprocessing [101] and standardisation, the MIMIC III dataset consists
of $36,296$ patients with 67 different laboratory tests. Focusing on the three
marginalised groups of interest, the population can be further divided into
marginalised subgroups as presented in Figure 16. This representation
underlines the importance of identifying subgroups at risk in the studied
population.

Figure 16: Venn diagram of the population distribution in the three
marginalised groups.

#### Experimental design.

For this real-world dataset, patients are split into three groups: 80% for
training, 10% for validation and 10% for hyper-parameters selection. The
hyper-parameter search consisted of the l2 penalty selection for the logistic
regression among $\lambda\in[0.1,1.,10.,100.]$.

We bootstrapped the test set 100 times and report the mean and 95% confidence
bounds.

#### Tabular results.

Table 7 presents the AUC for each group and imputation strategy. Similarly,
Table 8 details the false negative rate performances.

Table 7: AUC performance divided by group and imputation strategy - Bootstrapped mean (std). Bold indicates the highest AUC. | Imputation strategy
---|---
Group | Population Mean | Group Mean | MICE | Group MICE
Black |  0.807 (0.030) | 0.802 (0.030) | 0.785 (0.035) | 0.782 (0.036)
Non Black |  0.742 (0.010) |  0.742 (0.010) | 0.735 (0.010) | 0.735 (0.010)
Female |  0.736 (0.016) | 0.733 (0.016) | 0.729 (0.018) | 0.726 (0.018)
Male | 0.753 (0.016) |  0.756 (0.016) | 0.742 (0.015) | 0.738 (0.014)
Public |  0.735 (0.011) | 0.732 (0.012) | 0.722 (0.012) | 0.715 (0.013)
Private | 0.753 (0.021) | 0.755 (0.021) |  0.762 (0.019) | 0.751 (0.021)
Table 8: False Negative rate divided by group and imputation strategy - Bootstrapped mean (std). Bold indicates the smallest FNR. | Imputation strategy
---|---
Group | Population Mean | Group Mean | MICE | Group MICE
Black |  0.271 (0.074) | 0.458 (0.082) | 0.314 (0.078) | 0.336 (0.076)
Non Black | 0.366 (0.017) |  0.350 (0.017) | 0.371 (0.018) | 0.371 (0.018)
Female | 0.348 (0.030) |  0.330 (0.028) | 0.381 (0.030) | 0.361 (0.032)
Male | 0.367 (0.027) | 0.394 (0.027) |  0.357 (0.026) | 0.394 (0.026)
Public | 0.336 (0.019) |  0.310 (0.018) | 0.348 (0.019) | 0.344 (0.021)
Private | 0.437 (0.041) | 0.596 (0.041) |  0.423 (0.039) | 0.510 (0.036)

#### Missingness Indicators.

Similarly, Table 9 presents the AUC for each group and imputation strategy
when missingness indicators are added to the regression model. Table 10 shows
the false negative rate for the same pipelines. This analysis shows that the
missingness patterns are informative of the outcome of interest as adding the
missingness indicators as regressors improves performance. Note, however, that
MICE Group presents lower AUC performance than MICE in Table 9. This
observation echoes the criticism that group imputation may lead to suboptimal
performance. Finally, these tables underline how no strategy consistently
outperforms the others across groups.

Table 9: AUC performance divided by group and imputation strategy - Bootstrapped mean (std). Bold indicates the highest AUC. | Imputation strategy with missingness indicators
---|---
Group | Population Mean | Group Mean | MICE | Group MICE
Black |  0.827 (0.027) | 0.825 (0.027) | 0.818 (0.028) | 0.817 (0.029)
Non Black |  0.786 (0.010) | 0.785 (0.010) | 0.781 (0.010) | 0.783 (0.010)
Female | 0.770 (0.013) | 0.770 (0.013) |  0.772 (0.014) |  0.772 (0.014)
Male |  0.801(0.013) | 0.800 (0.013) | 0.793 (0.013) | 0.796 (0.013)
Public |  0.773 (0.010) |  0.773 (0.011) | 0.767 (0.011) | 0.766 (0.010)
Private | 0.816 (0.016) | 0.810 (0.017) |  0.819 (0.017) | 0.811 (0.018)
Table 10: False Negative rate divided by group and imputation strategy - Bootstrapped mean (std). Bold indicates the smallest FNR. | Imputation strategy with missingness indicators
---|---
Group | Population Mean | Group Mean | MICE | Group MICE
Black |  0.224 (0.063) | 0.459 (0.077) | 0.229 (0.063) | 0.294 (0.070)
Non Black | 0.299 (0.017) |  0.273 (0.017) | 0.294 (0.017) | 0.290 (0.017)
Female | 0.295 (0.024) |  0.255 (0.025) | 0.292 (0.025) | 0.278 (0.026)
Male | 0.296 (0.025) | 0.314 (0.024) |  0.285 (0.025) | 0.287 (0.026)
Public | 0.290 (0.020) |  0.250 (0.020) | 0.286 (0.019) | 0.280 (0.019)
Private | 0.302 (0.038) | 0.448 (0.043) |  0.295 (0.037) | 0.348 (0.040)

