# Light-induced phase crossovers in a quantum spin Hall system

Fang Qin<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117542, Singapore  Ching Hua Lee<EMAIL_ADDRESS>Department of Physics, National University of Singapore, Singapore 117542,
Singapore  Rui Chen<EMAIL_ADDRESS>Department of Physics, Hubei
University, Wuhan 430062, China  Department of Physics, The University of Hong
Kong, Pokfulam Road, Hong Kong 999077, China

###### Abstract

In this work, we theoretically investigate the light-induced topological
phases and finite-size crossovers in a paradigmatic quantum spin Hall (QSH)
system with high-frequency pumping optics. Taking the HgTe quantum well for an
example, our numerical results show that circularly polarized light can break
time-reversal symmetry and induce the quantum anomalous Hall (QAH) phase. In
particular, the coupling between the edge states is spin dependent and is
related not only to the size of the system, but also to the strength of the
polarized pumping optics. By tuning the two parameters (system width and
optical pumping strength), we obtain four transport regimes, namely, QSH, QAH,
edge conducting, and normal insulator. These four different transport regimes
have contrasting edge conducting properties, which will feature prominently in
transport experiments on various topological materials.

##  I Introduction

The quantum spin Hall (QSH) insulator is a member of the family of quantum
Hall insulators first proposed by Kane and Mele Kane and Mele (2005a, b). It
is an energy band insulator in bulk, but conducting at the edges Abergel
(2018). The edges carry two different charge currents with spin-up and spin-
down moving in opposite directions. A more realistic version of the QSH effect
in HgTe quantum wells was proposed by Bernevig, Hughes, and Zhang Bernevig
_et al._ (2006) and was soon confirmed by experiment Konig  _et al._ (2007);
König  _et al._ (2008). Since then, there have been intensive studies to
investigate the exotic properties in QSH systems Prodan (2009). They include
topological field theory Qi  _et al._ (2008); Yu  _et al._ (2011); Lee and Ye
(2015), finite-size crossovers in topological insulators Zhou  _et al._
(2008); Liu  _et al._ (2010), magnetic doping in QSH system (Mn-doped HgTe
quantum well) Fu  _et al._ (2014), disorder-induced topological Anderson
insulator in QSH systems Li  _et al._ (2009); Jiang  _et al._ (2009); Groth
_et al._ (2009); Xing  _et al._ (2011); Li  _et al._ (2011); Ning  _et al._
(2022), Floquet topological insulator states in QSH systems Calvo  _et al._
(2015); Zhu  _et al._ (2014); Qin  _et al._ (2022a); Liu  _et al._ (2021), as
well as QSH effects in three-dimensional topological insulators Li  _et al._
(2010); Dabiri  _et al._ (2021); Dabiri and Cheraghchi (2021); Pervishko  _et
al._ (2018); Zhu  _et al._ (2022).

The quantum anomalous Hall (QAH) insulator, which was first proposed by
Haldane Haldane (1988), is another key member of the family of quantum Hall
insulators. A QAH insulator can have a quantized Hall conductivity even
without an external magnetic field that breaks the time-reversal symmetry
Haldane (1988); Chang  _et al._ (2013); Yu  _et al._ (2010); Linder  _et al._
(2009); Lu  _et al._ (2013, 2010); Shan  _et al._ (2010). The QAH effect was
first experimentally observed in Ref. Chang  _et al._ (2013); in another work
on finite-size effects in QAH systems Fu  _et al._ (2014), they adopted the
magnetic doping (Mn doping) to break the time-reversal symmetry and introduce
the QAH phase in the HgTe quantum well. By appropriate tuning of the doping
concentration and the system size, they have obtained a variety of topological
phases Fu  _et al._ (2014).

Yet, this experimental success Fu  _et al._ (2014) in realizing QAH states
proved difficult to follow up since the doping concentration is difficult to
tune continuously in experiments. Therefore, departing from previous works Fu
_et al._ (2014), we shall provide a more experimentally accessible alternative
to realizing QAH states, using circularly polarized light instead of magnetic
doping to break the time-reversal symmetry in HgTe quantum well. This hinges
on the fact that the intensity of circularly polarized light can be easily
tuned continuously in experiments, and that Floquet driving can induce a host
of exotic phases in various media, such as disorder-induced Floquet
topological insulators Titum  _et al._ (2015), quenched topological boundary
modes induced by Floquet engineering Lee and Song (2021), Floquet semimetal
with exotic topological linkages Li  _et al._ (2018), Floquet mechanism for
non-Abelian fractional quantum Hall states Lee  _et al._ (2018), photoinduced
half-integer quantized conductance plateaus in topological insulators Yap  _et
al._ (2018), and tunable Floquet-Weyl semimetals driven from nodal line
semimetals Chan  _et al._ (2016); Bonasera  _et al._ (2022). In this work,
based on numerical calculations with high-frequency Floquet expansion, we find
that circularly polarized light can break the time-reversal symmetry and
introduce the QAH phase in a HgTe quantum well. Our findings complement other
works that utilize circularly polarized light to induce new topological phases
in graphene Oka and Aoki (2009); Sato  _et al._ (2019); McIver  _et al._
(2020); Sentef  _et al._ (2015); Broers and Mathey (2022); Candussio  _et al._
(2022); Qin  _et al._ (2022b); Cupo  _et al._ (2021); Assi  _et al._ (2021);
Luo (2021); Schüler  _et al._ (2020); Topp  _et al._ (2019); Rodriguez-Vega
_et al._ (2020); Li  _et al._ (2020); Berdakin  _et al._ (2021); Dal Lago  _et
al._ (2017); Huamán  _et al._ (2021), but with significant differences. In
particular, the coupling between the edge states is spin dependent and is
related not only to the width of the system, but also to the strength of the
polarized pumping optics. By appropriate tuning of the system size and
strength of the pumping optics, we obtain four transport regimes including
QSH, QAH, edge conducting (EC), and normal insulator (NI).

The paper is organized as follows. In Section II, we introduce the model
Hamiltonian that we use. In Section III, we derive the corresponding Floquet
effective Hamiltonian by using the high-frequency expansions. In Section IV,
we calculate the energy dispersions under the open boundary condition along
the $y$ direction and periodic boundary conditions along the $x$ direction
with fixed light intensity and different system size by numerically
diagonalizing the corresponding tight-binding Hamiltonian. In Section V, we
give and discuss our phase diagram. Finally, we summarize our results in
Section VI.

##  II Model

We start with the Bernevig-Hughes-Zhang (BHZ) model given by the low-energy
two-dimensional effective Hamiltonian near the $\Gamma$ point to describe the
electronic states in HgTe/(Hg,Cd)Te quantum wells Bernevig  _et al._ (2006);
Konig  _et al._ (2007); König  _et al._ (2008); Zhou  _et al._ (2008); Rothe
_et al._ (2010); Zhu  _et al._ (2014)

$\displaystyle{\cal H}({\bf k})$ $\displaystyle=\begin{pmatrix}h({\bf k})&0\\\
0&h^{*}(-{\bf k})\end{pmatrix}$ $\displaystyle=\epsilon_{\bf
k}\tau_{0}\otimes\sigma_{0}+Ak_{x}\tau_{z}\otimes\sigma_{x}+Ak_{y}\tau_{0}\otimes\sigma_{y}$
$\displaystyle+{\cal M}_{\bf k}\tau_{0}\otimes\sigma_{z},$ (1)

where $h({\bf k})=\epsilon_{\bf
k}\sigma_{0}+Ak_{x}\sigma_{x}+Ak_{y}\sigma_{y}+{\cal M}_{\bf k}\sigma_{z}$,
$h^{*}(-{\bf k})=\epsilon_{\bf
k}\sigma_{0}-Ak_{x}\sigma_{x}-Ak_{y}\sigma_{y}^{*}+{\cal M}_{\bf
k}\sigma_{z}$, $\epsilon_{\bf k}=C-Dk^{2}$, ${\cal M}_{\bf k}=M-Bk^{2}$,
$k^{2}=k_{x}^{2}+k_{y}^{2}$, $A$, $B$, $C$, $D$, and $M$ are the material
specific parameters, ${\bf k}=(k_{x},k_{y})$, $k_{x}$ and $k_{y}$ are the
momenta of two-dimensional electron gas, $\sigma_{x,y,z}$ ($\tau_{x,y,z}$) are
the Pauli matrices, and $\sigma_{0}$ ($\tau_{0}$) is the $2\times 2$ identity
matrix. The parameters are adopted as König  _et al._ (2008) $A=364.5$
meV$\cdot$nm, $B=-686$ meV$\cdot$nm2, $D=-512$ meV$\cdot$nm2, $M=-10$ meV.
Without loss of generality, we assume that $C=0$.

