††thanks: They contribute equally to this work. ††thanks: They contribute
equally to this work.

# Phase transitions in intrinsic magnetic topological insulator with high-
frequency pumping

Fang Qin  Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology
(SUSTech), Shenzhen 518055, China  CAS Key Laboratory of Quantum Information,
University of Science and Technology of China, Chinese Academy of Sciences,
Hefei, Anhui 230026, China  Department of Physics, National University of
Singapore, Singapore 117542  Rui Chen  Shenzhen Institute for Quantum Science
and Engineering and Department of Physics, Southern University of Science and
Technology (SUSTech), Shenzhen 518055, China  School of Physics, Southeast
University, Nanjing 211189, China  Department of Physics, The University of
Hong Kong, Pokfulam Road, Hong Kong, China  Hai-Zhou Lu  Corresponding author:
<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science and Engineering and
Department of Physics, Southern University of Science and Technology
(SUSTech), Shenzhen 518055, China  Shenzhen Key Laboratory of Quantum Science
and Engineering, Shenzhen 518055, China

###### Abstract

In this work, we investigate the topological phase transitions in an effective
model for a topological thin film with high-frequency pumping. In particular,
our results show that the circularly polarized light can break the time-
reversal symmetry and induce the quantum anomalous Hall insulator (QAHI)
phase. Meanwhile, the bulk magnetic moment can also break the time-reversal
symmetry. Therefore, it shows rich phase diagram by tunning the intensity of
the light and the thickness of the thin film. Using the parameters fitted by
experimental data, we give the topological phase diagram of the Cr-doped
Bi2Se3 thin film, showing that by modulating the strength of the polarized
optical field in an experimentally accessible range, there are four different
phases: the normal insulator phase, the time-reversal-symmetry-broken quantum
spin Hall insulator phase, and two different QAHI phases with opposite Chern
numbers. Comparing with the non-doped Bi2Se3, it is found that the interplay
between the light and bulk magnetic moment separates the two different QAHI
phases with opposite Chern numbers. The results show that an intrinsic
magnetic topological insulator with high-frequency pumping is an ideal
platform for further exploring various topological phenomena with a
spontaneously broken time-reversal symmetry.

##  I Introduction

Topological insulators are band insulators with topologically protected
boundary states and insulating bulk states Moore2010 ; Hasan2010 ; Qi2011 ;
Shen2012 ; Wang2017 ; Qin2020 ; Chen2021prl ; Chen2021prr . A well-known
topological insulator paradigm is the quantum Hall insulator (QHI) phase
ZhangExp2009 . Distinct from the QHI phase, there also exists a quantized
version of the Hall effects in a magnetically doped topological insulator
without applying any an external magnetic field. In an intrinsic magnetic
topological insulator XueExp2013 ; CheckelskyExp2014 ; QAHIexp2014 ;
QAHIexp2015 ; XueExp2015 ; YoshimiExp2015 ; ChangExp2016 ; GrauerExp2017 ;
XueExp2018 ; ZhangExp2020 ; QAHIexp2020 ; ZhaoExp2020 ; MogiExp2021 , a band
gap on the surface states of the topological materials is opened by the time-
reversal symmetry breaking, which is essential for the realization of the
quantum anomalous Hall insulator (QAHI) phase XueExp2013 ; Qi2008 ; Yu2010 ;
Chu2011 . Physically, how to realize the QAHI phase has attracted much
attention in the past few decades Qiao2010 ; Qi2006 ; Onoda2006 ; Nomura2011 ;
Checkelsky2012 . One of the promising physical schemes is on the basis of the
topological insulators doped with magnetic impurities Hor2010 ; Chen2010 ;
Wray2011 ; Wang2013 ; Wang2014 ; Law2017 ; Kim2018 ; Kawamura2018 , where the
interplay of the magnetic exchange interaction and the spin-orbit coupling
gives rise to the band inversion between the conduction and valence bands.
More importantly, the experimental realization of the QAHI phase has been
reported in the thin films of Cr-doped (Bi,Sb)2Te3, which is an intrinsic
magnetic topological insulator XueExp2013 .

The quantum spin Hall insulator (QSHI) phase is another family member of the
Hall effects Kane2005 ; Bernevig2006 ; Haldane2006 ; Prodan2009 ; Xing2010 ;
Xing2011 ; Chen2017 . To explain QSHI, we take the Kane and Mele model
Kane2005  for example. In this model, the spin-up and -down electrons exhibit
opposite Chern numbers due to the spin-orbit coupling, so that the total Chern
number vanishes but the spin Chern number is nonzero. Particularly, the QSHI
phase has been observed experimentally in HgTe QSHIexp2006 ; QSHIexp2007 .
Usually, the QSHI phase was considered to be protected by the time-reversal
symmetry. However, it has been found that the QSHI phase with nonzero spin
Chern numbers persists when the time-reversal symmetry is broken and this is
called the time-reversal-symmetry-broken QSHI phase Xing2011 ; Chen2017 .

Based on the Floquet theory that allows one to map a time-dependent problem
into a stationary one, it has been shown that a periodic perturbation can
induce topological phase transitions in a topological trivial insulator
Oka2009prb ; Calve2015prb ; Kitagawa2011prb ; Lindner2011np ;
Rechtsman2013nature ; Wang2018prl ; Xu2021as ; Zhu2014 ; lightexp2013 ;
lightexp2016 ; Leon2013 ; Katan2013 ; Inoue2010 ; Kitagawa2010 ; Yan2016 ;
Narayan2015 ; Saha2016 ; Chen20181 ; Chen20182 ; Pervishko2018 ; Kyriienko2019
; Huang20prl ; Hu20prl ; Du21arxiv ; Ning21arxiv . In particular, the quantum
Hall effect induced by circularly polarized field in Dirac materials was
predicted decade ago by Oka and Aoki Oka2009prb . Further, the theory given by
Oka and Aoki was generalized for various topological insulators in the
previous Refs. Calve2015prb ; Kitagawa2011prb ; Lindner2011np ;
Rechtsman2013nature ; Wang2018prl ; Xu2021as ; Zhu2014 ; lightexp2013 ;
lightexp2016 ; Leon2013 ; Katan2013 ; Inoue2010 ; Kitagawa2010 ; Yan2016 ;
Narayan2015 ; Saha2016 ; Chen20181 ; Chen20182 ; Pervishko2018 ; Kyriienko2019
; Huang20prl ; Hu20prl ; Du21arxiv ; Ning21arxiv . Later the corresponding
Floquet topological insulators have been observed by experiments lightexp2013
; lightexp2016 . Also, it has been shown that an intense high-frequency
linearly polarized light can be used to manipulate the value of a gap in of a
non-doped topological insulator thin film Pervishko2018 . However, the
linearly polarized light cannot include the contribution of $1/\omega$, so
that they only estimated the terms proportional to $1/\omega^{2}$
Pervishko2018 , where $\omega$ is the frequency of the polarized light.
Noteworthy, the circularly polarized light can include the contribution of
$1/\omega$. Therefore, the impact of high-frequency pumping with the
circularly polarized light on the thin films of topological insulators needs
to be studied comprehensively.

In this work, we investigate the topological phase transitions in the
atomically thin flakes of an intrinsic magnetic topological insulator with
high-frequency pumping. Based on the results, it is found that the intensity
of the circularly polarized light can be used as a knob to drive a topological
transition. Different from the situation in the absent of optical field, there
exist four different phases: the normal insulator (NI) phase, the time-
reversal-symmetry-broken QSHI phase, and two different QAHI phases with
opposite Chern numbers $C=\pm 1$, respectively. In particular, for a special
given layer thickness, one can apparently detect the tendency towards light-
induced band inversion upon increasing optical field intensity and passing
through the QSHI phase region. This means that the energy gap of the surface
states can be tuned by adjusting the intensity of the driving optical field in
an experimentally accessible range.

The paper is organized as the following: In Sec. II, we give the model
Hamiltonian. In Sec. III, we introduce the Floquet theory for a time-periodic
Hamiltonian. In Sec. B, we give the polarized light and Floquet Hamiltonian
which is used in the following calculations. Furthermore, we study the basis
states at the $\Gamma$ point in Sec. V. Moreover, we calculate the high-
frequency pumping-induced topological properties of the effective Floquet
Hamiltonian of the thin film in Sec. VI. In addition, we give the light-
induced topological phase diagram in Sec. VII. Finally, we summarize in Sec.
VIII.

