# Non-Hermitian Floquet Topological Matter – A Review

Longwen Zhou<EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Ocean University of China, Qingdao 266100, China  Da-Jian Zhang
<EMAIL_ADDRESS>Department of Physics, Shandong University, Jinan 250100, China

###### Abstract

Non-Hermitian Floquet topological phases appear in systems described by time-
periodic non-Hermitian Hamiltonians. This review presents a sum-up of our
studies on non-Hermitian Floquet topological matter in one and two spatial
dimensions. After a brief overview of the literature, we introduce our
theoretical framework for the study of non-Hermitian Floquet systems and the
topological characterization of non-Hermitian Floquet bands. Based on our
theories, we describe typical examples of non-Hermitian Floquet topological
insulators, superconductors and quasicrystals with a focus on their
topological invariants, bulk-edge correspondences, non-Hermitian skin effects,
dynamical properties and localization transitions. We conclude this review by
summarizing our main discoveries and discussing potential future directions.

###### Contents

  1. I Introduction
  2. II Backgrounds
    1. II.1 Floquet theorem
    2. II.2 Floquet eigenvalue equation
    3. II.3 Floquet effective Hamiltonian and high-frequency expansion
    4. II.4 Adiabatic perturbation theory
    5. II.5 Symmetry and topological characterization
    6. II.6 Dynamical indicators
      1. II.6.1 Dynamic winding number (DWN)
      2. II.6.2 Mean chiral displacement (MCD)
    7. II.7 Localization transition and mobility edge
  3. III Non-Hermitian Floquet phases of matter
    1. III.1 Non-Hermitian Floquet exceptional topology
    2. III.2 Non-Hermitian Floquet topological insulators
      1. III.2.1 First-order topological phase
      2. III.2.2 Second-order topological phase
      3. III.2.3 qth-root topological phase
    3. III.3 Non-Hermitian Floquet topological superconductors
    4. III.4 Non-Hermitian Floquet quasicrystals
  4. IV Conclusion and outlook

##  I Introduction

When a physical system is driven periodically in time, its properties could be
drastically modified, leading to new phases and phenomena beyond the static
limit FloBok01 ; FloBok02 ; FloBok03 ; FloBok04 ; FloBok05 ; FloBok06 . One
such example, which can be traced back to the early history of human
civilization, is the Archimedes screw pump. Under periodic operations, the
Archimedes screw could transfer water from low-lying rivers into high-lying
irrigation ditches, rather than letting the water to follow its natural
flowing tendency. Another example is the tide caused by the combined effects
of the gravitational forces exerted by the Moon and its periodic orbiting
around the Earth. In the quantum domain, rich nonequilibrium features have
been identified in periodically driven systems over the past decades, such as
the Rabi oscillation Rabi1937 ; RabiBok1 ; RabiBok2 , stimulated Raman
adiabatic passage STIRAP1 ; STIRAP2 ; STIRAP3 , dynamical localization DL1 ;
DL2 ; DL3 , Thouless pump Thouless1983 ; Thouless1984 ; ThoulessRev , time
crystal DTC1 ; DTC2 ; DTC3 , Floquet topological phase FTP1 ; FTP2 ; FTP3  and
integer quantum Hall effect from chaos ChaosIQHE1 ; ChaosIQHE2 ; ChaosIQHE3
(for more information, see the reviews FloRev01 ; FloRev02 ; FloRev03 ;
FloRev04 ; FloRev05 ; FloRev06 ; FloRev07 ; FloRev08 ; FloRev09 ; FloRev10 ;
FloRev11 ; FloRev12 ; FloRev13 ; FloRev14 ; FloRev15 ; FloRev16 ; FloRev17 ;
FloRev18 ; FloRev19 ; FloRev20 ; FloRev21 ; FloRev22 ; FloRev23 ; FloRev24 ;
FloRev25 ; FloRev26 ; FloRev27 ; FloRev28 ; FloRev29 ; DTCRev1 ; DTCRev2 ;
DTCRev3 ; DTCRev4  and the references therein). Among all these discoveries,
the Floquet topological matter stands out as one pivotal impetus in the study
of periodically driven quantum systems. Over the past twenty years, it has
attracted great attention in the context of quantum dynamics FloRev06 ;
FloRev09 ; FloRev10 , quantum simulation FloRev07 ; FloRev08 ; FloRev16 ,
condensed matter physics FloRev11 ; FloRev12 ; FloRev15 , and so on.

The topological properties of a periodically driven system are mainly coded in
its Floquet states, which are eigenstates of the evolution operator over a
driving period. This one-period propagator is called the Floquet operator. For
a system described by the Hamiltonian $\hat{H}(t)=\hat{H}(t+T)$, the Floquet
operator can be expressed as
$\hat{U}=\hat{\mathsf{T}}e^{-i\int_{t_{0}}^{t_{0}+T}\hat{H}(t)dt}$. Here $t$
denotes time, $T$ is the driving period, $t_{0}$ denotes the initial time of
the period, $\hat{\mathsf{T}}$ performs the time ordering, and we have set the
Planck constant $\hbar=1$. Solving the eigenvalue equation
$\hat{U}|\psi\rangle=e^{-iE}|\psi\rangle$ gives us the Floquet eigenstate
$|\psi\rangle$ and the quasienergy $E$. The latter is a phase factor and
defined modulus $2\pi$. Its range $E\in[-\pi,\pi)$ is usually referred to as
the Floquet quasienergy Brillouin zone. Moreover, the quasienergy is
stroboscopically conserved and plays the role of “energy” in driven systems
with only discrete-time translational symmetries. If the system also obeys the
spatial translational symmetry, the quasienergy dispersion $E({\bf k})$ with
respect to the conserved quasimomentum ${\bf k}$ could also group into bands,
which are thus called Floquet bands. Under appropriate conditions, these
quasienergy bands could show nontrivial topological properties, which can be
captured by topological invariants of the corresponding Floquet-Bloch states
or the Floquet operator itself. Periodically driven quantum systems could thus
support a new class of nonequilibrium phases, which is nowadays known as the
Floquet topological matter.

Similar to static topological phases, Floquet topological phases could also
possess symmetry-protected edge states under open boundary conditions and
exhibit quantized dynamical or transport signals. However, three key features
distinguish Floquet topological phases from their static counterparts. First,
suitable driving fields could break the symmetry dynamically and open gaps
around the touching points of static energy bands, yielding band inversions
and topologically nontrivial bandstructures FTP1 . This aspect of driving can
usually be taken into account by the Floquet effective Hamiltonians obtained
from high-frequency expansions of the driving potential FloHeff1 ; FloHeff2 ;
FloHeff3 ; FloHeff4 . Second, as the quasienergy $E$ is bounded from above
($E=\pi$) and below ($E=-\pi$), Floquet bands could meet with each other at
$E=\pm\pi$ and develop nontrivial windings around the whole quasienergy
Brillouin zone $E\in[-\pi,\pi)$. These spectral windings result in unique
states of matter in periodically driven systems, such as Floquet semimetals
with Floquet band holonomy FSM1 ; FSM2 ; FSM3 , degenerated edge modes at
$E=\pm\pi$ (anomalous Floquet $\pi$ modes) FPiMod1 ; FPiMod2 ; FPiMod3 ;
FPiMod4  and anomalous chiral edge states in Floquet topological insulators
AFTI1 ; AFTI2 ; AFTI3 , which have no counterparts in static systems. Third,
the driving field could assist in the formation of long-ranged, and even
spatially non-decaying coupling among different degrees of freedom in a
lattice FTP3 , leading to Floquet phases with large topological invariants,
rich topological transitions, and a substantial number of topological edge
states FTPLg01 ; FTPLg02 ; FTPLg03 ; FTPLg04 ; FTPLg05 ; FTPLg07 ; FTPLg08 .
These phases and boundary states go beyond the description of any static
tight-binding models in typical situations (i.e., with finite ranged or
spatially decaying hopping amplitudes). The investigation of the above-
mentioned features have not only led to new classification schemes of Floquet
matter in theories FTPCls1 ; FTPCls2 ; FTPCls3 ; FTPCls4  but also promoted
experimental realizations of numerous Floquet topological phases in solid-
state and quantum simulator setups FTPExp01 ; FTPExp02 ; FTPExp03 ; FTPExp04 ;
FTPExp05 ; FTPExp06 ; FTPExp07 ; FTPExp08 ; FTPExp09 ; FTPExp10 ; FTPExp11 ;
FTPExp12 ; FTPExp13 ; FTPExp14 ; FTPExp15 ; FTPExp16 , bringing new hope for
applications in ultrafast electronics FloRev11  and topological quantum
computing FloQC1 ; FloQC2 ; FloQC3 .

Non-Hermitian physics deals with classical and open quantum systems subject to
measurements, dissipation, gain and loss or nonreciprocal effects NHBok1 ;
NHBok2 ; NHBok3 ; NHBok4 . A non-Hermitian system can usually be modeled by a
Hamiltonian $\hat{H}$ that is not self-adjoint, i.e.,
$\hat{H}\neq\hat{H}^{\dagger}$. The spectrum of such a non-Hermitian
Hamiltonian is complex in general and the resulting dynamics is non-unitary.
In recent years, the study of topological phases in non-Hermitian systems have
attracted much attention from both theoretical and experimental sides (see
Refs. NHTPRev01 ; NHTPRev02 ; NHTPRev03 ; NHTPRev04 ; NHTPRev05 ; NHTPRev06 ;
NHTPRev07 ; NHTPRev08 ; NHTPRev09 ; NHTPRev10 ; NHTPRev11 ; NHTPRev12 ;
NHTPRev13 ; NHTPRev14 ; NHTPRev15 ; NHTPRev16 ; NHTPRev17 ; NHTPRev18 ;
NHTPRev19 ; NHTPRev20 ; NHTPRev21 ; NHTPRev22 ; NHTPRev23 ; NHTPRev24 ;
NHTPRev25  for reviews). The investigation of these nonequilibrium phases may
help us deepen our understanding of topological matter in open systems and
optimize the design of noise-resilient quantum devices.

The recent booming of non-Hermitian topological matter is driven by a couple
of key concepts, including the parity-time reversal (PT) symmetry, the
exceptional point (EP), and the non-Hermitian skin effect (NHSE). The PT
symmetry can appear in open systems with balanced gain and loss. A PT-
symmetric, non-Hermitian Hamiltonian has a real spectrum in the PT unbroken
regime Bender2002 ; Zhang2019 . The PT-symmetry was thus viewed as a possible
way to lift the restriction of Hermiticity for Hamiltonians in its early
studies NHBok4 . Recently, it was found that the topological and transport
properties of a system may also change when it undergoes a PT-symmetry-
breaking transition, yielding phases unique to non-Hermitian Hamiltonians
PTExp1 ; PTExp2 ; PTExp3 . Models with the PT symmetry thus become focuses in
the study of non-Hermitian topological matter NHTPRev03 ; NHTPRev04 ;
NHTPRev05 ; NHTPRev07 ; NHTPRev08 ; NHTPRev11 ; NHTPRev17 ; NHTPRev25 . The
combination of PT and other symmetries further results in various
classification schemes for non-Hermitian topological phases that go beyond
their Hermitian counterparts NHCls01 ; NHCls02 ; NHCls03 ; NHCls04 ; NHCls05 ;
NHCls06 ; NHCls07 ; NHCls08 ; NHCls09 . The EP is a class of level degeneracy
point unique to non-Hermitian operators. At an EP, the geometric and algebraic
multiplicities of a non-Hermitian matrix is different, leading to the
breakdown of its diagonalizability and the coalescence of its eigenvectors
NHBok1 ; NHTPRev01 ; NHTPRev02 . In recent years, various gapless band
structures (nodal points, lines, loops, surfaces, knots etc.) formed by EPs
were discovered, bringing about rich non-Hermitian topological (semi-)metallic
phases with intriguing transport properties NHTPRev09 ; NHTPRev11 ; NHTPRev15
; NHTPRev16 ; NHTPRev18 . Besides, EPs were also found to play key roles in
the topological energy transfer NHTET1 ; NHTET2 ; NHTET3 , high-precision
sensing EPSense1 ; EPSense2 ; EPSense3 ; EPSense4 , topological lasers EPLas1
; EPLas2 ; EPLas3  and strongly correlated phases NHTPRev15 ; NHTPRev16 . The
NHSE refers to the accumulation of bulk states around the boundaries of an
open-boundary non-Hermitian lattice. It highlights the extreme sensitivity of
non-Hermitian physics to the boundary condition of a system NHTPRev13 ;
NHTPRev19 ; NHTPRev24 . This phenomenon not only blurs the distinction between
bulk and edge states in non-Hermitian models, but also breaks the well-
established bulk-boundary correspondence in Hermitian topological matter
NHSE01 ; NHSE02 ; NHSE03 ; NHSE04 ; NHSE05 . In the past few years, a couple
of theoretical frameworks have been introduced to characterize the NHSE and
its related topological phenomena NHSE06 ; NHSE07 ; NHSE08 ; NHSE09 ; NHSE10 ;
NHSE11 ; NHSE12 ; NHSE13 ; NHSE14 ; NHSE15 ; NHSE16 , which are accompanied by
the experimental observations of NHSE in AMO systems, electrical circuits and
metamaterials SkinExp01 ; SkinExp02 ; SkinExp03 ; SkinExp04 ; SkinExp05 ;
SkinExp06 ; SkinExp07 . Entanglement transitions associated with the NHSE was
also identified in a recent study SkinESEE . Despite these, progresses have
been made in the study of non-Hermitian topological matter in randomly
disordered NHDsod01 ; NHDsod02 ; NHDsod03 ; NHDsod04 ; NHDsod05 ,
quasiperiodic NHQC01 ; NHQC02 ; NHQC03 ; NHQC04 ; NHQC05  and many-body
systems NHMBP01 ; NHMBP02 ; NHMBP03 ; NHMBP04 ; NHMBP05 ; NHMBP06 ; NHMBP07 ;
NHMBP08 ; NHMBP09 ; NHMBP10 ; NHMBP11 ; NHMBP12 ; NHMBP13 .

Figure 1: Non-Hermitian Floquet topological matter: a schematic diagram to
illustrate the concepts and several typical phases.

With all these developments, a natural follow-up is to consider the system in
a more general situation, in which it is subject to both time-periodic
drivings and non-Hermitian effects. Such non-Hermitian Floquet systems may
possess exotic dynamical phenomena and topological phases with no static or
Hermitian analogies. On the theoretical side, the investigation of driven non-
Hermitian systems may lead to the discovery of new topological states and
bring about the improvement of classification schemes for nonequilibrium
phases of matter in general. On the practical side, the exploration of non-
Hermitian Floquet matter is helpful to the design of new approaches for
preparing or stabilizing topologically nontrivial states and controlling
material properties. It also brings about new ideas for realizing quantum
devices and quantum computing protocols that are robust to perturbations
caused by the environment. Though still at the early state, many progresses
have been made in the realization and characterization of non-Hermitian
Floquet phases NHFTP01 ; NHFTP02 ; NHFTP03 ; NHFTP04 ; NHFTP05 ; NHFTP06 ;
NHFTP07 ; NHFTP08 ; NHFTP09 ; NHFTP10 ; NHFTP11 ; NHFTP12 ; NHFTP13 ; NHFTP14
; NHFTP15 ; NHFTP16 ; NHFTP17 ; NHFTP18 ; NHFTP19 ; NHFTP20 ; NHFTP21 ;
NHFTP22 ; NHFTP23 ; NHFTP24 ; NHFTP25 ; NHFTP26 ; NHFTP27 ; NHFTP28 ; NHFTP29
; NHFTP30 ; NHFTP31 ; NHFTP32 ; NHFTP33 ; NHFTP34 ; NHFTP35 ; NHFTP36 ;
NHFTP37 ; NHFTP38 ; NHFTP39 ; NHFTP40 ; NHFTP41 ; NHFTP42 . In this review, we
limit our scope to the discussion of several typical topological phases we
discovered in non-Hermitian Floquet systems ZhouNHFTP01 ; ZhouNHFTP02 ;
ZhouNHFTP03 ; ZhouNHFTP04 ; ZhouNHFTP05 ; ZhouNHFTP06 ; ZhouNHFTP07 ;
ZhouNHFTP08 ; ZhouNHFTP09 ; ZhouNHFTP10 ; ZhouNHFTP11 . In Sec. II, we give a
pedagogical introduction to some key aspects of Floquet systems, including
their dynamical and topological characterizations. In Sec. III, we present
typical examples of non-Hermitian Floquet topological insulators,
superconductors, quasicrystals and summarize their main physical properties,
with a focus on the features that are unique to driven non-Hermitian systems.
In Sec. IV, we conclude this review, briefly mention some relevant studies and
discuss potential future work.

##  II Backgrounds

We start with a recap of the basic description of a non-Hermitian Floquet
system. The Hamiltonian of such a system satisfies $\hat{H}(t)=\hat{H}(t+T)$,
and there exists $t\in[0,T]$ such that $\hat{H}(t)\neq[\hat{H}(t)]^{\dagger}$.
Here $t$ denotes time and $T$ is the driving period. We further assume that
the state of the system evolves following the Schrödinger equation

$i\frac{d}{dt}|\Psi(t)\rangle=\hat{H}(t)|\Psi(t)\rangle,$ (1)

where we have set $\hbar=1$. We first show that this equation can be solved by
Floquet states even though $\hat{H}(t)$ is non-Hermitian. This is followed by
different ways of obtaining the Floquet states in general situations, in the
high-frequency regime and in the adiabatic regime. We next discuss the
symmetry, topological invariants and dynamical characterizations of non-
Hermitian Floquet states, with a focus on the types of physical systems
explored in our previous study. We conclude this section by presenting some
tools for characterizing the spectrum and localization transitions in non-
Hermitian Floquet disordered system.

