# Entanglement spectrum and entropy in Floquet topological matter

Longwen Zhou<EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Ocean University of China, Qingdao, China 266100

###### Abstract

Entanglement is one of the most fundamental features of quantum systems. In
this work, we obtain the entanglement spectrum and entropy of Floquet
noninteracting fermionic lattice models and build their connections with
Floquet topological phases. Topological winding and Chern numbers are
introduced to characterize the entanglement spectrum and eigenmodes.
Correspondences between the spectrum and topology of entanglement Hamiltonians
under periodic boundary conditions and topological edge states under open
boundary conditions are further established. The theory is applied to Floquet
topological insulators in different symmetry classes and spatial dimensions.
Our work thus provides a useful framework for the study of rich entanglement
patterns in Floquet topological matter.

††preprint: APS/123-QED

##  I Introduction

Floquet topological phases are intrinsically nonequilibrium states in
periodically driven systems FloRev1 ; FloRev2 ; FloRev3 ; FloRev4 ; FloRev5 .
They are characterized by large topological invariants FloBigTN1 ; FloBigTN2 ;
FloBigTN3 ; FloBigTN4 ; FloBigTN5 ; FloBigTN6 ; FloBigTN7 ; FloBigTN8 ;
FloBigTN9 ; FloBigTN10 , unique symmetry classifications FloClass1 ; FloClass2
; FloClass3 ; FloClass4  and anomalous edge modes with no static analogies
FloAnoES1 ; FloAnoES2 ; FloAnoES3 . The realization of Floquet topological
matter in various experimental settings FloExp1 ; FloExp2 ; FloExp3 ; FloExp4
; FloExp5 ; FloExp6 ; FloExp7 ; FloExp8  further promotes the development of
new methods in quantum engineering FloRev2 , ultrafast electronics FloRev3
and topological quantum computing FloQC1 ; FloQC2 ; FloQC3 .

Entanglement is one of the most profound concepts in quantum physics
EntangleRev1 ; EntangleRev2 ; EntangleRev3 ; EntangleRev4 ; EntangleRev5 ;
EntangleRev6 ; EntangleRev7 . It characterizes the non-classical correlation
among different parts of a composite quantum system. Information theoretical
measures, such as the entanglement spectrum (ES) ES0  and entanglement entropy
(EE) EE0 , have been further shown to be able to provide important insights
for the understanding of topological phases in condensed matter systems
ESEETP1 ; ESEETP2 ; ESEETP3 ; ESEETP4 ; ESEETP42 ; ESEETP5 ; ESEETP52 ;
ESEETP6 ; ESEETP7 ; ESEETP8 ; ESEETP9 ; ESEETP10 ; ESEETP11 ; ESEETP12 ;
ESEETP13 . For example, in two-dimension, topology-induced subleading
corrections were found in the EE ESEETP3 ; ESEETP4 ; ESEETP42 . Moreover, the
ES of reduced density matrix has been shown to contain information about the
bulk-edge correspondence in topological insulators and superconductors ESEETP5
; ESEETP52 ; ESEETP6 ; ESEETP7 ; ESEETP8 ; ESEETP9 ; ESEETP10 ; ESEETP11 ;
ESEETP12 . In Floquet systems, the ES and EE have also been considered in
periodically driven Kitaev chain and graphene lattice, and unique bulk-edge
correspondences in the EE of Floquet ground states were identified
YatesPRB2016 ; YatesPRB2017 ; YatesPRL2018 ; JafariPRA2021 . However, a
systematic approach to reveal the ES and EE for general states of Floquet
systems in different physical dimensions still awaits to be developed.
Furthermore, connections between the topological properties of Floquet
entanglement Hamiltonians and Floquet topological edge states for models
belonging to different symmetry classes and with different numbers of
quasienergy bands have yet to be established.

Figure 1: Partitions for the bipartite ES and EE of 1D chain (a) and 2D
lattice (b) under periodic boundary conditions. Red dashed lines denote the
partition boundaries (entanglement cuts) between the subsystems A and B.

In this work, we introduce a framework to characterize the ES and EE of
Floquet systems consisting of noninteracting fermions in discrete lattices.
The recipe of our theory is developed in Sec. II, where we build the
connection between the ES and EE of a Floquet system and the spectrum of its
correlation matrix. The entanglement is introduced by first decomposing the
whole system S into two parts A and B in coordinate space, and then tracing
out all the degrees of freedom belonging to the subsystem B from the many-
particle density matrix $\hat{\rho}$ of the considered Floquet state in S. The
reduced density matrix $\hat{\rho}_{\rm A}={\rm Tr}_{\rm B}\hat{\rho}$ then
carries the information about the bipartite entanglement between subsystems A
and B. Illustrations of the considered partitions for one- and two-dimensional
systems are given in Fig. 1. We will refer to the entanglement Hamiltonian
$\hat{H}_{\rm A}$ of $\hat{\rho}_{\rm A}$, defined by $\hat{\rho}_{\rm
A}=e^{-\hat{H}_{\rm A}}/{\rm Tr}(e^{-\hat{H}_{\rm A}})$ as the Floquet
entanglement Hamiltonian, as it is originated from the many-particle Floquet
state. Real-space winding and Chern number invariants are further introduced
to depict the topological nature of Floquet entanglement Hamiltonians in one
and two spatial dimensions. In Sec. III, we apply our theory to typical one-
dimensional (1D) Floquet topological insulator models in different symmetry
classes and two-dimensional (2D) Floquet Chern insulator models with different
numbers of quasienergy bands. For each case, we establish concrete relations
between the topological properties of the ES under the periodic boundary
condition (PBC) and the topological edge states of the Floquet operator under
the open boundary condition (OBC). These entanglement bulk-edge
correspondences are generic and applicable to other systems within the same
symmetry classes as those considered in this work. We summarize our results,
compare them with related studies and discuss potential future directions in
Sec. IV. More details about the properties of EE around Floquet topological
phase transitions are provided in the Appendix A.

##  II Theory

In this section, we introduce the definitions of ES and EE for Floquet states
in periodically driven noninteracting fermionic systems. We further establish
their connection with the single-particle correlation matrix of the system.
The latter provides us with an efficient means to explore the ES and EE
theoretically and numerically. Finally, we introduce topological invariants
for the reduced density matrix, which will be used to characterize Floquet
topological phases at the level of entanglement Hamiltonian in one and two
spatial dimensions.

In the second quantization formalism, the time-dependent Hamiltonian for a
Floquet system of free fermions reads

$\hat{H}(t)=\sum_{m,n}\hat{c}_{m}^{\dagger}H_{mn}(t)\hat{c}_{n},$ (1)

where $\hat{H}(t)=\hat{H}(t+T)$ and $H_{mn}(t)=H_{mn}(t+T)$ are periodic in
time $t$ with the driving period $T$ and driving frequency $\omega=2\pi/T$.
$\hat{c}_{n}$ ($\hat{c}_{n}^{\dagger}$) is the annihilation (creation)
operator of a fermion in the basis $\\{|n\rangle\\}$ so that
$\hat{c}_{n}|n\rangle=|\emptyset\rangle$ and
$\hat{c}_{n}^{\dagger}|\emptyset\rangle=|n\rangle$, where $|\emptyset\rangle$
denotes the vacuum state. The fermionic creation and annihilation operators
satisfy the anticommutation relations

$\left\\{\hat{c}_{m},\hat{c}_{n}\right\\}=\left\\{\hat{c}_{m}^{\dagger},\hat{c}_{n}^{\dagger}\right\\}=0,\qquad\left\\{\hat{c}_{m},\hat{c}_{n}^{\dagger}\right\\}=\delta_{mn}.$
(2)

The Floquet operator of the system generates its dynamics over a complete
driving period. Consider the evolution from $t=0$ to $T$, it can be defined as

$\hat{U}=\hat{\mathsf{T}}e^{-\frac{i}{\hbar}\int_{0}^{T}\hat{H}(t)dt},$ (3)

where $\hat{\mathsf{T}}$ is the time-ordering operator. From now on we set
$\hbar=1$. Since $\hat{H}(t)$ is Hermitian, $\hat{U}$ is unitary and it
possesses a single-particle basis $\\{|\psi_{j}\rangle\\}$ called Floquet
eigenstates. In this basis, we can express $\hat{U}$ as

$\hat{U}=e^{-i\sum_{j}E_{j}\hat{\psi}_{j}^{\dagger}\hat{\psi}_{j}},$ (4)

where $E_{j}\in[-\pi,\pi)$ is the quasienergy. The operator
$\hat{\psi}_{j}^{\dagger}$ creates a particle in the Floquet state
$|\psi_{j}\rangle$ from vacuum, i.e.,
$\hat{\psi}_{j}^{\dagger}|\emptyset\rangle=|\psi_{j}\rangle$.
$\\{|n\rangle\\}$ and $\\{|\psi_{j}\rangle\\}$ are both orthonormal and
complete basis satisfying

$\langle m|n\rangle=\delta_{mn},\qquad\sum_{n}|n\rangle\langle n|=\hat{I},$
(5)
$\langle\psi_{i}|\psi_{j}\rangle=\delta_{ij},\qquad\sum_{j}|\psi_{j}\rangle\langle\psi_{j}|=\hat{I},$
(6)

with $\hat{I}$ being the identity operator. We can thus relate
$\hat{\psi}_{j}^{\dagger},\hat{\psi}_{j}$ and
$\hat{c}_{n}^{\dagger},\hat{c}_{n}$ by

$\hat{\psi}_{j}^{\dagger}|\emptyset\rangle=|\psi_{j}\rangle=\sum_{n}|n\rangle\langle
n|\psi_{j}\rangle=\sum_{n}a_{nj}\hat{c}_{n}^{\dagger}|\emptyset\rangle,$ (7)
$\hat{c}_{n}^{\dagger}|\emptyset\rangle=|n\rangle=\sum_{j}|\psi_{j}\rangle\langle\psi_{j}|n\rangle=\sum_{j}a_{nj}^{*}\hat{\psi}_{j}^{\dagger}|\emptyset\rangle,$
(8)

where $a_{nj}=\langle n|\psi_{j}\rangle$, or equivalently

$\hat{\psi}_{j}^{\dagger}=\sum_{n}a_{nj}\hat{c}_{n}^{\dagger},\qquad\hat{\psi}_{j}=\sum_{n}a_{nj}^{*}\hat{c}_{n},$
(9)
$\hat{c}_{n}^{\dagger}=\sum_{j}a_{nj}^{*}\hat{\psi}_{j}^{\dagger},\qquad\hat{c}_{n}=\sum_{j}a_{nj}\hat{\psi}_{j}.$
(10)

It can be verified that $\hat{\psi}_{j}$ and $\hat{\psi}_{j}^{\dagger}$ also
satisfy the anticommutation relation of fermions, i.e.,

$\\{\hat{\psi}_{i},\hat{\psi}_{j}\\}=\\{\hat{\psi}_{i}^{\dagger},\hat{\psi}_{j}^{\dagger}\\}=0,\qquad\\{\hat{\psi}_{i},\hat{\psi}_{j}^{\dagger}\\}=\delta_{ij}.$
(11)

We next construct the many-particle density operator for a set of occupied
Floquet states. Since the quasienergies $\\{E_{j}\\}$ of $\hat{U}$ are phase
factors, they do not have a natural order even though they are
stroboscopically conserved in Floquet dynamics. Therefore, the definition of a
“quasienergy Fermi surface” below which each state $|\psi_{j}\rangle$ is
occupied by one fermion is ambiguous. Under the conditions: (i) the
quasienergies $\\{E_{j}\\}$ are confined to the range of $[-\pi,\pi)$, (ii)
the configuration of quasienergies is symmetric with respect to $E=0$, and
(iii) the quasienergy spectrum is gapped at $E=0,\pm\pi$, it may look natural
to define a “Floquet ground state” by uniformly filling all states
$\\{|\psi_{j}\rangle\\}$ with quasienergies ranging from $-\pi$ to zero. We
make this choice for most of the numerical calculations presented in this
work. Here, to be general enough, we just assume that a collection Floquet
states $\\{|\psi_{j}\rangle|j\in{\rm occ.}\\}$ are filled initially, such that
the many-particle wave function of the system takes the form

$|\Psi\rangle=\prod_{j\in{\rm
occ.}}\hat{\psi}_{j}^{\dagger}|\emptyset\rangle,$ (12)

which is normalized as $\langle\Psi|\Psi\rangle=1$ and fulfills the eigenvalue
equation $\hat{U}|\Psi\rangle=e^{-i\sum_{j\in{\rm occ.}}E_{j}}|\Psi\rangle$.
The density operator corresponding to such a many-particle Floquet state takes
the form

$\hat{\rho}=|\Psi\rangle\langle\Psi|.$ (13)

It satisfies the general feature of a pure-state density matrix, i.e.,
$\hat{\rho}=\hat{\rho}^{\dagger}$, $\hat{\rho}=\hat{\rho}^{2}$ and ${\rm
Tr}\hat{\rho}=1$.