##  II Model

We take the periodic boundary conditions in the $x-y$ plane such that $k_{x}$
and $k_{y}$ are good quantum numbers, and denote the thickness of the thin
film along $z$ direction as $L$. In the basis
$(|p1_{z}^{+},\uparrow\rangle,|p2_{z}^{-},\uparrow\rangle,|p1_{z}^{+},\downarrow\rangle,|p2_{z}^{-},\downarrow\rangle)$
which are the hybridized states of Se $p_{z}$ orbital and Bi $p_{z}$ orbital,
with even ($+$) and odd ($-$) parities, up ($\uparrow$) and down
($\downarrow$) spins, the low-energy three-dimensional Hamiltonian for Cr-
doped Bi2Se3 is given by Shan2010 ; Lu2010 ; Xing2010 ; Lu2013 ; Liu2019 ;
Sun2020 ; Liu2010prb ; Dabiri2021prb1 ; Dabiri2021prb2

$\displaystyle H({\bf k})=H_{0}({\bf k})+H_{X}(z),$ (1)

where

$\displaystyle H_{0}({\bf k})$ $\displaystyle\\!=\\!\epsilon_{0}({\bf
k})I_{4}$ $\displaystyle~{}\\!+\\!\\!\left(\\!\begin{array}[]{cccc}\\!M({\bf
k})\sigma_{z}\\!-\\!iA_{1}\partial_{z}\sigma_{x}&A_{2}k_{-}\sigma_{x}\\\
A_{2}k_{+}\sigma_{x}&M({\bf
k})\sigma_{z}\\!+\\!iA_{1}\partial_{z}\sigma_{x}\\!\end{array}\\!\right)\\!$
(4) $\displaystyle\\!=\\!\epsilon_{0}({\bf
k})\tau_{0}\otimes\sigma_{0}\\!+\\!M({\bf
k})\tau_{0}\otimes\sigma_{z}\\!+\\!\\!A_{1}\\!k_{z}\tau_{z}\otimes\sigma_{x}$
$\displaystyle~{}\\!+\\!A_{2}k_{x}\tau_{x}\otimes\sigma_{x}\\!+\\!A_{2}k_{y}\tau_{y}\otimes\sigma_{x}.$
(5)

Here $\sigma_{x,y,z}$ are the Pauli matrices for the orbital degree of
freedom, $k_{z}=-i\partial_{z}$, $I_{4}$ is the $4\times 4$ identity matrix,
$k_{\pm}=k_{x}\pm ik_{y}$, $\epsilon_{0}({\bf
k})=C_{0}-D_{1}\partial_{z}^{2}+D_{2}(k_{x}^{2}+k_{y}^{2})$, $M({\bf
k})=M_{0}+B_{1}\partial_{z}^{2}-B_{2}(k_{x}^{2}+k_{y}^{2})$, $C_{0}$, $D_{i}$,
$M_{0}$, $B_{i}$, and $A_{i}$ are model parameters with $i=1,2$. The
parameters for Bi2Se3 are adopted as ZhangExp2009 ; XueExp2013 :
$C_{0}=-0.0068$ eV, $D_{1}=1.3$ eVÅ2, $D_{2}=19.6$ eVÅ2, $A_{1}=2.2$ eVÅ,
$A_{2}=4.1$ eVÅ, $M_{0}=0.28$ eV, $B_{1}=10$ eVÅ2, and $B_{2}=56.6$ eVÅ2. The
exchange field reads Lu2013 ; Liu2019

$\displaystyle H_{X}(z)=m_{0}\tau_{z}\otimes\sigma_{0},$ (6)

where $m_{0}$ is the magnitude of the bulk magnetic moment XueExp2013 ,
$\tau_{z}$ is the $z$ Pauli matrix for the spin degree of freedom,
$\sigma_{0}$ is a $2\times 2$ unit matrix, and the magnetization energy along
the $z$ direction is given by $m_{0}$, i.e., $m_{0}$ is the exchange field
from the magnetic dopants.

##  III Floquet formula

The Floquet theory can be applied to a time-periodic Hamiltonian $H(t)=H(t+T)$
with the period $T=2\pi/\omega$ and the frequency $\omega$ of the light. By
employing the Floquet theory, the wave function of the time-periodic
Schrödinger equation $i\partial_{t}\Psi(t)=H(t)\Psi(t)$, has the form
$\Psi(t)=\sum_{m}\psi_{m}e^{-i(\epsilon/\hbar+m\omega)t}$, where $\epsilon$ is
the quasienergy and $m$ is an integer. With a Fourier series expansion, we
find that $\sum_{m}H_{n,m}\psi_{m}=\epsilon\psi_{n}$, where

$\displaystyle
H_{n,m}=n\hbar\omega\delta_{n,m}+\frac{1}{T}\int_{0}^{T}H(t)e^{i(n-m)\omega
t}dt,$ (7)

which is a block Hamiltonian of the Floquet state, $n$ and $m$ are integers.
If $\Psi(t)$ is an eigenvector with the quasienergy $\epsilon$, $e^{in\omega
t}\Psi(t)$ is also an eigenvector of the system with the quasienergy
$\epsilon+n\hbar\omega$.

From Eq. (7), one can have

$\displaystyle
H_{n,m}=\left(\begin{array}[]{ccccc}\cdots&\cdots&\cdots&\cdots&\cdots\\\
\cdots&H_{-1,-1}&H_{-1,0}&H_{-1,1}&\cdots\\\
\cdots&H_{0,-1}&H_{0,0}&H_{0,1}&\cdots\\\
\cdots&H_{1,-1}&H_{1,0}&H_{1,1}&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots\end{array}\right),$ (13)

where $H_{n,m}=H_{-m,-n}$ with $n\neq m$.

##  IV Polarized light and Floquet Hamiltonian

The time-dependent Hamiltonian can be experimentally introduced by normally
illuminating with elliptically polarized light described by a time-varying
gauge field (or the vector potential) ${\bf A}(t)=A(\sin(\omega t),\sin(\omega
t+\varphi))$ Saha2016 ; Zhu2014 , which gives the optical field as ${\bf
E}(t)=\partial{\bf A}(t)/\partial t=E_{0}(\cos(\omega t),\cos(\omega
t+\varphi))$, where $E_{0}=A/\omega$ is the amplitude of the optical field,
$\omega$ is the frequency of the optical field, and the phase $\varphi$
controls the polarization: when $\varphi=0$ or $\pi$, the optical field is
linearly polarized; when $\varphi=\pm\pi/2$, the optical field is circularly
polarized; for example, ${\bf A}(t)=A(\sin(\omega t),\cos(\omega t))$ with
$\varphi=\pi/2$; when $\varphi$ takes other values, the optical field is
elliptically polarized. From the point of view of the source
wikipedia_Circular , with $\varphi=\pi/2$, the corresponding electric field
vector is ${\bf E}(t)=E_{0}(\cos(\omega t),\cos(\omega
t+\pi/2))=E_{0}(\cos(\omega t),-\sin(\omega t))$, which is a left-handed
circularly polarized wave and corresponds to ${\bf A}(t)=A(\sin(\omega
t),\cos(\omega t))$. For $\varphi=-\pi/2$, the corresponding electric field
vector is ${\bf E}(t)=E_{0}(\cos(\omega t),\cos(\omega
t-\pi/2))=E_{0}(\cos(\omega t),\sin(\omega t))$, which is a right-handed
circularly polarized wave and corresponds to ${\bf A}(t)=A(\sin(\omega
t),-\cos(\omega t))$. The vector potentials for right- and left-handed
circularly polarized lights are different from each other and different vector
potential will give different time-dependent Hamiltonian (14). In the
following calculations, we focus on the left-handed circularly polarized light
with $\varphi=\pi/2$. As a high-frequency laser light, we choose the photon
energy, for example, to be $\hbar\omega\approx 0.15$ eV ($\omega\sim
2.2789\times 10^{2}$ THz), which is close to the typical values in recent
optical pump-probe experiments. For example, the typical amplitude of light
$A_{0}=eA/\hbar=eE_{0}/(\hbar\omega)$ is about 0.03 Å-1, and the corresponding
electric field strength $E_{0}=\hbar\omega A_{0}/e$ is $4.5\times 10^{7}$ V/m,
which is within the experimental accessibility lightexp2013 ; lightexp2016 .