#### Controlling for all groups.

In Section 6, the group-specific results correspond to the imputation
associated with each respective group. As practitioners aim to reduce the
performance gap across all presented groups simultaneously, we propose
alternative group imputation strategies in which we control over all groups.
Specifically, group mean imputation associate to each patient the group mean
associated with its Ethnicity, Sex and Insurance. Similarly, Group MICE
regresses on the three groups simultaneously.

The use of these imputation strategies result in the updated Figure 17. These
experiments leads to identical conclusions.

Figure 17: Prioritisation performance gaps $\Delta$ across marginalised groups
in MIMIC III experiment when controlling on inter-sectional groups. If
$\Delta>0$, the marginalised group has a larger value of the given metric than
the rest of the population.

#### Threshold sensitivity.

In Section 6, we present results for a policy of 30% additional care. As we
arbitrarily chose this threshold, we propose to measure how the results vary
under two different thresholds: 5% and 50%. Additionally, we introduce an
additional metric to quantify equity in care delivery. When considering a
fixed resource threshold (30%), we measure each group’s prioritisation — the
proportion of all patients who would receive care under this policy,
regardless of their ground-truth condition. A gap in prioritisation quantifies
how care is delivered differently across groups while a gap in FNR between
groups illustrates how patients would be incorrectly de-prioritised across
groups.

Figures 18, 19 and 20 present the results at 5%, 30% and 50% thresholds.
First, note that the magnitude of the $\Delta$ in prioritisation increases
with larger thresholds, but similar trends are observed. This indicates that
members of the same group have similar risk scores. Increasing the threshold,
therefore, further penalises this whole group. Second, the $\Delta$ in false
positive rates demonstrates how the choice of imputation is sensitive to the
target task. In addition to validating the insights from Section 6, this set
of experiments demonstrates that the target task may also affect whether an
imputation strategy favours or penalises a given group.

Figure 18: Prioritisation performance gaps $\Delta$ across marginalised groups
in MIMIC III experiment for 5% additional care. If $\Delta>0$, the
marginalised group has a larger value of the given metric than the rest of the
population. Figure 19: Prioritisation performance gaps $\Delta$ across
marginalised groups in MIMIC III experiment for 30% additional care. If
$\Delta>0$, the marginalised group has a larger value of the given metric than
the rest of the population. Figure 20: Prioritisation performance gaps
$\Delta$ across marginalised groups in MIMIC III experiment for 50% additional
care. If $\Delta>0$, the marginalised group has a larger value of the given
metric than the rest of the population.


# A Framework and Dataset for Abstract Art Generation via CalligraphyGAN

Jinggang Zhuo  Design AI Lab, Tongji University, Shanghai, China  Ling Fan
Design AI Lab, Tongji University, Shanghai, China  Harry Jiannan Wang
University of Delaware, Newark, DE, USA
{zhuojg1519<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

With the advancement of deep learning, artificial intelligence (AI) has made
many breakthroughs in recent years and achieved superhuman performance in
various tasks such as object detection, reading comprehension, and video
games. Generative Modeling such as various Generative Adversarial Networks
(GAN) [4] models has been applied to generate paintings and music. Research in
Natural Language Processing (NLP) also had a leap forward in 2018 since the
release of the pre-trained contextual neural language models such as BERT [2]
and recently released GPT3 [1]. Despite the exciting AI applications
aforementioned, AI is still significantly lagging behind humans in creativity,
which is often considered the ultimate moonshot for AI. Our work is inspired
by Chinese calligraphy, which is a unique form of visual art where the
character itself is an aesthetic painting. We also draw inspirations from
paintings of the Abstract Expressionist movement in the 1940s and 1950s, such
as the work by American painter Franz Kline
111https://www.moma.org/collection/works/79234. In this paper, we present a
creative framework based on Conditional Generative Adversarial Networks [6]
and Contextual Neural Language Model to generate abstract artworks that have
intrinsic meaning and aesthetic value, which is different from the existing
work, such as image captioning and text-to-image generation, where the texts
are the descriptions of the images. In addition, we have publicly released a
Chinese calligraphy image dataset 222https://github.com/zhuojg/chinese-
calligraphy-dataset and demonstrate our framework using a prototype system and
a user study.