To investigate the ES and EE, we decompose the system into two parts A and B
with Hilbert spaces ${\cal H}_{A}$ and ${\cal H}_{B}$, such that the Hilbert
space of the whose system ${\cal H}={\cal H}_{A}\otimes{\cal H}_{B}$. For 1D
and 2D systems, such a bipartition is illustrated in Fig. 1. Tracing over all
degrees of freedom belonging to the subsystem B (${\rm Tr}_{{\rm B}}$) yields
the reduced density matrix $\hat{\rho}_{{\rm A}}$ of subsystem A, i.e.,

$\hat{\rho}_{{\rm A}}={\rm Tr}_{{\rm
B}}\hat{\rho}=\frac{1}{Z}e^{-\hat{H}_{{\rm A}}},\qquad Z\equiv{\rm
Tr}e^{-\hat{H}_{{\rm A}}}.$ (14)

In the second equality, we introduced the entanglement Hamiltonian
$\hat{H}_{{\rm A}}$, whose eigenvalues form the ES of the reduced density
matrix $\hat{\rho}_{{\rm A}}$. Note that even though the Hamiltonian of the
whole system $\hat{H}(t)$ is time-dependent, the $\hat{H}_{{\rm A}}$ as
constructed from Floquet eigenstates is time-independent in stroboscopic
dynamics. Due to the Hermiticity of $\hat{\rho}_{{\rm A}}$, $\hat{H}_{{\rm
A}}$ admits a spectral decomposition

$\hat{H}_{{\rm A}}=\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j},$
(15)

where $\hat{\phi}_{j}^{\dagger}$ creates an eigenstate $|\phi_{j}\rangle$ of
$\hat{H}_{{\rm A}}$ with the entanglement eigenvalue $\xi_{j}\in\mathbb{R}$,
i.e., $|\phi_{j}\rangle=\hat{\phi}_{j}^{\dagger}|\emptyset\rangle$ and
$\hat{H}_{{\rm A}}|\phi_{j}\rangle=\xi_{j}|\phi_{j}\rangle$. The ES is formed
by the collection of $\xi_{j}$ for all $j$. Despite the ES, we will also study
the EE of Floquet system. The von Neumann EE for $\hat{\rho}_{{\rm A}}$ is
defined as

$S=-{\rm Tr}\left(\hat{\rho}_{{\rm A}}\ln\hat{\rho}_{{\rm A}}\right).$ (16)

A larger value of $S$ indicates that the subsystems A and B are entangled more
strongly across their interface. Meanwhile, the configuration of ES may
provide more information about topological edge states in the system even
under the PBC.

For static noninteracting systems, due to the Wick’s theorem, both the ES and
EE can be obtained from the spectrum of single-particle correlation matrix of
the occupied state $|\Psi\rangle$ restricted to the subsystem A EntangleRev4 .
A similar relation can be derived for the reduced density matrix of Floquet
states. To see this, we first express the matrix elements of the single
particle correlator as

$C_{mn}=\langle\Psi|\hat{c}_{m}^{\dagger}\hat{c}_{n}|\Psi\rangle,$ (17)

where the indices $m$ and $n$ are restricted to the subsystem A. Using Eqs.
(10)–(12), $C_{mn}$ can be cast into the form

$C_{mn}=\sum_{j\in{\rm occ.}}\langle
n|\psi_{j}\rangle\langle\psi_{j}|m\rangle=\langle n|\hat{P}|m\rangle,$ (18)

where $\hat{P}=\sum_{j\in{\rm occ.}}|\psi_{j}\rangle\langle\psi_{j}|$ is the
single-particle projector onto the occupied Floquet states in $|\Psi\rangle$.
We can use Eq. (18) to obtain the correlation matrix in numerical
calculations.

We next establish the relation between the spectrum of correlator $C$ and the
entanglement Hamiltonian $\hat{H}_{{\rm A}}$. First, $C_{mn}$ can be
equivalently expressed as

$C_{mn}={\rm Tr}\left(\hat{c}_{m}^{\dagger}\hat{c}_{n}\hat{\rho}_{{\rm
A}}\right),$ (19)

since the operator $\hat{c}_{m}^{\dagger}\hat{c}_{n}$ belongs to the subsystem
A for all $m,n\in{\rm A}$. Plugging Eqs. (14) and (15) into Eq. (19), we
obtain

$C_{mn}=\frac{1}{Z}{\rm
Tr}\left(\hat{c}_{m}^{\dagger}\hat{c}_{n}e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right).$
(20)

When restricted to the subsystem A, both $\\{|n\rangle\\}$ and
$\\{|\phi_{j}\rangle\\}$ form complete bases of A. Therefore, similar to the
Eq. (10), we can express the relation between
$\hat{c}_{n}^{\dagger},\hat{c}_{n}$ and
$\hat{\phi}_{j}^{\dagger},\hat{\phi}_{j}$ as

$\hat{c}_{n}^{\dagger}=\sum_{j}b_{nj}^{*}\hat{\phi}_{j}^{\dagger},\qquad\hat{c}_{n}=\sum_{j}b_{nj}\hat{\phi}_{j},$
(21)

where $b_{nj}=\langle n|\phi_{j}\rangle$. Inserting Eq. (21) into Eq. (20)
yields

$C_{mn}=\frac{1}{Z}\sum_{j^{\prime},j^{\prime\prime}}b_{mj^{\prime}}^{*}b_{nj^{\prime\prime}}{\rm
Tr}\left(\hat{\phi}_{j^{\prime}}^{\dagger}\hat{\phi}_{j^{\prime\prime}}e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right),$
(22)

which will vanish once $j^{\prime}\neq j^{\prime\prime}$. Thus we find

$\displaystyle C_{mn}=$
$\displaystyle\sum_{j^{\prime}}b_{mj^{\prime}}^{*}b_{nj^{\prime}}\frac{1}{Z}{\rm
Tr}\left(\hat{\phi}_{j^{\prime}}^{\dagger}\hat{\phi}_{j^{\prime}}e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right)$
$\displaystyle=$
$\displaystyle\sum_{j^{\prime}}b_{mj^{\prime}}^{*}b_{nj^{\prime}}\frac{1}{Z}\left(-\frac{\partial}{\partial\xi_{j^{\prime}}}\right){\rm
Tr}\left(e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right).$
(23)

To proceed, we notice from Eqs. (14), (15) and the occupation nature of
fermions that

$Z={\rm
Tr}\left(e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right)=\prod_{j}\left(1+e^{-\xi_{j}}\right).$
(24)

Therefore, we have $\ln Z=\sum_{j}\ln(1+e^{-\xi_{j}})$ and

$\displaystyle\left(-\frac{\partial}{\partial\xi_{j^{\prime}}}\right)\ln Z=$
$\displaystyle\frac{1}{Z}\left(-\frac{\partial}{\partial\xi_{j^{\prime}}}\right){\rm
Tr}\left(e^{-\sum_{j}\xi_{j}\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{e^{\xi_{j^{\prime}}}+1}.$ (25)

Comparing Eq. (25) and the second line of Eq. (23), we arrive at

$C_{mn}=\sum_{j}\frac{b_{mj}^{*}b_{nj}}{e^{\xi_{j}}+1}=\sum_{j}\frac{\langle
n|\phi_{j}\rangle\langle\phi_{j}|m\rangle}{e^{\xi_{j}}+1}.$ (26)

The transpose of correlation matrix $C$ thus admits the spectral decomposition

$C^{\top}=\sum_{j}\zeta_{j}|\phi_{j}\rangle\langle\phi_{j}|,\qquad\zeta_{j}=\frac{1}{e^{\xi_{j}}+1}.$
(27)

Since $C$ and $C^{\top}$ share the same spectrum, we establish the one-to-one
correspondence between the spectrum $\\{\xi_{j}\\}$ of entanglement
Hamiltonian $H_{{\rm A}}$ and the spectrum $\\{\zeta_{j}\\}$ of correlation
matrix $C$, i.e.,

$\xi_{j}=\ln\left(\zeta_{j}^{-1}-1\right),\qquad j\in{\rm A}.$ (28)

Note in passing that the ranges of eigenvalues are $\xi_{j}\in\mathbb{R}$ and
$\zeta_{j}\in(0,1)$. Eq. (28) allows us to determine the ES from the spectrum
of single-particle correlation matrix for free fermions in Floquet systems.

The EE can also be extracted from the spectrum of correlation matrix. Plugging
Eqs. (14) and (24) into Eq. (16), we find

$\displaystyle S=$ $\displaystyle-{\rm Tr}\left[\frac{1}{Z}e^{-H_{{\rm
A}}}\ln\left(\frac{1}{Z}e^{-H_{{\rm A}}}\right)\right]$ $\displaystyle=$
$\displaystyle\sum_{j}\ln\left(1+e^{-\xi_{j}}\right)+\frac{1}{Z}{\rm
Tr}\left(H_{{\rm A}}e^{-H_{{\rm A}}}\right).$ (29)

With the help of Eqs. (15) and (23)–(25), we further obtain

$\displaystyle{\rm Tr}\left(H_{{\rm A}}e^{-H_{{\rm
A}}}\right)=\sum_{j}\xi_{j}{\rm
Tr}\left(\hat{\phi}_{j}^{\dagger}\hat{\phi}_{j}e^{-H_{{\rm A}}}\right)$
$\displaystyle=$
$\displaystyle\sum_{j}\xi_{j}\left(-\frac{\partial}{\partial\xi_{j}}\right){\rm
Tr}\left(e^{-H_{{\rm A}}}\right)=Z\sum_{j}\frac{\xi_{j}}{e^{\xi_{j}}+1}.$ (30)

The expression for $S$ in Eq. (29) can then be reduced to

$S=\sum_{j}\left[\ln\left(1+e^{-\xi_{j}}\right)+\frac{\xi_{j}}{e^{\xi_{j}}+1}\right].$
(31)

Replacing $\xi_{j}$ by $\zeta_{j}$ following Eq. (28), we arrive at the
expression of EE in terms of the eigenvalues of correlation matrix $C$, i.e.,

$S=-\sum_{j}\left[\zeta_{j}\ln\zeta_{j}+\left(1-\zeta_{j}\right)\ln\left(1-\zeta_{j}\right)\right].$
(32)

In previous studies, the Eqs. (28) and (32) for the ES and EE have been
derived for static systems of free fermions EntangleRev4 . Here we showed that
they can also be used to describe the ES and EE of noninteracting fermions in
Floquet systems for rather general partitions of the system and populations of
the initial Floquet state. In Sec. III, we will employ the connection between
the correlation matrix, ES and EE to further uncover the topological and
entanglement nature of typical Floquet systems in one and two spatial
dimensions.