By use of the Peierls substitution, the time-dependent Hamiltonian is obtained
as

$\displaystyle H(t)=H\left({\bf k}_{\perp}-\frac{e}{\hbar}{\bf
A}(t),-i\partial_{z}\right),$ (14)

where ${\bf k}_{\perp}=(k_{x},k_{y})$. Making use of the Floquet theory
Zhu2014 ; Pervishko2018 ; Chen20181 ; Chen20182  in the high-frequency limit,
the periodically driven system can be described by a static effective
Hamiltonian as high-frequency1982 ; high-frequency1988 ; high-frequency20031 ;
high-frequency20032 ; high-frequency2014 ; high-frequency20151 ; high-
frequency20152

$\displaystyle H^{(F)}=H_{0,0}+\frac{[H_{0,-1},H_{0,1}]}{\hbar\omega},$ (15)

where

$\displaystyle H_{0,0}=H({\bf
k})+D_{2}A_{0}^{2}-B_{2}A_{0}^{2}I_{2}\otimes\sigma_{z},$ (16) $\displaystyle
H_{0,-1}$ $\displaystyle\\!=\\!-iD_{2}A_{0}(k_{x}\\!+\\!e^{-i\varphi}k_{y})$
$\displaystyle\\!+\\!\\!\left(\\!\begin{array}[]{cccc}iB_{2}A_{0}(k_{x}\\!+\\!e^{-i\varphi}k_{y})\sigma_{z}&\\!-\\!\frac{i}{2}A_{2}A_{0}(1\\!-\\!ie^{-i\varphi})\sigma_{x}\\!\\\
\\!-\\!\frac{i}{2}A_{2}A_{0}(1\\!+\\!ie^{-i\varphi})\sigma_{x}&iB_{2}A_{0}(k_{x}\\!+\\!e^{-i\varphi}k_{y})\sigma_{z}\\!\end{array}\\!\right)\\!,$
(19) $\displaystyle H_{0,1}$
$\displaystyle\\!=\\!iD_{2}A_{0}(k_{x}+e^{i\varphi}k_{y})$
$\displaystyle\\!+\\!\\!\left(\\!\begin{array}[]{cccc}\\!-\\!iB_{2}A_{0}(k_{x}\\!+\\!e^{i\varphi}k_{y})\sigma_{z}&\frac{i}{2}A_{2}A_{0}(1\\!-\\!ie^{i\varphi})\sigma_{x}\\\
\frac{i}{2}A_{2}A_{0}(1\\!+\\!ie^{i\varphi})\sigma_{x}&\\!-\\!iB_{2}A_{0}(k_{x}\\!+\\!e^{i\varphi}k_{y})\sigma_{z}\\!\end{array}\\!\right)\\!.$
(22)

From Eq. (15), the Floquet Hamiltonian is

$\displaystyle H^{(F)}$
$\displaystyle\\!=\\!\\!\left(\\!\begin{array}[]{cccc}\tilde{M}({\bf
k})\sigma_{z}\\!-\\!iA_{1}\partial_{z}\sigma_{x}&A_{2}k_{-}\sigma_{x}\\\
A_{2}k_{+}\sigma_{x}&\tilde{M}({\bf
k})\sigma_{z}\\!+\\!iA_{1}\partial_{z}\sigma_{x}\\!\end{array}\\!\right)\\!$
(25)
$\displaystyle\\!+\\!\frac{A_{0}^{2}A_{2}\sin\varphi}{\hbar\omega}\\!\left(\\!\begin{array}[]{cccc}\\!-\\!A_{2}I_{2}&\\!-\\!B_{2}k_{-}(\sigma_{+}\\!-\\!\sigma_{-})\\!\\\
\\!B_{2}k_{+}(\sigma_{+}\\!-\\!\sigma_{-})&A_{2}I_{2}\end{array}\\!\right)\\!$
(28) $\displaystyle\\!+\\!\tilde{\epsilon}_{0}({\bf k})I_{4},$ (29)

where $\tilde{\epsilon}_{0}({\bf k})=\epsilon_{0}({\bf k})+A_{0}^{2}D_{2}$,
$\tilde{M}({\bf k})=M({\bf k})-A_{0}^{2}B_{2}$,
$\sigma_{+}=\sigma_{x}+i\sigma_{y}$, and $\sigma_{-}=\sigma_{x}-i\sigma_{y}$.

To give a simple quantitative estimation of the validity of the theoretical
formalism developed here, we evaluate the maximum instantaneous energy of the
time-dependent Hamiltonian (14) averaged over a period of the field
$\frac{1}{T}\int_{0}^{T}dt~{}\text{max}\left\\{\big{|}\big{|}H(t)\big{|}\big{|}\right\\}<\hbar\omega$.
Therefore, in the vicinity of the $\Gamma$ point, the field parameters have to
meet the condition $A_{0}A_{2}/(\hbar\omega)<1$. Particularly, in the high-
frequency regime for an external pumping $\hbar\omega\approx 0.4$ eV
($\omega\sim 6.07707\times 10^{2}$ THz), one can estimate $A_{0}\lesssim 0.09$
Å-1.

Experimentally, from the spectroscopic ellipsometry measurements of the bulk
Bi2Se3 McIver2012 , the penetration depth of the laser light is about $25$ nm
McIver2012 . At the same time, from the transient reflectivity/transmission
measurements of the bulk Bi2Se3 Glinka2013 , the penetration depth of the
laser light is about $21$ nm Glinka2013 . Both $21$ nm and $25$ nm are
reasonably larger than the film thickness ($2\sim 8$ nm) of the bulk Bi2Se3.
It is important that the penetration depth of the laser light should be
reasonably larger than the film thickness, so that the vector potential ${\bf
A}(t)$ of the laser light can properly change the momentum ${\bf k}_{\perp}$
as in the thin-film Hamiltonian (14). Otherwise, the effect of the vector
potential will not go into the bulk state of the thin film due to the skin
effect.

##  V Basis states at the $\Gamma$ point

To establish an effective model for the surface states, we first find the four
solutions to the surface states of the model in Eq. (1) at the $\Gamma$ point
($k_{x}=k_{y}=0$) as Shan2010 ; Lu2010

$\displaystyle H_{0}=\left(\begin{array}[]{cccc}h(A_{1})&0\\\
0&h(-A_{1})\end{array}\right),$ (32)

where

$\displaystyle h(A_{1})$
$\displaystyle\\!=\\!(C_{0}-D_{1}\partial_{z}^{2})I_{2}+(M_{0}+B_{1}\partial_{z}^{2})\sigma_{z}-iA_{1}\partial_{z}\sigma_{x}$
$\displaystyle\\!=\\!\left(\begin{array}[]{cccc}C_{0}+M_{0}-D_{-}\partial_{z}^{2}&-iA_{1}\partial_{z}\\\
-iA_{1}\partial_{z}&C_{0}-M_{0}-D_{+}\partial_{z}^{2}\end{array}\right),$ (35)

and $D_{\pm}=D_{1}\pm B_{1}$.

$H_{0}$ in Eq. (32) is block-diagonal and its solution can be found by solving
each block separately, i.e.,

$\displaystyle h(A_{1})\Psi_{\uparrow}(z)=E\Psi_{\uparrow}(z),$ (36)
$\displaystyle h(-A_{1})\Psi_{\downarrow}(z)=E\Psi_{\downarrow}(z).$ (37)

Because the lower block is the “time” reversal of the upper block, the
solutions satisfy $\Psi^{\downarrow}(z)=\Theta\Psi^{\uparrow}(z)$, where
$\Theta=-i\sigma_{y}\mathcal{K}$ is the time-reversal operator and
$\mathcal{K}$ is the complex conjugation operation. Equivalently, we can
replace $A_{1}$ by $-A_{1}$ in all the results for the upper block, to obtain
those for the lower block. Therefore, we only need to solve $h(A_{1})$.