###  II.1 Floquet theorem

We sketch a proof of the Floquet theorem in this subsection ZhouThesis2015 .
Applying the Fourier expansion to our time-periodic Hamiltonian $\hat{H}(t)$
in the time-frequency domain, we find

$\hat{H}(t)=\sum_{n}\hat{H}(\omega_{n})e^{i\omega_{n}t},\qquad\omega_{n}=n\omega=n\frac{2\pi}{T},\qquad
n\in\mathbb{Z}.$ (2)

Here $\omega$ is the driving frequency. For an infinite system, the set of
plane waves $\\{|\psi_{\varepsilon}(t)\rangle=|\varphi({\bf r})\rangle
e^{-i\varepsilon t}\\}$ can be chosen as a suitable basis. We can write down a
matrix expression for $\hat{H}(t)$ in the orthonormal and complete basis
$\\{|\psi_{\varepsilon}(t)\rangle\\}$. Acting $\hat{H}(t)$ on
$|\psi_{\varepsilon}(t)\rangle$, we obtain

$\hat{H}(t)|\psi_{\varepsilon}(t)\rangle=\sum_{n}\hat{H}(\omega_{n})|\varphi({\bf
r})\rangle
e^{-i(\varepsilon-\omega_{n})t}=\sum_{n}\hat{H}(\omega_{n})|\psi_{\varepsilon-\omega_{n}}(t)\rangle.$
(3)

For any given $n\in\mathbb{Z}$, the state
$|\psi_{\varepsilon-\omega_{n}}(t)\rangle$ belongs to the subspace

$S_{\varepsilon}=\\{|\psi_{\varepsilon}(t)\rangle,|\psi_{\varepsilon\pm\omega}(t)\rangle,|\psi_{\varepsilon\pm
2\omega}(t)\rangle,...,|\psi_{\varepsilon\pm n\omega}(t)\rangle,...\\}.$ (4)

It is clear that any two subspaces $S_{\varepsilon}$ and
$S_{\varepsilon^{\prime}}$ $(\varepsilon^{\prime}\neq\varepsilon)$ are
decoupled under the action of $\hat{H}(t)$ if
$\varepsilon-\varepsilon^{\prime}\in(-\omega,\omega)$. Moreover,
$S_{\varepsilon}$ and $S_{\varepsilon^{\prime}}$ are equivalent if there
exists an $n\in\mathbb{Z}$ such that
$\varepsilon-\varepsilon^{\prime}=n\omega$. Therefore, we can study the
dynamics of the system separately in each subspace $S_{\varepsilon}$ for ${\rm
Re}\varepsilon\in[-\pi/T,\pi/T)$, which is usually called the first
quasienergy Brillouin zone (BZ). The quasienergy $\varepsilon$ is thus a
conserved quantity due to the discrete-time translational symmetry of
$\hat{H}(t)$, similar to the conserved quasimomentum due to the discrete-space
translational symmetry of a static Hamiltonian. In the subspace
$S_{\varepsilon}$, a solution of the Schrödinger equation can be written as

$|\Psi_{\varepsilon}(t)\rangle=\sum_{n}c(\varepsilon+\omega_{n})|\psi_{\varepsilon+\omega_{n}}(t)\rangle=e^{-i\varepsilon
t}\sum_{n}c(\varepsilon+\omega_{n})|\varphi({\bf r})\rangle
e^{-i\omega_{n}t},$ (5)

where $\\{c(\varepsilon+\omega_{n})\\}$ are complex coefficients. It is clear
that the Eq. (5) possesses a time-periodic component

$|u_{\varepsilon}(t)\rangle\equiv\sum_{n}c(\varepsilon+\omega_{n})|\varphi({\bf
r})\rangle e^{-i\omega_{n}t}=|u_{\varepsilon}(t+T)\rangle.$ (6)

Therefore, we can express the general solution $|\Psi_{\varepsilon}(t)\rangle$
as the product of an oscillating phase factor $e^{-i\varepsilon t}$ and a
time-periodic Floquet mode $|u_{\varepsilon}(t)\rangle$, i.e.,

$|\Psi_{\varepsilon}(t)\rangle=e^{-i\varepsilon t}|u_{\varepsilon}(t)\rangle.$
(7)

It also implies that

$|\Psi_{\varepsilon}(t+T)\rangle=e^{-i\varepsilon
T}|\Psi_{\varepsilon}(t)\rangle.$ (8)

The latter equation implies that the only change of the state
$|\Psi_{\varepsilon}(t)\rangle$ after undergoing a one-period evolution is to
pick up an exponential factor $e^{-i\varepsilon T}$. We refer to the set
$\\{|\Psi_{\varepsilon}(t)\rangle|{\rm Re}\varepsilon\in[-\pi/T,\pi/T)\\}$ as
Floquet eigenstates of the system. They form a complete basis at each instant
of time $t$.

Since the evolution from $|\Psi_{\varepsilon}(t)\rangle$ to
$|\Psi_{\varepsilon}(t+T)\rangle$ is generated by the Schrödinger equation, we
can also express Eq. (8) as

$\hat{U}(t+T,t)|\Psi_{\varepsilon}(t)\rangle=e^{-i\varepsilon
T}|\Psi_{\varepsilon}(t)\rangle,$ (9)

where
$\hat{U}(t+T,t)=\hat{\mathsf{T}}e^{-i\int_{t}^{t+T}\hat{H}(t^{\prime})dt^{\prime}}$
is nothing but the Floquet operator (evolution operator over one driving
period) of the system. When we are only concerned with the stroboscopic
dynamics, the initial time dependence of Eq. (9) is not important. In this
case, we can set $t=0$ in Eq. (9) and express the Floquet eigenvalue equation
as

$\hat{U}|\Psi_{E}\rangle=e^{-iE}|\Psi_{E}\rangle,$ (10)

where $\hat{U}=\hat{\mathsf{T}}e^{-i\int_{0}^{T}\hat{H}(t)dt}$ and we have
introduced $E=\varepsilon T$ as the dimensionless quasienergy, whose first BZ
is given by $[-\pi,\pi)$. To sum up, we find that the solution of Eq. (1) with
a time-periodic $\hat{H}(t)$ has the form of Eq. (7) or (8), where
$|\Psi_{\varepsilon}(t)\rangle$ is an eigenstate of the Floquet operator of
the system. For stroboscopic observations, all the Floquet eigenstates can
thus be obtained by solving the eigenvalue equation (10) of Floquet operator
$\hat{U}$. Due to the completeness of Floquet eigenstate basis
$\\{|\Psi_{E}\rangle\\}$, we can expand an arbitrary initial state of the
system as $|\Psi(0)\rangle=\sum_{E}c_{E}|\Psi_{E}\rangle.$ The resulting state
after an evolution over $n$ driving periods is then given by

$|\Psi(t)\rangle=\sum_{E}c_{E}e^{-inE}|\Psi_{E}\rangle.$ (11)

Note that for a non-Hermitian $\hat{H}(t)$, $\hat{U}$ is generally non-unitary
and the quasienergy $E$ may possess a nonvanishing imaginary part. In this
case, the real part of $E$ still belongs to the range of $[-\pi,\pi)$ and our
arguments leading to the general solution (11) can be satisfied.

###  II.2 Floquet eigenvalue equation

In the most general case, we can solve the Eq. (10) numerically by the split-
operator method FloBok03 . Decomposing the evolution periodic $T$ into $N$
segments and assuming that $N$ is large enough, we can express the Floquet
operator $\hat{U}$ approximately as

$\hat{U}\simeq e^{-i\hat{H}((N-1)\Delta t)\Delta t}e^{-i\hat{H}((N-2)\Delta
t)\Delta t}\cdots e^{-i\hat{H}(\Delta t)\Delta t}e^{-i\hat{H}(0)\Delta t},$
(12)

where $\Delta t=T/N$. Over each small time interval $\Delta t$, $\hat{H}(t)$
is approximately time-independent and we can diagonalize it numerically at
$t=\ell\Delta t$ as

$\hat{H}(\ell\Delta t)=V_{\ell}D_{\ell}V_{\ell}^{-1}.$ (13)

Each column of $V_{\ell}$ represents a right eigenvector of
$\hat{H}(\ell\Delta t)$. The evolution operator over one time interval $\Delta
t$ then takes the form of

$e^{-i\hat{H}(\ell\Delta t)\Delta t}=V_{\ell}e^{-iD_{\ell}\Delta
t}V_{\ell}^{-1},\qquad\ell=0,1,...,N-1.$ (14)

The multiplication of all the $V_{\ell}e^{-iD_{\ell}\Delta t}V_{\ell}^{-1}$
from right to left for $\ell=0$ to $N-1$ yields the Eq. (12), which further
converges to the exact Floquet operator in the limit $N\rightarrow\infty$. We
can thus numerically solve the Floquet eigenvalue equation by diagonalizing
the approximated $\hat{U}$ in Eq. (12).

When the driving field takes the form of periodic kicking or quenching, the
series in Eq. (12) can be greatly simplified and even obtained exactly. Here
we give several examples. Consider a time-periodic Hamiltonian of the form

$\hat{H}(t)=\hat{H}_{0}+\sum_{\ell\in\mathbb{Z}}\delta(t/T-\ell)\hat{H}_{1},$
(15)

where $\delta(t/T-\ell)$ is the delta function peaked at $t=\ell T$, i.e.,
each integer multiple of the driving period. The dynamics over each driving
period thus constitutes a free evolution part controlled by $\hat{H}_{0}$ and
a delta kicking force controlled by $\hat{H}_{1}$. The quantum kicked rotor is
one representative example of such a system FloRev01 . Focusing on the one-
period evolution from $t=\ell T-0^{+}$ to $t=(\ell+1)T-0^{+}$, we find the
Floquet operator of $\hat{H}(t)$ to be

$\displaystyle\hat{U}=$ $\displaystyle e^{-i\int_{\ell
T+0^{+}}^{(\ell+1)T-0^{+}}\hat{H}(t)dt}e^{-i\int_{\ell T-0^{+}}^{\ell
T+0^{+}}\hat{H}(t)dt}$ $\displaystyle=$ $\displaystyle e^{-i\int_{\ell
T+0^{+}}^{(\ell+1)T-0^{+}}\hat{H}_{0}dt}e^{-i\int_{\ell T-0^{+}}^{\ell
T+0^{+}}\delta(t/T-\ell)\hat{H}_{1}dt}$ $\displaystyle=$ $\displaystyle
e^{-i\hat{H}_{0}T}e^{-i\hat{H}_{1}T}.$ (16)

Similarly, if there are two kicks separated by a time interval $\tau$ within
each driving period, the Hamiltonian could have the form of

$\hat{H}(t)=\hat{H}_{0}+\sum_{\ell\in\mathbb{Z}}\delta(t/T-\tau/T-\ell)\hat{H}_{2}+\sum_{\ell\in\mathbb{Z}}\delta(t/T-\ell)\hat{H}_{1}.$
(17)

For the evolution from time $t=\ell T-0^{+}$ to $t=(\ell+1)T-0^{+}$, the
Floquet operator now takes the form of

$\hat{U}=e^{-i\hat{H}_{0}(T-\tau)}e^{-i\hat{H}_{2}T}e^{-i\hat{H}_{0}\tau}e^{-i\hat{H}_{1}T}.$
(18)

One typical example of such a system is the double-kicked quantum rotor FTP3 .
For a periodically quenched Hamiltonian in the form of

$\hat{H}(t)=\begin{cases}\hat{H}_{1}&t\in[\ell T,\ell T+T_{1})\\\
\hat{H}_{2}&t\in[\ell T+T_{1},\ell T+T_{1}+T_{2})\end{cases},$ (19)

where $T=T_{1}+T_{2}$, we can also directly obtain the corresponding Floquet
operator from $t=\ell T-0^{+}$ to $t=(\ell+1)T-0^{+}$ as

$\displaystyle\hat{U}=$ $\displaystyle e^{-i\int_{\ell T+T_{1}}^{\ell
T+T-0^{+}}\hat{H}(t)dt}e^{-i\int_{\ell T-0^{+}}^{\ell
T+T_{1}-0^{+}}\hat{H}(t)dt}$ $\displaystyle=$ $\displaystyle e^{-i\int_{\ell
T+T_{1}}^{\ell T+T-0^{+}}\hat{H}_{2}dt}e^{-i\int_{\ell T-0^{+}}^{\ell
T+T_{1}-0^{+}}\hat{H}_{1}dt}$ $\displaystyle=$ $\displaystyle
e^{-i\hat{H}_{2}T_{2}}e^{-i\hat{H}_{1}T_{1}}.$ (20)

The discrete time quantum walk can be viewed as one example of such a
periodically quenched system FloRev04 . Time-periodic quenches are also
frequently implemented in the study of discrete time crystals DTC1 ; DTC2 ;
DTC3 .

For a continuously driven system, the solution of the Floquet eigenvalue
equation may also be obtained approximately in terms of the frequency (Sambe)
space formalism SambeSpace1 ; SambeSpace2 ; SambeSpace3 . Inserting the
Floquet state in Eq. (7) into the Schrödinger equation (1) and reorganizing
the terms, we find

$\left[\hat{H}(t)-i\frac{d}{dt}\right]|u_{\varepsilon}(t)\rangle=\varepsilon|u_{\varepsilon}(t)\rangle.$
(21)

Using the Fourier expansion of $\hat{H}(t)$ and
$|u_{\varepsilon}(t)\rangle=\sum_{n}e^{-i\omega_{n}t}|u_{\varepsilon}(\omega_{n})\rangle$,
we further obtain

$\sum_{m,n}\hat{H}(\omega_{m})e^{i(\omega_{m}-\omega_{n})t}|u_{\varepsilon}(\omega_{n})\rangle-\sum_{n}\omega_{n}e^{-i\omega_{n}t}|u_{\varepsilon}(\omega_{n})\rangle=\varepsilon\sum_{n}e^{-i\omega_{n}t}|u_{\varepsilon}(\omega_{n})\rangle.$
(22)

Multiplying $\frac{1}{T}e^{i\omega_{\ell}t}$ from the left on both sides and
integrating the time $t$ over a driving period $T$, we arrive at

$\sum_{m}(\hat{H}_{m-n}-\omega_{m}\delta_{m,n})|u_{\varepsilon}(\omega_{m})\rangle=\varepsilon|u_{\varepsilon}(\omega_{n})\rangle,$
(23)

where

$\hat{H}_{m-n}\equiv\hat{H}(\omega_{m-n})=\frac{1}{T}\int_{0}^{T}e^{-i\omega_{m-n}t}\hat{H}(t)dt,$
(24)

and $\omega_{m-n}=(m-n)\omega=(m-n)\frac{2\pi}{T}$. The Eq. (23) is an
infinite-dimensional matrix equation of the form

$\begin{bmatrix}\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\\
\ddots&\hat{H}_{0}+2\omega&\hat{H}_{1}&\hat{H}_{2}&\hat{H}_{3}&\hat{H}_{4}&\ddots\\\
\ddots&\hat{H}_{-1}&\hat{H}_{0}+\omega&\hat{H}_{1}&\hat{H}_{2}&\hat{H}_{3}&\ddots\\\
\ddots&\hat{H}_{-2}&\hat{H}_{-1}&\hat{H}_{0}&\hat{H}_{1}&\hat{H}_{2}&\ddots\\\
\ddots&\hat{H}_{-3}&\hat{H}_{-2}&\hat{H}_{-1}&\hat{H}_{0}-\omega&\hat{H}_{1}&\ddots\\\
\ddots&\hat{H}_{-4}&\hat{H}_{-3}&\hat{H}_{-2}&\hat{H}_{-1}&\hat{H}_{0}-2\omega&\ddots\\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\end{bmatrix}\begin{bmatrix}\vdots\\\
|u_{\varepsilon}(\omega_{-2})\rangle\\\
|u_{\varepsilon}(\omega_{-1})\rangle\\\ |u_{\varepsilon}(\omega_{0})\rangle\\\
|u_{\varepsilon}(\omega_{1})\rangle\\\ |u_{\varepsilon}(\omega_{2})\rangle\\\
\vdots\end{bmatrix}=\varepsilon\begin{bmatrix}\vdots\\\
|u_{\varepsilon}(\omega_{-2})\rangle\\\
|u_{\varepsilon}(\omega_{-1})\rangle\\\ |u_{\varepsilon}(\omega_{0})\rangle\\\
|u_{\varepsilon}(\omega_{1})\rangle\\\ |u_{\varepsilon}(\omega_{2})\rangle\\\
\vdots\end{bmatrix}.$ (25)

Note that each $\hat{H}_{m-n}$ has the same Hilbert space dimension $d$ as the
original $\hat{H}(t)$. As an example, for a harmonically driven system
described by the Hamiltonian

$\hat{H}(t)=\hat{H}_{0}+\hat{V}e^{i\omega t}+\hat{W}e^{-i\omega t},$ (26)

the above equation reduces to the following block tridiagonal form

$\begin{bmatrix}\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\\
\ddots&\hat{H}_{0}+2\omega&\hat{V}&0&0&0&\ddots\\\
\ddots&\hat{W}&\hat{H}_{0}+\omega&\hat{V}&0&0&\ddots\\\
\ddots&0&\hat{W}&\hat{H}_{0}&\hat{V}&0&\ddots\\\
\ddots&0&0&\hat{W}&\hat{H}_{0}-\omega&\hat{V}&\ddots\\\
\ddots&0&0&0&\hat{W}&\hat{H}_{0}-2\omega&\ddots\\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\end{bmatrix}\begin{bmatrix}\vdots\\\
|u_{\varepsilon}(\omega_{-2})\rangle\\\
|u_{\varepsilon}(\omega_{-1})\rangle\\\ |u_{\varepsilon}(\omega_{0})\rangle\\\
|u_{\varepsilon}(\omega_{1})\rangle\\\ |u_{\varepsilon}(\omega_{2})\rangle\\\
\vdots\end{bmatrix}=\varepsilon\begin{bmatrix}\vdots\\\
|u_{\varepsilon}(\omega_{-2})\rangle\\\
|u_{\varepsilon}(\omega_{-1})\rangle\\\ |u_{\varepsilon}(\omega_{0})\rangle\\\
|u_{\varepsilon}(\omega_{1})\rangle\\\ |u_{\varepsilon}(\omega_{2})\rangle\\\
\vdots\end{bmatrix}.$ (27)

In practical calculations, one should truncate the infinite-dimensional matrix
in Eq. (25) at a sufficiently high harmonics $N\omega$, leading to an
$Nd\times Nd$ dimensional Floquet effective Hamiltonian, whose eigenvalue
problem can be numerically solved. Assuming the characteristic energy scale of
$\hat{H}_{m-n}$ for all $m-n$ to be $\Omega$, we can take a smaller $N$ to do
the truncation for a larger ratio of $\omega/\Omega$, i.e., for a high
frequency driving field. All the discussions presented in this subsection
holds for both Hermitian and non-Hermitian Hamiltonians $\hat{H}(t)$.

###  II.3 Floquet effective Hamiltonian and high-frequency expansion

From the Floquet operator $\hat{U}$ of a periodically driven system, one can
formally define its Floquet effective Hamiltonian as

$\hat{H}_{{\rm
eff}}=\frac{i}{T}\ln\hat{U}\Leftrightarrow\hat{U}=e^{-i\hat{H}_{{\rm eff}}T}.$
(28)

Taking into account the fact that the quasienergies are defined modulus
$2\pi/T$, the $\hat{H}_{{\rm eff}}$ contains the same physical information as
$\hat{U}$. Yet it provides us with more room to treat the properties of
Floquet systems in analogy with static Hamiltonian models. For a continuously
driven system, the explicit form of $\hat{H}_{{\rm eff}}$ is usually involved.
This can be inspected from the Eq. (12), as in general the $\hat{H}(\ell\Delta
t)$ and $\hat{H}(m\Delta t)$ are not commute for $\ell\neq m$. When the
frequency of the driving field is high, an approximate series expression for
$\hat{H}_{{\rm eff}}$ can be obtained via high-frequency expansion methods
FloHeff1 ; FloHeff2 ; FloHeff3 ; FloHeff4 . Here we recap one such method in
its general form, which is applicable to both Hermitian and non-Hermitian
systems.

We first decompose the time-periodic Hamiltonian $\hat{H}(t)$ into a static
part $\hat{H}_{0}$ and a periodically modulated part $\hat{V}(t)$, such that

$\hat{H}(t)=\hat{H}_{0}+\hat{V}(t),\qquad\hat{V}(t)=\hat{V}(t+T).$ (29)

Here $T=2\pi/\omega$ is the driving period and $\omega$ is the driving
frequency. Next, we apply a similarity transformation to the evolved state
$|\Psi(t)\rangle$ in the Schrödinger equation (1), yielding a rotated state

$|\Phi(t)\rangle=\hat{{\cal
U}}(t)|\Psi(t)\rangle=e^{i\hat{K}(t)}|\Psi(t)\rangle.$ (30)

Here $\hat{K}(t)$ is sometimes called the kick operator. It encodes the
information regarding the micromotion dynamics of the system. Plugging
$|\Psi(t)\rangle=e^{-i\hat{K}(t)}|\Phi(t)\rangle$ into Eq. (1) leads to the
transformed Schrödinger equation

$i\frac{d}{dt}|\Phi(t)\rangle=\hat{H}_{{\rm eff}}|\Phi(t)\rangle,$ (31)

where

$\hat{H}_{{\rm
eff}}=e^{i\hat{K}(t)}\hat{H}(t)e^{-i\hat{K}(t)}+i\frac{de^{i\hat{K}(t)}}{dt}e^{-i\hat{K}(t)}.$
(32)

The aim of the high-frequency expansion method is to find a time-independent
$\hat{H}_{{\rm eff}}$ by transferring all time-dependent terms into the kick
operator $\hat{K}(t)$. When such a purpose is formally achieved, we can
express the Floquet evolution of a system as

$\hat{U}(t_{1},t_{0})|\Psi(t_{0})\rangle=e^{-i\hat{K}(t_{1})}e^{-i\hat{H}_{{\rm
eff}}(t_{1}-t_{0})}e^{i\hat{K}(t_{0})}|\Psi(t_{0})\rangle.$ (33)

That is, the system is subject to an initial kick associated with the operator
$\hat{K}(t_{0})$, then evolved by the time-independent $\hat{H}_{{\rm eff}}$,
and finally subject to a second kick carried out by the operator
$\hat{K}(t_{1})$.