In particular, the $h({\bf k})$ and $h^{*}(-{\bf k})$ in the matrix diagonal
blocks of Eq. (II) provide the simplest ansatz that ensures time inversion
symmetry in the model Hamiltonian. In addition, $h({\bf k})$ and $h^{*}(-{\bf
k})$ are equivalent to the two-dimensional Dirac model or Qi-Wu-Zhang (QWZ)
model Qi  _et al._ (2006); Qi and Zhang (2011); Chen  _et al._ (2019), which
has a quantized Hall conductance without Landau levels. Thus, $h({\bf k})$ and
$h^{*}(-{\bf k})$ correspond to two equal but opposite QAH Hamiltonians.
Therefore, the net Hall conductance of the Hamiltonian (II) is zero, but the
spin Hall conductance for $h({\bf k})$ or $h^{*}(-{\bf k})$ is nonzero. This
phenomenon corresponds to the QSH state. The BHZ model can be viewed as two
copies of QWZ models with opposite Hall conductance.

##  III Floquet Hamiltonian

To realize QAH states in the Hamiltonian (II), circularly polarized light can
be used to break the time-reversal symmetry between the two spin blocks of the
matrix (II) to result in nonzero net Hall conductance.

The optical driving field can be expressed as ${\bf E}(t)=\partial{\bf
A}(t)/\partial t=E(\cos(\omega t),\cos(\omega t+\varphi),0)$, where ${\bf
A}(t)={\bf A}(t+T)$ is the time periodic vector potential with period
$T=2\pi/\omega$, $\omega$ is the frequency of the optical field, and $E$ is
the amplitude of the optical field. The optical field is circularly polarized
with $\varphi=\pm\pi/2$ and ${\bf A}(t)=A_{a}(\sin(\omega t),\pm\cos(\omega
t),0)$, where $A_{a}=E/\omega$ and $\pm$ corresponds to right- and left-handed
circularly polarized waves, respectively. To guarantee the off-resonant regime
in which the central Floquet band is far away from other replicas, the driven
frequency can be set as $\hbar\omega=150$ meV, which is much larger than the
bandwidth Trevisan  _et al._ (2022); Qin  _et al._ (2022a); Dabiri  _et al._
(2021); Dabiri and Cheraghchi (2021); Pervishko  _et al._ (2018).

After irradiation with light, the photon-dressed effective Hamiltonian is
given by

$\displaystyle{\cal H}({\bf k},t)={\cal H}\left({\bf k}-\frac{e}{\hbar}{\bf
A}(t)\right).$ (2)

Within the Floquet theory Oka and Aoki (2009); Calvo  _et al._ (2015); Chen
_et al._ (2018a, b); Du  _et al._ (2022); Ning  _et al._ (2022); Qin  _et al._
(2022a); Wang  _et al._ (2022); Jangjan and Hosseini (2020, 2021) and
$A_{0}=eA_{a}/\hbar$ which denotes the light intensity, the effective static
Hamiltonian in the high-frequency regime can be expanded as

$\displaystyle{\cal H}^{(F)}={\cal H}_{0,0}+\frac{[{\cal H}_{0,-1},{\cal
H}_{0,1}]}{\hbar\omega},$ (3)

where we use ${\cal H}_{n,m}=\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{i(n-m)\omega t}dt$ with $n$ and $m$ as integers, and the concrete
analytical expressions for ${\cal H}_{0,0}$, ${\cal H}_{0,-1}$, and ${\cal
H}_{0,1}$ can be found in Appendix A.

From Eq. (3), the Floquet Hamiltonian is given by

$\displaystyle{\cal H}^{(F)}({\bf k})$
$\displaystyle\\!=\\!\begin{pmatrix}h_{\uparrow}({\bf k})&0\\\
0&h_{\downarrow}({\bf k})\end{pmatrix},$ (4)

where $h_{\uparrow}({\bf
k})\\!=\\!-D(A_{0}^{2}+k^{2})\sigma_{0}\\!+\\!\left(\\!\tilde{M}_{\uparrow}\\!-\\!Bk^{2}\\!\right)\\!\sigma_{z}\\!+\\!\tilde{A}_{\uparrow}(k_{x}\sigma_{x}\\!+\\!k_{y}\sigma_{y})$,
$h_{\downarrow}({\bf
k})\\!=\\!-D(A_{0}^{2}+k^{2})\sigma_{0}\\!+\\!\left(\\!\tilde{M}_{\downarrow}\\!-\\!Bk^{2}\\!\right)\\!\sigma_{z}\\!-\\!\tilde{A}_{\downarrow}(k_{x}\sigma_{x}\\!-\\!k_{y}\sigma_{y})$,
$\tilde{A}_{\uparrow}\\!=\\!\left[\\!1\\!-\\!2A_{0}^{2}B\sin\varphi/(\hbar\omega)\\!\right]A$,
$\tilde{A}_{\downarrow}\\!=\\!\left[\\!1\\!+\\!2A_{0}^{2}B\sin\varphi/(\hbar\omega)\\!\right]A$,
$\tilde{M}_{\uparrow}\\!=\\!M\\!-\\!BA_{0}^{2}\\!-\\!A_{0}^{2}A^{2}\sin\varphi/(\hbar\omega)$,
and
$\tilde{M}_{\downarrow}\\!=\\!M\\!-\\!BA_{0}^{2}\\!+\\!A_{0}^{2}A^{2}\sin\varphi/(\hbar\omega)$.
Note that it should not be taken for granted that optical driving will induce
the QAH phase: when $\varphi=0$, we have
$\tilde{M}_{\uparrow/\downarrow}\\!=\\!M\\!-\\!BA_{0}^{2}$ and
$\tilde{A}_{\uparrow/\downarrow}=A$, so that the Floquet Hamiltonian (4) still
satisfies the time-reversal symmetry, i.e., ${\cal T}{\cal H}^{(F)}({\bf
k}){\cal T}^{-1}\\!=\\!{\cal H}^{(F)}(-{\bf k})$ with the time-reversal
symmetry operator ${\cal T}\\!=\\!i\tau_{y}\otimes\sigma_{0}{\cal K}$ Michetti
_et al._ (2012), where ${\cal K}$ is the operator of complex conjugation.
However, when $\varphi\neq 0$, the terms containing $\varphi$ in
$\tilde{M}_{\uparrow/\downarrow}$ and $\tilde{A}_{\uparrow/\downarrow}$ lead
to ${\cal T}{\cal H}^{(F)}({\bf k}){\cal T}^{-1}\\!\neq\\!{\cal H}^{(F)}(-{\bf
k})$ which breaks time-reversal symmetry of the Hamiltonian (4). The
corresponding detailed derivations can be found in Appendix B. This is the so-
called time-reversal-symmetry-broken QSH insulator Yang  _et al._ (2011); Chen
_et al._ (2017).

The energy dispersions of the Floquet Hamiltonian (4) are given by

$\displaystyle E_{\uparrow,\pm}(k)$
$\displaystyle\\!=\\!-D(A_{0}^{2}+k^{2})\\!\pm\\!\sqrt{(\tilde{M}_{\uparrow}\\!-\\!Bk^{2})^{2}+\tilde{A}_{\uparrow}^{2}k^{2}},$
(5) $\displaystyle E_{\downarrow,\pm}(k)$
$\displaystyle\\!=\\!-D(A_{0}^{2}+k^{2})\\!\pm\\!\sqrt{(\tilde{M}_{\downarrow}\\!-\\!Bk^{2})^{2}+\tilde{A}_{\downarrow}^{2}k^{2}}.$
(6)

To estimate the validity of the high-frequency expansions developed here
quantitatively, we evaluate the maximum instantaneous energy of the time-
dependent Hamiltonian (2) averaged over a period of the field
$\frac{1}{T}\int_{0}^{T}dt~{}\text{max}\left\\{\big{|}\big{|}H({\bf
k},t)\big{|}\big{|}\right\\}<\hbar\omega$ at the $\Gamma$ point
($k_{x}=k_{y}=0$). The optical field parameters have to satisfy the condition
$A_{0}A/(\hbar\omega)<1$. In the high-frequency regime $\omega\sim
2.2789\times 10^{2}$ THz ($\hbar\omega=150$ meV), one can obtain $A_{0}\ll
0.41$ nm-1. In the following numerical calculations, we have regularized the
continuous model Hamiltonian (4) into the corresponding lattice Hamiltonian
where topological invariants are well-defined (the detailed derivations can be
found in Appendix C). All the numerical results are based on the lattice
regularized Hamiltonian, even though they all pertain to the region around the
$\Gamma$ point.