The solution of the block-diagonal $H_{0}$ can be found by putting a two-
component trial solution into the eigenequation (36) of the upper block with
$\Psi_{\uparrow}(z)=\Psi^{\uparrow}_{\lambda}(z)e^{\lambda z}$, where
$\lambda$ is the trial coefficients defining the behavior of the wave
functions and $E$ is the trial eigenenergy. Therefore, one can have

$\displaystyle E_{\pm}$ $\displaystyle=C_{0}-D_{1}\lambda^{2}$
$\displaystyle~{}~{}\pm\sqrt{(M_{0}+B_{1}\lambda^{2}-A_{1}\lambda)(M_{0}+B_{1}\lambda^{2}+A_{1}\lambda)},$
(38)

and

$\displaystyle\Psi^{\uparrow}_{\lambda}=\left(\begin{array}[]{cc}D_{+}\lambda^{2}-L_{-}+E_{\pm}\\\
-iA_{1}\lambda\end{array}\right).$ (41)

Note that the trial coefficients may have multiple solutions, the final
solution should be a linear superposition of these solutions with the
superposition coefficients determined by boundary conditions. Then the problem
becomes a straightforward calculation of the Schrödinger equation or the
secular equation $\text{det}|h(A_{1})-E|=0$ which gives four solutions of
$\lambda_{\alpha}(E)$, denoted as $\beta\lambda_{\alpha}(E)$, with
$\alpha\in\\{1,2\\}$, $\beta\in\\{+,-\\}$ and $\lambda_{\alpha}$ define the
behavior of the wave functions along $z$ axis and are functions of the energy
$E$ as

$\displaystyle\\!\lambda_{\alpha}(E)\\!=\\!\sqrt{\frac{\\!-\\!F\\!+\\!(-1)^{\alpha-1}\sqrt{R}}{2D_{+}D_{-}}},$
(42)

where we have defined

$\displaystyle F$ $\displaystyle=A_{1}^{2}+2D_{1}(E-C_{0})-2B_{1}M_{0}$
$\displaystyle=A_{1}^{2}+D_{+}(E-L_{+})+D_{-}(E-L_{-}),$ (43) $\displaystyle
R$ $\displaystyle=F^{2}-4(D_{1}^{2}-B_{1}^{2})[(E-C_{0})^{2}-M_{0}^{2}]$
$\displaystyle=F^{2}-4D_{+}D_{-}(E-L_{+})(E-L_{-}),$ (44)

$D_{\pm}=D_{1}\pm B_{1}$, and $L_{\pm}=C_{0}\pm M_{0}$. With Eq. (41), the
general solution is a linear combination of the four linearly independent two-
component vectors

$\displaystyle\Psi_{\uparrow}(z)$
$\displaystyle=\sum_{\alpha=1,2}\sum_{\beta=+,-}C_{\alpha\beta}\Psi^{\uparrow}_{\alpha\beta}e^{\beta\lambda_{\alpha}z}$
$\displaystyle=\sum_{\alpha=1,2}\sum_{\beta=+,-}C_{\alpha\beta}\left(\begin{array}[]{cc}D_{+}\lambda_{\alpha}^{2}-L_{-}+E_{\pm}\\\
-iA_{1}(\beta\lambda_{\alpha})\end{array}\right)e^{\beta\lambda_{\alpha}z},$
(47)

where the superposition coefficients $C_{\alpha\beta}$ are determined by
boundary conditions.

##  VI Effective model of the thin film

In this section, we derive the effective low-energy continuous model for the
thin film of the three-dimensional topological insulators.

###  VI.1 Finite-thickness boundary conditions

Now, we turn to discuss the gap in a thin film with both top and bottom open
surfaces. When the thickness $L$ of the film is comparable with the
characteristic length $1/\lambda$ of the surface states, there is a coupling
between the states on opposite surfaces. One has to consider the boundary
conditions at both surfaces simultaneously. Without loss of generality, we
will consider that the top surface is located at $z=L/2$ and the bottom
surface at $-L/2$. The boundary conditions are given as

$\displaystyle\Psi_{\uparrow}\left(z=\pm\frac{L}{2}\right)=0,$ (48)

where $L=N_{L}d$ is the thickness of the film with the number $N_{L}$ of the
layers and the thickness $d$ of a layer, and $-L/2\leqslant z\leqslant L/2$.

With the boundary conditions in Eq. (48), the trial wave function can be given
by

$\displaystyle\Psi_{\uparrow}(z)=\left(\begin{array}[]{cc}\psi_{1}(z)\\\
\psi_{2}(z)\end{array}\right),$ (51)

where

$\displaystyle\psi_{1}(z)$ $\displaystyle=c_{+}f_{+}(z)+c_{-}f_{-}(z),$ (52)
$\displaystyle\psi_{2}(z)$ $\displaystyle=d_{+}f_{+}(z)+d_{-}f_{-}(z),$ (53)
$\displaystyle f_{+}(z)$
$\displaystyle=\frac{\cosh(\lambda_{1}z)}{\cosh(\lambda_{1}L/2)}-\frac{\cosh(\lambda_{2}z)}{\cosh(\lambda_{2}L/2)},$
(54) $\displaystyle f_{-}(z)$
$\displaystyle=\frac{\sinh(\lambda_{1}z)}{\sinh(\lambda_{1}L/2)}-\frac{\sinh(\lambda_{2}z)}{\sinh(\lambda_{2}L/2)}.$
(55)

Substituting Eq. (51) into (36):
$h(A_{1})\Psi_{\uparrow}(z)\\!=\\!E\Psi_{\uparrow}(z)$, we have

$\displaystyle\\!\left(\\!\begin{array}[]{cccc}\\!C_{0}\\!+\\!M_{0}\\!-\\!E\\!-\\!D_{-}\partial_{z}^{2}\\!&-iA_{1}\partial_{z}\\\
-iA_{1}\partial_{z}&\\!C_{0}\\!-\\!M_{0}\\!-\\!E\\!-\\!D_{+}\partial_{z}^{2}\\!\end{array}\\!\right)\\!\\!\left(\\!\begin{array}[]{cc}\\!\psi_{1}(z)\\!\\\
\\!\psi_{2}(z)\\!\end{array}\\!\right)\\!=\\!0.$ (60)

With Eq. (60), we have

$\displaystyle\frac{c_{+}}{d_{-}}$
$\displaystyle=\frac{C_{0}-M_{0}-E-D_{+}\lambda_{1}^{2}}{iA_{1}\lambda_{1}}\frac{\cosh(\lambda_{1}L/2)}{\sinh(\lambda_{1}L/2)},$
(61) $\displaystyle\frac{c_{+}}{d_{-}}$
$\displaystyle=\frac{C_{0}-M_{0}-E-D_{+}\lambda_{2}^{2}}{iA_{1}\lambda_{2}}\frac{\cosh(\lambda_{2}L/2)}{\sinh(\lambda_{2}L/2)},$
(62) $\displaystyle\frac{c_{-}}{d_{+}}$
$\displaystyle=\frac{C_{0}-M_{0}-E-D_{+}\lambda_{1}^{2}}{iA_{1}\lambda_{1}}\frac{\sinh(\lambda_{1}L/2)}{\cosh(\lambda_{1}L/2)},$
(63) $\displaystyle\frac{c_{-}}{d_{+}}$
$\displaystyle=\frac{C_{0}-M_{0}-E-D_{+}\lambda_{2}^{2}}{iA_{1}\lambda_{2}}\frac{\sinh(\lambda_{2}L/2)}{\cosh(\lambda_{2}L/2)}.$
(64)

Furthermore, the secular equation of the nontrivial solution to the
superposition coefficients $C_{\alpha\beta}$ leads to the transcendental
equations

$\displaystyle\frac{[C_{0}\\!-\\!M_{0}\\!-\\!E_{+}^{0}\\!-\\!D_{+}\lambda_{1}^{2}(E_{+}^{0})]\lambda_{2}(E_{+}^{0})}{[C_{0}\\!-\\!M_{0}\\!-\\!E_{+}^{0}\\!-\\!D_{+}\lambda_{2}^{2}(E_{+}^{0})]\lambda_{1}(E_{+}^{0})}\\!=\\!\frac{\tanh(\lambda_{2}(E_{+}^{0})L/2)}{\tanh(\lambda_{1}(E_{+}^{0})L/2)},$
(65)
$\displaystyle\frac{[C_{0}\\!-\\!M_{0}\\!-\\!E_{-}^{0}\\!-\\!D_{+}\lambda_{2}^{2}(E_{-}^{0})]\lambda_{1}(E_{-}^{0})}{[C_{0}\\!-\\!M_{0}\\!-\\!E_{-}^{0}\\!-\\!D_{+}\lambda_{1}^{2}(E_{-}^{0})]\lambda_{2}(E_{-}^{0})}\\!=\\!\frac{\tanh(\lambda_{2}(E_{-}^{0})L/2)}{\tanh(\lambda_{1}(E_{-}^{0})L/2)}.$
(66)

The numerical solutions of the transcendental equations (65) and (66) are
shown in Fig. 1.