Assuming the driving frequency $\omega$ to be large, we can expand
$\hat{H}_{{\rm eff}}$ and $\hat{K}(t)$ into series of $1/\omega$ as

$\hat{H}_{{\rm eff}}=\sum_{n=0}^{\infty}\frac{1}{\omega^{n}}\hat{H}_{{\rm
eff}}^{(n)},\qquad\hat{K}=\sum_{n=1}^{\infty}\frac{1}{\omega^{n}}\hat{K}^{(n)}.$
(34)

In the meantime, we can apply Taylor expansions to the two terms in Eq. (32)
and express the $\hat{H}_{{\rm eff}}$ as

$\displaystyle\hat{H}_{{\rm eff}}=$
$\displaystyle\sum_{n=0}^{\infty}\frac{i^{n}}{n!}[\stackrel{{\scriptstyle
n\,{\rm
of}\,\hat{K}}}{{\overbrace{\hat{K},[\hat{K},...,\hat{K},[\hat{K}},}}\hat{H}]]]$
$\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}i\frac{i^{n}}{n!}[\stackrel{{\scriptstyle(n-1)\,{\rm
of}\,\hat{K}}}{{\overbrace{\hat{K},[\hat{K},...,\hat{K},[\hat{K}},}}\partial_{t}\hat{K}]]].$
(35)

Note that we have concealed the explicit time dependence of $\hat{K}(t)$ and
$\hat{K}^{(n)}(t)$ in the above two equations for brevity. To proceed, we
impose two further requirements for $\hat{K}(t)$, i.e.,

$\hat{K}(t)=\hat{K}(t+T),\qquad\int_{0}^{T}\hat{K}(t)dt=0.$ (36)

Inserting Eq. (34) and the Fourier expansion

$\hat{H}(t)=\hat{H}_{0}+\sum_{m\neq 0}\hat{V}_{m}e^{im\omega t}$ (37)

into Eq. (35), we arrive at

$\displaystyle\sum_{n=0}^{\infty}\frac{1}{\omega^{n}}\hat{H}_{{\rm
eff}}^{(n)}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{i^{n}}{n!}[\stackrel{{\scriptstyle
n\,{\rm
of}\,\hat{K}}}{{\overbrace{\sum_{p}\frac{1}{\omega^{p}}\hat{K}^{(p)},[\sum_{q}\frac{1}{\omega^{q}}\hat{K}^{(q)},...,\sum_{r}\frac{1}{\omega^{r}}\hat{K}^{(r)},[\sum_{s}\frac{1}{\omega^{s}}\hat{K}^{(s)}},}}\left(\hat{H}_{0}+\sum_{m\neq
0}\hat{V}_{m}e^{im\omega t}\right)]]]$ $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}\frac{i^{n+1}}{n!}[\stackrel{{\scriptstyle(n-1)\,{\rm
of}\,\hat{K}}}{{\overbrace{\sum_{p}\frac{1}{\omega^{p}}\hat{K}^{(p)},[\sum_{q}\frac{1}{\omega^{q}}\hat{K}^{(q)},...,\sum_{r}\frac{1}{\omega^{r}}\hat{K}^{(r)},[\sum_{s}\frac{1}{\omega^{s}}\hat{K}^{(s)}},}}\partial_{t}\sum_{m}\frac{1}{\omega^{m}}\hat{K}^{(m)}]]].$
(38)

The high-frequency expansions of $\hat{H}_{{\rm eff}}$ and $\hat{K}$ are
obtained by equating both sides the Eq. (38) at each order of $1/\omega$.

At the zeroth order, we could obtain

$\hat{H}_{{\rm
eff}}^{(0)}=\hat{H}_{0}+\hat{V}(t)-\frac{1}{\omega}\partial_{t}\hat{K}^{(1)}$
(39)

from the Eq. (38). To make sure that the $\hat{H}_{{\rm eff}}^{(0)}$ is time-
independent, we must choose

$\frac{1}{\omega}\partial_{t}\hat{K}^{(1)}=\hat{V}(t)=\sum_{m\neq
0}\hat{V}_{m}e^{im\omega t},$ (40)

such that

$\hat{K}^{(1)}=\sum_{m\neq 0}\frac{1}{im}\hat{V}_{m}e^{im\omega t}.$ (41)

Therefore, up to the zeroth order of $1/\omega$, we have the following
approximation for the effective Hamiltonian

$\hat{H}_{{\rm eff}}\approx\hat{H}_{{\rm eff}}^{(0)}=\hat{H}_{0}.$ (42)

Up to the first order of $1/\omega$, we have the following approximation for
the kick operator

$\hat{K}(t)\approx\frac{1}{\omega}\hat{K}^{(1)}=\frac{1}{\omega}\sum_{m\neq
0}\frac{\hat{V}_{m}}{im}e^{im\omega t}.$ (43)

At the first order, we could find from the Eq. (38) that

$\displaystyle\hat{H}_{{\rm eff}}^{(1)}=$ $\displaystyle
i\left[\hat{K}^{(1)},\hat{H}_{0}+\hat{V}(t)\right]-\frac{1}{\omega}\partial_{t}\hat{K}^{(2)}-\frac{i}{2}\left[\hat{K}^{(1)},\frac{1}{\omega}\partial_{t}\hat{K}^{(1)}\right]$
$\displaystyle=$ $\displaystyle
i\left[\hat{K}^{(1)},\hat{H}_{0}+\hat{V}(t)\right]-\frac{1}{\omega}\partial_{t}\hat{K}^{(2)}-\frac{i}{2}\left[\hat{K}^{(1)},\hat{V}(t)\right]$
$\displaystyle=$ $\displaystyle
i\left[\hat{K}^{(1)},\hat{H}_{0}\right]+\frac{i}{2}\left[\hat{K}^{(1)},\hat{V}(t)\right]-\frac{1}{\omega}\partial_{t}\hat{K}^{(2)}.$
(44)

By definition, any non-vanishing elements of
$i\left[\hat{K}^{(1)},\hat{H}_{0}\right]$ and
$-\frac{1}{\omega}\partial_{t}\hat{K}^{(2)}$ are time-dependent. The only
time-independent term that could contribute to $\hat{H}_{{\rm eff}}^{(1)}$
comes from the term $\frac{i}{2}\left[\hat{K}^{(1)},\hat{V}(t)\right]$, which
can be written explicitly as

$\displaystyle\frac{i}{2}\left[\hat{K}^{(1)},\hat{V}(t)\right]=$
$\displaystyle\frac{i}{2}\left[\sum_{m\neq
0}\frac{1}{im}\hat{V}_{m}e^{im\omega t},\sum_{m^{\prime}\neq
0}\hat{V}_{m^{\prime}}e^{im^{\prime}\omega t}\right]$ $\displaystyle=$
$\displaystyle\sum_{m,m^{\prime}\neq 0}\frac{e^{i(m+m^{\prime})\omega
t}}{2m}\left[\hat{V}_{m},\hat{V}_{m^{\prime}}\right].$ (45)

The time-independent terms are those with $m^{\prime}=-m$. Collecting these
terms together, we find

$\hat{H}_{{\rm eff}}^{(1)}=\sum_{m\neq
0}\frac{\left[\hat{V}_{m},\hat{V}_{-m}\right]}{2m}=\sum_{m\neq
0}\frac{\hat{V}_{m}\hat{V}_{-m}}{m}.$ (46)

Therefore, up to the first order of $1/\omega$, we have the following
approximation for the effective Hamiltonian

$\hat{H}_{{\rm eff}}\approx\hat{H}_{{\rm
eff}}^{(0)}+\frac{1}{\omega}\hat{H}_{{\rm
eff}}^{(1)}=\hat{H}_{0}+\frac{1}{\omega}\sum_{m\neq
0}\frac{\hat{V}_{m}\hat{V}_{-m}}{m}.$ (47)

The form of $\hat{K}^{(2)}$ is determined by the remaining time-dependent
terms. Performing the integration over time directly, we obtain

$\displaystyle\hat{K}^{(2)}=$ $\displaystyle\sum_{m\neq
0}\frac{1}{m}\left[\hat{V}_{m},\hat{H}_{0}\right]\int\omega e^{im\omega
t}dt+\sum_{m,m^{\prime}\neq
0,-m}\frac{\left[\hat{V}_{m},\hat{V}_{m^{\prime}}\right]}{2m}\int\omega
e^{i(m+m^{\prime})\omega t}dt$ $\displaystyle=$ $\displaystyle\sum_{m\neq
0}\frac{1}{im^{2}}\left[\hat{V}_{m},\hat{H}_{0}\right]e^{im\omega
t}+\sum_{m,m^{\prime}\neq
0,-m}\frac{\left[\hat{V}_{m},\hat{V}_{m^{\prime}}\right]}{2im(m+m^{\prime})}e^{i(m+m^{\prime})\omega
t}.$ (48)

Up to the second order of $1/\omega$, we can thus approximate the kick
operator by

$\displaystyle\hat{K}(t)\approx\frac{1}{\omega}\hat{K}^{(1)}+\frac{1}{\omega^{2}}\hat{K}^{(2)}$
$\displaystyle=$ $\displaystyle\frac{1}{\omega}\sum_{m\neq
0}\frac{\hat{V}_{m}}{im}e^{im\omega t}+\frac{1}{\omega^{2}}\sum_{m\neq
0}\frac{1}{im^{2}}\left[\hat{V}_{m},\hat{H}_{0}\right]e^{im\omega t}$
$\displaystyle+$ $\displaystyle\frac{1}{\omega^{2}}\sum_{m,m^{\prime}\neq
0,-m}\frac{\left[\hat{V}_{m},\hat{V}_{m^{\prime}}\right]}{2im(m+m^{\prime})}e^{i(m+m^{\prime})\omega
t}.$ (49)

We can continue these self-consistent calculations to obtain higher order
terms in the high-frequency expansion of $\hat{H}_{{\rm eff}}$ and
$\hat{K}(t)$. For example, for the second order component $\hat{H}_{{\rm
eff}}^{(2)}$, we have

$\displaystyle\hat{H}_{{\rm eff}}^{(2)}$
$\displaystyle=\frac{-1}{2}[\hat{K}^{(1)},[\hat{K}^{(1)},\hat{H}_{0}+\hat{V}(t)]]+i[\hat{K}^{(2)},\hat{H}_{0}+\hat{V}(t)]$
$\displaystyle+$
$\displaystyle\frac{-1}{\omega}\partial_{t}\hat{K}^{(3)}+\frac{-i}{2\omega}[\hat{K}^{(1)},\partial_{t}\hat{K}^{(2)}]+\frac{-i}{2\omega}[\hat{K}^{(2)},\partial_{t}\hat{K}^{(1)}]$
$\displaystyle+$
$\displaystyle\frac{1}{6\omega}[\hat{K}^{(1)},[\hat{K}^{(1)},\partial_{t}\hat{K}^{(1)}].$
(50)

Dropping the time-dependent terms, we are left with

$\hat{H}_{{\rm eff}}^{(2)}=\frac{1}{2}\sum_{m\neq
0}\frac{[[\hat{V}_{m},\hat{H}_{0}],\hat{V}_{-m}]}{m^{2}}+\frac{1}{3}\sum_{l,m\neq
0}\frac{[\hat{V}_{l},[\hat{V}_{m},\hat{V}_{-l-m}]]}{lm}.$ (51)

Therefore, up to the second order correction in $1/\omega$, the effective
Hamiltonian turns out to be

$\displaystyle\hat{H}_{{\rm
eff}}\approx\hat{H}_{0}+\frac{1}{\omega}\sum_{m\neq
0}\frac{\hat{V}_{m}\hat{V}_{-m}}{m}$ $\displaystyle+$
$\displaystyle\frac{1}{2\omega^{2}}\sum_{m\neq
0}\frac{[[\hat{V}_{m},\hat{H}_{0}],\hat{V}_{-m}]}{m^{2}}+\frac{1}{3\omega^{2}}\sum_{l,m\neq
0}\frac{[\hat{V}_{l},[\hat{V}_{m},\hat{V}_{-l-m}]]}{lm}.$ (52)

This approximation holds for both Hermitian and non-Hermitian Floquet systems
under high-frequency driving fields. Note that for stroboscopic evolutions, we
have $t_{1}-t_{0}=NT$ with $N\in\mathbb{Z}$ and
$\hat{K}(t_{1})=\hat{K}(t_{0})$. In this case, the term $\hat{K}(t_{0})$ in
Eq. (33) describes an initial phase, which can be set to zero as the expansion
of $\hat{K}(t)$ at every order of $1/\omega$ is only determined up to a
constant [see Eqs. (41) and (48)]. Therefore, for stroboscopic dynamics, we
can simply use the $\hat{H}_{{\rm eff}}$ to capture most of the essential
physics.

Here we illustrate the usage of $\hat{H}_{{\rm eff}}$ with a simple example.
Consider a harmonically driven two-level Hamiltonian of the form

$H(t)=(h_{x}+i\gamma/2)[\cos(\omega t)\sigma_{x}+\sin(\omega
t)\sigma_{y}]+h_{z}\sigma_{z},$ (53)

where $(h_{x},h_{z},\gamma)\in\mathbb{R}$ and $\sigma_{x,y,z}$ are Pauli
matrices. It is not hard to verify that by choosing the kick operator as

$e^{iK(t)}=\begin{pmatrix}1&0\\\ 0&e^{-i\omega t}\end{pmatrix},$ (54)

the Floquet effective Hamiltonian in the rotating frame can be exactly
obtained from the Eq. (32) as

$H_{{\rm
eff}}=\frac{\omega}{2}\sigma_{0}+(h_{x}+i\gamma/2)\sigma_{x}+(h_{z}-\omega/2)\sigma_{z}.$
(55)

We see that the two levels of $H(t)$ have the instantaneous energies

$E_{\pm}=\pm\sqrt{h_{x}^{2}+h_{z}^{2}-\gamma^{2}/4+ih_{x}\gamma},$ (56)

which are time-independent and in general complex. They could meet with each
other at a second-order EP with $E=0$ when $h_{x}=0$ and $h_{z}=\pm\gamma/2$.
Meanwhile, the $H_{{\rm eff}}$ has two quasienergy levels defined by

$\varepsilon_{\pm}=\frac{\omega}{2}\pm\sqrt{h_{x}^{2}+\left(h_{z}-\frac{\omega}{2}\right)^{2}-\gamma^{2}/4+ih_{x}\gamma}.$
(57)

They could meet with each other at a second-order Floquet EP with
$\varepsilon=\omega/2$ when $h_{x}=0$ and $h_{z}=\omega/2\pm\gamma/2$. It is
clear that the EP is shifted by the driving in both its energy and location in
the parameter space. If we define $\varepsilon T$ as the quasienergy $E$, the
Floquet EP of our two-level system appears at the quasienergy $E=\pi$. This
could lead to Floquet exceptional topology and anomalous $\pi$ edge modes,
which are unique to driven non-Hermitian systems as will be discussed in the
Sec. III.

###  II.4 Adiabatic perturbation theory

We now consider a non-Hermitian system that is subject to slow-in-time and
cyclic modulations. It can be viewed as the opposite limit of a high-frequency
driven system. Introducing a rescaled and dimensionless time variable $s=t/T$,
we can express the Schrödinger equation as

$i\frac{d}{ds}|\Psi(s)\rangle=T\hat{H}(s)|\Psi(s)\rangle.$ (58)

Our purpose is to find an expansion for $|\Psi(s)\rangle$ in the series of
$1/T$. It constitutes an adiabatic perturbation theory of the system APT01 ,
providing that different energy levels of $\hat{H}(s)$ are gapped for all
$s\in[0,1]$.

The Hamiltonian $\hat{H}(s)$ is periodic in the scaled time $s$ with
$\hat{H}(s)=\hat{H}(s+1)$. We denote its instantaneous right and left
biorthonormal eigenvectors as $\\{|n(s)\rangle\\}$ and
$\\{|\widetilde{n}(s)\rangle\\}$ BioQM01 , such that

$\hat{H}(s)|n(s)\rangle=E_{n}(s)|n(s)\rangle,\qquad\langle\widetilde{n}(s)|\hat{H}(s)=\langle\widetilde{n}(s)|E_{n}(s),$
(59)
$\langle\widetilde{m}(s)|n(s)\rangle=\delta_{mn},\qquad\sum_{n}|n(s)\rangle\langle\widetilde{n}(s)|=1.$
(60)

Here $E_{n}(s)$ is the instantaneous eigenenergy associated with
$|n(s)\rangle$, which could be complex if
$\hat{H}(s)\neq\hat{H}^{\dagger}(s)$. To proceed, we introduce an ansatz
solution for the evolved state $|\Psi(s)\rangle$ as

$|\Psi(s)\rangle=\sum_{n}e^{-i\Omega_{n}(s)}c_{n}(s)|n(s)\rangle,$ (61)

where

$\Omega_{n}(s)\equiv T\int_{0}^{s}E_{n}(s^{\prime})ds^{\prime}$ (62)

describes an accumulated dynamical phase from the scaled time $0$ to $s$. At
$s=0$, we have $\Omega_{n}(0)=0$ and

$c_{n}(0)=\langle\widetilde{n}(0)|\Psi(0)\rangle$ (63)

for the initial state. Plugging the Eq. (61) into Eq. (58), we obtain

$\sum_{n}e^{-i\Omega_{n}(s)}\dot{c}_{n}(s)|n(s)\rangle=-\sum_{n}e^{-i\Omega_{n}(s)}c_{n}(s)|\dot{n}(s)\rangle,$
(64)

where $\dot{c}_{n}(s)=dc_{n}/ds$ and $|\dot{n}(s)\rangle=d|n(s)\rangle/ds$.
Acting $\langle\widetilde{m}(s)|$ from the left on both sides of the above
equation and using the Eq. (60), we find

$\dot{c}_{m}(s)=-\sum_{n\neq
m}e^{i\Omega_{mn}(s)}c_{n}(s)\langle\widetilde{m}(s)|\dot{n}(s)\rangle,$ (65)

where $\Omega_{mn}(s)\equiv\Omega_{m}(s)-\Omega_{n}(s)$. We have also made the
parallel transport gauge choice so that
$\langle\widetilde{n}(s)|\dot{n}(s)\rangle=0$. Performing the integration over
$s$ and keeping the terms on the right-hand-side up to the first order of
$1/T$ (i.e., up to the first order non-adiabatic correction), we arrive at

$c_{m}(s)=c_{m}(0)+\frac{1}{T}\sum_{n\neq
m}\left.\left[\frac{i\langle\widetilde{m}(s^{\prime})|\dot{n}(s^{\prime})\rangle}{\Delta_{mn}(s^{\prime})}e^{i\Omega_{mn}(s^{\prime})}\right]\right|_{s^{\prime}=0}^{s^{\prime}=s}c_{n}(0),$
(66)

where $\Delta_{mn}(s)=E_{m}(s)-E_{n}(s)$. To reach the Eq. (66), we have
assumed $\Omega_{mn}(s)$ to be real for all $m\neq n$ and $s\in[0,1]$. This
can be achieved if every instantaneous energy level possesses the same
imaginary part $i\gamma(s)$. Other the other hand, the Eq. (66) is valid if
the $\hat{H}(s)$ is PT-invariant at every $s$, such that the
$E_{n}(s)\in\mathbb{R}$ for all $n$. The approximation in Eq. (66) does not
hold if $\Delta_{mn}(s)\notin\mathbb{R}$, which will cause the exponential
amplification or decay of the amplitude $c_{n}(s)$. Inserting the Eq. (66)
into Eq. (61) results in the solution to the Eq. (58) up to first order non-
adiabatic corrections, i.e.,

$|\Psi(s)\rangle=\sum_{m}e^{-i\Omega_{m}(s)}\left\\{c_{m}(0)+\frac{1}{T}\sum_{n\neq
m}\left.\left[\frac{i\langle\widetilde{m}(s^{\prime})|\dot{n}(s^{\prime})\rangle}{\Delta_{mn}(s^{\prime})}e^{i\Omega_{mn}(s^{\prime})}\right]\right|_{s^{\prime}=0}^{s^{\prime}=s}c_{n}(0)\right\\}|m(s)\rangle.$
(67)