##  IV Energy dispersions

Figure 1: (Color online) Energy bands for the Floquet Hamiltonian of the
HgTe/(Hg,Cd)Te with different size $L_{y}$ under open boundary condition along
the $y$ direction and periodic boundary conditions along the $x$ direction.
(a) $L_{y}=850$ nm corresponds to quantum spin Hall (QSH). (b) $L_{y}=400$ nm
corresponds to quantum anomalous Hall (QAH). (c) $L_{y}=150$ nm corresponds to
edge conducting (EC). (d) $L_{y}=80$ nm corresponds to normal insulator (NI).
Here, the definitions of $E_{\uparrow\pm}$ and $E_{\downarrow\pm}$ are the
energies of two spin-up and spin-down edge bands under open boundary condition
along the $y$ direction. Here, QSH is topological protected for both spin-up
and spin-down edge states, QAH is topological protected for one of them only,
and EC for none of them, although they still exist unless broken by disorder.
The threshold of the energy gap for calculating the phase boundaries is 0.01
meV. The other parameters are given as $A_{0}=0.06$ nm-1, $a_{x}=a_{y}=1$ nm,
$\varphi=\pi/2$, and $\hbar\omega=150$ meV. Here, $a_{x}$ and $a_{y}$ are the
lattice constants along the $x$ and $y$ directions, respectively. Notice that
we have mapped the continuous model Hamiltonian (4) into the lattice
Hamiltonian (the detailed derivations can be found in Appendix C). Figure 2:
(Color online) Schematic diagram (or physical picture) for the four phases
with the clockwise (red lines) and anticlockwise (blue lines) charge currents
of the spin-up and spin-down edge states, respectively. (a) QSH state, (b) QAH
state, (c) EC state, and (d) NI state. The solid red line denotes the current
for the spin-up edge states, the solid blue line denotes the current for the
spin-down edge states, and the dashed red and blue lines denote the currents
for the spin-up and spin-down edge states possible with the disorder,
respectively. It means that the dashed lines refer to states that are not
protected against disorder.

In order to explore how sufficiently small system sizes can lead to Floquet
phase transitions in a QSH system, we calculate the tight-binding energy
spectra of a HgTe/(Hg,Cd)Te quantum well with fixed light intensity in a
stripe geometry with sample width $L_{y}$ along the $y$ direction, and the
system length in the $x$ direction is infinite. The energy gap opening and
closing in the energy dispersions of the edge states is the most important
feature of finite-size effects with fixed light intensity.

As shown in Fig. 1, we calculate the energy spectra of the HgTe/(Hg,Cd)Te with
fixed light intensity $A_{0}=0.06$ nm-1 and different width $L_{y}$ under the
open boundary condition along the $y$ direction and periodic boundary
conditions along the $x$ direction (the detailed derivations can be found in
Appendix C). In addition, we give the detailed derivations of the tight-
binding model under the open boundary condition along the $x$ direction and
periodic boundary conditions along the $y$ direction in Appendix D. In Fig. 1,
the red thick line indicates the upper band $E_{\uparrow+}$ of the spin-up
edge states, the red thin line indicates the lower band $E_{\uparrow-}$ of the
spin-up edge states, the blue thick line indicates the upper band
$E_{\downarrow+}$ of the spin-down edge states, and the blue thin line
indicates the lower band $E_{\downarrow-}$ of the spin-down edge states.

To illustrate the physical picture for the four phases with different system
widths in Fig. 1, we provide a schematic diagram for the four phases with the
clockwise (red lines) and anticlockwise (blue lines) charge currents of the
spin-up and spin-down edge states, respectively, as shown in Fig. 2.

In Fig. 1(a), the system width is $L_{y}=850$ nm which is sufficiently large
that there is no coupling between edge states. It is found that the energy
dispersions for both spin-up and spin-down edge states are gapless and each
spin state provides a charge conductance $e^{2}/h$, as shown in Fig. 2(a).
This indicates that the system is in the QSH state which is time-reversal
symmetry broken due to circularly polarized light; the corresponding charge
conductance along the $x$ direction is $2e^{2}/h$, as shown in Fig. 2(a), with
the contributions from both spin-up and spin-down charge currents flowing from
higher potential to lower potential; and it is robust against the disorder.

In Fig. 1(b), the system width is decreased to $L_{y}=400$ nm, where the
coupling between the spin-down edge states is strong enough to open an energy
gap in the corresponding energy dispersions, while the energy dispersions of
the spin-up edge states remain gapless which provides a charge conductance
$e^{2}/h$. As a result, the spin-down edge states can easily be broken by the
disorder. This is because the transport electrons in the spin-down edge
channels can possibly be backscattered by impurities. Hence, the charge
conductance along the $x$ direction in this state ranges from $e^{2}/h$ to
$2e^{2}/h$ with disorder, as shown in Fig. 2(b). This state is called the QAH
state Fu  _et al._ (2014).

In Fig. 1(c), when the system width decreases to $L_{y}=150$ nm, all the
energy dispersions of the edge states become gapped. Particularly, the lower
band of the spin-up edge states is higher than the upper band of the spin-down
edge states near the $k_{x}=0$ point. Therefore, there are always conducting
edge channels near the Fermi level which is in the bulk gap. This state is
called the EC state Fu  _et al._ (2014), in which the charge currents through
the edge can be killed by disorder. So the corresponding charge conductance
along the $x$ direction ranges from $0$ to $2e^{2}/h$, as shown in Fig. 2(c).
In Fig. 1(d), by further decreasing the system width to $L_{y}=80$ nm, there
is a bulk energy gap in the energy dispersions and we get a NI state which
corresponds to Fig. 2(d) and is induced by finite-size effects.

Figure 3: (Color online) Probability distributions of the edge states for the
Floquet Hamiltonian of the HgTe/(Hg,Cd)Te with different size $L_{y}$ under
$k_{x}=0$. The other parameters are the same as those in Fig. 1. Notice that
the subscript $+$ denotes the edge states of $E_{\sigma+}$ with
$\sigma=\uparrow,\downarrow$. The red solid curve denotes the probability
distribution for the spin-$\uparrow$ edge state and the blue solid curve
denotes the one for the spin-$\downarrow$ edge state. The density
distributions of the edge states for $E_{\sigma-}$ can be found in Appendix E.

Meanwhile, along the $y$ direction, the edge states have a probability
distribution in the sense of the wave function of edge states. As shown in
Fig. 3, one can obtain a clear idea of how well-located edge states are forced
to be mixed by changing $L_{y}$. Here, the red solid curve denotes the
probability distribution for the spin-$\uparrow$ edge state and the blue solid
curve denotes the one for the spin-$\downarrow$ edge state. It is indicated
from Fig. 3(a) that the wave functions for both spin-$\uparrow$ and
spin-$\downarrow$ edge states are well localized at the left and right edges
of the system with large system width $L_{y}=850$ nm where there is no
coupling between the edge states. When the system width $L_{y}$ becomes
smaller and smaller, as shown in Figs. 3(b)–3(d), it is found that the edge
localizations of the wave functions of the edge states get worse and worse.

Figure 4: (Color online) Band energies $E_{\uparrow\pm}$ and
$E_{\downarrow\pm}$ at the zero momentum point $k_{x}=0$ as a function of the
size $L_{y}$ with fixed (a) $A_{0}=0.06$ nm-1 and (b) $A_{0}=0.03$ nm-1 under
the open boundary condition along the $y$ direction and periodic boundary
conditions along the $x$ direction. The threshold of the energy gap for
calculating the phase boundaries is $0.01$ meV. The other parameters are the
same as those in Fig. 1.

We furthermore investigate phase crossover due to adjusting the finite system
width, with light intensity fixed. In Fig. 4, the definitions of
$E_{\uparrow\pm}$ and $E_{\downarrow\pm}$ are the energies of two spin-up and
spin-down edge bands at the zero momentum point $k_{x}=0$ under the open
boundary condition along the $y$ direction. Meanwhile, the red thick line
indicates the upper band $E_{\uparrow+}$ of the spin-up edge states, the red
thin line indicates the lower band $E_{\uparrow-}$ of the spin-up edge states,
the blue thick line indicates the upper band $E_{\downarrow+}$ of the spin-
down edge states, and the blue thin line indicates the lower band
$E_{\downarrow-}$ of the spin-down edge states.