Figure 1: The eigen energies $E_{+}^{0}$ and $E_{-}^{0}$ as a function of the
finite thickness $L$ for Bi2Se3. The red solid star points are for $E_{+}^{0}$
, and the blue solid circle points are for $E_{-}^{0}$. The numerical
solutions are obtained by solving the transcendental equations (65) and (66).

In particular, for $L\rightarrow+\infty$ (infinitely thick case), we have the
analytic solutions for Eqs.(65) and (66) as (the detail derivations can be
found in the Appendix A)

$\displaystyle
E_{\pm}^{0}\xrightarrow{L\rightarrow+\infty}C_{0}+\frac{D_{1}M_{0}}{B_{1}}=0.0296~{}({\rm
eV}),$ (67)

which is consistent with the numerical results shown in Fig. 1.

For the infinitely thin case ($L\rightarrow 0$), there will be no surface
state which corresponds to no boundary condition. Besides, there is no wave
function in the bulk and surface of the thin film physically. Therefore, there
is no analytic solutions in the infinitely thin case.

Therefore, the eigen wavefunctions for $E_{+}^{0}$ and $E_{-}^{0}$ are,
respectively,

$\displaystyle\varphi(A_{1})\equiv\Psi_{\uparrow}^{+}$
$\displaystyle=\bar{C}_{+}\left(\begin{array}[]{cc}-D_{+}\eta_{1}^{+}f_{-}^{+}\\\
iA_{1}f_{+}^{+}\end{array}\right),$ (70)
$\displaystyle\chi(A_{1})\equiv\Psi_{\uparrow}^{-}$
$\displaystyle=\bar{C}_{-}\left(\begin{array}[]{cc}-D_{+}\eta_{2}^{-}f_{+}^{-}\\\
iA_{1}f_{-}^{-}\end{array}\right),$ (73)

where $\bar{C}_{\pm}$ is the normalization factor. The superscripts of
$f_{\pm}^{\pm}$ and $\eta_{1,2}^{\pm}$ stand for $E_{\pm}^{0}$ and the
subscripts of $f_{\pm}^{\pm}$ for parity, respectively. The expressions for
$f_{\pm}^{\pm}$ and $\eta_{1,2}^{\pm}$ are given by

$\displaystyle f_{+}^{\pm}(z)$
$\displaystyle=\left[\frac{\cosh(\lambda_{1}z)}{\cosh(\lambda_{1}L/2)}-\frac{\cosh(\lambda_{2}z)}{\cosh(\lambda_{2}L/2)}\right]_{E=E_{\pm}^{0}},$
(74) $\displaystyle f_{-}^{\pm}(z)$
$\displaystyle=\left[\frac{\sinh(\lambda_{1}z)}{\sinh(\lambda_{1}L/2)}-\frac{\sinh(\lambda_{2}z)}{\sinh(\lambda_{2}L/2)}\right]_{E=E_{\pm}^{0}},$
(75) $\displaystyle\eta_{1}^{\pm}$
$\displaystyle=\frac{\lambda_{1}^{2}-\lambda_{2}^{2}}{\lambda_{1}\coth(\lambda_{1}L/2)-\lambda_{2}\coth(\lambda_{2}L/2)}\bigg{|}_{E=E_{\pm}^{0}},$
(76) $\displaystyle\eta_{2}^{\pm}$
$\displaystyle=\frac{\lambda_{1}^{2}-\lambda_{2}^{2}}{\lambda_{1}\tanh(\lambda_{1}L/2)-\lambda_{2}\tanh(\lambda_{2}L/2)}\bigg{|}_{E=E_{\pm}^{0}}.$
(77)

###  VI.2 Effective Floquet Hamiltonian of the thin film

Figure 2: A thin film of Bi2Se3 pumped by a circularly polarized optical field
propagating to its surface. Here we suppose that $L\ll\lambda_{L}$ with the
thickness $L$ of the thin film and the wavelength $\lambda_{L}$ of the driving
optical field.

The energy spectra and wavefunctions of the lower block $h(A_{1})$ of $H_{0}$
can be obtained directly by replacing $A_{1}$ by $-A_{1}$. Based on the above
discussions, the four eigenstates of $H_{0}$ can be given by

$\displaystyle\Phi_{1}$
$\displaystyle=\left(\begin{array}[]{cc}\varphi(A_{1})\\\
0\end{array}\right),~{}~{}~{}~{}~{}~{}~{}\Phi_{2}=\left(\begin{array}[]{cc}\chi(A_{1})\\\
0\end{array}\right),$ (82) $\displaystyle\Phi_{3}$
$\displaystyle=\left(\begin{array}[]{cc}0\\\
\varphi(-A_{1})\end{array}\right),~{}~{}~{}~{}~{}\Phi_{4}=\left(\begin{array}[]{cc}0\\\
\chi(-A_{1})\end{array}\right),$ (87)

with $\Phi_{1}\rightarrow\Phi_{3}$ and $\Phi_{2}\rightarrow\Phi_{4}$ under the
time-reversal operation. We should emphasize that these four solutions are for
the surface states, and we use the four states as the basis states.

With the help of the four states, at the $\Gamma$ point, we can expand the
Hamiltonian Eq. (25) to obtain a new Floquet Hamiltonian of the thin film

$\displaystyle\\!H_{\text{film}}^{(F)}$
$\displaystyle\\!=\\!\int_{\\!-\\!L/2}^{L/2}\\!dz\\!\\{\Phi_{1},\Phi_{4},\Phi_{2},\Phi_{3}\\}^{\dagger}[\\!H^{(F)}\\!(\\!{\bf
k}\\!)\\!]\\{\Phi_{1},\Phi_{4},\Phi_{2},\Phi_{3}\\}$
$\displaystyle\\!=\\!\left(\begin{array}[]{cccc}h_{f1}(k_{\perp})&0\\\
0&h_{f2}(k_{\perp})\end{array}\right),$ (90)

where

$\displaystyle\\!h_{f1}(k_{\perp})$
$\displaystyle\\!=\\!\tilde{E}_{1}^{0}\\!-\\!Dk_{\perp}^{2}\\!+\\!\\!\left(\\!\begin{array}[]{cccc}\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}&i\gamma_{1}k_{-}\\\
-i\gamma_{1}k_{+}&-\frac{\tilde{\Delta}}{2}\\!+\\!Bk_{\perp}^{2}\\!-\\!\tilde{m}\end{array}\\!\right)\\!$
(93)
$\displaystyle\\!=\\!(\tilde{E}_{1}^{0}\\!-\\!Dk_{\perp}^{2})\sigma_{0}\\!+\\!\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}\right)\sigma_{z}$
$\displaystyle~{}\\!+\\!i\gamma_{1}k_{-}\sigma_{+}\\!-\\!i\gamma_{1}k_{+}\sigma_{-},$
(94) $\displaystyle\\!h_{f2}(k_{\perp})$
$\displaystyle\\!=\\!\tilde{E}_{2}^{0}\\!-\\!Dk_{\perp}^{2}\\!+\\!\\!\left(\\!\begin{array}[]{cccc}-\frac{\tilde{\Delta}}{2}\\!+\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}&i\gamma_{2}k_{-}\\\
-i\gamma_{2}k_{+}&\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!-\\!\tilde{m}\end{array}\\!\right)\\!$
(97)
$\displaystyle\\!=\\!(\tilde{E}_{2}^{0}\\!-\\!Dk_{\perp}^{2})\sigma_{0}\\!+\\!\left(-\frac{\tilde{\Delta}}{2}\\!+\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}\right)\sigma_{z}$
$\displaystyle~{}\\!+\\!i\gamma_{2}k_{-}\sigma_{+}\\!-\\!i\gamma_{2}k_{+}\sigma_{-},$
(98)