Since we have employed the formalism of biorthonormal eigenvectors, we may
consider another ansatz solution for the left eigenvector

$\langle\widetilde{\Psi}(s)|=\sum_{n}e^{i\Omega_{n}(s)}c_{n}^{*}(s)\langle\widetilde{n}(s)|,$
(68)

which follows a conjugate Schrödinger equation

$-i\frac{d}{ds}\langle\widetilde{\Psi}(s)|=\langle\widetilde{\Psi}(s)|\hat{H}.$
(69)

Following the steps of deriving the Eq. (67), we find up to the first order of
$1/T$ that

$\langle\widetilde{\Psi}(s)|=\sum_{m}e^{+i\Omega_{m}(s)}\left\\{c_{m}^{*}(0)+\frac{1}{T}\sum_{n\neq
m}\left[\frac{i\langle\dot{\widetilde{n}}(s^{\prime})|m(s^{\prime})\rangle}{\Delta_{nm}(s^{\prime})}e^{i\Omega_{nm}(s^{\prime})}\right]_{s^{\prime}=0}^{s^{\prime}=s}c_{n}^{*}(0)\right\\}\langle\widetilde{m}(s)|.$
(70)

Assuming that initially only the state with $m=\ell$ is occupied, such that
$c_{m}(0)=\delta_{m\ell}$, we can further simplify the Eqs. (67) and (70) to

$|\Psi_{\ell}(s)\rangle=e^{-i\Omega_{\ell}(s)}\left[|\ell(s)\rangle+\frac{1}{T}\sum_{m\neq\ell}\frac{i\langle\widetilde{m}(s)|\dot{\ell}(s)\rangle}{\Delta_{m\ell}(s)}|m(s)\rangle\right]-\frac{1}{T}\sum_{m\neq\ell}e^{-i\Omega_{m}(s)}\frac{i\langle\widetilde{m}(0)|\dot{\ell}(0)\rangle}{\Delta_{m\ell}(0)}|m(s)\rangle,$
(71)
$\langle\widetilde{\Psi}_{\ell}(s)|=e^{+i\Omega_{\ell}(s)}\left[\langle\widetilde{\ell}(s)|+\frac{1}{T}\sum_{m\neq\ell}\frac{i\langle\dot{\widetilde{\ell}}(s)|m(s)\rangle}{\Delta_{\ell
m}(s)}\langle\widetilde{m}(s)|\right]-\frac{1}{T}\sum_{m\neq\ell}e^{+i\Omega_{m}(s)}\frac{i\langle\dot{\widetilde{\ell}}(0)|m(0)\rangle}{\Delta_{\ell
m}(0)}\langle\widetilde{m}(s)|.$ (72)

For any observable $\hat{O}$, up to the correction of order $1/T$, we can now
express its biorthogonal average over the states
$\\{|\Psi_{\ell}(s)\rangle,|\widetilde{\Psi}_{\ell}(s)\rangle\\}$ as

$\displaystyle\langle\widetilde{\Psi}_{\ell}(s)|\hat{O}|\Psi_{\ell}(s)\rangle$
$\displaystyle=\langle\widetilde{\ell}(s)|\hat{O}|\ell(s)\rangle+\frac{1}{T}\sum_{m\neq\ell}\frac{i\langle\widetilde{m}(s)|\dot{\ell}(s)\rangle}{\Delta_{m\ell}(s)}\langle\widetilde{\ell}(s)|\hat{O}|m(s)\rangle$
$\displaystyle+\frac{1}{T}\sum_{m\neq\ell}\frac{i\langle\dot{\widetilde{\ell}}(s)|m(s)\rangle}{\Delta_{\ell
m}(s)}\langle\widetilde{m}(s)|\hat{O}|\ell(s)\rangle.$ (73)

We now illustrate this adiabatic perturbation theory with an application. Let
us consider noninteracting particles in a one-dimensional (1D) periodic
lattice, whose onsite potential is also varied slowly and periodically in
time. This is the typical situation encountered in the topological Thouless
pump Thouless1983 ; Thouless1984 ; ThoulessRev . The group velocity of the
particle can be expressed as $\hat{v}=\partial_{k}\hat{H}$, where $k$ is the
quasimomentum. At the initial time $s=0$, we assume that the band $\ell$ is
uniformly filled, and it is separated from the other bands at all $k$ and $s$.
The pumped number of particles over one adiabatic cycle due to this initially
filled band is then given by

$N_{\ell}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}\int_{0}^{1}ds\langle\widetilde{\Psi}_{\ell}(s)|\partial_{k}\hat{H}|\Psi_{\ell}(s)\rangle.$
(74)

With the Eq. (59), it is not hard to identify that

$\langle\widetilde{\ell}(s)|\partial_{k}\hat{H}|\ell(s)\rangle=\partial_{k}E_{\ell}(k,s),$
(75)
$\frac{\langle\widetilde{\ell}(s)|\partial_{k}\hat{H}|m(s)\rangle}{\Delta_{m\ell}(s)}=\langle\widetilde{\ell}(s)|\partial_{k}|m(s)\rangle,$
(76)

where $E_{\ell}(k,s)$ denotes the energy dispersion of the $\ell$th adiabatic
Bloch band. Plugging the Eqs. (75), (76) and (73) into the Eq. (74), we obtain

$\displaystyle N_{\ell}=$
$\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}\int_{0}^{1}ds\partial_{k}E_{\ell}(k,s)$
(77) $\displaystyle+$
$\displaystyle\sum_{m\neq\ell}\int_{-\pi}^{\pi}\frac{dk}{2\pi
i}\int_{0}^{T}dt\langle\partial_{k}\widetilde{\ell}(k,t)|m(k,t)\rangle\langle\widetilde{m}(k,t)|\partial_{t}\ell(k,t)\rangle$
$\displaystyle-$ $\displaystyle\sum_{m\neq\ell}\int_{-\pi}^{\pi}\frac{dk}{2\pi
i}\int_{0}^{T}dt\langle\partial_{t}\widetilde{\ell}(k,t)|m(k,t)\rangle\langle\widetilde{m}(k,t)|\partial_{k}\ell(k,t)\rangle.$

Noting that $E_{\ell}(k=-\pi,s)=E_{\ell}(k=\pi,s)$ and
$\sum_{m}|m(k,t)\rangle\langle\widetilde{m}(k,t)|=1$, we can simplify the
above equation and arrive at the pumped number of particles over an adiabatic
cycle

$N_{\ell}=\int_{-\pi}^{\pi}\frac{dk}{2\pi
i}\int_{0}^{T}dt\left[\langle\partial_{k}\widetilde{\ell}(k,t)|\partial_{t}\ell(k,t)\rangle-\langle\partial_{t}\widetilde{\ell}(k,t)|\partial_{k}\ell(k,t)\rangle\right].$
(78)

Eq. (78) describes nothing but the Chern number of the adiabatic Bloch band
$\ell$, whose energy dispersion is defined on a two dimensional (2D) torus
$(k,t)\in[-\pi,\pi)\times[0,T)$. The conditions for the Eq. (78) to hold are
as follows. First, the Hamiltonian of the system should be quasi-Hermitian
with a real spectrum (e.g., PT-invariant) in our considered parameter regime.
Second, the band $E_{\ell}$ should be well-gapped from the other bands
throughout the 2D torus $(k,t)\in[-\pi,\pi)\times[0,T)$, and $\hbar/T$ should
be much larger than $|\Delta_{\ell m}|$ for all $m\neq\ell$. Third, the
evolution of left and right vectors of the system should follow the Eqs. (58)
and (69). Note here that the expression of Chern number $N_{\ell}$ is not
sensitive to the choice of biorthonormal basis. We will get the same
$N_{\ell}$ after changing $\widetilde{\ell}\rightarrow\ell$ or
$\ell\rightarrow\widetilde{\ell}$ in the Eq. (78), as proved before for non-
Hermitian Chern bands NHCls01 .

###  II.5 Symmetry and topological characterization

In the past few years, rich symmetry classes and topological invariants have
been identified for non-Hermitian topological matter NHCls01 ; NHCls02 ;
NHCls03 ; NHCls04 ; NHCls05 ; NHCls06 ; NHCls07 ; NHCls08 ; NHCls09 . In this
subsection, we mainly recap two symmetries together with their associated
topological numbers. They are the most relevant ones for the characterization
of non-Hermitian Floquet topological phases reviewed in this work.

We first discuss the PT-symmetry, which is denoted by the operator ${\cal
PT}$. If the Hamiltonian of a non-Hermitian system $\hat{H}$ possesses the PT-
symmetry, we will have $[{\cal PT},\hat{H}]=0$. In this case, the system could
have a real spectrum in its PT-invariant regime, where the eigenstates
$|\psi\rangle$ and ${\cal PT}|\psi\rangle$ of $\hat{H}$ are coincident up to a
global phase. To see this, let us consider a non-degenerate eigenstate
$|\psi\rangle$ of $\hat{H}$ that satisfies the eigenvalue equation

$\hat{H}|\psi\rangle=E|\psi\rangle.$ (79)

The PT-symmetry of $\hat{H}$ then implies that

$\hat{H}({\cal PT}|\psi\rangle)=E^{*}({\cal PT}|\psi\rangle).$ (80)

Therefore, ${\cal PT}|\psi\rangle$ is also an eigenstate of $\hat{H}$ with the
energy $E^{*}$. If $|\psi\rangle$ and $\hat{H}$ share the same PT-symmetry,
$|\psi\rangle$ should be the common eigenstate of $\hat{H}$ and ${\cal PT}$.
${\cal PT}|\psi\rangle$ can thus only be differ from $|\psi\rangle$ up to a
global phase, which means that $E=E^{*}\in\mathbb{R}$. However, with the
change of system parameters (e.g., the strengths of gain and loss), the PT-
symmetry of $|\psi\rangle$ could be spontaneously broken and the spectrum of
$\hat{H}$ could switch from real to complex after undergoing a PT-symmetry
breaking transition. A topological invariant, defined as NHQC01

$w=\int_{0}^{2\pi}\frac{d\theta}{2\pi
i}\partial_{\theta}\ln\det[\hat{H}(\theta)-E_{0}],$ (81)

might be employed to characterize such a real-to-complex spectral transition.
Here $E_{0}$ is a base energy chosen appropriately on the complex plane. The
parametrized Hamiltonian $\hat{H}(\theta)=\hat{H}(\theta+2\pi)$, where
$\theta$ can be viewed as the quasimomentum along an artificial dimension. We
have also taken the periodic boundary condition (PBC) for $\hat{H}$ before
implementing its $\theta$-parametrization. The $w$ in Eq. (81) thus depicts a
spectral winding number with respect to the base energy $E_{0}$, i.e., it
counts the number of times that the spectrum of $\hat{H}(\theta)$ winds around
$E_{0}$ on the complex plane when the synthetic quasimomentum $\theta$ is
varied over a cycle. When the spectrum of $\hat{H}(\theta)$ is real, we must
have $w=0$, as a spectral loop cannot be formed on the complex plane in this
case. When the spectrum of $\hat{H}(\theta)$ is complex, $w$ may take an
integer-quantized value if $\hat{H}(\theta)$ possesses spectral loops around
$E_{0}$. A suitably chosen $E_{0}$ could then yield a nonzero $w$ when the
first spectral loop appears on the complex-$E$ plane, and thus detecting the
topological changes in the spectrum of $\hat{H}$ across the PT-breaking
transition. For a Floquet system, if the time-periodic Hamiltonian
$\hat{H}(t)$ possesses the PT-symmetry at each instant $t$, the resulting
Floquet operator also has the PT-symmetry. In this case, we can define the
spectral winding number $w$ as in Eq. (81) for the stroboscopic Floquet
effective Hamiltonian so as to capturing the PT-breaking transition in the
quasienergy spectrum. We will provide explicit examples for this usage in the
Sec. III.4, where we consider PT transitions, localization transitions and
topological transitions in non-Hermitian Floquet quasicrystals.

We next consider the chiral (or sublattice) symmetry, whose associated
operator will be denoted by ${\cal S}$. If the Hamiltonian $\hat{H}$ of a
system possesses the chiral symmetry ${\cal S}$, we will have ${\cal
S}\hat{H}{\cal S}=-\hat{H}$, where ${\cal S}$ is Hermitian and unitary
Tenfold01 . The implication of this symmetry on the spectrum of $\hat{H}$ is
as follows. Suppose that $|\psi\rangle$ is an eigenstate of a chiral-symmetric
$\hat{H}$ with the energy $E$, i.e., $\hat{H}|\psi\rangle=E|\psi\rangle$, we
will have

$\hat{H}({\cal S}|\psi\rangle)=-E({\cal S}|\psi\rangle).$ (82)

Therefore, ${\cal S}|\psi\rangle$ is also an eigenstate of $\hat{H}$ with the
energy $-E$. The eigenstates of a chiral-symmetric $\hat{H}$ should then come
in pairs of $\\{|\psi\rangle,{\cal S}|\psi\rangle\\}$ with the energies
$\\{E,-E\\}$ that are symmetric with respect to $E=0$. This further leads to a
chiral-symmetry protected degeneracy for any eigenstate with $E=0$. When the
spectrum of $\hat{H}$ is gapped at $E=0$, we can group its energy levels into
two clusters with $E<0$ and $E>0$. If these two clusters could meet with each
other at $E=0$ and then separate again with the change of certain system
parameters, the system may undergo a phase transition. In one dimension, the
change of band topology of the system before and after such a transition could
be characterized by a winding number $w_{0}$, which is defined as Tenfold01 ;
WN01 ; WN02

$w_{0}=\int_{-\pi}^{\pi}\frac{dk}{4\pi}{\rm Tr}[{\cal S}{\cal
Q}(k)i\partial_{k}{\cal Q}(k)].$ (83)

Here the PBC has been assumed and the quasimomentum $k\in[-\pi,\pi)$. The
sign-resolved projector ${\cal Q}(k)$ is obtained from the spectral
decomposition $H(k)=\sum_{n}E_{n}(k)|n(k)\rangle\langle\widetilde{n}(k)|$ of
$\hat{H}=\sum_{k}|k\rangle H(k)\langle k|$ at each $k$ by attributing $+1$
($-1$) to every energy band $n$ with ${\rm Re}E_{n}>0$ (${\rm Re}E_{n}<0$).
${\cal Q}(k)$ can thus be expressed as ZhouNHFTP08

${\cal Q}(k)=\sum_{n}{\rm sgn}\\{{\rm
Re}[E_{n}(k)]\\}|n(k)\rangle\langle\widetilde{n}(k)|.$ (84)

Here $H(k)|n(k)\rangle=E_{n}(k)|n(k)\rangle$,
$\langle\widetilde{n}(k)|H(k)=\langle\widetilde{n}(k)|E_{n}(k)$, and the basis
set $\\{|n(k)\rangle,|\widetilde{n}(k)\rangle\\}$ satisfies the biorthonormal
relations in Eqs. (60). Under the PBC, the winding number $w_{0}$ could
characterize the bulk topological properties of a 1D chiral-symmetric
Hamiltonian $\hat{H}$, either Hermitian or non-Hermitian. It could further
distinguish different bulk topological insulating phases by showing a
quantized jump at the transition point, where the two band clusters of
$\hat{H}$ meet with each other at $E=0$. However, under the open boundary
condition (OBC), due to the possible existence of NHSE, the $w_{0}$ defined as
in Eq. (83) may not be able to correctly predict the gap-closing points of the
spectrum in the parameter space and determine the number of degenerate edge
modes at $E=0$ in different parameter regions. This non-Hermiticity induced
breakdown of bulk-edge correspondence may be recovered by introducing a real-
space counterpart of $w_{0}$, which is defined as NHSE10

$W_{0}=-\frac{1}{L_{{\rm B}}}{\rm Tr}_{{\rm B}}({\cal S}{\cal Q}[{\cal
Q},\hat{N}]).$ (85)

Here ${\cal S}$ is the chiral symmetry operator of $\hat{H}$. $\hat{N}$ is the
position operator in real-space. ${\cal Q}$ is again the flat band projector
defined as

${\cal Q}=\sum_{|\psi_{j}\rangle\in{\rm bulk}}{\rm sgn}[{\rm
Re}(E_{j})]|\psi_{j}\rangle\langle\widetilde{\psi}_{j}|,$ (86)

where the summation is now taken over all the bulk eigenstates
$\\{|\psi_{j}\rangle\\}$ of $\hat{H}$ under the OBC. The whole lattice of
length $L=L_{{\rm B}}+2L_{{\rm E}}$ is decomposed into three segments, with a
bulk region of length $L_{{\rm B}}$ in the middle and two edge regions of the
same length $L_{{\rm E}}$ at the left and right boundaries of the open chain.
The trace ${\rm Tr}_{{\rm B}}(\cdot)$ is only taken over the bulk region,
which excludes all possible interruptions caused by the NHSE in the edge
regions. The resulting $W_{0}$ was found to be able to faithfully capture the
topological phase transitions and bulk-edge correspondence in 1D, chiral
symmetric non-Hermitian systems even in the presence of NHSE NHSE10 . It was
also suggested to be equivalent to the topological winding number defined
through the non-Bloch Brillouin zone of non-Hermitian systems. By definition,
the winding number $W_{0}$ in Eq. (85) is also robust to perturbations induced
by symmetry-preserved disorders and impurities, making it applicable to more
general situations. In the clean, Hermitian and thermodynamic limit, the
$W_{0}$ in Eq. (85) can be further reduced to the $w_{0}$ in Eq. (83)
Tenfold01 ; WN01 ; WN02 .