We calculate the energies ($E_{\uparrow\pm}$ and $E_{\downarrow\pm}$) of the
spin-up and spin-down edge states at $k_{x}=0$ as a function of the system
width $L_{y}$ as shown in Fig. 4(a) with the same parameters as those in Fig.
1. When $L_{y}$ is very small, the energy gap for both bulk and edge states is
large and the system is insulating in the NI state. By gradually increasing
the width $L_{y}$, once the lower energy band of the spin-up edge states
touches the upper energy band of the spin-down edge states, the system enters
into the EC state. Particularly, we notice that there is a critical point
which separates the EC state from the NI state, as shown in Fig. 4. In the EC
state, the lower energy band of the spin-up edge states is higher than the
upper energy band of the spin-down edge states, and the energy dispersions for
both spin-up and spin-down edge states are gapped. By further increasing the
width $L_{y}$, the energy gap of the spin-up edge states vanishes but the
energy dispersion for the spin-down edge states is still gapped. At this time,
the system enters into a QAH state. When $L_{y}$ is very large, the system
enters into a QSH state, where the energy dispersions for both spin-up and
spin-down edge states are gapless.

Different phases are determined by the edge states and energy gaps. Here, we
consider the finite-size effect on the edge states and energy gaps. However,
the Chern number and Bott index are used to describe the topological
properties of an infinite-size system under periodic boundary conditions both
along the $x$ and $y$ directions. Therefore, for a small-size system under
open boundary condition along the $y$ direction here, both the Chern number
and Bott index are inapplicable. Concretely, when the system size is very
small, the coupling between the edge states becomes very strong and the edge
states are destroyed by opening a gap, as shown in Fig. 3(d). In addition,
Ref. Zhou  _et al._ (2008) also mentioned that the finite-size effect can
destroy the quantum spin Hall effect phase without a topological index
transition. Next, we will reply to the applicability of our approach to the
phase transition regime. For finite small sizes, our phase transitions belong
to continuous phase transitions which are in general crossovers, and we can
only use the energy gaps of the edge states to determine the phase transitions
quantitatively. Meanwhile, as mentioned in Ref. Fu  _et al._ (2014), another
applicable method is to calculate the transport or conductance of the finite-
size system by using the Landauer-Büttiker formalism Landauer (1970); Büttiker
(1988). For example, Ref. Li  _et al._ (2011) has calculated the transport
properties of a finite-size system in HgTe/CdTe quantum wells without optical
pumping. We will investigate the transport properties of a finite-size Floquet
system in the future.

##  V Overall phase diagram

In this section, we will investigate the overall phase diagram of the system.

Figure 5: (Color online) Phase diagram in the system width vs. light intensity
($L_{y}$, $A_{0}$) plane under the open boundary condition along the $y$
direction and periodic boundary conditions along the $x$ direction, as
determined by the long-$x$-wavelength (small-$k_{x}$) limit. There are four
kinds of phase regimes, i.e., NI, EC, QAH, and QSH are shown. The white zone
denotes the NI regime, the green zone denotes the EC regime, the yellow zone
denotes the QAH regime, and the blue zone denotes the QAH regime. The
threshold of the energy gap for calculating the phase boundaries is $0.01$
meV, which is also used as a criterion of the energy gap opening. The other
parameters are the same as those in Fig. 1.

Based on the energy gap for spin-up and spin-down edge states, we plot the
whole phase diagram with two tunable parameters: light intensity $A_{0}$ and
system width $L_{y}$ as shown in Fig. 5. There are four kinds of phase
regimes, i.e., NI, EC, QAH, and QSH, as shown in Fig. 5. An important feature
is that each transport regime has its own permitted width range with fixed
light intensity. For example, the width for the EC state is in the range from
$116$ to $391$ nm with fixed $A_{0}=0.06$ nm-1. The permitted width range for
different transport states is important because it can help us to find the
desired transport regime by choosing an appropriate device width and intensity
of light.

##  VI Summary

Even though Floquet engineering had been extensively used to engineer various
topological phases that are elusive in the static regime, our proposal instead
engineers QAH and other phases in a realistic QSH setup that takes into
account the hybridization effects due to the finite size of the realistic
samples. We numerically investigate the light-induced topological phases and
finite-size topological phase crossovers in a QSH system with high-frequency
pumping optics, taking the HgTe quantum well as the paradigmatic realistic
model. We find that the coupling of edge states depends on both the system
width and the light intensity. Furthermore, we can get four different
topological transport regimes by choosing the proper width and light
intensity. Since the intensity of light can easily be continuously tuned in
experiments, our proposal that uses circularly polarized light instead of
magnetic doping provides a feasible route towards realizing QAH states in
experiments.

With state-of-the-art nanofabrication, it is possible to engineer realistic
quantum materials to reduce its size in the length scale presented in our
figures. For example, in Ref. Qiu  _et al._ (2022), it is reported that the
quantum anomalous Hall (QAH) devices can be fabricated as miniaturized with
channel widths down to 600 nm in six quintuple layers of
Cr0.12(Bi0.26Sb0.62)2Te3 film which were epitaxially deposited onto GaAs (111)
substrates. Furthermore, they measured the longitudinal and transverse
resistivities in four devices with a channel width of 0.6, 1.5, 3, and 5
$\mu$m Qiu  _et al._ (2022). Meanwhile, Ref. Zhou  _et al._ (2022)
demonstrates that the QAH effect persists in a Hall bar device with width 72
nm in three quintuple layers of Cr-doped (Bi,Sb)2Te3, four quintuple layers of
(Bi,Sb)2Te3, and three quintuple layers of Cr-doped (Bi,Sb)2Te3 Zhao  _et al._
(2020, 2022), and they measured the transverse resistance and longitudinal
resistance in Hall bar devices with widths from 300 nm to 10 $\mu$m.
Technologically, first, by using photolithography, the samples can be
patterned into a Hall bar with width 10 $\mu$m Qiu  _et al._ (2022); Zhou  _et
al._ (2022). Then by using another experimental technology, i.e., electron-
beam lithography, the realization of the QAH bar devices with widths from 10
$\mu$m down to 72 nm are further fabricated Zhou  _et al._ (2022).

###### Acknowledgements.

We acknowledge helpful discussions with Hao-Jie Lin and Xiao-Bin Qiang. This
work is supported by the Singapore National Research Foundation (Grant No.
NRF2021-QEP2-02-P09). F.Q. acknowledges support from the National Natural
Science Foundation of China (Grants No. 11404106), and the project funded by
the China Postdoctoral Science Foundation (Grants No. 2019M662150 and No.
2020T130635) before he joined NUS. R.C. acknowledges support from the project
funded by the China Postdoctoral Science Foundation (Grant No. 2019M661678).

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##  Appendix A Expressions for ${\cal H}_{0,0}$, ${\cal H}_{0,-1}$, and ${\cal
H}_{0,1}$

The motivation for Appendix A is to show the concrete analytical expressions
for ${\cal H}_{0,0}$, ${\cal H}_{0,-1}$, and ${\cal H}_{0,1}$ in the Floquet
Hamiltonian (3) of the main text. The concrete analytical expressions for
${\cal H}_{0,0}$, ${\cal H}_{0,-1}$, and ${\cal H}_{0,1}$ in the Floquet
Hamiltonian (3) are given as