$\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$,
$\tilde{E}_{1}^{0}\\!=\\!E^{0}\\!-\\!A_{0}^{2}D\\!+\\!\frac{m_{1}\\!-\\!m_{3}}{2}$,
$\tilde{E}_{2}^{0}\\!=\\!E^{0}\\!-\\!A_{0}^{2}D\\!-\\!\frac{m_{1}\\!-\\!m_{3}}{2}$,
$\frac{\tilde{\Delta}}{2}\\!=\\!\frac{\Delta}{2}\\!-\\!A_{0}^{2}B$,
$\tilde{m}\\!=\\!\frac{m_{1}\\!+\\!m_{3}}{2}\\!-\\!A_{0}^{2}A_{2}^{2}\sin\varphi/(\hbar\omega)$,
$E^{0}\\!=\\!(E_{+}^{0}\\!+\\!E_{-}^{0})/2$,
$\Delta\\!=\\!E_{+}^{0}\\!-\\!E_{-}^{0}$,
$B\\!=\\!(\tilde{B}_{1}\\!-\\!\tilde{B}_{2})/2$,
$D\\!=\\!(\tilde{B}_{1}\\!+\\!\tilde{B}_{2})/2\\!-\\!D_{2}$,

$\tilde{B}_{1}\\!=\\!B_{2}|\bar{C}_{+}|^{2}\int_{-L/2}^{L/2}dz\left(D_{+}^{2}|\eta_{1}^{+}f_{-}^{+}|^{2}\\!-\\!A_{1}^{2}|f_{+}^{+}|^{2}\right),$
$\tilde{B}_{2}\\!=\\!B_{2}|\bar{C}_{-}|^{2}\int_{-L/2}^{L/2}dz\left(D_{+}^{2}|\eta_{2}^{-}f_{+}^{-}|^{2}\\!-\\!A_{1}^{2}|f_{-}^{-}|^{2}\right),$

$\gamma_{1}\\!=\\!\gamma\\!-\\!\gamma_{f}A_{0}^{2}\sin\varphi/(\hbar\omega)$,
$\gamma_{2}\\!=\\!\gamma\\!+\\!\gamma_{f}A_{0}^{2}\sin\varphi/(\hbar\omega)$,
$\gamma\\!=\\!A_{1}A_{2}\bar{C}_{+}^{*}\bar{C}_{-}D_{+}\int_{-L/2}^{L/2}dz\left(\eta_{1}^{+}f_{-}^{+}f_{-}^{-}\\!+\\!\eta_{2}^{-}f_{+}^{+}f_{+}^{-}\right)$,
$\gamma_{f}=2A_{1}A_{2}B_{2}\bar{C}_{+}^{*}\bar{C}_{-}D_{+}\int_{-L/2}^{L/2}dz\left(\eta_{1}^{+}f_{-}^{+}f_{-}^{-}-\eta_{2}^{-}f_{+}^{-}f_{+}^{+}\right)$,
$m_{1}\\!=\\!m_{0}|\bar{C}_{+}|^{2}\int_{-L/2}^{L/2}dz~{}\left(D_{+}^{2}|\eta_{1}^{+}f_{-}^{+}|^{2}+A_{1}^{2}|f_{+}^{+}|^{2}\right)\\!=\\!m_{0}$,
and
$m_{3}\\!=\\!m_{0}|\bar{C}_{-}|^{2}\int_{-L/2}^{L/2}dz~{}\left(D_{+}^{2}|\eta_{2}^{-}f_{+}^{-}|^{2}\\!+\\!A_{1}^{2}|f_{-}^{-}|^{2}\right)\\!=\\!m_{0}$.

From the time-dependent Hamiltonian (14), it is found that the circularly
polarized light is acting as an effective “magnetic field”. Furthermore, from
the Floquet thin-film Hamiltonian (VI.2), we find that the “effective field”
is causing kind of a Zeeman effect with the term
$\tilde{m}\tau_{0}\otimes\sigma_{z}$ in (VI.2) and an effective Zeeman field
$\tilde{m}\\!=\\!m_{0}\\!-\\!A_{0}^{2}A_{2}^{2}\sin\varphi/(\hbar\omega)$.

The dispersions are

$\displaystyle E_{f1\pm}$
$\displaystyle\\!=\\!\tilde{E}_{1}^{0}\\!-\\!Dk_{\perp}^{2}\\!\pm\\!\sqrt{\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}\right)^{2}\\!+\\!\gamma_{1}^{2}k_{\perp}^{2}},$
(99) $\displaystyle E_{f2\pm}$
$\displaystyle\\!=\\!\tilde{E}_{2}^{0}\\!-\\!Dk_{\perp}^{2}\\!\pm\\!\sqrt{\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!-\\!\tilde{m}\right)^{2}\\!+\\!\gamma_{2}^{2}k_{\perp}^{2}}.$
(100)

At the $\Gamma$ point with $k_{\perp}=0$, the surface gap
$\Delta_{\text{sur}}$ is defined as the minimum energy gap between the
conduction and valence bands, i.e.,

$\displaystyle\Delta_{\text{sur}}\\!=\\!\text{Min}\\!\left[\\!2\bigg{|}\frac{\tilde{\Delta}}{2}\\!+\\!\tilde{m}\bigg{|},2\bigg{|}\frac{\tilde{\Delta}}{2}\\!-\\!\tilde{m}\bigg{|},\left(\\!\bigg{|}\frac{\tilde{\Delta}}{2}\\!+\\!\tilde{m}\bigg{|}\\!+\\!\bigg{|}\frac{\tilde{\Delta}}{2}\\!-\\!\tilde{m}\bigg{|}\\!\right)\\!\right]\\!,$
(101)

where we use $m_{1}=m_{3}=m_{0}$. Further, the wavefunctions of the two
valence bands $E_{f1-}$ and $E_{f2-}$ are found to be

$\displaystyle\psi_{f1-}\\!=\\!G_{f1}\\!\left(\\!\begin{array}[]{cccc}b_{f1}(k_{\perp})\\\
-i\gamma_{1}k_{+}\\\ 0\\\
0\end{array}\\!\right),~{}\psi_{f2-}\\!=\\!G_{f2}\\!\left(\\!\begin{array}[]{cccc}0\\\
0\\\ b_{f2}(k_{\perp})\\\ i\gamma_{2}k_{+}\end{array}\\!\right)\\!,$ (110)

where

$\displaystyle\\!b_{f1}(\\!k_{\perp}\\!)\\!$
$\displaystyle\\!=\\!\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}\\!-\\!\sqrt{\left(\\!\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!+\\!\tilde{m}\\!\right)^{2}\\!+\\!\gamma_{1}^{2}k_{\perp}^{2}},$
(111) $\displaystyle\\!b_{f2}(\\!k_{\perp}\\!)\\!$
$\displaystyle\\!=\\!\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!-\\!\tilde{m}\\!+\\!\sqrt{\left(\\!\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\\!-\\!\tilde{m}\\!\right)^{2}\\!+\\!\gamma_{2}^{2}k_{\perp}^{2}},$
(112) $\displaystyle\\!G_{f1}$
$\displaystyle\\!=\\!\frac{1}{\sqrt{b_{f1}^{2}(k_{\perp})\\!+\\!\gamma_{1}^{2}k_{\perp}^{2}}},~{}\\!G_{f2}\\!=\\!\frac{1}{\sqrt{b_{f2}^{2}(k_{\perp})\\!+\\!\gamma_{2}^{2}k_{\perp}^{2}}}.$
(113)

###  VI.3 Chern number of a thin film with high-frequency pumping

In principle, we can find the Hall conductance for each
$h_{f1/f2}(k_{\perp})$. Note that $h_{f1/f2}(k_{\perp})$ in Eq. (VI.2) can be
explicitly written as