##  2 Research Framework and Dataset

Inspired by Pablo Picasso, art washes away from the soul and dust of everyday
life. Our current target application is to generate abstract arts drawing from
a most typical daily scene: we will use dish names and their images to
generate abstract arts. Figure 1 shows our proposed framework. We want to use
Chinese calligraphy to link the dish name text with the final abstract art to
enable its intrinsic meaning. We collected 138,499 images of Chinese
calligraphy characters written by 19 calligraphers from the Internet, which
cover 7328 different characters in total. We chose 1000 characters with more
than 25 different images each to train a slightly revised conditional GAN [6]
named CalligraphyGAN using each character as the control condition. Due to the
dataset limitation, we developed a simple algorithm based on BERT [2] to map
the input text, i.e., Chinese dish names, with arbitrary number of characters
into five characters from the 1000 characters : 1) generate the embeddings for
the input text and each of the 1000 characters using the pre-trained BERT
model; 2) calculate the pairwise similarity between the input text embedding
and character embedding; 3) choose the top 5 characters with the highest
similarity scores. Then, we use the five chosen characters as the controlling
conditions to generate a new character using CalligraphyGAN, which has never
been seen before but embeds the shape characteristics from all five
characters. Note that we use equal weights for each character but can adjust
weights to draw more attention to any specific character. In the prototype
system, we feed the dish names with some random noises so that the same dish
can generate different artwork each time. Fréchet Inception Distance (FID) [5]
is often used to measure the quality and diversity of generated images and we
generate 50 images and choose the one with lowest FID score. Next, we conduct
a number of algorithmic aesthetic control: 1) denoising the image to clean up
the background; 2) applying an algorithm to transfer the image to oil-painting
style using key colors automatically extracted from the dish image; 3)
providing optional additional style transfer [3] based on paintings from
famous painters such as Picasso, Rothko, Pollock, and de Kooning. The last
step is to generate the final artwork by combining multiple elements, such as
the generated image, a piece of text description, a company logo, etc. with a
smart layout algorithm.

Figure 1: Our proposed framework

##  3 Prototype and Evaluation

We partner with a restaurant in Shanghai, China to build a prototype system
333http://harrywang.me/calligraphy/ to demonstrate our framework. The users
can customize various parameters, such as style transfer sample, number of key
colors extracted, and the ratio of white spaces, to control the generation. We
asked 8 users to try the prototype systems using different dish samples for at
least 8 times each and asked them to pick their favorite result from the
generated arts. Figure 2 shows the system UI and some favorite generation
results. We are in the process of deploying a pilot system to project the
generated artworks onto the table cloth via a gesture-controlled projector,
using which the restaurant guests can interact with our artwork, such as
generating a new artwork, choosing different style transfer options, and
giving rating and feedback. To the best of our knowledge, this is the first of
its kind of applying AI creativity in a restaurant setting to create a unique
customer dining experience.

Figure 2: Prototype System and Sample Generated Art

## References

  * [1] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al.  Language models are few-shot learners.  arXiv preprint arXiv:2005.14165, 2020.
  * [2] J. Devlin, M. Chang, K. Lee, and K. Toutanova.  BERT: pre-training of deep bidirectional transformers for language understanding.  CoRR, abs/1810.04805, 2018.
  * [3] G. Ghiasi, H. Lee, M. Kudlur, V. Dumoulin, and J. Shlens.  Exploring the structure of a real-time, arbitrary neural artistic stylization network.  arXiv preprint arXiv:1705.06830, 2017.
  * [4] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.  Generative adversarial nets.  In Advances in neural information processing systems, pages 2672–2680, 2014.
  * [5] M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter.  Gans trained by a two time-scale update rule converge to a local nash equilibrium.  In Advances in neural information processing systems, pages 6626–6637, 2017.
  * [6] M. Mirza and S. Osindero.  Conditional generative adversarial nets.  CoRR, abs/1411.1784, 2014.