$\displaystyle{\cal H}_{0,0}$
$\displaystyle\\!=\\![C-D(A_{0}^{2}+k_{x}^{2}+k_{y}^{2})]\tau_{0}\otimes\sigma_{0}+Ak_{x}\tau_{z}\otimes\sigma_{x}+[M-B(A_{0}^{2}+k_{x}^{2}+k_{y}^{2})]\tau_{0}\otimes\sigma_{z}+Ak_{y}\tau_{0}\otimes\sigma_{y}$
$\displaystyle\\!=\\!{\cal H}({\bf
k})-DA_{0}^{2}\tau_{0}\otimes\sigma_{0}-BA_{0}^{2}\tau_{0}\otimes\sigma_{z},$
(7) $\displaystyle\\!{\cal H}_{0,-1}$
$\displaystyle\\!=\\!iDA_{0}(k_{x}\\!+\\!e^{-i\varphi}k_{y})\tau_{0}\otimes\sigma_{0}\\!+\\!iBA_{0}(k_{x}\\!+\\!e^{-i\varphi}k_{y})\tau_{0}\otimes\sigma_{z}\\!-\\!\frac{i}{2}AA_{0}\tau_{z}\otimes\sigma_{x}\\!-\\!\frac{i}{2}AA_{0}e^{-i\varphi}\tau_{0}\otimes\sigma_{y},$
(8) $\displaystyle\\!{\cal H}_{0,1}$
$\displaystyle\\!=\\!\\!-\\!iDA_{0}(k_{x}\\!+\\!e^{i\varphi}k_{y})\tau_{0}\otimes\sigma_{0}\\!-\\!iBA_{0}(k_{x}\\!+\\!e^{i\varphi}k_{y})\tau_{0}\otimes\sigma_{z}\\!+\\!\frac{i}{2}AA_{0}\tau_{z}\otimes\sigma_{x}\\!+\\!\frac{i}{2}AA_{0}e^{i\varphi}\tau_{0}\otimes\sigma_{y}.$
(9)

##  Appendix B Time-reversal symmetry

The motivation for Appendix B is to prove that when $\varphi=0$, the Floquet
Hamiltonian (4) in the main text still satisfies the time-reversal symmetry.
However, when $\varphi\neq 0$, the time-reversal symmetry of the Hamiltonian
(4) is broken.

The Floquet Hamiltonian (4) under time-reversal transformation becomes

$\displaystyle~{}~{}{\cal T}{\cal H}^{(F)}({\bf k}){\cal T}^{-1}$
$\displaystyle\\!=\\!i\tau_{y}\otimes\sigma_{0}{\cal
K}\left[-D(A_{0}^{2}+k^{2})\tau_{0}\otimes\sigma_{0}+A(k_{x}\tau_{z}\otimes\sigma_{x}+k_{y}\tau_{0}\otimes\sigma_{y})+(\\!M\\!-\\!BA_{0}^{2}\\!-\\!Bk^{2})\tau_{0}\otimes\sigma_{z}\right]{\cal
K}^{-1}(i\tau_{y}\otimes\sigma_{0})^{-1}$
$\displaystyle\\!-\\!\frac{A_{0}^{2}\sin\varphi}{\hbar\omega}i\tau_{y}\otimes\sigma_{0}{\cal
K}\left[2AB(k_{x}\tau_{0}\otimes\sigma_{x}+k_{y}\tau_{z}\otimes\sigma_{y})+A^{2}\tau_{z}\otimes\sigma_{z}\right]{\cal
K}^{-1}(i\tau_{y}\otimes\sigma_{0})^{-1}$
$\displaystyle\\!=\\!-D(A_{0}^{2}+k^{2})\tau_{0}\otimes\sigma_{0}-A(k_{x}\tau_{z}\otimes\sigma_{x}+k_{y}\tau_{0}\otimes\sigma_{y})+(\\!M\\!-\\!BA_{0}^{2}\\!-\\!Bk^{2})\tau_{0}\otimes\sigma_{z}$
$\displaystyle\\!-\\!\frac{A_{0}^{2}\sin\varphi}{\hbar\omega}\left[2AB(k_{x}\tau_{0}\otimes\sigma_{x}+k_{y}\tau_{z}\otimes\sigma_{y})-A^{2}\tau_{z}\otimes\sigma_{z}\right]$
$\displaystyle\\!=\\!{\cal H}^{(F)}(-{\bf
k})\\!-\\!\frac{2A_{0}^{2}\sin\varphi}{\hbar\omega}\left[2AB(k_{x}\tau_{0}\otimes\sigma_{x}+k_{y}\tau_{z}\otimes\sigma_{y})-A^{2}\tau_{z}\otimes\sigma_{z}\right],$
(10)

where ${\cal T}=i\tau_{y}\otimes\sigma_{0}{\cal K}$ Michetti  _et al._ (2012)
is the time-reversal operator with the complex conjugate operator ${\cal K}$,
we have ${\cal K}{\cal H}^{(F)}({\bf k}){\cal K}^{-1}={\cal H}^{(F)*}({\bf
k})$, and we use

$\displaystyle(i\tau_{y}\otimes\sigma_{0})(\tau_{0}\otimes\sigma_{0})(i\tau_{y}\otimes\sigma_{0})^{-1}=\tau_{0}\otimes\sigma_{0},$
(11)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(\tau_{z}\otimes\sigma_{x})(i\tau_{y}\otimes\sigma_{0})^{-1}=-\tau_{z}\otimes\sigma_{x},$
(12)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(K\tau_{0}\otimes\sigma_{y}K^{-1})(i\tau_{y}\otimes\sigma_{0})^{-1}=-\tau_{0}\otimes\sigma_{y},$
(13)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(\tau_{0}\otimes\sigma_{z})(i\tau_{y}\otimes\sigma_{0})^{-1}=\tau_{0}\otimes\sigma_{z},$
(14)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(\tau_{0}\otimes\sigma_{x})(i\tau_{y}\otimes\sigma_{0})^{-1}=\tau_{0}\otimes\sigma_{x},$
(15)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(K\tau_{z}\otimes\sigma_{y}K^{-1})(i\tau_{y}\otimes\sigma_{0})^{-1}=\tau_{z}\otimes\sigma_{y},$
(16)
$\displaystyle(i\tau_{y}\otimes\sigma_{0})(\tau_{z}\otimes\sigma_{z})(i\tau_{y}\otimes\sigma_{0})^{-1}=-\tau_{z}\otimes\sigma_{z}.$
(17)

As a result of Eq. (B), if $\varphi=0$, we have ${\cal T}{\cal H}^{(F)}({\bf
k}){\cal T}^{-1}={\cal H}^{(F)}(-{\bf k})$ which shows a time-reversal
symmetry. However, for $\varphi\neq 0$, Eq. (B) shows that the time-reversal
symmetry is broken.

##  Appendix C Tight-binding model under the open boundary condition along the
$y$ direction and periodic boundary conditions along the $x$ direction

The motivation of Appendix C is to derive the analytical expression of the
tight-binding model Hamiltonian under the open boundary condition along the
$y$ direction and periodic boundary conditions along the $x$ direction for the
further numerical calculations in the main text.

In a lattice, one makes the following replacements Shen (2012):

$\displaystyle k_{j}\rightarrow\frac{1}{a_{j}}\sin(k_{j}a_{j}),$ (18)
$\displaystyle k_{j}^{2}\rightarrow\frac{2}{a_{j}^{2}}[1-\cos(k_{j}a_{j})],$
(19)

where $j=x,y,z$, $a_{j}$ is the lattice constant along the $j$ direction,
$\sin(k_{j}a_{j})=\frac{e^{ik_{j}a_{j}}-e^{-ik_{j}a_{j}}}{2i}$, and
$\cos(k_{j}a_{j})=\frac{e^{ik_{j}a_{j}}+e^{-ik_{j}a_{j}}}{2}$,

$\displaystyle
k^{2}\rightarrow\frac{2}{a_{x}^{2}}[1-\cos(k_{x}a_{x})]+\frac{2}{a_{y}^{2}}[1-\cos(k_{y}a_{y})],$
(20)

where $k^{2}=k_{x}^{2}+k_{y}^{2}$. In this way, the hopping terms in the
lattice model only exist between the nearest-neighbor sites. With this
mapping, one obtains the following tight-binding model with high-frequency
pumping in the basis $(c_{{\bf k},+,\uparrow},c_{{\bf k},-,\uparrow},c_{{\bf
k},+,\downarrow},c_{{\bf k},-,\downarrow})^{T}$ as

$\displaystyle\\!{\cal H}^{(F)}({\bf
k})\\!=\\!\left(\begin{array}[]{cccc}h_{\uparrow}({\bf k})&0\\\
0&h_{\downarrow}({\bf k})\end{array}\right),$ (23)

where

$\displaystyle h_{\uparrow}({\bf k})$
$\displaystyle\\!=\\!\\!-\\!D\left\\{A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\sigma_{0}$
$\displaystyle\\!+\\!\left\\{M\\!-\\!B\left[A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right]\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\right\\}\sigma_{z}$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{+}$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{-},$
(24) $\displaystyle h_{\downarrow}({\bf k})$
$\displaystyle\\!=\\!\\!-\\!D\left\\{A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\sigma_{0}$
$\displaystyle\\!+\\!\left\\{M\\!-\\!B\left[A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right]\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\right\\}\sigma_{z}$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{+}$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{-}.$
(25)