Besides the ES and EE, we can also characterize the nature of reduced density
matrix (or entanglement Hamiltonian) of Floquet states through topological
invariants, whose definitions are insensitive to the choice of boundary
conditions. In one spatial dimension, if the Floquet operator of the system
possesses the chiral (sublattice) symmetry (i.e., for systems belonging to the
symmetry classes AIII, BDI and CII), we can introduce an open-bulk winding
number OBWN1 ; OBWN2 ; OBWN3 ; OBWN4  for the resulting correlation matrix
$C$, which is defined as

$W=\frac{1}{L^{\prime}}{\rm Tr}^{\prime}(\Gamma_{{\rm A}}Q[Q,N_{{\rm A}}]).$
(33)

Here $\Gamma_{{\rm A}}$ and $N_{{\rm A}}$ are the chiral symmetry operator and
position operator of the subsystem A. In the definition, the subsystem A is
decomposed into a bulk region and two edge regions around its left and right
boundaries. The trace ${\rm Tr}^{\prime}$ is taken over the bulk region of A,
whose size is $L^{\prime}=L_{{\rm A}}-2L_{{\rm E}}$, where $L_{{\rm A}}$ is
the length of subsystem A and $L_{{\rm E}}$ counts the number of lattice sites
belonging to the left and right edge regions of A. The projector matrix $Q$ is
defined as

$Q=\sum_{j}[\Theta(\zeta_{j}-1/2)-\Theta(1/2-\zeta_{j})]|\phi_{j}\rangle\langle\phi_{j}|,$
(34)

where the step function $\Theta(\zeta)=1$ ($=0$) if $\zeta>0$ ($\zeta<0$).
$\\{\text{$\zeta_{j}$}\\}$ and $\\{|\phi_{j}\rangle\\}$ are eigenvalues and
eigenvectors of the correlation matrix $C$. In Sec. III.1, we will demonstrate
that when working in symmetric time frames, the winding number $W$ introduced
here could fully capture the topological phases of 1D Floquet systems in the
symmetry classes BDI and CII.

In two spatial dimensions, we will focus on the ES and EE of Floquet Chern
insulators. For that purpose, we introduce the local Chern marker (LCM) RSCN1
; RSCN2 ; RSCN3  for the entanglement Hamiltonian, i.e.,

$\mathfrak{C}(x,y)=-4\pi{\rm Im}\langle x,y|P\hat{x}P\hat{y}P|x,y\rangle,$
(35)

where $\hat{x},\hat{y}$ are position operators along the $x,y$ directions, and
$\\{|x,y\rangle\\}$ refers to position eigenbasis of the 2D plane. $P$ is the
projector to the eigenstates of entanglement Hamiltonian at half filling,
i.e.,

$P=\sum_{j}\Theta(1/2-\zeta_{j})|\phi_{j}\rangle\langle\phi_{j}|.$ (36)

From $\mathfrak{C}(x,y)$, one can extract a real-space Chern number that works
equally well for uniform and non-uniform samples, which is defined as

${\rm Ch}=\frac{1}{L_{x}L_{y}}\sum_{x,y}\mathfrak{C}(x,y),$ (37)

where the coordinates $(x,y)$ belong to a region of the subsystem A with size
$L_{x}\times L_{y}$ that is chosen away from the boundaries of A. The real-
space Chern number ${\rm Ch}$ has been used to characterize the topology of
Chern insulators in the presence of junctions and various types of onsite
potential RSCN1 ; RSCN2 ; RSCN3 ; RSCN4 ; RSCN5 . In Sec. III.2, we will
demonstrate that the ${\rm Ch}$ in Eq. (37) can also be used to describe the
topological properties of entanglement Hamiltonians for 2D Floquet systems.

##  III Results

In this section, we use the entanglement tools introduced in Sec. II to study
topological phases and phase transitions in 1D and 2D Floquet systems. In one
dimension, we consider two prototypical Floquet topological insulator models
in the symmetry classes BDI and CII, which are characterized by
$\mathbb{Z}\times\mathbb{Z}$ and $2\mathbb{Z}\times 2\mathbb{Z}$ topological
invariants. Under the OBC, they possess twofold and fourfold degenerate edge
modes at the quasienergies zero and $\pi$. We discuss how the numbers of these
topological edge modes and the transition points where these numbers change
could be related to the ES and EE of the corresponding Floquet systems under
the PBC in Sec. III.1. In two dimensions, we demonstrate the connection
between the topological property of a uniformly filled Floquet band and the
gapless chiral edge modes in their related ES for both two- and three-band
Floquet Chern insulator models. The EE and the real-space Chern number of the
entanglement Hamiltonian are further shown to be able to capture the
topological phase transitions in these systems.

###  III.1 ES and EE in 1D Floquet systems

####  III.1.1 Floquet topological insulators in BDI symmetry class

We first consider a model of Floquet topological insulator in the symmetry
class BDI, which is characterized by $\mathbb{Z}\times\mathbb{Z}$ topological
invariants. It was first proposed as a spin-$1/2$ extension of the on-
resonance double kicked rotor (ORDKR) in Ref. KRSZhou2018 , and later realized
experimentally by an NV center in diamond in Ref. KRSChen2021 . The
possibility of realizing this model in cold atom systems is also discussed in
detail in Ref. KRSBolik2022 . The Floquet operator of the model takes the form

$\hat{U}=e^{-i\hat{H}_{2}}e^{-i\hat{H}_{1}},$ (38)

where

$\hat{H}_{1}=\frac{K_{1}}{2}\sum_{n}(\hat{{\bf
c}}_{n}^{\dagger}\sigma_{x}\hat{{\bf c}}_{n+1}+{\rm H.c.}),$ (39)
$\hat{H}_{2}=\frac{K_{2}}{2i}\sum_{n}(\hat{{\bf
c}}_{n}^{\dagger}\sigma_{y}\hat{{\bf c}}_{n+1}-{\rm H.c.}).$ (40)

Here $K_{1}$ and $K_{2}$ are kicking strengths KRSZhou2018 .
$\sigma_{x}=\left|\uparrow\right\rangle\left\langle\downarrow\right|+\left|\downarrow\right\rangle\left\langle\uparrow\right|$
and
$\sigma_{y}=i(\left|\downarrow\right\rangle\left\langle\uparrow\right|-\left|\uparrow\right\rangle\left\langle\downarrow\right|)$
are Pauli matrices acting on the spin-$1/2$ degree of freedom. $\hat{{\bf
c}}_{n}\equiv(\hat{c}_{n,\uparrow},\hat{c}_{n,\downarrow})^{\top}$, and
$\hat{c}_{n,\sigma}$ is the annihilation operator of a fermion with the
momentum index $n\in\mathbb{Z}$ and spin $\sigma\in\\{\uparrow,\downarrow\\}$
KRSZhou2018 . $\top$ denotes the matrix transpose and ${\rm H.c.}$ means the
Hermitian conjugate. To describe the symmetry and topology of the system, one
can introduce symmetric time frames by performing unitary transformations to
$\hat{U}$, so that the transformed Floquet operators possess the chiral
symmetry explicitly and a topological winding number can be defined for each
of them appropriately AsbothSTM . For 1D systems, there are two such time
frames available in the usual case, and the Floquet operators in these time
frames can be obtained by shifting the initial time of the evolution forward
and backward over a quarter of the driving period. For our model, the
resulting Floquet operators in a pair of complementary symmetric time frames
read

$\hat{U}_{1}=e^{-i\frac{\hat{H}_{1}}{2}}e^{-i\hat{H}_{2}}e^{-i\frac{\hat{H}_{1}}{2}},\qquad\hat{U}_{2}=e^{-i\frac{\hat{H}_{2}}{2}}e^{-i\hat{H}_{1}}e^{-i\frac{\hat{H}_{2}}{2}}.$
(41)

In these time frames, the Floquet operators possess the chiral symmetry
$\Gamma=\sigma_{z}$ in the sense that
$\Gamma\hat{U}_{\alpha}\Gamma^{-1}=\hat{U}_{\alpha}^{\dagger}$ ($\alpha=1,2$).
This implies that for any Floquet eigenstate $|\psi_{j}\rangle$ of
$\hat{U}_{\alpha}$ with the quasienergy $E_{j}$ (i.e.,
$\hat{U}_{\alpha}|\psi_{j}\rangle=e^{-iE_{j}}|\psi_{j}\rangle$), there must be
another eigenstate $\Gamma|\psi_{j}\rangle$ of $\hat{U}_{\alpha}$ with the
quasienergy $-E_{j}$. Floquet states with $E_{j}=0$ or $E_{j}=\pi$ then must
come as degenerate pairs. The same spectral symmetry also holds for the
Floquet operator $\hat{U}$ in the original time frame [Eq. (38)], as unitary
transformations do not change the Floquet spectrum obtained by solving
$\hat{U}|\psi\rangle=e^{-iE}|\psi\rangle$ AsbothSTM .

Besides the chiral symmetry, $\hat{U}_{\alpha}$ also possesses the time-
reversal symmetry ${\cal K}$ and particle-hole symmetry $\sigma_{z}{\cal K}$,
where ${\cal K}$ takes the complex conjugation. It thus belongs to the
symmetry class BDI AsbothSTM . In Ref. KRSZhou2018 , two integer-quantized
topological winding numbers $(w_{1},w_{2})$, defined for the $\hat{U}_{1}$ and
$\hat{U}_{2}$ in symmetric time frames, were used to characterize the Floquet
topological phases and bulk-edge correspondence of the spin-$1/2$ ORDKR. With
the increase of kicking strengths $(K_{1},K_{2})$, rich Floquet phases
featured by large winding numbers
$(w_{0},w_{\pi})=(w_{1}+w_{2},w_{1}-w_{2})/2$ and many topological edge states
at zero and $\pi$ quasienergies can be induced KRSZhou2018 . Different Floquet
topological phases are separated by the phase transition lines

$\frac{\mu^{2}}{K_{1}^{2}}+\frac{\nu^{2}}{K_{2}^{2}}=\frac{1}{\pi^{2}},\qquad\mu,\nu\in\mathbb{Z}$
(42)

in the parameter space KRSZhou2018 . Across each phase boundary line, $w_{0}$
or $w_{\pi}$ gets a quantized change. Within each topological phase, the
winding numbers $(w_{0},w_{\pi})$ are related to the number of Floquet edge
modes at zero and $\pi$ quasieneriges $(n_{0},n_{\pi})$ through the bulk-edge
correspondence relation $(n_{0},n_{\pi})=2(|w_{0}|,|w_{\pi}|)$ KRSZhou2018
$w_{0}$ ($w_{\pi}$) thus counts the number of degenerate Floquet edge modes at
the quasienergy zero ($\pi$).