For a non-Hermitian Floquet system, we can state its chiral symmetry as
follows ZhouNHFTP01 . From the Eq. (28), we can express the Floquet operator
as $\hat{U}=e^{-i\hat{H}_{{\rm eff}}}$, where we have set the driving period
$T=1$ for brevity. Viewing the $\hat{H}_{{\rm eff}}$ as a static Hamiltonian,
we say that it possesses the chiral symmetry if there exists a unitary
Hermitian operator ${\cal S}$ such that ${\cal S}\hat{H}_{{\rm eff}}{\cal
S}=-\hat{H}_{{\rm eff}}$. At the level of $\hat{U}$, the chiral symmetry then
implies that

${\cal S}\hat{U}{\cal S}=\hat{U}^{-1}.$ (87)

Similar to the case of static systems, the chiral symmetry of $\hat{U}$ has a
direct implication to the symmetry of its spectrum. If $|\Psi_{E}\rangle$ is
an eigenstate of $\hat{U}$ with the quasienergy $E$, i.e.,
$\hat{U}|\Psi_{E}\rangle=e^{-iE}|\Psi_{E}\rangle$, we will immediately have

$\hat{U}({\cal S}|\Psi_{E}\rangle)=e^{-i(-E)}({\cal S}|\Psi_{E}\rangle),$ (88)

which means that ${\cal S}|\Psi_{E}\rangle$ is also an eigenstate of $\hat{U}$
with the quasienergy $-E$. The Floquet spectrum of $\hat{U}$ is then symmetric
with respect to both the quasienergies $E=0$ and $E=\pi$. The latter is
because $-E$ and $E$ are identified at the quasienergy $\pi$. When the
spectrum is gapped at $E=0$ and $\pi$, the quasienergy levels of $\hat{U}$
could be grouped into two clusters. One of them has the quasienergy ${\rm
Re}E\in(-\pi,0)$ and then other one has ${\rm Re}E\in(0,\pi)$. They could meet
with each other at either the quasienergy zero or $\pi$, leading to two
possible phase transitions. This implies that a complete topological
characterization of a chiral symmetric Floquet system should require at least
two winding numbers, which is rather different from the case of static systems
where a single winding number is sufficient. To identify these winding
numbers, let us consider the example of a periodically kicked 1D system, whose
Hamiltonian and Floquet operator take the forms of the Eqs. (15) and (16). We
also assume that the two parts of Hamiltonians $\hat{H}_{0}$ and $\hat{H}_{1}$
in the Eq. (15) have the same chiral symmetry ${\cal S}$. At the level of
$\hat{U}=e^{-i\hat{H}_{0}}e^{-i\hat{H}_{1}}$ (assuming $T=1$), it is not
straightforward to identify the form of a chiral symmetry. However, we can
apply similarity transformations to $\hat{U}$ and express it in two symmetric
time frames FPiMod2  as

$\hat{U}_{1}=e^{-\frac{i}{2}\hat{H}_{1}}e^{-i\hat{H}_{0}}e^{-\frac{i}{2}\hat{H}_{1}},$
(89)
$\hat{U}_{2}=e^{-\frac{i}{2}\hat{H}_{0}}e^{-i\hat{H}_{1}}e^{-\frac{i}{2}\hat{H}_{0}}.$
(90)

It is then clear that ${\cal S}\hat{U}_{\alpha}{\cal S}=\hat{U}_{\alpha}^{-1}$
for $\alpha=1,2$, that is, the Floquet operators $\hat{U}_{1}$ and
$\hat{U}_{2}$ in the two symmetric time frames possess the chiral symmetry
${\cal S}$. We can thus introduce a winding number for each of them under the
PBC as ZhouNHFTP01

$w_{\alpha}=\int_{-\pi}^{\pi}\frac{dk}{4\pi}{\rm Tr}[{\cal S}{\cal
Q}_{\alpha}(k)i\partial_{k}{\cal Q}_{\alpha}(k)],$ (91)

where the Bloch quasimomentum $k\in[-\pi,\pi)$, $\alpha=1,2$, and

${\cal Q}_{\alpha}(k)=\sum_{n}{\rm sgn}\\{{\rm
Re}[E_{n}(k)]\\}|n_{\alpha}(k)\rangle\langle\widetilde{n}_{\alpha}(k)|.$ (92)

The $E_{n}(k)$ now denotes the quasienergy of the Floquet eigenstate
$|n_{\alpha}(k)\rangle$ of $\hat{U}_{\alpha}$ at the quasimomentum $k$ under
the PBC. Using the $w_{1}$ and $w_{2}$, we can construct another pair of
winding numbers $w_{0}$ and $w_{\pi}$, which are given by ZhouNHFTP01

$w_{0}=\frac{w_{1}+w_{2}}{2},\qquad w_{\pi}=\frac{w_{1}-w_{2}}{2}.$ (93)

In the Secs. III.2 and III.3, we will demonstrate with explicit examples that
the $w_{0}$ ($w_{\pi}$) could correctly capture the bulk topological
transition of non-Hermitian Floquet bands through the gap closing/reopening at
the quasienergy $E=0$ ($E=\pi$) in various chiral symmetric, non-Hermitian
Floquet insulating and superconducting models. Furthermore, in the absence of
NHSE, the $w_{0}$ and $w_{\pi}$ could also capture the numbers of Floquet edge
modes at zero and $\pi$ quasienergies under the OBC, and thus describing the
bulk-edge correspondence of the related models. In the presence of NHSE, we
can retrieve the characterization of topological transitions and bulk-edge
correspondence in chiral symmetric, non-Hermitian Floquet systems under the
OBC through the open-bulk winding numbers, in analogy with the Eq. (85). For
the $\hat{U}_{1}$ and $\hat{U}_{2}$ in the Eqs. (89) and (90), we can define a
winding number for each of them under the OBC as ZhouNHFTP08

$W_{\alpha}=-\frac{1}{L_{{\rm B}}}{\rm Tr}_{{\rm B}}({\cal S}{\cal
Q}_{\alpha}[{\cal Q}_{\alpha},\hat{N}]),\qquad\alpha=1,2.$ (94)

Here, the meanings of $L_{{\rm B}}$, ${\rm Tr}_{{\rm B}}$, ${\cal S}$ and
$\hat{N}$ are the same as those in the Eq. (85). The Floquet band projector in
the time frame $\alpha$ is given by

${\cal Q}_{\alpha}=\sum_{|\psi_{j}^{\alpha}\rangle\in{\rm bulk}}{\rm sgn}[{\rm
Re}(E_{j})]|\psi_{j}^{\alpha}\rangle\langle\widetilde{\psi}_{j}^{\alpha}|,$
(95)

where $|\psi_{j}^{\alpha}\rangle$ is the $j$th bulk eigenstate of
$\hat{U}_{\alpha}$ ($\alpha=1,2$) with the quasienergy $E_{j}$ under the OBC.
The linear combinations of $W_{1}$ and $W_{2}$ lead to another pair of winding
numbers ZhouNHFTP08

$W_{0}=\frac{W_{1}+W_{2}}{2},\qquad W_{\pi}=\frac{W_{1}-W_{2}}{2}.$ (96)

In Sec. III.2, we will illustrate that with the help of winding numbers
$(w_{0},w_{\pi})$ and $(W_{0},W_{\pi})$, a dual topological characterization
of the phase transitions, edge states and bulk-edge correspondence can be
established for 1D, chiral symmetric non-Hermitian Floquet systems under
different boundary conditions, regardless of whether there is the NHSE
ZhouNHFTP08 .

###  II.6 Dynamical indicators

In this section, we review two complementary dynamical probes in position and
momentum spaces. Both of them can be used to characterize the topological
properties of 1D non-Hermitian Floquet systems with chiral symmetry. These
indicators allow us to extract the topological winding numbers of the system
from its long-time stroboscopic dynamics ZhouNHFTP01 ; ZhouNHFTP02 ;
ZhouNHFTP03 .

####  II.6.1 Dynamic winding number (DWN)

The DWN DWN01 ; DWN02 , obtained from the long-time stroboscopic average of
spin textures, could provide us with information about the bulk topological
properties of non-Hermitian Floquet systems ZhouNHFTP03 . Let us consider a
1D, chiral symmetric non-Hermitian Floquet system with two quasienergy bands.
Under the PBC, we can express its Floquet operator in the symmetric time frame
$\alpha$ ($=1,2$) as $\hat{U}_{\alpha}=\sum_{k\in{\rm BZ}}|k\rangle
e^{-iH_{\alpha}(k)}\langle k|$. Here $H_{\alpha}(k)$ is the effective
Hamiltonian in time frame $\alpha$ and $k\in[-\pi,\pi)$ is the quasimomentum.
$\hat{U}_{\alpha}$ and $H_{\alpha}(k)$ share the same chiral symmetry ${\cal
S}$, i.e., ${\cal S}H_{\alpha}(k){\cal S}=-H_{\alpha}(k)$. For a non-Hermitian
system, the $\hat{U}_{\alpha}$ is generally not unitary and $H_{\alpha}(k)$ is
also not Hermitian. The right and left biorthonormal eigenvectors
$\\{|n_{\alpha}(k)\rangle\\}$ and $\\{|\widetilde{n}_{\alpha}(k)\rangle\\}$ of
$H_{\alpha}(k)$ satisfy the eigenvalue equations

$H_{\alpha}(k)|n_{\alpha}(k)\rangle=E_{n}(k)|n_{\alpha}(k)\rangle,$ (97)
$\langle\widetilde{n}_{\alpha}(k)|H_{\alpha}(k)=\langle\widetilde{n}_{\alpha}(k)|E_{n}(k).$
(98)

Here $n=\pm$ are the indices of the two Floquet bands with the quasienergies
$E_{\pm}(k)\equiv\pm E(k)$. The biorthonormal relationship requires

$\langle\widetilde{n}_{\alpha}(k)|n^{\prime}_{\alpha}(k)\rangle=\delta_{nn^{\prime}},\quad\sum_{n=\pm}|n_{\alpha}(k)\rangle\langle\widetilde{n}_{\alpha}(k)|=1,\quad\alpha=1,2.$
(99)

We consider the case in which the system is prepared in a general initial
state $|\psi_{\alpha}(k,0)\rangle=\sum_{n=\pm}c_{n}(k)|n_{\alpha}(k)\rangle$.
The corresponding initial state in the left Hilbert space reads
$|\widetilde{\psi}_{\alpha}(k,0)\rangle=\sum_{n\pm}c_{n}(k)|\widetilde{n}_{\alpha}(k)\rangle$,
such that initially $\sum_{n=\pm}|c_{n}(k)|^{2}=1$ at each $k$. After the
stroboscopic evolution over a number of $\ell$ driving periods, the right
initial state becomes

$|\psi_{\alpha}(k,\ell)\rangle=\sum_{n=\pm}c_{n}(k)e^{-i\ell
E_{n}(k)}|n_{\alpha}(k)\rangle.$ (100)

For the left initial state, we assume it to be evolved by a different
effective Hamiltonian
$\widetilde{H}_{\alpha}(k)=\sum_{\pm}E_{n}(k)|\widetilde{n}_{\alpha}(k)\rangle\langle
n_{\alpha}(k)|$, so that after the evolution over $\ell$ driving periods it
reaches the state

$|\widetilde{\psi}_{\alpha}(k,\ell)\rangle=\sum_{n=\pm}c_{n}(k)e^{-i\ell
E_{n}(k)}|\widetilde{n}_{\alpha}(k)\rangle.$ (101)

Note that the dynamical equation of $|\widetilde{\psi}_{\alpha}(k,0)\rangle$
we used here is different from that employed in the study of the adiabatic
perturbation theory in Sec. II.4.

The stroboscopic average of an observable $\hat{O}$ over
$|\widetilde{\psi}_{\alpha}(k,t)\rangle$ at the time $t=\ell T$ is then given
by

$\langle\hat{O}(k,\ell)\rangle_{\alpha}=\frac{\langle\widetilde{\psi}_{\alpha}(k,\ell)|\hat{O}|\psi_{\alpha}(k,\ell)\rangle}{\langle\widetilde{\psi}_{\alpha}(k,\ell)|\psi_{\alpha}(k,\ell)\rangle}.$
(102)

Without the loss of generality, we can consider the chiral symmetric
$H_{\alpha}(k)$ to be in the form of

$H_{\alpha}(k)=h_{\alpha x}(k)\sigma_{x}+h_{\alpha y}(k)\sigma_{y},$ (103)

where $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices. Any two out of the
three Pauli matrices $\sigma_{x,y,z}$ can be chosen to enter the
$H_{\alpha}(k)$, and the rest Pauli matrix [$\sigma_{z}$ for the
$H_{\alpha}(k)$ in the Eq. (103)] plays the role of the chiral symmetry
operator ${\cal S}$. By diagonalizing $H_{\alpha}(k)$, we obtain the
biorthonormal eigenvectors as

$|n_{\alpha}(k)\rangle=\frac{1}{\sqrt{2}E_{n}(k)}\begin{bmatrix}h_{\alpha
x}(k)-ih_{\alpha y}(k)\\\
E_{n}(k)\end{bmatrix},\qquad|\widetilde{n}_{\alpha}(k)\rangle=\frac{1}{\sqrt{2}E_{n}^{*}(k)}\begin{bmatrix}h_{\alpha
x}^{*}(k)-ih_{\alpha y}^{*}(k)\\\ E_{n}^{*}(k)\end{bmatrix},$ (104)

where $E_{n}(k)=n\sqrt{h_{\alpha x}^{2}(k)+h_{\alpha y}^{2}(k)}$ for $n=\pm$.
We can now compute the stroboscopic-averaged spin textures in the long-time
limit. For the $H_{\alpha}(k)$ in Eq. (103), this means that we need to find
the multi-cycle averages of $\sigma_{x}$ and $\sigma_{y}$. According to the
Eq. (102), they are given by

$r_{j}^{\alpha}(k)\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{\ell=1}^{N}\langle\sigma_{j}(k,\ell)\rangle_{\alpha},$
(105)

where $j=x,y$ and $\alpha=1,2$. $N$ counts the total number of driving
periods. From the averaged spin textures
$[r_{x}^{\alpha}(k),r_{y}^{\alpha}(k)]$, we can define the dynamic winding
angle as

$\theta_{yx}^{\alpha}(k)\equiv\arctan\left[\frac{r_{y}^{\alpha}(k)}{r_{x}^{\alpha}(k)}\right].$
(106)

The net winding number of $\theta_{yx}^{\alpha}(k)$ over a cycle in $k$-space
defines the dynamic winding number in the time frame $\alpha$, i.e.,

$\nu_{\alpha}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}\partial_{k}\theta_{yx}^{\alpha}(k),\qquad\alpha=1,2.$
(107)

In Ref. ZhouNHFTP03 , it was proved with straightforward calculations that in
the limit $N\rightarrow\infty$, the $\nu_{\alpha}$ will converge to the
$w_{\alpha}$ in the Eq. (91) if the initial condition satisfies
$c_{\pm}(k)\neq 0$ at each $k$. Therefore, by preparing the initial state at
different $k$ under this condition and measuring the averaged spin textures
over a long stroboscopic time, we can obtain the winding numbers
$(w_{0},w_{\pi})$ of a chiral symmetric 1D Floquet system (either Hermitian or
non-Hermitian) through the following combinations of $(\nu_{1},\nu_{2})$ in
two symmetric time frames ZhouNHFTP03

$w_{0}=\frac{\nu_{1}+\nu_{2}}{2},\qquad w_{\pi}=\frac{\nu_{1}-\nu_{2}}{2}.$
(108)

In the Sec. III, we will illustrate the application of this dynamical-
topological correspondence to non-Hermitian Floquet topological insulators in
one dimension.

####  II.6.2 Mean chiral displacement (MCD)

The MCD allows us to detect the winding numbers of a Floquet system from the
long-time averaged chiral displacement of an initially localized wavepacket.
It was first proposed as a way to probe the topological invariants of chiral
symmetric topological insulators in one dimension MCD01 ; MCD02 ; MCD03 .
Later, the MCD was used to obtain the winding numbers of Floquet systems
FTPLg04 ; FTPLg08  and also generalized to 2D higher-order topological
insulators FTPLg07 . Its applicability was demonstrated both theoretically and
experimentally MCD01 ; MCD03 .

For a chiral symmetric non-Hermitian system, the chiral displacement in the
symmetric time frame $\alpha$ ($=1,2$) can be defined as

$C_{\alpha}(t)={\rm
Tr}[\rho_{0}(\hat{\widetilde{U}}_{\alpha}^{\dagger})^{t}(\hat{N}\otimes{\cal
S})\hat{U}_{\alpha}^{t}].$ (109)

Here $t=\ell T$ denotes the stroboscopic time, with $T$ being the driving
period. $\hat{N}$ is the unit cell position operator and ${\cal S}$ is the
chiral symmetry operator. The initial state $\rho_{0}$ can be chosen as a
state localized in the middle of the lattice. For example, $\rho_{0}$ may take
the form of $(|0\rangle\langle 0|\otimes\sigma_{0})/2$ for a 1D bipartite
lattice, where $|0\rangle$ is the eigenbasis of the central unit cell and
$\sigma_{0}$ is an identity acting on the internal space of the two
sublattices. Both the Floquet operator $\hat{U}_{\alpha}$ and its dual
$\hat{\widetilde{U}}_{\alpha}$ have the chiral symmetry ${\cal S}$. In the
lattice representation, they can be expressed as

$\hat{U}_{\alpha}=\sum_{j=1}^{L}e^{-iE_{j}}|\psi_{j}^{\alpha}\rangle\langle\widetilde{\psi}_{j}^{\alpha}|,\qquad\hat{\widetilde{U}}_{\alpha}=\sum_{j=1}^{L}e^{-iE_{j}}|\widetilde{\psi}_{j}^{\alpha}\rangle\langle\psi_{j}^{\alpha}|.$
(110)

Here $L$ denotes the total number of degrees of freedom of the lattice.
$|\psi_{j}^{\alpha}\rangle$ and $\langle\widetilde{\psi}_{j}^{\alpha}|$
denotes the right and left eigenvector of $\hat{U}_{\alpha}$ with the
quasienergy $E_{j}$. They form a biorthonormal basis such that

$\langle\widetilde{\psi}_{j}^{\alpha}|\psi_{m}^{\alpha}\rangle=\delta_{jm},\qquad\sum_{j=1}^{L}|\psi_{j}^{\alpha}\rangle\langle\widetilde{\psi}_{j}^{\alpha}|=1.$
(111)

We now consider the stroboscopic long-time average of $C_{\alpha}(t)$ in the
time frames $\alpha=1,2$ for a 1D non-Hermitian Floquet system with chiral
symmetry. Under the PBC, taking the initial state to be
$\rho_{0}=(|0\rangle\langle 0|\otimes\sigma_{0})/2$ and performing the Fourier
transformation from position to momentum representations, we find the
$C_{\alpha}(t)$ in Eq. (109) to be ZhouNHFTP02

$C_{\alpha}(t)=\frac{1}{2}\int_{-\pi}^{\pi}\frac{dk}{2\pi}{\rm
Tr}[\widetilde{U}_{\alpha}^{\dagger t}(k){\cal
S}i\partial_{k}U_{\alpha}^{t}(k)].$ (112)

Here $k\in[-\pi,\pi)$ is the quasimomentum. $U_{\alpha}(k)=\langle
k|\hat{U}_{\alpha}|k\rangle$ and $\widetilde{U}_{\alpha}(k)=\langle
k|\hat{\widetilde{U}}_{\alpha}|k\rangle$ act on the internal degrees of
freedom (spins and/or sublattices) of the system at a fixed $k$. Taking the
long-time stroboscopic average and incorporating the normalization factor
${\rm
Tr}[\rho_{0}(\hat{\widetilde{U}}_{\alpha}^{\dagger})^{t}\hat{U}_{\alpha}^{t}]$,
we find the expression of MCD as ZhouNHFTP02

$\overline{C}_{\alpha}=\lim_{\ell\rightarrow\infty}\frac{1}{\ell
T}\sum_{t=T}^{\ell T}\int_{-\pi}^{\pi}\frac{dk}{2\pi}\frac{{\rm
Tr}\left[\widetilde{U}_{\alpha}^{\dagger t}(k){\cal
S}i\partial_{k}U_{\alpha}^{t}(k)\right]}{{\rm
Tr}\left[\widetilde{U}_{\alpha}^{\dagger t}(k)U_{\alpha}^{t}(k)\right]}.$
(113)

For a given $H_{\alpha}(k)$ of the form in the Eq. (II.6.1), we have ${\cal
S}=\sigma_{z}$, $U_{\alpha}(k)=e^{-iH_{\alpha}(k)}$ and
$\widetilde{U}_{\alpha}(k)=e^{-iE(k)H_{\alpha}^{\dagger}(k)/E^{*}(k)}$, where
$E(k)=\frac{1}{2}{\rm Tr}[H_{\alpha}^{2}(k)]$. One can then work out the Eq.
(113) explicitly and obtain ZhouNHFTP02

$\overline{C}_{\alpha}=\frac{w_{\alpha}}{2},\qquad\alpha=1,2.$ (114)

Here the $w_{\alpha}$ is nothing but the winding number of $\hat{U}_{\alpha}$
as defined in the Eq. (91). Therefore, by measuring the MCDs
$(\overline{C}_{1},\overline{C}_{2})$ in two symmetric time frames, we can
determine the topological winding numbers $(w_{0},w_{\pi})$ of a 1D chiral
symmetric Floquet system through the relations ZhouNHFTP02

$w_{0}=C_{0}\equiv\overline{C}_{1}+\overline{C}_{2},\qquad
w_{\pi}=C_{\pi}\equiv\overline{C}_{1}-\overline{C}_{2}.$ (115)

In the Sec. III, we will demonstrate the usage of MCD to dynamically probing
the winding numbers of first- and second-order non-Hermitian Floquet
topological insulators in one ZhouNHFTP02  and two ZhouNHFTP06  spatial
dimensions.