Performing the Fourier transformation, one obtains

$\displaystyle c_{{\bf k},s,\sigma}$
$\displaystyle=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{-ik_{y}j_{y}a_{y}}c_{k_{x},j_{y},s,\sigma},$
(26) $\displaystyle c_{{\bf k},s,\sigma}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{ik_{y}j_{y}a_{y}}c_{k_{x},j_{y},s,\sigma}^{\dagger}.$
(27) $\displaystyle C_{{\bf k},\sigma}$ $\displaystyle=\begin{pmatrix}c_{{\bf
k},+,\sigma}\\\ c_{{\bf
k},-,\sigma}\end{pmatrix}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{-ik_{y}j_{y}a_{y}}C_{k_{x},j_{y},\sigma},$
(28) $\displaystyle C_{{\bf k},\sigma}^{\dagger}$
$\displaystyle=\begin{pmatrix}c_{{\bf k},+,\sigma}^{\dagger}&c_{{\bf
k},-,\sigma}^{\dagger}\end{pmatrix}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{ik_{y}j_{y}a_{y}}C_{k_{x},j_{y},\sigma}^{\dagger}.$
(29)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\uparrow}^{\dagger}\left[h_{\uparrow}({\bf k})\right]C_{{\bf k},\uparrow}$
$\displaystyle\\!=\\!\\!-\\!D\sum_{k_{x},j_{y}}\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{D}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{k_{x},j_{y}}\\!\left[\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{B}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{x}}\sum_{k_{x},j_{y}}\sin(k_{x}a_{x})\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{k_{x},j_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{-}-\sigma_{+})C_{k_{x},j_{y}+1,\uparrow}\\!+\\!C_{k_{x},j_{y}+1,\uparrow}^{\dagger}(\sigma_{+}-\sigma_{-})C_{k_{x},j_{y},\uparrow}\\!\right],$
(30)

where

$\displaystyle
C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{-}-\sigma_{+})C_{k_{x},j_{y}+1,\uparrow}$
$\displaystyle=\begin{pmatrix}c_{k_{x},+,j_{y},\uparrow}^{\dagger}&c_{k_{x},-,j_{y},\uparrow}^{\dagger}\end{pmatrix}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\begin{pmatrix}c_{k_{x},+,j_{y}+1,\uparrow}\\\
c_{k_{x},-,j_{y}+1,\uparrow}\end{pmatrix}=\begin{pmatrix}c_{k_{x},-,j_{y},\uparrow}^{\dagger}&-c_{k_{x},+,j_{y},\uparrow}^{\dagger}\end{pmatrix}\begin{pmatrix}c_{k_{x},+,j_{y}+1,\uparrow}\\\
c_{k_{x},-,j_{y}+1,\uparrow}\end{pmatrix}$
$\displaystyle=c_{k_{x},-,j_{y},\uparrow}^{\dagger}c_{k_{x},+,j_{y}+1,\uparrow}-c_{k_{x},+,j_{y},\uparrow}^{\dagger}c_{k_{x},-,j_{y}+1,\uparrow},$
(31)

$\displaystyle
C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}-\sigma_{-})C_{k_{x},j_{y}+1,\uparrow}$
$\displaystyle=\begin{pmatrix}c_{k_{x},+,j_{y},\uparrow}^{\dagger}&c_{k_{x},-,j_{y},\uparrow}^{\dagger}\end{pmatrix}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}c_{k_{x},+,j_{y}+1,\uparrow}\\\
c_{k_{x},-,j_{y}+1,\uparrow}\end{pmatrix}=\begin{pmatrix}-c_{k_{x},-,j_{y},\uparrow}^{\dagger}&c_{k_{x},+,j_{y},\uparrow}^{\dagger}\end{pmatrix}\begin{pmatrix}c_{k_{x},+,j_{y}+1,\uparrow}\\\
c_{k_{x},-,j_{y}+1,\uparrow}\end{pmatrix}$
$\displaystyle=-c_{k_{x},-,j_{y},\uparrow}^{\dagger}c_{k_{x},+,j_{y}+1,\uparrow}+c_{k_{x},+,j_{y},\uparrow}^{\dagger}c_{k_{x},-,j_{y}+1,\uparrow}=c_{k_{x},+,j_{y},\uparrow}^{\dagger}c_{k_{x},-,j_{y}+1,\uparrow}-c_{k_{x},-,j_{y},\uparrow}^{\dagger}c_{k_{x},+,j_{y}+1,\uparrow}$
$\displaystyle=(c_{k_{x},-,j_{y},\uparrow}^{\dagger}c_{k_{x},+,j_{y}+1,\uparrow}-c_{k_{x},+,j_{y},\uparrow}^{\dagger}c_{k_{x},-,j_{y}+1,\uparrow})^{\dagger}.$
(32)

Therefore, we have

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\uparrow}^{\dagger}\left[h_{\uparrow}({\bf k})\right]C_{{\bf k},\uparrow}$
$\displaystyle\\!=\\!\\!-\\!D\sum_{k_{x},j_{y}}\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{D}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{k_{x},j_{y}}\\!\left[\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{B}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{x}}\sum_{k_{x},j_{y}}\sin(k_{x}a_{x})\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{k_{x},j_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{-}-\sigma_{+})C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right].$ (33)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\downarrow}^{\dagger}\left[h_{\downarrow}({\bf k})\right]C_{{\bf
k},\downarrow}$
$\displaystyle\\!=\\!\\!-\\!D\sum_{k_{x},j_{y}}\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{D}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{k_{x},j_{y}}\\!\left[\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y},\uparrow}\\!+\\!\frac{B}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{x}}\sum_{k_{x},j_{y}}\sin(k_{x}a_{x})\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{k_{x},j_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}-\sigma_{-})C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right].$ (34)

Therefore, the tight-binding Hamiltonian under $x$ periodic boundary
conditions (PBCs) and $y$ open boundary conditions (OBCs) in the basis
$(C_{k_{x},j_{y},\uparrow},C_{k_{x},j_{y},\downarrow},C_{k_{x},j_{y}+1,\uparrow},C_{k_{x},j_{y}+1,\downarrow},\cdots,C_{k_{x},j_{y}+N_{y}-1,\uparrow},C_{k_{x},j_{y}+N_{y}-1,\downarrow})^{T}$
is given by

$\displaystyle{\cal H}_{y-OBC}^{(F)}(k_{x})=\begin{pmatrix}h&T&0&\cdots&0\\\
T^{\dagger}&h&T&\cdots&0\\\ 0&T^{\dagger}&h&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T\\\
0&\cdots&0&T^{\dagger}&h\end{pmatrix}_{(4N_{y})\times(4N_{y})},$ (35)

where

$\displaystyle h\\!=\\!\begin{pmatrix}\\!h_{\uparrow}&0\\\
0&h_{\downarrow}\\!\end{pmatrix},$ (36) $\displaystyle
h_{\uparrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}(\\!\sigma_{+}\\!+\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}\sigma_{x},~{}~{}h_{\downarrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}(\\!\sigma_{+}\\!+\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}\sigma_{x},$
(37) $\displaystyle
E_{0}\\!=\\!-\\!D\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!=\\!-DA_{0}^{2}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{x}^{2}}\cos(k_{x}a_{x}),$
(38)
$\displaystyle\Delta_{\uparrow}\\!=\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega},~{}~{}\Delta_{\downarrow}\\!=\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega},$
(39)
$\displaystyle\Gamma_{\uparrow}\\!=\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{x}}\sin(k_{x}a_{x}),~{}~{}\Gamma_{\downarrow}\\!=\\!-\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{x}}\sin(k_{x}a_{x}),$
(40) $\displaystyle T\\!=\\!\begin{pmatrix}\\!T_{\uparrow}&0\\\
0&T_{\downarrow}\\!\end{pmatrix}\\!,~{}~{}T^{\dagger}\\!=\\!\begin{pmatrix}\\!T_{\uparrow}^{\dagger}&0\\\
0&T_{\downarrow}^{\dagger}\\!\end{pmatrix}\\!,$ (41) $\displaystyle
T_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{-}-\sigma_{+}),~{}~{}T^{\dagger}_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{+}-\sigma_{-}),$
(42) $\displaystyle
T_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{-}-\sigma_{+}),~{}~{}T^{\dagger}_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{+}-\sigma_{-}),$
(43)

where we use $\sigma_{+}\\!+\\!\sigma_{-}=\sigma_{x}$, and
$\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ are the spin-raising and spin-
lowering operators, i.e.,