In the following, we reveal the topology, phase transitions and bulk-edge
correspondence of the spin-$1/2$ ORDKR by investigating its ES, EE and the
winding numbers of entanglement Hamiltonian. Starting with the Floquet
operators $\hat{U}_{1}$ and $\hat{U}_{2}$ in Eq. (41), we consider a half-
filled system where all Floquet states with quasienergies $E\in(-\pi,0)$ are
uniformly occupied. Following Eqs. (4)–(27), the correlation matrix reads

$C_{\alpha}^{\top}=\sum_{j}\zeta_{j}^{\alpha}|\phi_{j}^{\alpha}\rangle\langle\phi_{j}^{\alpha}|,$
(43)

where $\alpha=1,2$ are indices of the two symmetric time frames. The ES and EE
are further obtained according to Eqs. (28) and (32), i.e., for $\alpha=1,2$
we have

$\xi_{j}^{\alpha}=\ln\left(1/\zeta_{j}^{\alpha}-1\right),\qquad j\in{\rm A},$
(44)
$S_{\alpha}=-\sum_{j}\left[\zeta_{j}^{\alpha}\ln\zeta_{j}^{\alpha}+\left(1-\zeta_{j}^{\alpha}\right)\ln\left(1-\zeta_{j}^{\alpha}\right)\right].$
(45)

Meanwhile, we find the Floquet spectrum $E$ of $\hat{U}$ in Eq. (38) by
solving the eigenvalue equation $\hat{U}|\psi\rangle=e^{-iE}|\psi\rangle$. All
calculations are performed under the PBC by setting $\hat{{\bf
c}}_{n}=\hat{{\bf c}}_{n+L}$ in Eqs. (39) and (40), with $L$ being the length
of lattice. A direct advantage of the entanglement approach deserves to be
mentioned. For the spin-$1/2$ ORDKR, the lattice structures in Eqs. (39) and
(40) are defined in momentum space KRSZhou2018 , where the realization of an
OBC for detecting the edge states there is practically challenging. As will be
seen in the following, the entanglement approach allows us to get access to
the information about Floquet edge states even under the PBC, which can be
implemented more naturally for the momentum space of a rotor.

Figure 2: Quasienergy and correlation matrix spectra of the spin-$1/2$ ORDKR
versus the kicking strength $K_{2}$. We set the kicking strength at
$K_{1}=0.5\pi$. The length of lattice is $L=300$ for all panels. (a) shows the
Floquet spectrum under the PBC. (b) shows the Floquet spectrum under the OBC,
where the red crosses denote the edge states with quasienergies $E=0,\pm\pi$.
(c) and (d) show the spectrum of correlation matrices $C_{1}$ and $C_{2}$ in
two symmetric time frames. Red crosses in (c) and (d) denote eigenstates of
$C_{1}$ and $C_{2}$ whose profiles are localized around the cuts between
subsystems A and B.

In Fig. 2, we report the quasienergy spectrum and ES (presented by the
spectrum of correlation matrices in symmetric time frames $\alpha=1,2$) of the
spin-$1/2$ ORDKR versus the kicking strength $K_{2}$. In Fig. 2(a), we see
that the gap of Floquet spectrum closes/reopens alternately at $E=\pi$ and
$E=0$ for $K_{2}=\nu\pi$ ($\nu=1,2,...$), as predicted by Eq. (42). New
Floquet edge states emerge at the quasienergy $E=0$ or $\pi$ after each
transition, as shown in Fig. 2(b). With the increase of $K_{2}$, the number of
zero ($\pi$) Floquet edge modes changes in the sequence $n_{0}=2\rightarrow
6\rightarrow 10$ ($n_{\pi}=0\rightarrow 4\rightarrow 8$) following each
transition at $E=0$ ($E=\pi$) KRSZhou2018 .

Near the critical kicking strengths $K_{2}=\nu\pi$ ($\nu=1,2,...$), the ES in
Figs. 2(c) and 2(d) show discontinuous changes around the correlation matrix
eigenvalue $\zeta=1/2$. In Fig. 3, the EE in both symmetric time frames also
show nonanalytic cusps around $K_{2}=\nu\pi$, implying the presence of phase
transitions around these critical kicking strengths. Referring to Figs. 2(a)
and (b), we confirm that the ES and EE indeed provide signatures for Floquet
topological phase transitions in the spin-$1/2$ ORDKR.

Figure 3: The number of maximally entangled states, winding numbers of the
entanglement Hamiltonian and the EE evaluated in symmetric time frames
$\alpha=1,2$. The kicking strength $K_{1}$ is set to $0.5\pi$ and the length
of lattice is $L=300$. $N_{\alpha}$ denotes the number of eigenstates of the
correlation matrix $C_{\alpha}$ whose eigenvalues are equal to $1/2$.
$W_{\alpha}$ denotes the open-bulk winding number of the entanglement
Hamiltonian [Eq. (15)]. $S_{\alpha}$ denotes the EE [Eq. (45)] in the time
frame $\alpha$.

Away from these transition points, the eigenvalues of $C_{\alpha}$ appear
around $0$, $1/2$ and $1$. According to the Eq. (45), the eigenvalues
$\zeta_{j}^{\alpha}\simeq 0,1$ have vanishing contributions to $S_{\alpha}$.
Meanwhile, each eigenvalue $\zeta_{j}^{\alpha}=1/2$ has the largest
contribution $\ln 2$ to the EE in Fig. 3. The corresponding eigenstate
$|\phi_{j}^{\alpha}\rangle$ should be maximally entangled, which means that it
should appear around the spatial cuts between the subsystems A and B. We thus
refer to the eigenstates $|\phi_{j}^{\alpha}\rangle$ of $C_{\alpha}$ with
eigenvalues $\zeta_{j}^{\alpha}=1/2$ as “edge states” of the entanglement
Hamiltonian. In Fig. 3, we also observe a steady growth in the global profile
of $S_{1}$, which implies an increase of the number $N_{1}$ of entanglement
edge states with $\zeta_{j}^{1}=1/2$ in the ES of $\hat{U}_{1}$. Note in
passing that the global profile of $S_{2}$ also shows a slow growth with the
increase of $K_{2}$. Since we have $N_{2}=2$ at all $K_{2}\neq\nu\pi$, the
$\zeta_{j}^{2}=1/2$ modes cannot contribute and this increase should be due to
the appearance of eigenmodes whose ES deviates further from $\zeta=0,1$ but
not reaching $1/2$, as observed in Fig. 2(d).

To further decode the topological nature of entanglement Hamiltonian, we
report the numbers of localized modes $(N_{1},N_{2})$ with eigenvalues
$(\zeta_{j}^{1},\zeta_{j}^{2})=(1/2,1/2)$ in the ES [Figs. 2(c)–(d)] and the
winding numbers $(W_{1},W_{2})$ with respect to the kicking strength $K_{2}$.
$(W_{1},W_{2})$ are defined following Eqs. (33) and (34). For the spin-$1/2$
ORDKR, we have $\Gamma_{{\rm A}}=\Gamma=\sigma_{z}$ and the projector in Eq.
(34) takes the form

$Q_{\alpha}=\sum_{j}[\Theta(\zeta_{j}^{\alpha}-1/2)-\Theta(1/2-\zeta_{j}^{\alpha})]|\phi_{j}^{\alpha}\rangle\langle\phi_{j}^{\alpha}|,$
(46)

where $\\{\zeta_{j}^{\alpha}\\}$ and $\\{|\phi_{j}^{\alpha}\rangle\\}$ are the
eigenvalues and eigenbasis of the correlation matrices [Eq. (43)] in the time
frames $\alpha=1,2$. The resulting topological winding numbers [Eq. (33)] of
the entanglement Hamiltonian reads

$W_{\alpha}=\frac{1}{L^{\prime}}{\rm
Tr}^{\prime}(\sigma_{z}Q_{\alpha}[Q_{\alpha},N_{{\rm A}}]),\quad\alpha=1,2.$
(47)

We now establish the correspondence between the ES and Floquet topological
phases. First, by comparing $(N_{1},N_{2})$ in the ES of Fig. 3 and the number
of edge modes $(n_{0},n_{\pi})$ with quasienergies zero and $\pi$ in Fig.
2(b), we arrive at the relations

$N_{1}=n_{0}+n_{\pi},\qquad N_{2}=|n_{0}-n_{\pi}|.$ (48)

That is, in the symmetric time frame $1$ ($2$), the number of modes with
eigenvalue $\zeta=1/2$ in the ES under the PBC is equal to the sum of (the
absolute difference between) the number of zero and $\pi$ Floquet edge modes
in the quasienergy spectrum under the OBC. Notably, the total number of
Floquet topological edge modes $n_{0}+n_{\pi}$ in the quasienergy spectrum can
be captured by the number of entanglement eigenmodes $N_{1}$ in a single time
frame. This indicates that the ES of a Floquet system indeed contain rich
information about its nonequilibrium topological properties.

Second, according to Fig. 3, the $(N_{1},N_{2})$ and winding numbers
$(W_{1},W_{2})$ are related by the relation

$N_{1}=2|W_{1}|,\qquad N_{2}=2|W_{2}|.$ (49)

Though obtained under the PBC, it may be viewed as a “bulk-edge
correspondence”, as $N_{\alpha}$ ($\alpha=1,2$) counts the number of modes
with degenerate eigenvalue $\zeta=1/2$ that are localized at the spatial
entanglement cuts between subsystems A and B. Finally, combining Eqs. (48) and
(49), we find the relation between the topology of entanglement Hamiltonian
under the PBC and the Floquet zero/$\pi$ edge modes under the OBC, i.e.,

$\displaystyle n_{0}=$ $\displaystyle\frac{1}{2}|{\rm sgn}(W_{1})N_{1}+{\rm
sgn}(W_{2})N_{2}|=|W_{1}+W_{2}|,$ $\displaystyle n_{\pi}=$
$\displaystyle\frac{1}{2}|{\rm sgn}(W_{1})N_{1}-{\rm
sgn}(W_{2})N_{2}|=|W_{1}-W_{2}|.$ (50)

Eq. (50) allows us to determine the numbers of Floquet topological edge modes
in an arbitrary time frame under the OBC from the entanglement information of
the system under the PBC. We can thus regard it as an entanglement bulk-edge
correspondence for Floquet systems. We have also checked the Eqs. (48)–(50) in
other parameter regions of the spin-$1/2$ ORDKR and confirmed their validness,
which implies that they are generic for 1D Floquet topological insulators in
the BDI symmetry class. The Eqs. (48)–(50) thus form one set of key results in
this work. The generality of this topology-entanglement connection will be
further demonstrated in the next subsection by considering Floquet topological
insulators in another symmetry class.

####  III.1.2 Floquet topological insulators in CII symmetry class

We now study a spin-$1/2$ periodically quenched ladder (PQL) model SCLZhou2020
, whose Hamiltonian takes the form $\hat{{\cal H}}(t)=\hat{{\cal H}}_{1}$ in
the time duration $t\in[\ell,\ell+1/2)$ and $\hat{{\cal H}}(t)=\hat{{\cal
H}}_{2}$ in the time duration $t\in[\ell+1/2,\ell+1)$, where
$\ell\in\mathbb{Z}$ and

$\displaystyle\hat{{\cal H}}_{1}=$ $\displaystyle\sum_{n}J_{x}(\hat{{\bf
c}}_{n}^{\dagger}\sigma_{0}\otimes\tau_{z}\hat{{\bf c}}_{n+1}+{\rm H.c.})$
$\displaystyle-$ $\displaystyle\sum_{n}iV(\hat{{\bf
c}}_{n}^{\dagger}\sigma_{y}\otimes\tau_{0}\hat{{\bf c}}_{n+1}-{\rm H.c.}),$
(51) $\displaystyle\hat{{\cal H}}_{2}=$ $\displaystyle\sum_{n}J_{y}\hat{{\bf
c}}_{n}^{\dagger}\sigma_{0}\otimes\tau_{x}\hat{{\bf c}}_{n}$ $\displaystyle+$
$\displaystyle\sum_{n}iJ_{d}(\hat{{\bf
c}}_{n}^{\dagger}\sigma_{z}\otimes\tau_{x}\hat{{\bf c}}_{n+1}-{\rm H.c.}).$
(52)