$\displaystyle h_{f1}(k_{\perp})$
$\displaystyle\\!=\\!h_{f+}(k_{\perp})\\!=\\!\tilde{E}_{1}^{0}\\!-\\!Dk_{\perp}^{2}\\!+\\!\sum_{i=x,y,z}d_{i}\sigma_{i},$
(114) $\displaystyle h_{f2}(k_{\perp})$
$\displaystyle\\!=\\!h_{f-}(k_{\perp})\\!=\\!\tilde{E}_{2}^{0}\\!-\\!Dk_{\perp}^{2}\\!+\\!\sum_{i=x,y,z}d_{i}\sigma_{i},$
(115)

where the subscripts are $f1=f+$, $f2=f-$, $\sigma_{i}$ are the Pauli matrices
and the $\textbf{d}(k_{\perp})$ vectors are

$\displaystyle d_{x}$
$\displaystyle\\!=\\!\\!-\\!\left(\\!-\\!\gamma\\!\pm\\!\frac{A_{0}^{2}}{\hbar\omega}\gamma_{f}\sin\varphi\right)k_{y},$
(116) $\displaystyle d_{y}$
$\displaystyle\\!=\\!\left(\\!-\\!\gamma\pm\frac{A_{0}^{2}}{\hbar\omega}\gamma_{f}\sin\varphi\right)k_{x},$
(117) $\displaystyle d_{z}$
$\displaystyle\\!=\\!\pm\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk_{\perp}^{2}\right)\\!+\\!\tilde{m}.$
(118)

For the $2\times 2$ Hamiltonian in terms of the $\textbf{d}(k_{\perp})$
vectors and Pauli matrices, the Kubo formula for the Hall conductance can be
generally expressed as Lu2010 ; Qi2006

$\displaystyle\sigma_{xy}=\frac{e^{2}}{2\hbar}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{(f_{k,c}-f_{k,\nu})}{d^{3}}\epsilon_{\alpha\beta\gamma}\frac{\partial
d_{\alpha}}{\partial k_{x}}\frac{\partial d_{\beta}}{\partial
k_{y}}d_{\gamma},$ (119)

where we use $\textbf{k}=(k_{x},k_{y})$, $k=k_{\perp}$,
$\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita anti-symmetric tensor,
$d=\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}$ is the norm of $(d_{x},d_{y},d_{z})$,
$\hbar=h/(2\pi)$ is the reduced Planck’s constant, $-e$ is the electron
charge, $f_{k,c/\nu}=1/\\{\exp[(\varepsilon_{c/\nu}(k)-\mu)/(k_{B}T)]+1\\}$ is
the Fermi distribution function of the conduction ($c$) and valence ($\nu$)
bands with the chemical potential $\mu$, the Boltzmann constant $k_{B}$, and
the temperature $T$ .

At zero temperature and when the chemical potential $\mu$ lies between
$(-|\Delta_{\text{sur}}|/2,|\Delta_{\text{sur}}|/2)$, the Fermi functions
reduce to $f_{k,c}=0$ and $f_{k,\nu}=1$. By substituting Eqs. (116)-(118) into
(119), we arrive at

$\displaystyle\\!\sigma_{xy}^{(\pm)}$
$\displaystyle\\!=\\!\\!-\\!\left(\\!\pm\frac{e^{2}}{2h}\\!\right)$
$\displaystyle\\!\times\\!\int_{0}^{\infty}\\!\frac{\left(\\!-\\!\gamma\\!\pm\\!\frac{A_{0}^{2}}{\hbar\omega}\gamma_{f}\sin\varphi\\!\right)^{2}\\!\left(\\!\frac{\tilde{\Delta}}{2}\\!+\\!Bk^{2}\pm\tilde{m}\\!\right)\\!kdk}{\left[\\!\left(\\!\\!-\\!\gamma\pm\frac{A_{0}^{2}}{\hbar\omega}\gamma_{f}\sin\varphi\\!\right)^{2}\\!k^{2}\\!+\\!\left(\\!\frac{\tilde{\Delta}}{2}\\!-\\!Bk^{2}\\!\pm\\!\tilde{m}\\!\right)^{2}\\!\right]^{3/2}},$
(120)

where $+$ ($-$) corresponds to pseudo-spin-$\uparrow$ (pseudo-
spin-$\downarrow$) or $h_{f+}(k_{\perp})$ ($h_{f-}(k_{\perp})$). By defining

$\displaystyle\\!\cos\theta\\!=\\!\frac{\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk^{2}\\!\pm\\!\tilde{m}\right)}{\\!\left[\\!\left(\\!-\\!\gamma\\!\pm\\!\frac{A_{0}^{2}}{\hbar\omega}\gamma_{f}\sin\varphi\right)^{2}k^{2}\\!+\\!\left(\frac{\tilde{\Delta}}{2}\\!-\\!Bk^{2}\\!\pm\\!\tilde{m}\\!\right)^{2}\\!\right]^{1/2}\\!},$
(121)

Eq. (VI.3) can be transformed into

$\displaystyle\sigma_{xy}^{(\pm)}=\pm\frac{e^{2}}{2h}\int_{0}^{\infty}dk^{2}\frac{\partial\cos\theta}{\partial
k^{2}}.$ (122)

The value of $\cos\theta$ at $k=0$ and $k\rightarrow\infty$ only depends on
the signs of $\frac{\tilde{\Delta}}{2}\pm\tilde{m}$ and $B$, respectively. As
a result, in the insulating regime
$-|\Delta_{\text{sur}}|/2\leqslant\mu\leqslant|\Delta_{\text{sur}}|/2$, we
find that the anomalous Hall conductance for each hyperbola has the form

$\displaystyle\sigma_{xy}^{(\pm)}=-\left(\pm\frac{e^{2}}{2h}\right)[\text{sgn}(\Delta_{\pm})+\text{sgn}(B)],$
(123)

where $\Delta_{\pm}=\Delta_{c}\pm\tilde{m}$,
$\Delta_{c}=\frac{\tilde{\Delta}}{2}=\frac{\Delta}{2}-A_{0}^{2}B$, and
$\tilde{m}=\frac{m_{1}+m_{3}}{2}-A_{0}^{2}A_{2}^{2}\sin\varphi/(\hbar\omega)$.
In particular, the information of the interplay between the circularly
polarized light and the bulk magnetic moment can be extracted from the
analytical expression for the anomalous Hall conductance Eq. (123). The
competition between the polarized light and the bulk magnetic moment can
change the first sign function [$\text{sgn}(\Delta_{\pm})$] in Eq. (123). This
interplay has significant impact on the anomalous Hall conductance or the
topological phase diagram of Cr-doped Bi2Se3 thin film as shown in Fig. 3(b).

Therefore, the spin Chern numbers of the valence bands are given by

$\displaystyle
C_{\pm}=-\left(\pm\frac{1}{2}\right)[\text{sgn}(\Delta_{\pm})+\text{sgn}(B)].$
(124)

Further, the total Chern number is given by

$\displaystyle C=C_{+}+C_{-}.$ (125)

Since a net magnetic moment should also break time-reversal symmetry, it is
important that the excitation decreases the magnetic moment as an effective
magnetic moment
$\tilde{m}\\!=\\!m_{0}\\!-\\!A_{0}^{2}A_{2}^{2}\sin\varphi/(\hbar\omega)$,
where we choose $\varphi=\pi/2$. Therefore, the effective magnetic moment
decreases with the increasing of the light intensity $A_{0}$.

##  VII Phase diagram

Figure 3: The total Chern number as functions of the thin-film thickness and
the light intensity. (a) Topological phase diagram for the non-doped Bi2Se3
with $m_{0}=0$. (b) Topological phase diagram for the Cr-doped Bi2Se3 with
$m_{0}=30$ meV. The other parameters are given as $\varphi=\pi/2$ and
$\hbar\omega\approx 150$ meV ($\omega\sim 2.2789\times 10^{2}$ THz).