$\displaystyle\sigma_{+}\\!=\\!\begin{pmatrix}0&1\\\
0&0\end{pmatrix},~{}~{}\sigma_{-}\\!=\\!\begin{pmatrix}0&0\\\
1&0\end{pmatrix}.$ (44)

In addition, we can also get the tight-binding Hamiltonian with pseudo-
spin-$\uparrow$ under $x$-PBCs and $y$-OBCs in the basis
$(C_{k_{x},j_{y},\uparrow},C_{k_{x},j_{y}+1,\uparrow},\cdots,C_{k_{x},j_{y}+N_{y}-1,\uparrow})^{T}$
as

$\displaystyle{\cal
H}_{y-OBC}^{(F)\uparrow}(k_{x})=\begin{pmatrix}h_{\uparrow}&T_{\uparrow}&0&\cdots&0\\\
T_{\uparrow}^{\dagger}&h_{\uparrow}&T_{\uparrow}&\cdots&0\\\
0&T_{\uparrow}^{\dagger}&h_{\uparrow}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T_{\uparrow}\\\
0&\cdots&0&T_{\uparrow}^{\dagger}&h_{\uparrow}\end{pmatrix}_{(2N_{y})\times(2N_{y})},$
(45)

where

$\displaystyle
h_{\uparrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}(\\!\sigma_{+}\\!+\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}\sigma_{x},$
(46) $\displaystyle
T_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{-}-\sigma_{+}),~{}~{}T^{\dagger}_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{+}-\sigma_{-}).$
(47)

Further, we can get the tight-binding Hamiltonian with pseudo-
spin-$\downarrow$ under $x$-PBCs and $y$-OBCs in the basis
$(C_{k_{x},j_{y},\downarrow},C_{k_{x},j_{y}+1,\downarrow},\cdots,C_{k_{x},j_{y}+N_{y}-1,\downarrow})^{T}$
as

$\displaystyle{\cal
H}_{y-OBC}^{(F)\downarrow}(k_{x})=\begin{pmatrix}h_{\downarrow}&T_{\downarrow}&0&\cdots&0\\\
T_{\downarrow}^{\dagger}&h_{\downarrow}&T_{\downarrow}&\cdots&0\\\
0&T_{\downarrow}^{\dagger}&h_{\downarrow}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T_{\downarrow}\\\
0&\cdots&0&T_{\downarrow}^{\dagger}&h_{\downarrow}\end{pmatrix}_{(2N_{y})\times(2N_{y})},$
(48)

where

$\displaystyle
h_{\downarrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}(\\!\sigma_{+}\\!+\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}\sigma_{x},$
(49) $\displaystyle
T_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{-}-\sigma_{+}),~{}~{}T^{\dagger}_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{y}}(\sigma_{+}-\sigma_{-}).$
(50)

##  Appendix D Tight-binding model under the open boundary condition along the
$x$ direction and periodic boundary conditions along the $y$ direction

The motivation for Appendix D is to derive the analytical expression for the
tight-binding model Hamiltonian under the open boundary condition along the
$x$ direction and periodic boundary conditions along the $y$ direction as a
comparison to Appendix C.

In a lattice, one makes the following replacements Shen (2012):

$\displaystyle k_{j}\rightarrow\frac{1}{a_{j}}\sin(k_{j}a_{j}),$ (51)
$\displaystyle k_{j}^{2}\rightarrow\frac{2}{a_{j}^{2}}[1-\cos(k_{j}a_{j})],$
(52)

where $j=x,y,z$, $a_{j}$ is the lattice constant along the $j$ direction,
$\sin(k_{j}a_{j})=\frac{e^{ik_{j}a_{j}}-e^{-ik_{j}a_{j}}}{2i}$, and
$\cos(k_{j}a_{j})=\frac{e^{ik_{j}a_{j}}+e^{-ik_{j}a_{j}}}{2}$,

$\displaystyle
k^{2}=k_{x}^{2}+k_{y}^{2}\rightarrow\frac{2}{a_{x}^{2}}[1-\cos(k_{x}a_{x})]+\frac{2}{a_{y}^{2}}[1-\cos(k_{y}a_{y})].$
(53)

In this way, the hopping terms in the lattice model only exist between the
nearest-neighbor sites. With this mapping, one obtains the following tight-
binding model with high-frequency pumping in the basis $(c_{{\bf
k},+,\uparrow},c_{{\bf k},-,\uparrow},c_{{\bf k},+,\downarrow},c_{{\bf
k},-,\downarrow})^{T}$ as

$\displaystyle\\!{\cal H}^{(F)}({\bf
k})\\!=\\!\left(\begin{array}[]{cccc}h_{\uparrow}({\bf k})&0\\\
0&h_{\downarrow}({\bf k})\end{array}\right),$ (56)

where

$\displaystyle h_{\uparrow}({\bf k})$
$\displaystyle\\!=\\!\\!-\\!D\left\\{A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\sigma_{0}$
$\displaystyle\\!+\\!\left\\{M\\!-\\!B\left[A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right]\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\right\\}\sigma_{z}$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{+}$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{-},$
(57) $\displaystyle h_{\downarrow}({\bf k})$
$\displaystyle\\!=\\!\\!-\\!D\left\\{A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\sigma_{0}$
$\displaystyle\\!+\\!\left\\{M\\!-\\!B\left[A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right]\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\right\\}\sigma_{z}$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{+}$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)A\left[\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right]\sigma_{-}.$
(58)

Performing the Fourier transformation, one obtains

$\displaystyle c_{{\bf k},s,\sigma}$
$\displaystyle=\frac{1}{\sqrt{N_{x}}}\sum_{j_{x},k_{y}}^{N_{x}}e^{-ik_{x}j_{x}a_{x}}c_{j_{x},k_{y},s,\sigma},$
(59) $\displaystyle c_{{\bf k},s,\sigma}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{N_{x}}}\sum_{j_{x},k_{y}}^{N_{x}}e^{ik_{x}j_{x}a_{x}}c_{j_{x},k_{y},s,\sigma}^{\dagger}.$
(60) $\displaystyle C_{{\bf k},\sigma}$ $\displaystyle=\begin{pmatrix}c_{{\bf
k},+,\sigma}\\\ c_{{\bf
k},-,\sigma}\end{pmatrix}=\frac{1}{\sqrt{N_{x}}}\sum_{j_{x},k_{y}}^{N_{x}}e^{-ik_{x}j_{x}a_{x}}C_{j_{x},k_{y},\sigma},$
(61) $\displaystyle C_{{\bf k},\sigma}^{\dagger}$
$\displaystyle=\begin{pmatrix}c_{{\bf k},+,\sigma}^{\dagger}&c_{{\bf
k},-,\sigma}^{\dagger}\end{pmatrix}=\frac{1}{\sqrt{N_{x}}}\sum_{j_{x},k_{y}}^{N_{x}}e^{ik_{x}j_{x}a_{x}}C_{j_{x},k_{y},\sigma}^{\dagger}.$
(62)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\uparrow}^{\dagger}\left[h_{\uparrow}({\bf k})\right]C_{{\bf k},\uparrow}$
$\displaystyle\\!=\\!\\!-\\!D\sum_{j_{x},k_{y}}\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{y}^{2}}\cos(k_{y}a_{y})\\!\right]\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{0}C_{j_{x},k_{y},\uparrow}\\!+\\!\frac{D}{a_{x}^{2}}\sum_{j_{x},k_{y}}\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{0}C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{j_{x},k_{y}}\\!\left[\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\cos(k_{y}a_{y})\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\\!\right]\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{z}C_{j_{x},k_{y},\uparrow}\\!+\\!\frac{B}{a_{x}^{2}}\sum_{j_{x},k_{y}}\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{y}}\sum_{j_{x},k_{y}}\sin(k_{y}a_{y})\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}(-i\sigma_{+}+i\sigma_{-})C_{j_{x},k_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{x}}\sum_{j_{x},k_{y}}\\!\left[\\!-iC_{j_{x},k_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right].$ (63)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\downarrow}^{\dagger}\left[h_{\downarrow}({\bf k})\right]C_{{\bf
k},\downarrow}$
$\displaystyle\\!=\\!\\!-\\!D\sum_{j_{x},k_{y}}\\!\left[\\!A_{0}^{2}\\!+\\!\frac{2}{a_{x}^{2}}\\!+\\!\frac{2}{a_{y}^{2}}\\!-\\!\frac{2}{a_{y}^{2}}\cos(k_{y}a_{y})\\!\right]\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{0}C_{j_{x},k_{y},\uparrow}\\!+\\!\frac{D}{a_{x}^{2}}\sum_{j_{x},k_{y}}\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{0}C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{j_{x},k_{y}}\\!\left[\\!M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\cos(k_{y}a_{y})\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega}\\!\right]\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\sigma_{z}C_{j_{x},k_{y},\uparrow}\\!+\\!\frac{B}{a_{x}^{2}}\sum_{j_{x},k_{y}}\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{y}}\sum_{j_{x},k_{y}}\sin(k_{y}a_{y})\\!\left[\\!C_{j_{x},k_{y},\uparrow}^{\dagger}(i\sigma_{+}-i\sigma_{-})C_{j_{x},k_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{2a_{x}}\sum_{j_{x},k_{y}}\\!\left[\\!-iC_{j_{x},k_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{j_{x}+1,k_{y},\uparrow}\\!+\\!{\rm
h.c.}\\!\right].$ (64)