Here have we set the driving period $T=1$. $J_{x}$, $J_{y}$ and $J_{d}$ denote
the hopping amplitudes along the $x$, $y$, and diagonal directions of the
ladder. $V$ denotes the spin-orbit coupling between fermions with opposite
spins in adjacent unit cells SCLZhou2020 . $\sigma_{x,y,z}$ and $\tau_{x,y,z}$
are Pauli matrices acting on the spin-$1/2$ degree of freedom and the two
sublattices $(a,b)$, respectively. $\sigma_{0}$ and $\tau_{0}$ are $2\times 2$
identity matrices. The vector annihilation operator $\hat{{\bf
c}}_{n}\equiv(\hat{c}_{n,\uparrow,a},\hat{c}_{n,\uparrow,b},\hat{c}_{n,\downarrow,a},\hat{c}_{n,\downarrow,b})^{\top}$,
where $\hat{c}_{n,\sigma,\tau}$ annihilates a fermion with spin $\sigma$
($=\uparrow,\downarrow$) at the sublattice $\tau$ ($=a,b$) in the $n$th unit
cell. The Floquet operator of the system takes the form

$\hat{{\cal U}}=e^{-\frac{i}{2}\hat{{\cal H}}_{2}}e^{-\frac{i}{2}\hat{{\cal
H}}_{1}}.$ (53)

To characterize the symmetry and topology of the system, we also transform
$\hat{{\cal U}}$ into symmetric time frames in parallel with what we did in
Sec. III.1.1, yielding the Floquet operators

$\hat{{\cal U}}_{1}=e^{-\frac{i}{4}\hat{{\cal
H}}_{1}}e^{-\frac{i}{2}\hat{{\cal H}}_{2}}e^{-\frac{i}{4}\hat{{\cal
H}}_{1}},\quad\hat{{\cal U}}_{2}=e^{-\frac{i}{4}\hat{{\cal
H}}_{2}}e^{-\frac{i}{2}\hat{{\cal H}}_{1}}e^{-\frac{i}{4}\hat{{\cal H}}_{2}}.$
(54)

$\hat{{\cal U}}_{1}$ and $\hat{{\cal U}}_{2}$ have been shown to possess the
time-reversal symmetry $i\sigma_{y}\otimes\tau_{0}{\cal K}$, particle-hole
symmetry $\sigma_{x}\otimes\tau_{y}{\cal K}$, and chiral symmetry
$\Gamma=-\sigma_{z}\otimes\tau_{y}$ with $\Gamma^{2}=1$ SCLZhou2020 . As the
time-reversal and particle-hole symmetries both square to $-1$, the spin-$1/2$
PQL belongs to the symmetry class CII. The Floquet topological phases of
$\hat{{\cal U}}$ are thus characterized by a pair of even-integer quantized
winding numbers SCLZhou2020 . In Figs. 4(a) and 4(b), we show the quasienergy
spectrum of the system by solving the eigenvalue equation $\hat{{\cal
U}}|\psi\rangle=e^{-iE}|\psi\rangle$ under the PBC and OBC, respectively. With
the increase of $J_{d}$, a series of gap closing/reopening processes are found
at $E=0$ and $E=\pm\pi$. Fourfold degenerate edge modes are observed at the
quasienergies zero and $\pi$ (denoted by red stars). Both of their numbers
$(n_{0},n_{\pi})$ change in the sequence $0\rightarrow 4\rightarrow
8\rightarrow 12\rightarrow 16$ following the topological phase transitions
induced by the increase of $J_{d}$ SCLZhou2020 .

Figure 4: Quasienergy and correlation matrix spectrum of the spin-$1/2$ PQL
versus the diagonal hopping amplitude $J_{d}$. Other system parameters are set
as $(J_{x},J_{y},V)=(0.5\pi,0.6\pi,0.2\pi)$, and the length of lattice is
$L=250$ for all panels. (a) shows the Floquet spectrum under the PBC. (b)
shows the Floquet spectrum under the OBC, where the red crosses denote edge
states with quasienergies $E=0,\pm\pi$. (c) and (d) show the spectrum of
correlation matrices $C_{\alpha}$ in symmetric time frames $\alpha=1,2$. Red
crosses in (c) and (d) highlight eigenstates of $C_{\alpha}$ for $\alpha=1,2$
with localized profiles around the cuts between subsystems A and B.

We now characterize the topology and phase transitions in this spin-$1/2$ PQL
from the quantum entanglement perspective. We again focus on the case with all
Floquet states whose quasienergies $E\in(-\pi,0)$ are uniformly filled. The
spin-$1/2$ PQL possesses four quasienergy bands, and the density operator of
our consideration then refers to the occupation of the two lower Floquet bands
in the first quasienergy Brillouin zone. In the $\alpha$’s time frame, the
correlation matrix, ES and EE of the spin-$1/2$ PQL share the same form with
the spin-$1/2$ ORDKR, as described by Eqs. (43)–(45). In Figs. 4(c) and 4(d),
we show the spectrum of correlation matrices $C_{1}$ and $C_{2}$ versus
$J_{d}$ under the PBC. Other parameters of $\hat{{\cal U}}_{1}$ and
$\hat{{\cal U}}_{2}$ are chosen to be the same as those used in Figs. 4(a) and
4(b). We again observe discontinuous behaviors in the ES whenever the
quasienergy spectrum gap of the spin-$1/2$ PQL closes at $E=0$ or $E=\pm\pi$.
Meanwhile, the numbers of localized modes in the ES with eigenvalue
$\zeta_{j}^{\alpha}=1/2$ change following different patterns in the two time
frames $\alpha=1$ and $2$. This situation is similar to what we encountered in
the spin-$1/2$ ORDKR. It implies that the entanglement characteristics of
Floquet reduced density matrices in both two symmetric time frames are needed
in order to capture the full topological information of the system. We regard
this as a generic feature of 1D Floquet topological phases protected by chiral
symmetry.

Figure 5: The number of the maximally entangled states, winding numbers of the
entanglement Hamiltonian and the EE of the spin-$1/2$ PQL, evaluated in
symmetric time frames $\alpha=1,2$. Other system parameters are
$(J_{x},J_{y},V)=(0.5\pi,0.6\pi,0.2\pi)$, and the length of lattice is
$L=250$. $N_{\alpha}$ denotes the number of eigenstates of the correlation
matrix $C_{\alpha}$ with eigenvalue $1/2$. $W_{\alpha}$ is the open-bulk
winding number of $C_{\alpha}$ [Eq. (43)]. $S_{\alpha}$ denotes the EE in the
time frame $\alpha$ [Eq. (45)].

In Fig. 5, we first notice that in both time frames, the EE shows a non-
analytic cusp right at each gap closing point in Figs. 4(a) and 4(b).
Therefore, the EE can also be used to signify transitions between different
Floquet topological phases of the spin-$1/2$ PQL. Moreover, a steady growth of
$S_{1}$ is observed with the increase of $J_{d}$. It goes in parallel with the
monotonic raise of the number of $\zeta_{j}^{1}=1/2$ eigenmodes by four after
each transition. Every such eigenmode makes a largest contribution $\ln 2$ to
$S_{1}$. Meanwhile, the $S_{2}$ also shows a growth with a much smaller rate
in its global profile. Since the number of $\zeta_{j}^{2}=1/2$ eigenmodes
$N_{2}$ only oscillates between zero and four following the transitions
induced by $J_{d}$, instead of showing a monotonic increase as observed in
$N_{1}$, the global increase in $S_{2}$ cannot be originated from these
maximally entanglement eigenmodes. Then there must be some other states whose
ES deviates from zero and one further with the increase of $J_{d}$, as we
indeed observe in Fig. 4(d).

To relate the topology of Floquet entanglement Hamiltonian under the PBC and
$\zeta=1/2$ eigenmodes with Floquet edge states under the OBC, we construct
the projector of correlation matrix eigenmodes in the $\alpha$’s time frame as
the form in Eq. (46). The open-bulk winding number [Eq. (33)] of entanglement
Hamiltonian in the $\alpha$’s time frame is then given by

$W_{\alpha}=\frac{1}{L^{\prime}}{\rm Tr}^{\prime}(\Gamma
Q_{\alpha}[Q_{\alpha},N_{{\rm A}}]),$ (55)

where $\alpha=1,2$. $\Gamma=-\sigma_{z}\otimes\tau_{y}$ is the chiral symmetry
operator of the spin-$1/2$ PQL. Other quantities share the same meanings with
those in Eq. (47). Referring to the results reported in Figs. 4 and 5, we
conclude that the relation between the numbers of $\zeta=1/2$ eigenmodes
$(N_{1},N_{2})$ in the ES and the numbers of Floquet edge modes
$(n_{0},n_{\pi})$ with quasienergies zero and $\pi$ in Eq. (48) holds also for
the spin-$1/2$ PQL. This observation implies the generality of Eq. (48) in
describing the relationship between entanglement and topological edge states
for 1D Floquet topological insulators in the symmetry class CII. Next, for the
spin-$1/2$ PQL, the relation between the numbers of $\zeta=1/2$ eigenmodes
$(N_{1},N_{2})$ and the winding numbers $(W_{1},W_{2})$ in Eq. (49) now takes
the form

$N_{1}=4|W_{1}|,\qquad N_{2}=4|W_{2}|,$ (56)

as verified by the data reported in Fig. 5. The prefactor $4$ comes from the
fact that the $\zeta=1/2$ eigenmodes are fourfold degenerate. Eq. (56)
describes the bulk-edge correspondence of ES in the two symmetric time frames.
It is generic for CII class Floquet topological phases in one-dimension.
Finally, the combination of Eqs. (48) and (56) yields the relation between the
topology of Floquet entanglement Hamiltonian under the PBC and the zero/$\pi$
Floquet edge modes of the spin-$1/2$ PQL under the OBC following Eq. (50), or
explicitly expressed as

$n_{0}=2|W_{1}+W_{2}|,\qquad n_{\pi}=2|W_{1}-W_{2}|.$ (57)

Eq. (57) allows us to know the exact numbers of Floquet zero and $\pi$
quasienergy edge modes under the OBC from the topology of ES under the PBC. We
regard it as an entanglement bulk-edge correspondence for 1D Floquet systems
in the CII symmetry class. We have tested these equations in other parameter
regions of the spin-$1/2$ PQL and confirmed their correctness. They are
therefore general for CII class 1D Floquet topological matter. Eqs. (48), (56)
and (57) form another set of key results in this work. In the next subsection,
we discuss the universality of the topology-entanglement connection in 2D
Floquet topological insulators.