###  II.7 Localization transition and mobility edge

In the last part of this section, we recap some tools that can be used to
characterize the real-to-complex spectral transitions, localization
transitions and mobility edges in disordered non-Hermitian Floquet systems
ZhouNHFTP09 ; ZhouNHFTP11 . Considering the relevance to this review, we focus
on a 1D quadratic lattice model of the form

$\hat{H}(t)=\sum_{\langle
n,n^{\prime}\rangle}[J_{n}(t)e^{\gamma_{n}}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}+J_{n}^{*}(t)e^{-\gamma_{n}}\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}]+\sum_{n}V_{n}(t)\hat{c}_{n}^{\dagger}\hat{c}_{n}.$
(116)

Here $\langle n,n^{\prime}\rangle$ includes the lattice site indices $n$ and
$n^{\prime}$ with $n^{\prime}>n$. $\hat{c}_{n}^{\dagger}$ ($\hat{c}_{n}$)
creates (annihilates) a particle on the site $n$ and the parameter
$\gamma\in\mathbb{R}$. The Hamiltonian $\hat{H}(t)$ is non-Hermitian if
$\gamma_{n}\neq 0$ for some $n$ (nonreciprocal hopping) or $V_{n}(t)\neq
V_{n}^{*}(t)$ (onsite gain and loss). $\hat{H}(t)$ is further time-periodic if
$J_{n}(t)=J_{n}(t+T)$ or $V_{n}(t)=V_{n}(t+T)$ for all $n$, with $T$ being the
driving period. The disorder terms may be included within $V_{n}(t)$ (diagonal
disorder) or $J_{n}(t)$ (off-diagonal disorder). As an example, for a 1D non-
Hermitian quasicrystal (NHQC) with correlated onsite disorder, the $V_{n}(t)$
may take the form of $V_{n}(t)=V(t)\cos(2\pi\alpha n+i\beta)$. Here $\alpha$
is irrational and $\beta\in\mathbb{R}$, with $i\beta$ describing an imaginary
phase shift in the superlattice potential $V_{n}$.

The Floquet operator of the system described by the $\hat{H}(t)$ in Eq. (116)
takes the general form of
$\hat{U}=\hat{\mathsf{T}}e^{-i\int_{0}^{T}\hat{H}(t)dt}=e^{-i\hat{H}_{{\rm
eff}}T}$. If $\hat{H}(t)$ possesses the PT-symmetry, $\hat{H}_{{\rm eff}}$
could have the same symmetry and the quasienergy spectrum of $\hat{U}$ could
be real. With the increase of the non-Hermitian parameter of $\hat{H}(t)$, the
Floquet spectrum of $\hat{U}$ may undergo a PT-breaking transition, after
which certain quasienergies of $\hat{U}$ (obtained by solving
$\hat{U}|\Psi\rangle=e^{-iE}|\Psi\rangle$) may acquire nonvanishing imaginary
parts. To take into account such a spectral transition, we introduce the
following quantities

$\max|{\rm Im}E|\equiv\max_{j\in\\{1,...,L\\}}(|{\rm Im}E_{j}|),$ (117)
$\rho\equiv N({\rm Im}E\neq 0)/L.$ (118)

Here $E_{j}$ is the $j$th quasienergy eigenvalue of $\hat{U}$. $L$ counts the
Hilbert space dimension of $\hat{U}$, i.e., the total number of degrees of
freedom of the lattice. $N({\rm Im}E\neq 0)$ denotes the number of quasienergy
eigenvalues whose imaginary parts are nonzero. $\rho$ thus describes the
density of states with complex quasienergies in the system. In the PT-
invariant phase, we would have $\max|{\rm Im}E|=\rho=0$, implying that all the
quasienergies of $\hat{U}$ are real. In the PT-broken phase, we would have
$\max|{\rm Im}E|>0$ and $\rho\in(0,1]$, meaning that there is a finite number
of eigenstates of $\hat{U}$ whose quasienergies are complex. Specially, we
would have $\rho\simeq 1$ if almost all the Floquet eigenstates of $\hat{U}$
possess complex quasienergies. Therefore, by locating the positions where both
the $\max|{\rm Im}E|$ and $\rho$ start to deviate from zero in the parameter
space, we can identify the PT transition points for the Floquet spectrum of
$\hat{U}$. We will see that due to the interplay between periodic drivings and
non-Hermitian effects, both PT-symmetry breaking and restoring transitions
could be induced by tuning a single parameter in the Floquet NHQC.

The localization nature of the Floquet eigenstates of $\hat{U}$ can be
characterized by the statistics of its quasienergy levels and the inverse
participation ratios (IPRs). Let us denote the normalized right eigenvectors
of $\hat{U}$ and their corresponding quasienergies as
$\\{|\psi_{j}\rangle|j=1,...,L\\}$ and $\\{E_{j}|j=1,...,L\\}$. Along the real
axis, the spacing between the $j$th and the $(j-1)$th quasienergies of
$\hat{U}$ is given by $\epsilon_{j}={\rm Re}E_{j}-{\rm Re}E_{j-1}$, from which
we obtain the ratio between two adjacent spacings of quasienergy levels as

$g_{j}=\frac{\min(\epsilon_{j},\epsilon_{j+1})}{\max(\epsilon_{j},\epsilon_{j+1})},\qquad
j=2,...,L-1.$ (119)

Here the $\max(\epsilon_{j},\epsilon_{j+1})$ and
$\min(\epsilon_{j},\epsilon_{j+1})$ find the maximum and minimum between the
$\epsilon_{j}$ and $\epsilon_{j+1}$, respectively. The statistical property of
adjacent gap ratios can then be obtained by averaging over all $g_{j}$ in the
thermodynamic limit, i.e.,

$\overline{g}=\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{j}g_{j}.$ (120)

We will have $\overline{g}\rightarrow 0$ if all the bulk Floquet eigenstates
of $\hat{U}$ are extended. Comparatively, we expect $\overline{g}$ to approach
a constant $\overline{g}_{\max}>0$ if all the bulk eigenstates of $\hat{U}$
are localized. If we find $\overline{g}\in(0,\overline{g}_{\max})$, extended
and localized eigenstates of $\hat{U}$ should coexist and be separated by some
mobility edges in the quasienergy spectrum. The $\overline{g}$ can thus be
utilized to distinguish phases with different localization nature in 1D non-
Hermitian Floquet systems from the perspective of level statistics. In the
lattice representation, we can expand the right eigenvector $|\psi_{j}\rangle$
as $|\psi_{j}\rangle=\sum_{n=1}^{L}\psi_{n}^{j}|n\rangle$, where
$\psi_{n}^{j}=\langle n|\psi_{j}\rangle$ and $\sum_{n}|\psi_{n}^{j}|^{2}=1$.
The inverse and normalized participation ratios of $|\psi_{j}\rangle$ in the
real space can then be defined as

${\rm IPR}_{j}=\sum_{n=1}^{L}|\psi_{n}^{j}|^{4},\qquad{\rm
NPR}_{j}=\frac{1}{L}{\rm IPR}_{j}^{-1}.$ (121)

For a localized Floquet eigenstate $|\psi_{j}\rangle$, we will have ${\rm
IPR}_{j}\rightarrow\lambda_{j}$ and ${\rm NPR}_{j}\rightarrow 0$ in the
thermodynamic limit, where the Lyapunov exponent $\lambda_{j}$ of
$|\psi_{j}\rangle$ could be a function of its quasienergy $E_{j}$. If
$|\psi_{j}\rangle$ represents an extended state, we are going to have ${\rm
IPR}_{j}\rightarrow 0$ and ${\rm NPR}_{j}\rightarrow 1$. The global
localization features of the Floquet system described by $\hat{U}$ can then be
extracted from the combined information of $\\{{\rm IPR}_{j}|j=1,...,L\\}$ and
$\\{{\rm NPR}_{j}|j=1,...,L\\}$. For the ease of usage, we introduce the
following localization indicators

${\rm IPR}_{\max}=\max_{j\in\\{1,...,L\\}}({\rm IPR}_{j}),$ (122) ${\rm
IPR}_{\min}=\min_{j\in\\{1,...,L\\}}({\rm IPR}_{j}),$ (123)
$\zeta=\log_{10}({\rm IPR}_{{\rm ave}}\cdot{\rm NPR}_{{\rm ave}}),$ (124)

where the ${\rm IPR}_{{\rm ave}}=\frac{1}{L}\sum_{j=1}^{L}{\rm IPR}_{j}$ and
${\rm NPR}_{{\rm ave}}=\frac{1}{L}\sum_{j=1}^{L}{\rm NPR}_{j}$ are the
averages of ${\rm IPR}_{j}$ and ${\rm NPR}_{j}$ over all Floquet eigenstates.
It is not hard to realize that in the metallic phase where all Floquet
eigenstates are extended, we will have ${\rm IPR}_{\max}\rightarrow 0$, ${\rm
IPR}_{\min}\rightarrow 0$ and $\zeta\sim-\log_{10}(L)\rightarrow-\infty$ in
the thermodynamic limit $L\rightarrow\infty$. In the insulating phase with all
Floquet eigenstates being localized, we will instead have ${\rm
IPR}_{\max}>0$, ${\rm IPR}_{\min}>0$ and $\zeta\rightarrow-\infty$. In the
critical phase where extended and localized Floquet eigenstates are
coexistent, we will have ${\rm IPR}_{\max}>0$, ${\rm IPR}_{\min}\rightarrow 0$
together with a finite $\zeta$. Assembling the information obtained from ${\rm
IPR}_{\max}$, ${\rm IPR}_{\min}$ and $\zeta$ thus allow us to distinguish the
extended, localized and critical mobility edge phases of a non-Hermitian
Floquet system with disorder. These tools will be applied to our study of the
Floquet NHQC in the Sec. III.4.

We can also probe the transport nature of distinct non-Hermitian Floquet
phases from the wave packet dynamics. Let us consider a generic and normalized
initial state $|\Psi(0)\rangle$ in the lattice representation. After the
stroboscopic evolution over a number of $\ell$ driving periods by $\hat{U}$,
the final state turns out to be $|\Psi^{\prime}(t=\ell
T)\rangle=\hat{U}^{\ell}|\Psi(0)\rangle$. Since the Floquet operator $\hat{U}$
is not unitary for a non-Hermitian $\hat{H}(t)$ in general, the norm of
$|\Psi(0)\rangle$ cannot be preserved during the evolution. We can express the
normalized state at $t=\ell T$ as
$|\Psi(t)\rangle=|\Psi^{\prime}(t)\rangle/\sqrt{\langle\Psi^{\prime}(t)|\Psi^{\prime}(t)\rangle}$.
The real space expansion $|\Psi(t)\rangle=\sum_{n=1}^{L}\psi_{n}(t)|n\rangle$
then provides us with the probability amplitude $\psi_{n}(t)=\langle
n|\Psi(t)\rangle$ of the normalized final state $|\Psi(t)\rangle$ on the
lattice site $n$ at time $t$. Using the collection of amplitudes
$\\{\psi_{n}(t)|n=1,...,L\\}$, we can define the following dynamical
quantities

$\langle x(t)\rangle=\sum_{n=1}^{L}n|\psi_{n}(t)|^{2},\qquad\langle
x^{2}(t)\rangle=\sum_{n=1}^{L}n^{2}|\psi_{n}(t)|^{2},$ (125) $\Delta
x(t)=\sqrt{\langle x^{2}(t)\rangle-\langle x(t)\rangle^{2}},\qquad
v(t)=\frac{1}{t}\sqrt{\langle x^{2}(t)\rangle},$ (126)

where the stroboscopic time $t=\ell T$ and $\ell\in\mathbb{Z}$. It is clear
that the $\langle x(t)\rangle$, $\Delta x(t)$ and $v(t)$ describe the center,
standard deviation and spreading speed of the wavepacket $|\Psi(t)\rangle$ in
the lattice representation, respectively. For simplicity, we usually choose
the initial state $|\Psi(0)\rangle$ to be exponentially localized at a single
site that is deep inside the bulk of the lattice. If the Floquet system
described by $\hat{U}$ resides in a localized phase, we expect the $\langle
x(t)\rangle$ and $\Delta x(t)$ to stay around their initial values, which
means that the wavepacket almost does not move and spread. In this case, the
speed $v(t)$ should also tend to zero for a long-time evolution ($\ell\gg 1$).
If the system stays in an extended phase, we expect the increasing of $\Delta
x(t)$ with time due to the spreading of the initial wavepacket. Meanwhile, the
$\langle x(t)\rangle$ may or may not change with time depending on whether the
hopping amplitudes are symmetric. For a nearest-neighbor, nonreciprocal
hopping as in Eq. (116), we may have $\langle x(t)\rangle\propto t$ and
$\Delta x(t)\propto\sqrt{t}$ for a metallic phase of the system. The $v(t)$
should take a maximal possible value $v_{\max}(t)$ in this case. If the system
is prepared in a critical mobility edge phase, we expect both the $\langle
x(t)\rangle$ and $\Delta x(t)$ to show intervening behaviors, while the
averaged spreading speed $v(t)$ should satisfy $0<v(t)<v_{\max}(t)$.
Therefore, we can exploit the dynamical quantities $\langle x(t)\rangle$,
$\Delta x(t)$ and $v(t)$ to discriminate phases with different localization
nature in disordered non-Hermitian Floquet systems. We will illustrate such an
application for Floquet NHQC in the Sec. III.4.

##  III Non-Hermitian Floquet phases of matter

This section collects and treats typical examples of non-Hermitian Floquet
phases discovered in our previous work ZhouNHFTP01 ; ZhouNHFTP02 ; ZhouNHFTP03
; ZhouNHFTP04 ; ZhouNHFTP05 ; ZhouNHFTP06 ; ZhouNHFTP07 ; ZhouNHFTP08 ;
ZhouNHFTP09 ; ZhouNHFTP10 ; ZhouNHFTP11 . We will see that the collaboration
between periodic driving fields and gain/loss or nonreciprocal effects could
induce rich phases and transitions in non-Hermitian Floquet systems.

###  III.1 Non-Hermitian Floquet exceptional topology

Let us start with a simple model, whose Floquet effective Hamiltonian and
quasienergy dispersion are given by the Eqs. (55) and (57). To be explicit, we
choose

$h_{x}=\sin k,\qquad h_{z}=\mu+\cos k,$ (127)

where $k\in(-\pi,\pi]$ denotes the quasimomentum in the first Brillouin zone,
and the system parameter $\mu\in\mathbb{R}$. The $H_{{\rm eff}}$ in Eq. (55)
thus describes the Bloch Hamiltonian of a 1D two-band lattice under the PBC in
the rotating frame, where $\mu$ corresponds to the amplitude of onsite
potential and the nearest-neighbor hopping amplitude has been chosen to be the
unit of energy. From the Eqs. (57) and (127), we find that the two quasienergy
bands of $H_{{\rm eff}}$ could meet with each other at the quasienergy
$\omega/2$ when

$\mu=\begin{cases}\frac{\omega}{2}\pm\frac{\gamma}{2}-1,&{\rm for}\quad
k=0,\\\ \frac{\omega}{2}\pm\frac{\gamma}{2}+1,&{\rm for}\quad
k=\pi.\end{cases}$ (128)

If $\mu$ satisfies one of the above two equalities, there will be a second-
order Floquet EP at $k=0$ or $k=\pi$ in the conventional Brillouin zone. Note
that both the quasienergy of this Floquet EP and its location in the parameter
space depend on the frequency $\omega$, which highlights the impact of the
harmonic driving field on phase transitions in the system. Moreover, the
appearance of an EP in $k$-space usually implies the breakdown of bulk-edge
correspondence in conventional topological phases. This issue might be
overcome by incorporating the formalism of generalized Brillouin zone (GBZ)
and non-Bloch band theory NHSE05 ; NHSE07 . Following the standard recipe, we
first make the substitutions $e^{ik}\rightarrow\beta$ and
$e^{-ik}\rightarrow\beta^{-1}$, where the complex number $\beta\in{\rm GBZ}$.
The effective Hamiltonian in Eq. (55) now takes the form of

$H_{{\rm
eff}}(\beta)=\frac{\omega}{2}\sigma_{0}+\left(\frac{\beta-\beta^{-1}}{2i}+i\frac{\gamma}{2}\right)\sigma_{x}+\left(\mu+\frac{\beta+\beta^{-1}}{2}-\frac{\omega}{2}\right)\sigma_{z},$
(129)

with the quasienergies bands

$\varepsilon_{\pm}(\beta)=\frac{\omega}{2}\pm\sqrt{\left(\frac{\beta-\beta^{-1}}{2i}+\frac{i\gamma}{2}\right)^{2}+\left(\mu-\frac{\omega}{2}+\frac{\beta+\beta^{-1}}{2}\right)^{2}}.$
(130)

We next focus on the non-constant part of $\varepsilon_{\pm}(\beta)$, whose
square takes the form

$\epsilon_{\pm}^{2}(\beta)=\frac{a\beta^{2}+b\beta+c}{\beta},$ (131)

where

$a=\mu-\frac{\omega}{2}+\frac{\gamma}{2},\qquad
b=1+\left(\mu-\frac{\omega}{2}\right)^{2}-\frac{\gamma^{2}}{4},\qquad
c=\mu-\frac{\omega}{2}-\frac{\gamma}{2}.$ (132)

The $\varepsilon_{\pm}^{2}(\beta)$ is a Laurent polynomial of $\beta$.
According to Ref. NHSE07  (see also Ref. NHCI01 ), $\beta\in{\rm GBZ}$ for the
$H_{{\rm eff}}(\beta)$ if $\epsilon_{\pm}^{2}(\beta)=\epsilon_{\pm}^{2}(\beta
e^{i\theta})$, with $\theta$ being a phase factor. For the Eq. (131), this
means that $\beta^{2}=ce^{-i\theta}/a$. The GBZ is thus a circle of the radius

$r=|\beta|=\left|\frac{\mu-\omega/2-\gamma/2}{\mu-\omega/2+\gamma/2}\right|^{1/2}.$
(133)

It is clear that the GBZ radius is controlled by both the driving field
(through $\omega$) and the non-Hermitian effect (through $\gamma$) . In the
Hermitian limit ($\gamma\rightarrow 0$), we have $r\rightarrow 1$ and the GBZ
is reduced to the conventional BZ, as expected. The GBZ becomes ill defined if
$\mu-\omega/2=\pm\gamma/2$, where the radius $r$ is zero or infinity. Finally,
we can use the GBZ to determine the gap closing (phase transition) points of
the system under the OBC. Setting the $\epsilon_{\pm}^{2}(\beta)=0$ in the Eq.
(131), we find

$\mu=\frac{\omega}{2}\pm\begin{cases}\sqrt{\gamma^{2}/4+1},&|\gamma|\leq 2,\\\
\sqrt{\gamma^{2}/4\pm 1},&|\gamma|>2.\end{cases}$ (134)

These bulk gap-closing conditions are clearly different from those found in
conventional BZ under the PBC in the Eq. (128). Using the transfer matrix
method NHSE09 , we can further obtain the parameter regions in which
degenerate Floquet edge states appear at the quasienergy $\omega/2$ (anomalous
Floquet $\pi$ edge modes), i.e.,

$\mu-\frac{\omega}{2}\in\begin{cases}\left(-\sqrt{\gamma^{2}/4+1},\sqrt{\gamma^{2}/4+1}\right),&|\gamma|\leq
2,\\\
\left(-\sqrt{\gamma^{2}/4+1},-\sqrt{\gamma^{2}/4-1}\right)\cup\left(\sqrt{\gamma^{2}/4-1},\sqrt{\gamma^{2}/4+1}\right),&|\gamma|>2.\end{cases}$
(135)

Finally, we can characterize the different topological phases of the system by
a non-Bloch winding number NHSE10

$W=\int_{{\rm GBZ}}\frac{d\beta}{4\pi i}{\rm Tr}[\sigma_{y}{\cal
Q}(\beta)\partial_{\beta}{\cal Q}(\beta)],$ (136)

where $\sigma_{y}$ is the chiral symmetry operator of the shifted effective
Hamiltonian $H_{{\rm eff}}(\beta)-\frac{\omega}{2}\sigma_{0}$. The Q matrix is
defined as ${\cal
Q}(\beta)=|\psi_{+}(\beta)\rangle\langle\widetilde{\psi}_{+}(\beta)|-|\psi_{-}(\beta)\rangle\langle\widetilde{\psi}_{-}(\beta)|$.
$|\psi_{\pm}(\beta)\rangle$ are the right eigenvectors of $H_{{\rm
eff}}(\beta)$ with the quasienergies $\varepsilon_{\pm}(\beta)$.
$\langle\widetilde{\psi}_{\pm}(\beta)|$ are the corresponding left
eigenvectors. For our model, their explicit expressions are given by

$|\psi_{\pm}(\beta)\rangle=\frac{1}{\sqrt{2(\varepsilon_{\pm}-\frac{\omega}{2})(\varepsilon_{\pm}-h_{z})}}\begin{pmatrix}h_{x}+i\gamma/2\\\
\varepsilon_{\pm}-h_{z}\end{pmatrix},$ (137)
$\langle\widetilde{\psi}_{\pm}(\beta)|=\frac{1}{\sqrt{2(\varepsilon_{\pm}-\frac{\omega}{2})(\varepsilon_{\pm}-h_{z})}}\begin{pmatrix}h_{x}+i\gamma/2&\varepsilon_{\pm}-h_{z}\end{pmatrix}.$
(138)

Figure 2: Floquet spectra and winding numbers of the harmonically driven 1D
non-Hermitian lattice model. The gray dots at each $\mu$ denote the real [in
(a) and (b)] and imaginary [in (c) and (d)] parts of quasienergies obtained
under the PBC in the conventional BZ. The blue dots at each $\mu$ denote the
real [in (a) and (b)] and imaginary [in (c) and (d)] parts of quasienergies
obtained under the OBC in the GBZ. The yellow circles denote the winding
number $W$, which is equal to $1$ ($0$) in the topological (trivial) phase
with (without) anomalous Floquet edge modes at the quasienergies $\pm\pi$,
which are highlighted by the red solid lines. Other system parameters are
chosen as $(\omega,\gamma)=(2\pi,1)$ [$(\omega,\gamma)=(2\pi,3)$] for the
panels (a) and (c) [(b) and (d)].