Therefore, the tight-binding Hamiltonian under $y$-PBCs and $x$-OBCs in the
basis
$(C_{j_{x},k_{y},\uparrow},C_{j_{x},k_{y},\downarrow},C_{j_{x}+1,k_{y},\uparrow},C_{j_{x}+1,k_{y},\downarrow},\cdots,C_{j_{x}+N_{x}-1,k_{y},\uparrow},C_{j_{x}+N_{x}-1,k_{y},\downarrow})^{T}$
is given by

$\displaystyle{\cal H}_{x-OBC}^{(F)}(k_{y})=\begin{pmatrix}h&T&0&\cdots&0\\\
T^{\dagger}&h&T&\cdots&0\\\ 0&T^{\dagger}&h&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T\\\
0&\cdots&0&T^{\dagger}&h\end{pmatrix}_{(4N_{x})\times(4N_{x})},$ (65)

where

$\displaystyle h\\!=\\!\begin{pmatrix}\\!h_{\uparrow}&0\\\
0&h_{\downarrow}\\!\end{pmatrix},$ (66) $\displaystyle
h_{\uparrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}(\\!-i\sigma_{+}\\!+\\!i\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}\sigma_{y},~{}~{}h_{\downarrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}(\\!i\sigma_{+}\\!-\\!i\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!-\\!\Gamma_{\downarrow}\sigma_{y},$
(67) $\displaystyle
E_{0}=\\!\\!-DA_{0}^{2}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{y}^{2}}\cos(k_{y}a_{y}),$
(68)
$\displaystyle\Delta_{\uparrow}=M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\cos(k_{y}a_{y})\\!-\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega},~{}~{}\Delta_{\downarrow}=M\\!-\\!BA_{0}^{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\cos(k_{y}a_{y})\\!+\\!\frac{A_{0}^{2}A^{2}\sin\varphi}{\hbar\omega},$
(69)
$\displaystyle\Gamma_{\uparrow}=\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{y}}\sin(k_{y}a_{y}),~{}~{}\Gamma_{\downarrow}=-\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{A}{a_{y}}\sin(k_{y}a_{y}),$
(70) $\displaystyle T\\!=\\!\begin{pmatrix}\\!T_{\uparrow}&0\\\
0&T_{\downarrow}\\!\end{pmatrix}\\!,~{}~{}T^{\dagger}\\!=\\!\begin{pmatrix}\\!T_{\uparrow}^{\dagger}&0\\\
0&T_{\downarrow}^{\dagger}\\!\end{pmatrix}\\!,$ (71) $\displaystyle
T_{\uparrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{(-iA)}{2a_{x}}(\sigma_{+}+\sigma_{-}),~{}~{}T^{\dagger}_{\uparrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{iA}{2a_{x}}(\sigma_{+}+\sigma_{-}),$
(72) $\displaystyle
T_{\downarrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{(-iA)}{2a_{x}}(\sigma_{+}+\sigma_{-}),~{}~{}T^{\dagger}_{\downarrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{iA}{2a_{x}}(\sigma_{+}+\sigma_{-}),$
(73)

where we use $\sigma_{+}\\!+\\!\sigma_{-}=\sigma_{x}$.

In addition, we can also get the tight-binding Hamiltonian with pseudo-
spin-$\uparrow$ under $y$-PBCs and $x$-OBCs in the basis
$(C_{j_{x},k_{y},\uparrow},C_{j_{x}+1,k_{y},\uparrow},\cdots,C_{j_{x}+N_{x}-1,k_{y},\uparrow})^{T}$
as

$\displaystyle{\cal
H}_{x-OBC}^{(F)\uparrow}(k_{y})=\begin{pmatrix}h_{\uparrow}&T_{\uparrow}&0&\cdots&0\\\
T_{\uparrow}^{\dagger}&h_{\uparrow}&T_{\uparrow}&\cdots&0\\\
0&T_{\uparrow}^{\dagger}&h_{\uparrow}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T_{\uparrow}\\\
0&\cdots&0&T_{\uparrow}^{\dagger}&h_{\uparrow}\end{pmatrix}_{(2N_{x})\times(2N_{x})},$
(74)

where

$\displaystyle
h_{\uparrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}(\\!-i\sigma_{+}\\!+\\!i\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!-\\!i\Gamma_{\uparrow}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\uparrow}\sigma_{z}\\!+\\!\Gamma_{\uparrow}\sigma_{y},$
(75) $\displaystyle
T_{\uparrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{(-iA)}{2a_{x}}(\sigma_{+}+\sigma_{-}),~{}~{}T^{\dagger}_{\uparrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!+\\!\left(\\!1\\!-\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{iA}{2a_{x}}(\sigma_{+}+\sigma_{-}).$
(76)

Further, we can get the tight-binding Hamiltonian with pseudo-
spin-$\downarrow$ under $y$-PBCs and $x$-OBCs in the basis
$(C_{j_{x},k_{y},\downarrow},C_{j_{x}+1,k_{y},\downarrow},\cdots,C_{j_{x}+N_{x}-1,k_{y},\downarrow})^{T}$
as

$\displaystyle{\cal
H}_{x-OBC}^{(F)\downarrow}(k_{y})=\begin{pmatrix}h_{\downarrow}&T_{\downarrow}&0&\cdots&0\\\
T_{\downarrow}^{\dagger}&h_{\downarrow}&T_{\downarrow}&\cdots&0\\\
0&T_{\downarrow}^{\dagger}&h_{\downarrow}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&T_{\downarrow}\\\
0&\cdots&0&T_{\downarrow}^{\dagger}&h_{\downarrow}\end{pmatrix}_{(2N_{x})\times(2N_{x})},$
(77)

where

$\displaystyle
h_{\downarrow}\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!\Gamma_{\downarrow}(\\!i\sigma_{+}\\!-\\!i\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!+\\!i\Gamma_{\downarrow}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!)\\!=\\!E_{0}\sigma_{0}\\!+\\!\Delta_{\downarrow}\sigma_{z}\\!-\\!\Gamma_{\downarrow}\sigma_{y},$
(78) $\displaystyle
T_{\downarrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{(-iA)}{2a_{x}}(\sigma_{+}+\sigma_{-}),~{}~{}T^{\dagger}_{\downarrow}\\!=\\!\frac{D}{a_{x}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{x}^{2}}\sigma_{z}\\!-\\!\left(\\!1\\!+\\!\frac{2A_{0}^{2}B\sin\varphi}{\hbar\omega}\\!\right)\frac{iA}{2a_{x}}(\sigma_{+}+\sigma_{-}).$
(79)

##  Appendix E Probability distributions of the edge states for $E_{\sigma-}$

The motivation for Appendix E is to give the probability distributions of the
edge states for $E_{\sigma-}$ as supplemental material for Fig. 3 in the main
text.

Figure 6: (Color online) Probability distributions of the edge states for the
Floquet Hamiltonian of the HgTe/(Hg,Cd)Te with different sizes $L_{y}$ under
$k_{x}=0$. The other parameters are the same as those in Fig. 1. Notice that
the subscript $-$ denotes the edge states of $E_{\sigma-}$ with
$\sigma=\uparrow,\downarrow$.

It is indicated in Fig. 6 that the probability distributions of the edge
states for the $-$ bands $E_{\sigma-}$ have similar properties as those for
the $+$ bands $E_{\sigma+}$ with changing system size $L_{y}$.