###  III.2 ES and EE in 2D Floquet systems

####  III.2.1 Periodically quenched Chern insulator

We now investigate 2D Floquet Chern insulators, which could possess
topological phases with large Chern numbers. We start with spinless
noninteracting fermions in a periodically quenched generalized Haldane model
(PQGHM) with third neighbor hopping FloBigTN7 ; Haldane1 ; Haldane2 ; Haldane3
. In the absence of quenches and under the PBC along two spatial dimensions,
the momentum-space Hamiltonian of the system takes the form
$\hat{H}=\sum_{{\bf k}\in{\rm BZ}}\hat{{\bf c}}_{{\bf k}}^{\dagger}H({\bf
k})\hat{{\bf c}}_{{\bf k}}$. Here ${\bf
k}=(k_{1},k_{2})\in[-\pi,\pi)\times[-\pi,\pi)$ is the quasimomentum.
$\hat{{\bf c}}_{{\bf k}}\equiv(\hat{c}_{{\bf k},a},\hat{c}_{{\bf
k},b})^{\top}$, where $\hat{c}_{{\bf k},a}$ ($\hat{c}_{{\bf k},b}$)
annihilates a fermion with quasimomentum ${\bf k}$ on the sublattice $a$ ($b$)
of the honeycomb lattice. The Bloch Hamiltonian is $H({\bf k})=h_{{\bf
k}}^{x}\sigma_{x}+h_{{\bf k}}^{y}\sigma_{y}+h_{{\bf k}}^{z}\sigma_{z}$, where
$h_{{\bf k}}^{x}=t_{1}(1+\cos k_{1}+\cos
k_{2})+t_{3}[2\cos(k_{1}-k_{2})+\cos(k_{1}+k_{2})]$, $h_{{\bf
k}}^{y}=t_{1}(\sin k_{1}+\sin k_{2})+t_{3}\sin(k_{1}+k_{2})$, and $h_{{\bf
k}}^{z}=2t_{2}\sin\phi[\sin k_{1}-\sin k_{2}-\sin(k_{1}-k_{2})]$ FloBigTN7 .
The Pauli matrices $\sigma_{x,y,z}$ act on the sublattice degrees of freedom.
The quench is applied by decomposing each driving period $T$ into two episodes
$T_{1}$ and $T_{2}$ ($T=T_{1}+T_{2}$), and setting the parameters
$(t_{3},\phi)=(t_{31},\phi_{1})$ ($(t_{3},\phi)=(t_{32},\phi_{2})$) in the
time duration $T_{1}$ ($T_{2}$). We denote the system’s Bloch Hamiltonians
during $T_{1}$ and $T_{2}$ as $H_{1}({\bf k})$ and $H_{2}({\bf k})$. The
Floquet operator then takes the form $\hat{U}=\sum_{{\bf k}\in{\rm
BZ}}\hat{{\bf c}}_{{\bf k}}^{\dagger}U({\bf k})\hat{{\bf c}}_{{\bf k}}$, where
$U({\bf k})=e^{-iH_{2}({\bf k})T_{2}}e^{-iH_{1}({\bf k})T_{1}}$. In Ref.
FloBigTN7 , it was found that the $U({\bf k})$ possesses quasienergy bands
with large Chern numbers and rich topological phase diagrams due to the
Floquet driving, even though the energy bands of underlying static
Hamiltonians $H_{1}({\bf k})$ and $H_{2}({\bf k})$ only carry relatively small
Chern numbers. The experimental simulation of $U({\bf k})$ and the detection
of its Floquet-band Chern numbers were also conducted recently in a setup
containing a nitrogen-vacancy center in diamond LargeCNExp .

Figure 6: Quasienergy spectrum of the PQGHM versus the quasimomentum $k_{1}$.
The PBC (OBC) is taken along the direction $1$ ($2$) of the lattice. Other
system parameters are set as $t_{1}=1$, $t_{2}=0.8$, $t_{31}=0.75$,
$t_{32}=-0.75$, $\phi_{1}=-\pi/6$, $\phi_{2}=-\pi/2$, and the length of
lattice along the direction $2$ is $L=200$ for all panels. Blue stars (red
circles) denote states localized around the left (right) edge of the lattice.

Performing a Fourier transformation $\hat{{\bf c}}_{{\bf
k}}=\frac{1}{\sqrt{L_{2}}}\sum_{n=1}^{L_{2}}e^{-ik_{2}n}\hat{{\bf
c}}_{k_{1},n}$ from momentum to position representation along the
$k_{2}$-direction, we obtain the Hamiltonian
$\hat{H}=\sum_{k_{1}}(\hat{h}_{k_{1}}^{x}\sigma_{x}+\hat{h}_{k_{1}}^{y}\sigma_{y}+\hat{h}_{k_{1}}^{z}\sigma_{z})$,
where

$\displaystyle\hat{h}_{k_{1}}^{x}=$ $\displaystyle\sum_{n}t_{1}\left[(1+\cos
k_{1})\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n}+\frac{\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}+{\rm H.c.}}{2}\right]$
$\displaystyle+$ $\displaystyle\sum_{n}\frac{3t_{3}}{2}\cos
k_{1}\left(\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}+{\rm
H.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\frac{t_{3}}{2i}\sin
k_{1}\left(\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}-{\rm
H.c.}\right),$ (58)

$\displaystyle\hat{h}_{k_{1}}^{y}=$ $\displaystyle\sum_{n}t_{1}\left[\sin
k_{1}\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf
c}}_{k_{1},n}-\frac{i}{2}\left(\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf
c}}_{k_{1},n+1}-{\rm H.c.}\right)\right]$ $\displaystyle+$
$\displaystyle\sum_{n}\frac{t_{3}}{2}\sin k_{1}\left(\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}+{\rm H.c.}\right)$
$\displaystyle+$ $\displaystyle\sum_{n}\frac{t_{3}}{2i}\cos
k_{1}\left(\hat{{\bf c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}-{\rm
H.c.}\right),$ (59)

$\displaystyle\hat{h}_{k_{1}}^{z}=$
$\displaystyle\sum_{n}t_{2}\sin\phi\left[2\sin k_{1}\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n}+i(\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}-{\rm H.c.})\right]$
$\displaystyle-$ $\displaystyle\sum_{n}t_{2}\sin\phi\sin k_{1}\left(\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}+{\rm H.c.}\right)$
$\displaystyle-$ $\displaystyle\sum_{n}it_{2}\sin\phi\cos k_{1}\left(\hat{{\bf
c}}_{k_{1},n}^{\dagger}\hat{{\bf c}}_{k_{1},n+1}-{\rm H.c.}\right).$ (60)

Following the above mentioned quench protocol, we obtain the quasienergy
spectrum of Floquet operator
$\hat{U}=e^{-i\hat{H}_{2}T_{2}}e^{-i\hat{H}_{1}T_{1}}$ under the PBC (OBC)
along the direction $1$ ($2$) of the lattice, as shown in Fig. 6. For all
considered cases, we find two Floquet bulk bands connected by gap-traversing
chiral edge states. In the first quasienergy Brillouin zone, the lower and
upper bands have Chern numbers $\pm 1$, $\pm 2$, $\pm 4$ and $\pm 7$ in Figs.
6(a), (b), (c) and (d), which are equal to the net difference between the
numbers of chiral edge states above and below a band while residing at the
same edge. This is consistent with the bulk-edge correspondence of 2D Floquet
Chern insulators revealed at the spectral level FloBigTN7 . The emerging
Floquet bands with large Chern numbers $\pm 4$ and $\pm 7$ are unreachable in
the static limit of the model LargeCNExp . It reflects one key advantage of
Floquet engineering, i.e., to create phases with large topological invariants
and many edge states by simply tuning the time durations of the drive.

Figure 7: Correlation matrix spectra of the PQGHM versus the quasimomentum
$k_{1}$. PBCs are taken along both directions of the lattice. Other system
parameters and the length of lattice are chosen to be the same as in Fig. 6.
Blue stars (red circles) denote eigenmodes localized around the left (right)
entanglement cuts between the subsystems A and B.

We now uncover the topology, phase transitions and bulk-edge correspondence of
the PQGHM from its ES, EE and the related topological numbers. We first take
the PBC along direction $2$ by setting $\hat{{\bf
c}}_{k_{1},n+L_{2}}=\hat{{\bf c}}_{k_{1},n}$, where $L_{2}$ is the number of
unit cells along the second dimension of the lattice. Then we decompose the
system into two equal segments A and B, and obtain the ES of reduced density
matrix $\hat{\rho}_{{\rm A}}$ following the Eqs. (12)–(28) in Sec. II. Here
the state $|\Psi\rangle$ in Eq. (12) describes the uniformly filled lower
Floquet band in the first quasienergy Brillouin zone. The resulting ES is
presented in Figs. 7(a)–(d), with the same system parameters as those used in
Figs. 6(a)–(d). We observe that large amounts of the correlation matrix
eigenvalues $\zeta_{j}$ [Eq. (27)] reside near $\zeta=0$ and $\zeta=1$, making
vanishing contributions to the EE in Eq. (32). In the meantime, chiral ES
flows across the ES gap are observed in each panel of Fig. 7. These chiral
bands are formed by eigenstates of the Floquet entanglement Hamiltonian
$\hat{H}_{{\rm A}}$ [Eq. (15)] that are localized around the entanglement cuts
between subsystems A and B. Furthermore, the number of chiral edge bands
$n_{{\rm c}}$ at the same edge in the ES is found to be equal to the Chern
number ${\cal C}_{0}$ of the Floquet band used the in the construction of the
initial density matrix $\hat{\rho}$ [Eq. (12)], i.e.,

$n_{{\rm c}}=|{\cal C}_{0}|.$ (61)

This relation establishes the correspondence between the Chern topology of
Floquet bands and the number of chiral edge modes in the ES for 2D Floquet
Chern insulators with two bands. As demonstrated in Sec. III.2.2, it is also
valid for systems with multiple bands.

Figure 8: Quasienergy spectra [in (a), (b)], EE [in (c), (d)] and real-space
Chern numbers [in (e), (f)] of the PQGHM versus the driving time durations
$T_{1}$ and $T_{2}$. The size of lattice is $L_{1}=L_{2}=60$. PBCs are taken
along both directions of the lattice. We set $T_{2}=1.2$ for (a), (c), (e) and
$T_{1}=0.9$ for (b), (d), (f). Other system parameters are chosen to be the
same as those used in Fig. 6.

We next consider topological phase transitions in the system. In Figs. 8(a)
and 8(b), we show the Floquet spectra under the PBC versus the quench time
spans $T_{1}$ and $T_{2}$. The gaps between the two quasienergy bands close
when $T_{1}=T_{1{\rm c}}$ and $T_{2}=T_{2{\rm c}}$ in Figs. 8(a) and 8(b),
respectively. The critical values $T_{1{\rm c}}$ and $T_{2{\rm c}}$ can be
obtained by solving the equations of gapless condition, i.e., ${\bf
h}_{1}/|{\bf h}_{1}|={\bf h}_{2}/|{\bf h}_{2}|$ and $T_{1}|{\bf
h}_{1}|+T_{2}|{\bf h}_{2}|=\ell\pi$ ($\ell\in\mathbb{Z}$). Here ${\bf h}_{1}$
and ${\bf h}_{2}$ are ${\bf k}$-dependent vectors formed by the three
components in front of the Pauli matrices in $H_{1}({\bf k})$ and $H_{2}({\bf
k})$, respectively LargeCNExp . With the increase of $T_{1}$ ($T_{2}$) from
$T_{1}-0^{+}$ ($T_{2}-0^{+}$) to $T_{1}+0^{+}$ ($T_{2}+0^{+}$), the PQGHM
undergoes a topological phase transition accompanied by the quantized change
of Floquet band Chern numbers from $\pm 4$ ($\pm 2$) to $\pm 7$ ($\pm 4$)
LargeCNExp . The EE curves in Figs. 8(c) and 8(d) are obtained following Eqs.
(27)–(32). We find that the derivatives $\partial_{T_{1}}S$ and
$\partial_{T_{2}}S$ become discontinuous at $T_{1}=T_{1{\rm c}}$ and
$T_{2}=T_{2{\rm c}}$, signifying the transitions between different Floquet
Chern insulator phases from the perspective of EE.

Finally, we characterize the topology of Floquet entanglement Hamiltonian
$H_{{\rm A}}$ [Eq. (15)] and relate it to the edge states observed in the
quasienergy spectrum. To this end, we consider the real-space Chern number Ch
defined through Eqs. (35)–(37). In Figs. 8(e) and 8(f), we show the Ch thus
computed at different values of quenching time intervals $T_{1}$ and $T_{2}$
for the PQGHM. Quantized jumps are observed in the Ch around $T_{1}=T_{1{\rm
c}}$ and $T_{2}=T_{2{\rm c}}$, i.e., at the expected transition points between
different Floquet Chern insulator phases. Away from the transition points, the
real-space Chern numbers are found to take quantized values, which are further
equal to the Chern numbers ${\cal C}_{0}$ of the occupied Floquet bands
employed in the construction of reduced density matrix $\hat{\rho}_{{\rm A}}$.
Therefore, we arrive at the relationship between the Chern numbers of
entanglement Hamiltonian and the chiral edge modes in the quasienergy
spectrum, i.e.,

$|n_{{\rm L}}-n^{\prime}_{{\rm L}}|=|{\rm Ch}|,$ (62)

where $n_{{\rm L}}$ and $n^{\prime}_{{\rm L}}$ denote the net number of chiral
edge bands leaving and entering the occupied Floquet band at the left edge in
the lattice (see Fig. 6). We refer to this relation as an entanglement bulk-
edge correspondence of 2D Floquet Chern insulators. As one key result of this
work, it establishes the connection between the Chern topology of Floquet
entanglement Hamiltonian under the PBC and Floquet chiral edge states under
the OBC. All the topological information of a Floquet Chern band can thus be
extracted from the entanglement nature of its related reduced density matrix.
Eq. (62) can be further generalized to Floquet Chern insulators with multiple
quasienergy bands, as shown in the following subsection.