In Fig. 2, we show the Floquet spectra of $H_{{\rm eff}}$ under different
boundary conditions and the non-Bloch winding numbers of $H_{{\rm eff}}$
versus $\mu$ for some typical cases. We see that the Floquet spectra and the
gap closing points at the quasienergies $E=\pm\pi$ could indeed be very
different under the PBC and the OBC. Some high-order EPs can be identified
from the bulk Floquet spectra under the OBC, which clearly signify the
emergence of Floquet exceptional topology. Besides, the non-Bloch winding
number $W$ can correctly discriminate different topological phases and
characterize topological phase transitions accompanied by quasienergy-band
touchings under the OBC at $E=\pi$. We find $W=1$ ($W=0$) in the topologically
nontrivial (trivial) phases with (without) anomalous Floquet $\pi$ edge modes.
The bulk-edge correspondence is then recovered. The gap-closing points and the
parameter regions with Floquet $\pi$ modes are found to be perfectly
coincident with the predictions of the Eqs. (134) and (135). These
observations demonstrate the applicability of non-Bloch band theories to non-
Hermitian Floquet systems. It is also straightforward to check that the system
possesses Floquet NHSEs in a broad parameter regime under the OBC. The simple
model introduced in this subsection thus allows us to get a bird’s-eye view on
the nontrivial topological phenomena that could be brought about by the
interplay between Floquet driving fields and non-Hermitian effects. Further
examples with more striking features will be reviewed in the following
subsections.

###  III.2 Non-Hermitian Floquet topological insulators

In this subsection, we review three types of non-Hermitian Floquet topological
insulators in one and two spatial dimensions. For all the cases, we uncover
that the collaboration between driving and gain/loss or nonreciprocal effects
could induce topological insulating states unique to non-Hermitian Floquet
systems. They are featured by large topological invariants, many topological
edge or corner modes, and separated by rich topological phase transitions. The
bulk-edge (or corner) correspondence will also be established for each class
of systems considered in this subsection.

####  III.2.1 First-order topological phase

We start with the characterization of 1D non-Hermitian Floquet topological
insulators. One typical model that incorporates their rich topological
properties is given by a periodically quenched dimerized tight-binding
lattice, whose time-dependent Hamiltonian reads ZhouNHFTP01

$\hat{H}(t)=\begin{cases}\hat{H}_{1},&t\in[\ell T,\ell T+T/2),\\\
\hat{H}_{2},&t\in[\ell T+T/2,\ell T+T),\end{cases}$ (139)

where

$\hat{H}_{1}=\sum_{n}(ir_{y}\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+{\rm
H.c.}+2i\gamma\hat{c}_{n}^{\dagger}\hat{c}_{n})\otimes\sigma_{y},$ (140)
$\hat{H}_{2}=\sum_{n}(r_{x}\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+\mu\hat{c}_{n}^{\dagger}\hat{c}_{n}+{\rm
H.c.})\otimes\sigma_{x}.$ (141)

Here $T$ is the driving period and we will set $\hbar=T=1$ in our
calculations. $\hat{c}_{n}^{\dagger}$ $(\hat{c}_{n})$ creates (annihilates) a
particle in the unit cell $n$ of the lattice. $\sigma_{x,y,z}$ are Pauli
matrices acting on the two sublattice degrees of freedom A and B within each
unit cell. The system parameters $r_{x},r_{y},\mu,\gamma$ are all real. The
non-Hermitian effect is introduced by the nonreciprocal intracell coupling
term $2i\gamma\sigma_{y}$ applied over the first half of each driving period
[see Fig. 3(a) for an illustration of the model]. Experimentally, such a term
might be realized by coupled-resonator optical waveguide with asymmetric
internal scattering MalzardPRL2015 .

Considering the one-period evolution of the system from $t=\ell+0^{-}$ to
$t=\ell+1+0^{-}$, the Floquet operator of $\hat{H}(t)$ takes the form of
$\hat{U}=e^{-\frac{i}{2}\hat{H}_{2}}e^{-\frac{i}{2}\hat{H}_{1}}$. Its
quasienergy spectrum and Floquet eigenstates are obtained by solving the
eigenvalue equation $\hat{U}|\psi\rangle=e^{-iE}|\psi\rangle$. Under the OBC,
we show the Floquet spectrum of $\hat{U}$ on the complex quasienergy plane
${\rm Re}E-{\rm Im}E$ for some typical cases in Figs. 3(b)–(e) [with
$(\mu,r_{y},\gamma)=(0,\pi/2,{\rm arccosh}(\sqrt{2}))$,
$r_{x}=\pi/2,\pi,3\pi/2,2\pi$, and the number of unit cells $N=150$]. The
numbers of degenerate Floquet edge modes at the quasienergies $E=0$ and
$E=\pi$ are denoted by $n_{0}$ and $n_{\pi}$ in the corresponding figure
captions. We observe one or multiple pairs of edge modes at both the center
($E=0$) and boundary ($E=\pi$) of the first quasienergy Brillouin zone.
Interestingly, when the numbers of these edge modes are the same, i.e.,
$n_{0}=n_{\pi}$, we find them to appear in different types of quasienergy gaps
[see Figs. 3(c) and 3(e)]. For example, we find a pair of Floquet zero modes
in the line gap at $E=0$, while another pair of Floquet $\pi$ modes are found
in the point gap at $E=\pi$. This is rather different from the situation in
Hermitian Floquet systems, where the quasienergy is real and there is no
distinction between and point and line gaps. It is also different from the
cases encountered in non-Hermitian static systems with two bands, where
topological edge modes can only appear in either a point gap or a line gap at
$E=0$. The Floquet spectra with hybrid (point plus line) quasienergy gaps are
thus unique to non-Hermitian Floquet systems.

Figure 3: Non-Hermitian Floquet topological insulator: (a) a schematic diagram
of the lattice model, (b)–(e) quasienergy spectra on the complex plane under
the OBC, (f)–(g) topological phase diagrams under the PBC, and (h)–(j) Floquet
spectra and edge states under the OBC ZhouNHFTP01 .

To understand the origin of the non-Hermitian Floquet edge modes at $E=0$ and
$E=\pi$, we study the bulk topological properties of the system. Under the
PBC, the Floquet operator of the system reads

$U(k)=e^{-i(\mu+r_{x}\cos k)\sigma_{x}}e^{-i(r_{y}\sin k+i\gamma)\sigma_{y}},$
(142)

where $k\in[-\pi,\pi)$ is the quasimomentum. The quasienergy spectrum of
$U(k)$ is given by

$\pm E(k)=\pm\arccos[\cos(\mu+r_{x}\cos k)\cos(r_{y}\sin k+i\gamma)],$ (143)

and the gap-closing conditions can be obtained analytically by setting
$\cos[E(k)]=\pm 1$ ZhouNHFTP01 . Transforming $U(k)$ to symmetric time frames,
we obtain

$U_{1}(k)=e^{-\frac{i}{2}(\mu+r_{x}\cos k)\sigma_{x}}e^{-i(r_{y}\sin
k+i\gamma)\sigma_{y}}e^{-\frac{i}{2}(\mu+r_{x}\cos k)\sigma_{x}},$ (144)
$U_{2}(k)=e^{-\frac{i}{2}(r_{y}\sin k+i\gamma)\sigma_{y}}e^{-i(\mu+r_{x}\cos
k)\sigma_{x}}e^{-\frac{i}{2}(r_{y}\sin k+i\gamma)\sigma_{y}}.$ (145)

It is clear that both $U_{1}(k)$ and $U_{2}(k)$ possesses the chiral
(sublattice) symmetry ${\cal S}=\sigma_{z}$. We can thus characterize the non-
Hermitian Floquet topological phases of $U(k)$ by the winding numbers
$(w_{0},w_{\pi})$ according to the Eq. (93). In Figs. 3(f) and 3(g), we
present the $w_{0}$ and $w_{\pi}$ of $U(k)$ versus $(r_{x},r_{y})$
respectively, yielding the topological phase diagram of the system. We find
various non-Hermitian Floquet insulating phases. They are characterized by
large integer winding numbers and separated by a series of topological phase
transitions with quasienergy level crossings at $E=0$ [white solid lines in
Fig. 3(f)] and $E=\pi$ [white dashed lines in Fig. 3(g)]. The winding numbers
$(w_{0},w_{\pi})$ could become arbitrarily large with the increase of system
parameters (e.g., the intercell hopping amplitude $r_{x}$), which implies that
Floquet phases with large topological invariants could indeed survive even
with finite non-Hermitian effects [$\mu=0$ and $\gamma={\rm
arccosh}(\sqrt{2})$ in Figs. 3(f)–(g)]. Moreover, these winding numbers
correctly count the numbers of degenerate Floquet edge modes $n_{0}$ and
$n_{\pi}$ at $E=0$ and $E=\pi$ under the OBC. That is, we have the bulk-edge
correspondence for our 1D chiral symmetric non-Hermitian Floquet topological
insulator as ZhouNHFTP01

$n_{0}=2|w_{0}|,\qquad n_{\pi}=2|w_{\pi}|.$ (146)

Representative examples of the Floquet spectra under the OBC are shown in
Figs. 3(h)–(j) [with $\mu=0$, $r_{y}=\pi/2$, $\gamma={\rm arccosh}(\sqrt{2})$
and the number of unit cells $N=150$], which provide verifications for the
found bulk-edge correspondence in Eq. (146). Note in passing that the system
does not show NHSE under the OBC for $\mu=0$, even though the Floquet spectra
possess point quasienergy gaps under the PBC. Meanwhile, many pairs of
degenerate edge modes at the quasienergies $E=0$ and $E=\pi$ are found in the
topological phases with large winding numbers. As these topological edge modes
could survive in the presence of non-Hermitian effects, they may provide more
resources for the realization of robust quantum state transfer and quantum
computing schemes in open systems.

Figure 4: Dynamical characterization of non-Hermitian Floquet topological
phases: (a) winding numbers ($w_{0},w_{\pi}$), MCDs ($C_{0},C_{\pi}$), and
(b)–(e) dynamic winding numbers from stroboscopic time-averaged spin textures
ZhouNHFTP03 .

The winding numbers of non-Hermitian Floquet topological insulators can be
experimentally probed by measuring the stroboscopic time-averaged spin
textures ZhouNHFTP01  or the MCDs ZhouNHFTP02 , as introduced in the Secs.
II.6.1 and II.6.2. Following Ref. ZhouNHFTP03 , we consider a periodically
quenched non-Hermitian lattice model, whose Floquet operator in the momentum
space reads $U(k)=e^{-iJ_{2}\sin k\sigma_{y}}e^{-iJ_{1}\cos k\sigma_{x}}$.
Here $J_{1}=u_{1}+iv_{1}$, $J_{2}=u_{2}+iv_{2}$, and
$u_{1},u_{2},v_{1},v_{2}\in\mathbb{R}$. The winding numbers $(w_{0},w_{\pi})$
of $U(k)$ can be obtained using the approach of symmetric time frames [see the
Eqs. (89)–(93)]. Following the recipes outline in the Secs. II.6.1 and II.6.2,
we obtain the MCDs in Fig. 4(a) [with $v=v_{1}=v_{2}$ and
$(u_{1},u_{2})=(4.5\pi,0.5\pi)$] and the dynamic winding angles of time-
averaged spin textures in Figs. 4(b)–(e) [with $(u_{1},u_{2})=(4.5\pi,0.5\pi)$
and $v=v_{1}=v_{2}=0.2\pi,0.35\pi,0.6\pi,0.9\pi$]. In all the cases, the
obtained MCDs and dynamic winding numbers are coincident with the winding
numbers $(w_{0},w_{\pi})$ of Floquet operator $U(k)$ even when there are
finite non-Hermitian effects ($v\neq 0$). The dynamic winding numbers and the
MCDs thus provide us with two complementary approaches to dynamically
characterize 1D non-Hermitian Floquet topological phases with chiral
(sublattice) symmetry in momentum and in real spaces, respectively. They can
be applied to detect non-Hermitian Floquet matter in different types of
experimental setups NHFTP06 ; NHFTP15 .

Figure 5: Non-Hermitian Floquet topological insulator with NHSE: a schematic
diagram of the lattice model, (a)–(b) topological phase diagrams, (c)–(f)
static and dynamic winding angles, (g) quasienergy gap functions under the
OBC, and (h)–(k) Floquet topological corner modes at zero, $\pi$ quasienergies
and skin-localized bulk modes. Under the OBC, the numbers of Floquet zero and
$\pi$ edge modes in each topological phase are shown below the horizontal axis
of (g). System parameters are
$(J_{1},J_{2},\mu,\lambda)=(\pi,0.5\pi,0.4\pi,0.25)$ and
$(J_{1},J_{2},\mu,\lambda)=(2\pi,0.5\pi,0.4\pi,0.25)$ for (h, i) and (j, k)
ZhouNHFTP08 .

The models we considered above in this subsection do not possess the NHSE. It
remains unclear whether the rich non-Hermitian Floquet topology could coexist
with NHSEs, and how to characterize the topological bulk-edge correspondence
in the presence of Floquet NHSE. To address this issue, we consider a
periodically quenched non-Hermitian SSH model ZhouNHFTP08 , whose $k$-space
Hamiltonian under the PBC takes the form of

$H(k,t)=\begin{cases}(\mu+J_{1}\cos k+i\lambda\sin k)\sigma_{x},&t\in[\ell
T,\ell T+T/2),\\\ (J_{2}\sin k+i\lambda\cos k)\sigma_{y},&t\in[\ell T+T/2,\ell
T+T).\end{cases}$ (147)

Here $k\in[-\pi,\pi)$ is the quasimomentum, $T$ is the driving period, and
$\sigma_{x,y,z}$ are Pauli matrices. $\mu$ and $J_{1}/2+J_{2}/2$ are the
intracell and intercell hopping amplitudes. The non-Hermiticity is introduced
by the asymmetric parts of hopping amplitudes $\pm i\lambda/2$ between the two
sublattices. A sketch of the model is shown in Fig. 5. The Floquet operator
associated with $H(k,t)$ (for $\hbar=1$ and $T=2$) reads

$U(k)=e^{-i(J_{2}\sin k+i\lambda\cos k)\sigma_{y}}e^{-i(\mu+J_{1}\cos
k+i\lambda\sin k)\sigma_{x}}.$ (148)

In symmetric time frames, it possesses the chiral symmetry ${\cal
S}=\sigma_{z}$ ZhouNHFTP08 . We may thus use the winding numbers
$(w_{0},w_{\pi})$ to characterize the Floquet topological phases of $U(k)$.
The calculation of such winding numbers following the Eqs. (89)–(93) leads to
the topological phase diagrams in Fig. 5(a)–(b). Interestingly, we find that
despite Floquet topological insulators with large integer winding numbers,
there are also various phases with half-integer quantized topological
invariants. These unique phase are absent in the Hermitian limit of the model.
Their appearance is due to the Floquet exceptional topology induced by non-
Hermitian effects in the system. The parameter regions which accommodate half-
integer topological phases may further show Floquet NHSE under the OBC. In
Figs. 5(c)–(f), we compare the dynamic winding angles of $U(k)$ with the
winding patterns of its Floquet Hamiltonian vectors in different time frames
[with $(J_{1},J_{2},\mu,\lambda)=(2.4\pi,0.5\pi,0.4\pi,0.25)$] ZhouNHFTP08 .
Their coincidence implies that we can use the dynamic winding numbers
introduced in the Sec. II.6.1 to probe the half-integer quantized topological
invariants of 1D non-Hermitian Floquet topological insulators with chiral
symmetry.

Under the OBC, the bulk Floquet spectra and the gap-closing points of
quasienergies are found to be different from those under the PBC, as shown in
Fig. 5(g) [with $(J_{2},\mu,\lambda)=(0.5\pi,0.4\pi,0.25)$] ZhouNHFTP08 . This
means that the bulk winding numbers $w_{0}$ (diamonds) and $w_{\pi}$
(pentagrams) of $U(k)$ cannot be used to characterize the edge states and
topological phase transitions under the OBC. To overcome this issue, we switch
to the OBC winding numbers $(\nu_{0},\nu_{\pi})$ [defined as the
$(W_{0},W_{\pi})$ in Eq. (96)]. These winding numbers are found to be integer
quantized. They change their values only when the Floquet spectrum of the
system close its gap at $E=0$ or $E=\pi$ under the OBC. Furthermore, in each
gapped phase, the winding numbers $(\nu_{0},\nu_{\pi})$ [circles and crosses
in Fig. 5(g)] are related to the numbers of zero and $\pi$ Floquet edge modes
$(n_{0},n_{\pi})$ through the bulk-edge correspondence
$(n_{0},n_{\pi})=2(|\nu_{0}|,|\nu_{\pi}|)$. This relation holds even with
NHSEs in our system ZhouNHFTP08 . Therefore, our work provides a dual
topological characterization of non-Hermitian Floquet topological insulators
under different boundary conditions. In Figs. 5(h)–(k), we present examples of
the skin-localized bulk states [in 5(h) and 5(j)] and two types of Floquet
edge states [in 5(i) and 5(k)] in our system, which are also consistent with
the theoretical predictions ZhouNHFTP08 .

Putting together, we found different types of 1D non-Hermitian Floquet
topological insulators in a series of studies ZhouNHFTP01 ; ZhouNHFTP02 ;
ZhouNHFTP03 ; ZhouNHFTP05 ; ZhouNHFTP08 . All the models considered in these
studies exhibit rich non-Hermitian Floquet phases with large topological
invariants, many topological edge states and unique topological transitions
induced by the interplay between non-Hermitian effects and time-periodic
driving fields. The bulk-edge correspondence and dynamic topological
characterizations of these intriguing new phases are also established, leading
to an explicit and all-round physical picture of 1D non-Hermitian Floquet
topological matter. In the following subsections, we uncover the essential
role of Floquet engineering in other types of non-Hermitian topological
matter.

####  III.2.2 Second-order topological phase

We next consider the example of a non-Hermitian Floquet second-order
topological insulator (SOTI) in two dimensions. An $n$th-order topological
insulator in $d$ ($d\geq n$) spatial dimensions possesses localized eigenmodes
along its $(d-n)$-dimensional boundaries, which are topologically nontrivial
and protected by the symmetries of the $d$-dimensional bulk. For example, an
SOTI in two dimensions usually own localized topological states around its
zero-dimensional geometric corners. Assisted by time-periodic drivings,
degenerate corner modes can further appear at different quasienergies in
Floquet second-order topological phases FTPLg07 ; ZhouFSOTSC .