####  III.2.2 Kicked quantum Hall insulator

We finally consider fermions hopping in a square lattice with perpendicular
magnetic fluxes. The system Hamiltonian is made time-dependent by adding
periodic delta kicks to the hopping amplitudes along $y$-direction of the
lattice, yielding

$\displaystyle\hat{H}(t)=$
$\displaystyle\frac{J}{2}\sum_{x,y}\left(\hat{c}_{x+1,y}^{\dagger}\hat{c}_{x,y}+{\rm
H.c.}\right)$ $\displaystyle+$
$\displaystyle\frac{V}{2}\sum_{x,y}\left(e^{i\lambda
x}\hat{c}_{x,y+1}^{\dagger}\hat{c}_{x,y}+{\rm
H.c.}\right)\sum_{\ell\in\mathbb{Z}}\delta(t-\ell).$ (63)

Here $\hat{c}_{x,y}$ ($\hat{c}_{x,y}^{\dagger}$) annihilates (creates) a
fermion on the lattice site $(x,y)$. $J$ and $V$ are hopping amplitudes along
$x$ and $y$ directions. $\lambda$ determines the magnetic flux over each cell
of the lattice. The driving period is set to $T=1$. The Floquet operator of
the system then takes the form

$\displaystyle\hat{U}$
$\displaystyle=e^{-\frac{iV}{2}\sum_{x,y}\left(e^{i\lambda
x}\hat{c}_{x,y+1}^{\dagger}\hat{c}_{x,y}+{\rm H.c.}\right)}$
$\displaystyle\times
e^{-\frac{iJ}{2}\sum_{x,y}\left(\hat{c}_{x+1,y}^{\dagger}\hat{c}_{x,y}+{\rm
H.c.}\right)}.$ (64)

It is usually called the kicked Harper model (KHM) or kicked quantum Hall
system FloBigTN3 ; FloBigTN4 ; FloBigTN5 . For $\lambda=2\pi p/q$, with $p$
and $q$ being coprime integers, the Floquet spectrum of $\hat{U}$ groups into
$q$ quasienergy bands under the PBC. In this work, we focus on the case with
$p/q=1/3$, so that there are three Floquet bands in the quasienergy spectrum.
The topology of each Floquet band is characterized by an integer-quantized
Chern number FloBigTN3 . With the change of $J$ and/or $V$, the KHM can
undergo topological phase transitions between different Floquet Chern
insulator phases, which are accompanied by quantized changes of band Chern
numbers FloBigTN3 . Floquet bands with large Chern numbers were found at large
hopping amplitudes $J$ and/or $V$ FloBigTN3 , showing the advantage of Floquet
engineering in creating phases with strong topological signals.

Taking the OBC (PBC) along the $x$ ($y$) direction and performing the Fourier
transformation
$\hat{c}_{x,y}=\frac{1}{\sqrt{L_{y}}}\sum_{k_{y}}e^{ik_{y}y}\hat{c}_{x,k_{y}}$
along $y$ direction, we can express the Floquet operator of the KHM as
$\hat{U}=\sum_{k_{y}}\hat{U}(k_{y})$, where

$\displaystyle\hat{U}(k_{y})$ $\displaystyle=e^{-iV\sum_{x}\cos(\lambda
x-k_{y})\hat{c}_{x,k_{y}}^{\dagger}\hat{c}_{x,k_{y}}}$ $\displaystyle\times
e^{-\frac{iJ}{2}\sum_{x}\left(\hat{c}_{x+1,k_{y}}^{\dagger}\hat{c}_{x,k_{y}}+{\rm
H.c.}\right)}.$ (65)

$k_{y}\in[-\pi,\pi)$ is the quasimomentum along $y$. Typical examples of the
Floquet spectrum of $\hat{U}(k_{y})$ are shown in Figs. 9(a) and 9(d). The
Chern numbers of the three Floquet bands from bottom to above in Figs. 9(a)
and 9(d) are $(1,-2,1)$ and $(-2,4,-2)$ FloBigTN3 . This is coincide with the
difference between the number of chiral edge bands at the same edge and with
the same chirality leaving and entering a given Floquet band, which verifies
the bulk-edge correspondence of Floquet Chern insulators from the aspect of
quasienergy spectrum FloBigTN3 .

Figure 9: Quasienergy spectrum $E$ and correlation matrix spectrum $\zeta$ of
the KHM versus the quasimomentum $k_{y}$. The length of lattice is chosen to
be $L=300$ along the $x$-direction for all panels. System parameters are set
as $(J,V)=(2\pi/3,\pi)$ for (a)–(c) and $(J,V)=(2\pi/3,2\pi)$ for (d)–(f). The
OBC (PBC) is taken along the $x$-direction in the calculation of $E$
($\zeta$). Blue stars (red circles) denote states localized around the left
(right) edges in (a), (d), and states localized around the left (right)
entanglement cuts between the subsystems A and B in (b), (c), (e), (f),
respectively. For (b) and (e), the states $|\Psi\rangle$ correspond to the
uniform filling of the bottom Floquet bands in (a) and (d). For (c) and (f),
the states $|\Psi\rangle$ correspond to the uniform filling of the middle
Floquet bands in (a) and (d).

We now unveil the topological nature and bulk-edge correspondence of the KHM
by investigating its ES and EE. Starting with the Floquet operator
$\hat{U}(k_{y})$ in Eq. (65), we take the PBC along $x$-direction and let the
density matrix in Eq. (13) to describe the many-particle Floquet state that
fills the bottom or the middle Floquet band of $\hat{U}(k_{y})$ uniformly.
Bisecting the system into two equal parts and following the steps in Eqs.
(14)–(28), we obtain the ES of KHM as shown in Figs. 9(b), 9(c), 9(e), and
9(f). In each case, we find that most of the correlation matrix eigenvalues
are pinned around $\zeta=0$ and $\zeta=1$, yielding vanishing contributions to
the EE in Eq. (32). Meanwhile, eigenmodes traversing the gap of ES are found
and they are localized at the entanglement cuts between the subsystems A and
B. Moreover, the ES of these eigenmodes are chiral, and the number of such
chiral entanglement bands at each edge is equal to the Chern number of the
filled Floquet band, which verifies the Eq. (61). This observation allows us
to inspect the Chern topology of a Floquet band and the related edge states
from its ES. It also demonstrates that our approach to the ES of Floquet Chern
bands are not limited to two-band models, but works for Floquet Chern
insulators with any number of gapped quasienergy bands.

Figure 10: The EE and real-space Chern numbers of the entanglement Hamiltonian
for the KHM. The hopping amplitude along $x$ direction is set to $J=2\pi/3$.
The size of lattice is $L_{x}=L_{y}=72$ with PBCs taken along both $x$ and $y$
directions. In (a) and (b), $S_{1}$ ($S_{2}$) and ${\rm Ch}_{1}$ (${\rm
Ch}_{2}$) refer to the EE and the real-space Chern number of the entanglement
Hamiltonian ${\hat{H}}_{{\rm A}}$ when the bottom (middle) quasienergy band is
filled in the construction of $\hat{\rho}_{{\rm A}}$.

The difference between the band Chern numbers in Figs. 9(a) and 9(d) suggests
that there is at least a transition from one Floquet Chern insulator phase to
another when the hopping amplitude $V$ goes from $\pi$ to $2\pi$. In Fig.
10(a), we show the EE of the bottom and middle filled Floquet band of the KHM
versus $V$ under the PBC along both two spatial dimensions [Eq. (32)]. We
indeed observe nonanalytic cusps in $S_{1}$ and $S_{2}$ near $V=V_{\rm c}$,
which is coincident with the topological phase transition point of the KHM
WangPRE2013 . To characterize the topology of this transition and the bulk-
edge correspondence directly at the level of Floquet entanglement Hamiltonian
[Eq. (15)], we compute the real-space Chern number of $\hat{H}_{{\rm A}}$ at
half-filling under the PBC from the local Chern marker following Eqs.
(35)–(37). The results for the cases of filled bottom and middle Floquet bands
of the KHM are displayed by ${\rm Ch}_{1}$ (stars) and ${\rm Ch}_{2}$
(circles) in Fig. 10(b). We find that both the Chern numbers show quantized
jumps at the transition point $V=V_{\rm c}$, implying that they can correctly
capture the topological phase transition in the system. Moreover, away from
the transition point, ${\rm Ch}_{1}$ and ${\rm Ch}_{2}$ take quantized values
(up to fluctuations due to finite-size effects) that are equal to the Chern
numbers of the bottom and middle Floquet bands in Figs. 9(a) and 9(d).
Therefore, we arrive at the following entanglement bulk-edge correspondence
for Floquet Chern insulators

$|n_{{\rm L}}-n^{\prime}_{{\rm L}}|=|{\rm Ch}_{j}|.$ (66)

Here $n_{{\rm L}}$ and $n^{\prime}_{{\rm L}}$ represent the net number of
chiral edge bands leaving and entering the $j$th Floquet quasienergy band at
the left edge of the lattice. ${\rm Ch}_{j}$ denotes the real-space Chern
number of the Floquet entanglement Hamiltonian, which is originated from the
reduced density operator of the $j$th Floquet band. Eq. (66) generalizes the
Eq. (62) to Floquet Chern insulators with more than two bands. We have
explored other parameter regions of the KHM and obtain results that are
consistent with those reported in Figs. 9, 10 and by Eq. (66). Therefore, we
conclude that the entanglement tools introduced in this work could
characterize the topology, edge states and bulk-edge correspondence of 2D
Floquet Chern insulators with an arbitrary number of quasienergy bands.

##  IV Discussion and Conclusion

It is interesting to discuss the connections and differences between our work
and some previous studies. First, in 1D Floquet topological insulators, both
the zero and $\pi$ quasienergy edge modes show their signatures in the ES. The
$\pi$ modes do not exist in usual static topological systems. Their
entanglement signatures are thus unique to Floquet systems showcased by the
first and second models in Sec. III. Second, in some former studies ESEETP52 ;
ESEETP6 ; ESEETP7 ; ESEETP8 ; ESEETP9 ; ESEETP10 , the ES of usual Chern and
quantum Hall insulators were investigated. The connection between the chiral
edge states in the ES and that in the energy spectrum was demonstrated.
Topological invariants were also introduced to characterize the ES in momentum
space. In our work, we considered the ES and EE of periodically driven Chern
and quantum Hall insulators, whose topological properties are carried by the
underlying Floquet states instead of a ground state in usual static
topological systems. Even though the chiral edge states observed in the
Floquet ES are similar to those found in static topological models, we have
introduced topological invariants to characterize the bulk-edge correspondence
between these edge modes and the topology of Floquet entanglement Hamiltonian
directly in real space, which is different from previous approaches. Third,
non-analytical properties of EE across quantum phase transitions have been
identified in early studies EntangleRev1 . The behaviors of EE around the
transition points between different topological phases were also explored in
usual static systems EETPT1 ; EETPT2 ; EETPT3 ; EETPT4 . There, non-analytic
signatures in the EE were observed and the scaling properties of EE were
investigated. In our study, we also observed non-analytic signatures in the EE
of Floquet states when the underlying system goes from one Floquet topological
insulator phase to another. Compared with previous results, we tend to believe
that in general, the EE possesses similar features around a transition point
between two topological insulator/superconductor phases either in usual static
systems or in Floquet driven systems. In our case studies, we focus more on
the topological characterization of ES and its related bulk-edge
correspondence in Floquet systems. The EE are presented mainly to provide
complementary information for the signatures of Floquet topological phase
transitions. We have supplemented our results with further details about EE in
the Appendix A.