In this subsection, we focus on one typical model of non-Hermitian Floquet
SOTI, which is obtained following the recipe of coupled-wire construction
FTPLg07 . A schematic diagram of the static lattice model is shown in Fig.
6(a). The system is formed by a stack of SSH chains along the $y$-direction,
with dimerized couplings $(J_{1},J_{2})$ between adjacent chains. We consider
a Floquet variant of the model by applying time-periodic quenches to the
coupling and onsite parameters $(J_{1},J_{2},\mu)$ along the $y$-direction.
Under the PBC along both $x$ and $y$ directions, the time-dependent Bloch
Hamiltonian of the Floquet model takes the form of

$H(k_{x},k_{y})=H_{x}(k_{x})\otimes\tau_{0}+\sigma_{0}\otimes H_{y}(k_{y},t),$
(149)

where $(k_{x},k_{y})\in[-\pi,\pi)\times[-\pi,\pi)$ are the quasimomenta.
$\sigma_{0}$ and $\tau_{0}$ are two by two identity matrices. The Hamiltonians
of the 1D subsystems $H_{x}(k_{x})$ and $H_{y}(k_{y},t)$ are explicitly given
by

$H_{x}(k_{x})=[(J-\delta)+(J+\delta)\cos k_{x}]\sigma_{x}+(J+\delta)\sin
k_{x}\sigma_{y},$ (150) $H_{y}(k_{y},t)=\begin{cases}2J_{1}\cos
k_{y}\tau_{x}&t\in[\ell T,\ell T+T/2)\\\ 2(\mu+J_{2}\sin
k_{y})\tau_{z}&t\in[\ell T+T/2,\ell T+T)\end{cases}.$ (151)

Here $T$ is the driving period and $\ell\in\mathbb{Z}$. $\sigma_{x,y,z}$ and
$\tau_{x,y,z}$ are Pauli matrices acting on the sublattice degrees of freedom
in $x$ and $y$ directions, respectively. Setting $\hbar=T=1$,
$J=\delta=\Delta/2$, and choosing $\mu=u+iv$ ($u,v\in\mathbb{R}$), the Floquet
operator of the system in ${\bf k}$-space is found to be ZhouNHFTP06

$U(k_{x},k_{y})=e^{-i\Delta(\cos k_{x}\sigma_{x}+\sin k_{x}\sigma_{y})}\otimes
e^{-ih_{z}(k_{y})\tau_{z}}e^{-ih_{x}(k_{y})\tau_{x}},$ (152)

where

$h_{x}(k_{y})=J_{1}\cos k_{y},\qquad h_{z}(k_{y})=u+iv+J_{2}\sin k_{y}.$ (153)

The non-Hermitian effect is brought about by the onsite gain and loss encoded
in the term $iv\tau_{z}$.

In symmetric time frames, the system has the chiral (sublattice) symmetry
${\cal S}=\sigma_{z}\otimes\tau_{y}$ ZhouNHFTP06 . One can thus characterize
its Floquet second-order topological phases by integer winding numbers
ZhouNHFTP06

$\nu_{0}=\frac{\nu_{1}+\nu_{2}}{2},\qquad\nu_{\pi}=\frac{\nu_{1}-\nu_{2}}{2}.$
(154)

Here $\nu_{\alpha}=ww_{\alpha}$ for $\alpha=1,2$. $w$ is the winding number of
the static Hamiltonian $\Delta(\cos k_{x}\sigma_{x}+\sin k_{x}\sigma_{y})$,
which is always equal to one for the topological flat-band limit of the SSH
model. $(w_{1},w_{2})$ are the winding numbers of the subsystem Floquet
operator $e^{-ih_{z}(k_{y})\tau_{z}}e^{-ih_{x}(k_{y})\tau_{x}}$, which can be
defined following the recipe of the Sec. II.5. They both take integer-
quantized values even with the non-Hermitian effects considered in our system
ZhouNHFTP06 .

Figure 6: Non-Hermitian Floquet SOTI: (a) a schematic diagram of the static
lattice model, (b)–(c) topological phase diagrams, (d)–(e) quasienergy gap
functions under the OBC, (f)–(h) Floquet topological corner modes at the
quasienergy zero, and (i)–(j) Floquet topological corner modes at the
quasienergy $\pi$ ZhouNHFTP06 .

In Fig. 6(b), we show the topological phase diagrams of $U(k_{x},k_{y})$ in
the parameter space $J_{1}-v$ [with $(\Delta,J_{2},u)=(\pi/20,3\pi,0)$]. We
find that both the winding numbers $(\nu_{0},\nu_{\pi})$ can take large
values, which indicates the presence of non-Hermitian Floquet SOTI phases with
large topological invariants in our setting. Furthermore, with the increase of
the gain and loss strength $v$, the system can undergo topological phase
transitions and even enter non-Hermitian Floquet SOTI phases with larger
winding numbers. These non-Hermiticity boosted topological signatures are also
not expected in non-driven systems. Their appearance is thus due to the
corporation between Floquet driving fields and non-Hermitian effects.

Under the OBC along both $x$ and $y$ directions, localized Floquet eigenmodes
with the quasienergies zero and $\pi$ appear at the four corners of the square
lattice. Some spatial profiles of these corner modes are shown in Figs.
6(f)–(j) [with $(\Delta,J_{1},J_{2},u,v)=(\pi/20,2.5\pi,3\pi,0,2)$]. There are
twelve (eight) Floquet corner modes at the quasienergy zero ($\pi$). More
generally, their numbers $(n_{0},n_{\pi})$ are related to the winding numbers
$(\nu_{0},\nu_{\pi})$ through the algebraic relations ZhouNHFTP06

$(n_{0},n_{\pi})=4(|\nu_{0}|,|\nu_{\pi}|).$ (155)

We thus established the bulk-corner correspondence for a class of non-
Hermitian Floquet SOTI with chiral symmetry. With the growth of the non-
Hermitian parameter $v$, topological phase transitions accompanied by the
increase of bulk winding numbers $(\nu_{0},\nu_{\pi})$ and corner mode numbers
$(n_{0},n_{\pi})$ are observed from the gap functions
$(F_{0},F_{\pi})\equiv(\frac{1}{\pi}|\varepsilon|,\frac{1}{\pi}\sqrt{(|{\rm
Re}\varepsilon|-\pi)^{2}+({\rm Im}\varepsilon)^{2}})$ under the OBC (here
$\varepsilon$ denotes the quasienergy), as showcased in Figs. 6(d)–(e). The
topological phase transition points $v_{1}$ and $v_{2}$ under the OBC are
precisely consistent with the gap closing points of the 2D bulk under the PBC,
which can be found analytically ZhouNHFTP06 . Finally, even with non-Hermitian
effects, the bulk winding numbers $(\nu_{0},\nu_{\pi})$ can be dynamically
probed by 2D generalizations of the MCD as introduced in the Sec. II.6.2 (see
Ref. ZhouNHFTP06  for more details).

Overall, we discovered rich non-Hermitian Floquet SOTI phases in a
periodically quenched 2D lattice with balanced gain and loss. Different from
the static case, each SOTI phase is now depicted by two topological invariants
$(\nu_{0},\nu_{\pi})$. Many non-Hermitian Floquet SOTIs with large topological
invariants and gain- or loss-induced topological phase transitions were
identified. Under the OBC, the winding numbers $(\nu_{0},\nu_{\pi})$ determine
the numbers of symmetry protected Floquet corner modes at zero and $\pi$
quasienergies. The interplay between driving and dissipation thus results in a
series of non-Hermitian Floquet SOTI phases with multiple zero and $\pi$
corner modes, which may find applications in topological state preparations
and quantum computations. The work of Ref. ZhouNHFTP06  offers one of the
earliest findings of rich non-Hermitian Floquet topological phases beyond one
spatial dimension. In the meantime, it introduces a generic scheme to
construct non-Hermitian Floquet higher-order topological phases across
different physical dimensions. Note in passing that some latter studies also
considered the NHSEs and anomalous $\pi$ modes in Floquet higher-order
topological phases with somewhat different focuses NHFTP28 ; NHFTP31 ; NHFTP38
.

####  III.2.3 qth-root topological phase

We now consider an exotic class of non-Hermitian Floquet topological phase,
which could have symmetry-protected boundary states at the quasienergies
$p\pi/q$, with $p,q\in\mathbb{Z}^{\pm}$ and $p/q\neq 0,1$. A systematic recipe
of obtaining these new phases is to take the nontrivial $q$th-root of a
propagator $\hat{U}$ that describes Floquet topological matter. The general
procedure is developed in the Ref. ZhouNHFTP10 , which generalizes the
previous schemes of taking $2^{n}$th and $3^{n}$th roots for static
Hamiltonians RootTP01 ; RootTP02 ; RootTP03 ; RootTP04 ; RootTP05 ; RootTP06 ;
RootTP07 ; RootTP08 ; RootTP09 ; RootTP10  to Floquet systems. The basic idea
is first outlined in the Ref. RootRaditya , and then expanded by utilizing
$\mathbb{Z}_{q}$ generalizations of Pauli matrices as ancillary degrees of
freedom before taking the $q$th-root of a Floquet operator in an enlarged
Hilbert space. It is in parallel with Dirac’s original idea by adding internal
degrees of freedom for electrons before taking the square-root to get their
relativistic wave equation RootDirac .

To be explicit, we consider the construction of a cubic-root non-Hermitian
Floquet topological insulator. An illustration of the scheme in given in Fig.
7. Let us consider a three-step periodically quenched parent system, whose
time-periodic Hamiltonian takes the form of ZhouNHFTP10

$\hat{H}(t)=\begin{cases}\hat{H}_{3},&t\in[\ell,\ell+1/3),\\\
\hat{H}_{2},&t\in[\ell+1/3,\ell+2/3),\\\
\hat{H}_{1},&t\in[\ell+2/3,\ell+1),\end{cases}$ (156)

where $\ell\in\mathbb{Z}$ and we have assumed $\hbar=T=1$. The Floquet
operator of the system is then given by
$\hat{U}=e^{-\frac{i}{3}\hat{H}_{1}}e^{-\frac{i}{3}\hat{H}_{2}}e^{-\frac{i}{3}\hat{H}_{3}}$.
Following the methodology of Ref. ZhouNHFTP10 , the cubic-root $\hat{U}_{1/3}$
of $\hat{U}$ is found to be

$\hat{U}_{1/3}=\begin{pmatrix}0&e^{-i\frac{\hat{H}_{1}}{3}}&0\\\
0&0&e^{-i\frac{\hat{H}_{2}}{3}}\\\
e^{-i\frac{\hat{H}_{3}}{3}}&0&0\end{pmatrix}.$ (157)

The cube of $\hat{U}_{1/3}$ then gives

$\hat{U}_{1/3}^{3}=\begin{pmatrix}e^{-\frac{i}{3}\hat{H}_{1}}e^{-\frac{i}{3}\hat{H}_{2}}e^{-\frac{i}{3}\hat{H}_{3}}&0&0\\\
0&e^{-\frac{i}{3}\hat{H}_{2}}e^{-\frac{i}{3}\hat{H}_{3}}e^{-\frac{i}{3}\hat{H}_{1}}&0\\\
0&0&e^{-\frac{i}{3}\hat{H}_{3}}e^{-\frac{i}{3}\hat{H}_{1}}e^{-\frac{i}{3}\hat{H}_{2}}\end{pmatrix}.$
(158)

It is clear that $\hat{U}_{1/3}^{3}$ contains three identical copies of
$\hat{U}$, in the sense that its three diagonal blocks are all related by
similarity transformations and thus sharing the same spectrum and topological
properties with $\hat{U}$ ZhouNHFTP10 .

Figure 7: Cubic-root non-Hermitian Floquet topological insulator: a schematic
diagram of taking the cubit-root for a Floquet operator, (a) quasienergy
spectrum under the PBC (in grey) and OBC (in blue), (b) gap functions under
the PBC (in grey) and OBC (in red and blue), (c) Floquet topological edge
modes with $E=\pi/3,2\pi/3$, and (d)–(f) spatial profiles of Floquet edge
modes and bulk skin modes ZhouNHFTP10 .

As an example, we choose the Floquet model introduced in the Ref. ZhouNHFTP08
as the parent system, whose Hamiltonian can now be expressed as

$\hat{H}(t)=\begin{cases}\frac{3}{4}\hat{H}_{1}&t\in[\ell,\ell+1/3)\\\
\frac{3}{2}\hat{H}_{2}&t\in[\ell+1/3,\ell+2/3)\\\
\frac{3}{4}\hat{H}_{1}&t\in[\ell+2/3,\ell+1)\end{cases},$ (159)

where

$\hat{H}_{1}=J_{2}\sum_{n}(i\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+{\rm
H.c.})\otimes\sigma_{y}+i\lambda\sum_{n}(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
H.c.})\otimes\sigma_{y},$ (160)
$\hat{H}_{2}=\sum_{n}(\mu\hat{c}_{n}^{\dagger}\hat{c}_{n}+J_{1}\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
H.c.})\otimes\sigma_{x}+i\lambda\sum_{n}(i\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+{\rm
H.c.})\otimes\sigma_{x}.$ (161)

The cubit-root Floquet operator $\hat{U}_{1/3}$ of the system then takes the
form of Eq. (157). The non-Hermitian effect is introduced by the asymmetric
hopping amplitude $i\lambda$. The Floquet quasienergy spectrum and gap
functions of $\hat{U}_{1/3}$ versus $J_{1}$ under the PBC (grey dots) and OBC
(blue dots, red and blue lines) are shown in Figs. 7(a)–(b) [with
$(J_{2},\mu,\lambda)=(0.5\pi,0.4\pi,0.25)$ and the length of lattice $L=400$].
Here the gap function

$F_{\varepsilon}\equiv\sqrt{({\rm Re}E-\varepsilon)^{2}+({\rm Im}E)^{2}},$
(162)

and the blue (red) lines in the Fig. 7(b) denote $F_{\pi/3}$ ($F_{2\pi/3}$).
We observe a series of phase transitions accompanied by quasienergy-gap
closings at $E=p\pi/q$ for $p=0,1,2,3$ and $q=3$. After these transitions,
more and more degenerate Floquet edge modes emerge at the fractional
quasienergies $E=\pi/3$ and $2\pi/3$, which are impossible in conventional
Floquet topological insulators with two quasienergy bands in one dimension.
These fractional-quasienergy edge modes are thus unique to cubit-root Floquet
topological insulators, either Hermitian or non-Hermitian. A zoom in of their
quasienergies for one exemplary case is shown in Fig. 7(c) [with
$(J_{1},J_{2},\mu,\lambda)=(\pi,0.5\pi,0.4\pi,0.25)$ and the length of lattice
$L=400$], and their spatial profiles are shown in Figs. 7(d) [for the
$E=\pi/3$ eigenmodes] and 7(e) [for the $E=2\pi/3$ eigenmodes]. Moreover, the
numbers of these fractional-quasienergy edge modes $(n_{\pi/3},n_{2\pi/3})$
are determined by the open-bulk winding numbers $(\nu_{0},\nu_{\pi})$
ZhouNHFTP08  of the parent system
$\hat{U}=e^{-i\frac{\hat{H}_{1}}{4}}e^{-i\frac{\hat{H}_{2}}{2}}e^{-i\frac{\hat{H}_{1}}{4}}$.
More precisely, we have the following bulk-edge correspondence for our cubic-
root non-Hermitian Floquet topological insulator $\hat{U}_{1/3}$, i.e.,
ZhouNHFTP10

$n_{0}=n_{2\pi/3}=2|\nu_{0}|,\qquad n_{\pi/3}=n_{\pi}=2|\nu_{\pi}|,$ (163)

where $n_{0}$ and $n_{\pi}$ are the numbers of zero and $\pi$ degenerate
Floquet edge modes. Finally, we notice that the system also possesses NHSE
under the OBC, as reflected by the profiles of its Floquet bulk states in Fig.
7(f) [with $(J_{1},J_{2},\mu,\lambda)=(\pi,0.5\pi,0.4\pi,0.25)$ and the length
of lattice $L=400$]. In Figs. 7(a) and 7(b), we also notice the discrepancy
between the gap-closing points of the system under the PBC and OBC.
Nevertheless, the open-bulk winding numbers $(\nu_{0},\nu_{\pi})$ correctly
capture the bulk-edge correspondence of the system under the OBC even with
NHSEs. Therefore, the work conducted in Ref. ZhouNHFTP10  also established a
dual topological characterization for $q$th-root non-Hermitian Floquet
topological insulators. The formalism developed in the Ref. ZhouNHFTP10  is
equally applicable to the construction of $q$th-root Floquet topological
insulators, superconductors and semimetals in other (non-)Hermitian systems
and across different physical dimensions. The fractional quasienergy edge
modes at $E=p\pi/q$ might also be employed to generate boundary time crystals
with different temporal periodicity and topological properties FTPExp13 .

###  III.3 Non-Hermitian Floquet topological superconductors

Similar to the static case, non-Hermitian Floquet topological phases could
also appear in superconducting systems. We review one such example in this
subsection, which admits many Floquet Majorana zero and $\pi$ edge modes even
with non-Hermitian effects.

The model we are going to consider describes a Floquet Kitaev chain, whose
superconducting pairing terms are subject to time-periodic kicks. The
Hamiltonian of the model takes the form of

$\hat{H}(t)=\frac{1}{2}\sum_{n}[J(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
H.c.})+\Delta\delta_{T}(t)(\hat{c}_{n}\hat{c}_{n+1}+{\rm
H.c.})+\mu(2\hat{c}_{n}^{\dagger}\hat{c}_{n}-1)],$ (164)

where $J$ is the nearest-neighbor hopping amplitude, $\Delta$ is the p-wave
superconducting pairing amplitude and $\mu$ is the chemical potential.
$\delta_{T}(t)\equiv\sum_{\ell\in\mathbb{Z}}\delta(t/T-\ell)$ implements delta
kickings periodically with the period $T$. The system is made non-Hermitian by
setting $J=J_{r}+iJ_{i}$ or $\mu=\mu_{r}+i\mu_{i}$ with $J_{i}\neq 0$ or
$\mu_{i}\neq 0$, respectively. Possible experimental realizations of the
kickings and non-Hermitian terms in this model are discussed in the Ref.
ZhouNHFTP04 .

Figure 8: Periodically kicked non-Hermitian Kitaev chain: topological phase
diagrams under the PBC [in (a) and (b)], Floquet spectra under the OBC [in (c)
and (d)] and quasienergy gap functions under the OBC [in (e) and (f)]. The
length of lattice is $L=1000$ for (c)–(f). In (e) and (f), the solid and
dashed lines at $F_{0}=0$ and $F_{\pi}=0$ denote non-Hermitian Floquet
Majorana zero and $\pi$ edge modes ZhouNHFTP04 .

Under the PBC and in symmetric time frames, the Floquet operator of $H(t)$
possesses the chiral (sublattice) symmetry ${\cal S}=\sigma_{x}$. Its bulk
topological phases can then be characterized by the winding numbers
$(w_{0},w_{\pi})$ following the the Eqs. (89)–(93), in which we choose

$\hat{H}_{0}=\frac{1}{2}\sum_{n}[J(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
H.c.})+\mu(2\hat{c}_{n}^{\dagger}\hat{c}_{n}-1)],$ (165)
$\hat{H}_{1}=\frac{1}{2}\sum_{n}\Delta(\hat{c}_{n}\hat{c}_{n+1}+{\rm H.c.}),$
(166)

and assuming $\hbar=T=1$. Typical topological phase diagrams of the system are
shown in Figs. 8(a) [for $J\in\mathbb{R}$, $\mu\in\mathbb{C}$,
$(\mu_{r},\Delta)=(0.3\pi,0.5\pi)$] and 8(b) [for $\mu\in\mathbb{R}$,
$J\in\mathbb{C}$, $(\mu,\Delta)=(0.4\pi,0.9\pi)$]. Different topological
phases correspond to different uniformly colored regions, and the winding
numbers $(w_{0},w_{\pi})$ of each phase are labeled therein. The boundaries
between different phases can be analytically determined from the gap closing