In Ref. YatesPRB2016 , the ES and EE of a graphene lattice under different
Floquet driving protocols were explored with a theoretical formalism tailored
to two-band models and a focus on the time-dependent properties of the
entanglement measures. Signatures of chiral edge states in the ES were also
identified there. In our work, we instead focus on the ES and EE of filled
Floquet bands and present a framework that is applicable to study them in
noninteracting fermionic systems with more than two quasienergy bands and
large Chern numbers, as illustrated by the case studies in Sec. III.2.
Moreover, we characterized the changes of ES, EE and topological property of
entanglement Hamiltonian when the system undergoes transitions between
different Floquet Chern or quantum Hall insulator phases, which were not
considered in Ref. YatesPRB2016 . In Refs. YatesPRB2017  and YatesPRL2018 ,
the ES, EE and central charge of a 1D two-band Floquet topological
superconductor were investigated by a theory presented in terms of Majorana
operators. Non-analytic behaviors in EE when the system undergoes transitions
between different Floquet superconducting phases were studied. In our work, we
formalized our theory in terms of normal fermions and demonstrated it in two
1D Floquet topological insulator models with different numbers of quasienergy
bands and belonging to symmetry classes that are different from those explored
in Refs. YatesPRB2017  and YatesPRL2018 . We also clarified the bulk-edge
correspondence by comparing the ES and the winding numbers of Floquet
entanglement Hamiltonians in different time frames. Furthermore, the real-
space Chern and winding numbers employed in our work allows us to directly
characterize the topology, phase transitions and bulk-edge correspondence
residing in the bulk Floquet ES, which were not considered in Refs.
YatesPRB2016 –YatesPRL2018 . Besides helping us to build the connection
between the topological properties of quasienergy and entanglement spectra,
these invariants are also robust to the change of boundary conditions and the
presence of symmetry-preserving impurities, making them generalizable to the
characterization of Floquet entanglement topology in more complicated
situations.

In summary, we propose a framework to describe the ES and EE of noninteracting
fermions in Floquet systems. The theory is applicable to Floquet lattice
models with an arbitrary number of quasienergy bands and at arbitrary fillings
for each band. Open-bulk winding numbers and real-space Chern numbers are
further constructed to characterize the topology of Floquet entanglement
Hamiltonians directly for 1D and 2D systems. In one dimension, the
correspondences between the topology of bulk ES and the Floquet edge states
are established for Floquet topological insulators in the symmetry classes BDI
and CII. In two dimensions, the total Chern number of filled Floquet bands
determines the number of chiral edge modes traversing the gap of ES with the
same chirality, which is also consistent with the real-space Chern number of
the Floquet entanglement Hamiltonian. The EE further shows non-analytic
behaviors around the transition points between different Floquet topological
phases. Our work thus revealed the topological phase transition and bulk-edge
correspondence of Floquet topological matter from quantum entanglement
perspectives, and provided efficient means to characterize the topological
nature of Floquet entanglement Hamiltonians (or reduced density matrices of
Floquet states). It further demonstrated the generality and usefulness of
quantum information measures in characterizing Floquet topological phases.

In future work, the extensions of our entanglement framework to Floquet
systems with disorder, in high spatial dimensions, with other symmetry
constraints and in gapless systems (e.g., the Floquet Weyl semimetal WSM1 ;
WSM2 ) deserve to be investigated. For example, beyond the on-resonance
condition, the kicked rotor considered in Sec. III.1.1 does not possess the
translational symmetry in momentum space, and its topological properties may
have distinct and deeper physical origins KRSOther1 ; KRSOther2 , such as the
emergence of integer quantum Hall effect from chaos KRSTian1 ; KRSTian2 .
Meanwhile, the application of our entanglement approach should be relatively
straightforward. One can first express the Floquet operator of the kicked
rotor in a given basis and diagonalize it upon appropriate truncations. One
may then pick up a set of Floquet eigenstates of the kicked rotor to form its
many-particle density matrix. By decomposing the momentum space of the rotor
into two parts A, B and tracing out the degrees of freedom belonging to B, one
would obtain a reduced density matrix for which the ES and EE can be computed
by the formulas presented in this work. Moreover, the open-bulk winding number
is directly defined in the discrete lattice or momentum space, and it is
applicable to systems with disorder or other types of translational symmetry
breaking so long as the chiral symmetry is preserved. When the chiral symmetry
is also broken, other topological invariants might be needed to characterize
the Floquet entanglement Hamiltonian of the kicked rotor. Finding such an
invariant should constitute an interesting future study. As our theory is
tailored to deal with many-particle Floquet states, we can further incorporate
inter-particle interactions and decode the entanglement nature of the emerging
correlated phases in Floquet systems with potential topological features, such
as the coupled or mean-field interacting kicked rotor INTQKR1 ; INTQKR2 . For
the coupled kicked rotor INTQKR2 , one may treat the coordinates of two rotors
as the coordinates along two different spatial dimensions. The interaction
term in the Lieb-Liniger form can then be interpreted as an onsite potential
along the diagonal direction of the effective 2D lattice. We are now left with
a noninteracting problem in a 2D kicked lattice. The quasienergy eigenstates
of the system may then be obtained by diagonalizing the Floquet operator of
the effective 2D model. Following these treatments, our entanglement approach
might be implemented in parallel with what we have done for the 2D models in
this work. Finally, it deserves to mention that similar to the work of Peschel
EntangleRev4 , our approach of obtaining the ES and EE from the single-
particle correlation matrix is restricted to many-body states in Gaussian
form. Finding a clear path to extend this approach to non-Gaussian states or
other more complicated situations is a challenging yet very intriguing topic
for future explorations.

###### Acknowledgements.

We acknowledge Jiangbin Gong and Chushun Tian for helpful comments. This work
is supported by the National Natural Science Foundation of China (Grants No.
12275260 and No. 11905211), the Young Talents Project at Ocean University of
China (Grant No. 861801013196), and the Applied Research Project of
Postdoctoral Fellows in Qingdao (Grant No. 861905040009).

##  Appendix A EE around Floquet topological phase transition points

Figure 11: Floquet bands of the spin-$1/2$ ORDKR under the PBC. We set
$K_{1}=0.5\pi$ for all cases. The solid, dashed, dash-dotted and dotted curves
represent the quasienergy bands for $K_{2}=\pi$, $2\pi$, $3\pi$, and $4\pi$,
respectively. In each case, the lower band is filled in the calculation of EE.

In this Appendix, we elaborate a bit more on the EE of the spin-$1/2$ ORDKR in
Sec. III.1.1. Taking the PBC and performing the Fourier transformations
$\hat{\bf c}_{n}^{(\dagger)}=\frac{1}{\sqrt{L}}\sum_{k}e^{(-)ikn}\hat{\bf
c}_{k}^{(\dagger)}$ with the quasiposition $k\in[-\pi,\pi)$, we find the
Floquet operator Eq. (38) in $k$-space to be $U(k)=e^{-iK_{2}\sin
k\sigma_{y}}e^{-iK_{1}\cos k\sigma_{x}}$ KRSZhou2018 . Solving the eigenvalue
equation $U(k)|\psi\rangle=e^{-iE(k)}|\psi\rangle$, we find two Floquet
quasienergy bands with the dispersions $E_{\pm}(k)=\pm\arccos[\cos(K_{1}\cos
k)\cos(K_{2}\sin k)]$. These bands are defined modulus $2\pi$ and symmetric
with respect to $E=0$. So they can only meet with each other at $E_{\pm}=0$ or
$E_{\pm}=\pm\pi$. We thus find the gapless condition of the Floquet spectrum
to be $\cos(K_{1}\cos k)\cos(K_{2}\sin k)=\pm 1$. In the main text, we focus
on the case with the kicking strength $K_{1}=0.5\pi$. In this case, the
gapless condition can be satisfied only if $K_{1}\cos k=0$, i.e.,
$k=\pm\pi/2$, enforcing $K_{2}$ to be integer multiples of $\pi$. More
specifically, when $K_{1}=0.5\pi$ and $K_{2}=(2{\mathbb{Z}})\pi$
[$K_{2}=(2{\mathbb{Z}}-1)\pi$], the two Floquet bands touch at
$E_{\pm}(k=\pm\pi/2)=0$ [$E_{\pm}(k=\pm\pi/2)=\pm\pi$]. In both cases, the
quasienergy gap closes at two points $k=\pm\pi/2$ in $k$-space when a
transition happens. The Floquet spectra of the spin-$1/2$ ORDKR for
$K_{1}=0.5\pi$ and $K_{2}=\pi,2\pi,3\pi,4\pi$ are presented in Fig. 11 in
order to verify our analysis.

Figure 12: EE of the spin-$1/2$ ORDKR $S$ versus $K_{2}$ at different lattice
sizes $L$. The size of subsystem A is $L/2$. We set $K_{1}=0.5\pi$ for all
cases. In each panel, the blue solid, red dashed and black dotted lines
represent the EE at the lattice sizes $L=1000$, $2000$ and $4000$,
respectively. Figure 13: EE of the spin-$1/2$ ORDKR $S$ versus the lattice
size $L$. The size of subsystem A is $L/2$. We set $K_{1}=0.5\pi$ for all
cases and let $L$ goes from $200$ up to $4000$ for each data set. In panel
(a), $c$ refers to the central charge extracted from the linear fitting of
each data set of $S$ at different $K_{2}$. In panel (b), the data sets from
bottom to top have the values of $K_{2}=0.95\pi$, $1.05\pi$, $1.95\pi$,
$2.05\pi$, $2.95\pi$, $3.05\pi$, $3.95\pi$ and $4.05\pi$.

We now discuss the properties of EE around these topological phase transition
points. Let us first consider the change of EE with system size. In Fig. 12,
we show the EE around the four topological transition points in Fig. 11. We
observe that away from these transition points, the EE almost has no changes
with the increse of lattice size. Meanwhile, right at each transition point,
the EE raises with the increase of the length of lattice. These observations
suggest that the EE has different scaling behaviors versus the system size
away from and right at each topological phase transition point. To further
clarify these scaling properties, we report the EE at different lattice sizes
together with their fitting curves in Fig. 13. Away from the transition
points, we observe that indeed the EE is almost invariant with the increase of
the lattice size $L$. Right at each transition point, the leading term in EE
is found to scale linearly with $\ln L$. The gradient has the form of $c/3$,
with $c$ usually being called the central charge. For the transition with a
gap-closing at either $E=0$ or $E=\pm\pi$, we find $c=2$ from Fig. 13(a). This
is consistent with what one may expect in a usual static system with a gapless
spectrum EEGapless1 ; EEGapless2 ; EEGapless3 , since at half-filling, there
are only two occupied states at the two band-touching points for all the cases
shown in Fig. 11. A similar result for the finite-size scaling of EE was
reported before in a 1D Floquet topological superconductor YatesPRL2018 . The
scaling relation of EE observed here for our 1D Floquet topological insulator
coincides with that found in the superconducting setup, where the topological
edge states exhibit themselves as Majorana modes YatesPRL2018  instead of
normal fermionic zero and $\pi$ modes in our case.

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