As shown in Fig. 3, we calculate the total Chern number as functions of the
thin-film thickness and the light intensity for Bi2Se3. For the non-doped
Bi2Se3 with $m_{0}=0$ as shown in Fig. 3(a), by modulating the strength of the
polarized optical field, there are three different regions: the NI phase with
the spin Chern numbers $C_{\pm}=0$, the QAHI phase with the spin Chern numbers
$C_{+}=1$ and $C_{-}=0$, and the QSHI phase with the spin Chern numbers
$C_{\pm}=\pm 1$. In the absent of optical field with $A_{0}=0$, there are only
two phases: NI phase and QSHI phase. The result reveals that the circularly
polarized light breaks the time-reversal symmetry and induces the QAHI phase.
Moreover, the light-induced time-reversal-symmetry-broken QSHI phase is
different from the time-reversal-symmetry QSHI phase in the absent of optical
field.

For the Cr-doped Bi2Se3 with $m_{0}=30$ meV, it is indicated from Fig. 3(b)
that there are four different regions: the NI phase, the time-reversal-
symmetry-broken QSHI phase, and two different QAHI phases with opposite
nonzero Chern numbers. Here the interplay between the light and bulk magnetic
moment can separate two different QAHI phases with opposite Chern numbers. For
a given layer thickness $L=3.2$ nm, one can apparently detect the tendency
towards light-induced band inversion upon increasing optical field intensity
and passing through the QSHI phase region marked by gray. In particular, the
gray regions between the fixed points are the QSHI phases with the Chern
number $C=0$. With increasing the thickness $L$, the gray regions will vanish.

Figure 4: Dispersions of the thin film for the Cr-doped Bi2Se3 under open
boundary condition along $y$ direction and periodic boundary conditions along
$x$ direction. (a) Bands for the Floquet Hamiltonian. (b) Bands for the
pseudo-spin-$\uparrow$ (“+”) Floquet Hamiltonian. (c) Bands for the pseudo-
spin-$\downarrow$ (“-”) Floquet Hamiltonian. The other parameters are given as
$A_{0}=0.036$ Å-1, $L=3$ nm, $a_{x}=a_{y}=1$ Å, $N_{y}=200$, $m_{0}=30$ meV,
$\varphi=\pi/2$, and $\hbar\omega\approx 0.15$ eV ($\omega\sim 2.2789\times
10^{2}$ THz) for chromium-doped Bi2Se3. Here, $a_{x}$ and $a_{y}$ are the
lattice constants along $x$ and $y$ directions respectively.

Furthermore, we calculate the energy dispersions of the thin film for the Cr-
doped Bi2Se3 with $A_{0}=0.036$ Å-1 and $L=3$ nm under open boundary condition
along $y$ direction and periodic boundary conditions along $x$ direction (the
detail derivations can be found in the Appendix B). As shown in Fig. 4, it is
found that there exist the edge states in the thin film as shown in Fig. 4(a),
and the edge states come from the pseudo-spin-$\uparrow$ (“$+$”) Floquet
Hamiltonian in the chosen parameters here as shown in Fig. 4(b). However,
there is no edge states in the energy spectra of the pseudo-spin-$\downarrow$
(“$-$”) Floquet Hamiltonian as shown in Fig. 4(c).

##  VIII Probation of the topological transitions

Figure 5:  (a) QAHI with $C_{-}=0$ and $C_{+}=1$ which corresponds to pseudo-
spin-$\uparrow$ (“$+$”) current on the side edges. Here, the arrow denotes the
direction of the current. (b) QAHI with $C_{+}=0$ and $C_{-}=-1$ which
corresponds to pseudo-spin-$\downarrow$ (“$-$”) current on the side edges. (c)
QSHI has opposite currents with different pseudo-spins on the side edges. (d)
NI with no pseudo-spin currents. (e) The proposed device for nonlocal edge
measurements with the electrodes (green cuboids). (f) Schematics of the
experimental setup with the electrodes (green rectangles). Here, $n_{L}$ and
$n_{R}$ are the coefficients of the transmission probability matrix, and the
number of the clockwise and anti-clockwise pseudo-spin currents are determined
by $n_{L}$ and $n_{R}$. (g)-(i) Electric-circuit diagrams with a bias voltage
leads on electrodes 1 and 4, and the currents which leads on electrodes 2, 3,
5, and 6 are set to zero. In particular, the electric-circuit diagram (i) is
used to measure the Hall resistance.

Physically, as shown in Fig. 5(a)-(d), the quantum anomalous Hall insulator
with the Chern number $C=1$ has a pseudo-spin-$\uparrow$ (“$+$”) current on
the side edges XueExp2013 ; Qi2008 ; Yu2010 ; Chu2011 ; Qiao2010 ; Qi2006 ;
Onoda2006 ; Nomura2011 ; Checkelsky2012 ; Hor2010 ; Chen2010 ; Wray2011 ;
Wang2013 ; Wang2014 ; Law2017 ; Kim2018 ; Kawamura2018 , the quantum anomalous
Hall insulator with the Chern number $C=-1$ has a pseudo-spin-$\downarrow$
(“$-$”) current on the side edges, and the quantum spin Hall insulator has
antiparallel currents with different pseudo-spins on the side edges Roth2009 ;
Chen2021prb ; Piskunow2021prl . Here, each current is responsible for the
$e^{2}/h$ Hall conductance as shown in Eq. (123). However, the normal
insulator has no pseudo-spin currents which correspond to a zero Hall
conductance.

Detecting the pseudo-spin current can help to identify the topological
transitions. An experimental set up to measure the pseudo-spin current is
given in Fig. 5(d): The side surface of the device has six electrodes as 1-6
(green cuboids). Further, one needs to measure the nonlocal edge transport
Roth2009 ; Chen2021prb ; Piskunow2021prl ; Chen2012prb , i.e., the nonlocal
resistances. As shown in Fig. 5(e) and (f), a current $I_{ij}$ is applied
between electrodes $i$ and $j$ on the side surface, and a nonlocal voltage
$V_{ij}$ between other two electrodes $i^{\prime}$ and $j^{\prime}$ is
measured to define a nonlocal resistance
$R_{ij,i^{\prime}j^{\prime}}=V_{i^{\prime}j^{\prime}}/I_{ij}$ Roth2009 ;
Chen2021prb ; Piskunow2021prl ; Chen2012prb ; Datta1997book .

Within the Landauer-Büttiker-Fisher-Lee formalism Landauer1970pm ;
Buttiker1988prb ; Buttiker1986prl ; Fisher1981prb , the current $I_{i}$
flowing out of the $i$th electrode into the sample region is expressed as
Roth2009 ; Chen2021prb ; Piskunow2021prl ; Chen2012prb ; Datta1997book

$\displaystyle I_{i}=\frac{e^{2}}{h}\sum_{j,j\neq
i}T_{ij}(E_{F})(V_{i}-V_{j}),$ (126)

where $V_{i}$ is the voltage on the $i$th electrode and $T_{ij}(E_{F})$ is the
transmission probability matrix from the $i$th to the $j$th electrode at Fermi
energy $E_{F}$. Notice that, in a time-reversal-invariant system, the
transmission coefficients satisfy the condition $T_{ij}=T_{ji}$ Roth2009 . For
the present case here, there are six electrodes on the side surface, so
$T_{ij}$ is a $6\times 6$ matrix. Concretely, Eq. (126) can be written in a
matrix form

$\displaystyle\begin{pmatrix}I_{1}\\\ I_{2}\\\ I_{3}\\\ I_{4}\\\ I_{5}\\\
I_{6}\end{pmatrix}=\frac{e^{2}}{h}\begin{pmatrix}\sum_{j=1}^{6}T_{1j}&-T_{12}&-T_{13}&-T_{14}&-T_{15}&-T_{16}\\\
-T_{21}&\sum_{j=1}^{6}T_{2j}&-T_{23}&-T_{24}&-T_{25}&-T_{26}\\\
-T_{31}&-T_{32}&\sum_{j=1}^{6}T_{3j}&-T_{34}&-T_{35}&-T_{36}\\\
-T_{41}&-T_{42}&-T_{43}&\sum_{j=1}^{6}T_{4j}&-T_{45}&-T_{46}\\\
-T_{51}&-T_{52}&-T_{53}&-T_{54}&\sum_{j=2}^{6}T_{5j}&-T_{56}\\\
-T_{61}&-T_{62}&-T_{63}&-T_{64}&-T_{65}&\sum_{j=1}^{6}T_{6j}\end{pmatrix}\begin{pmatrix}V_{1}\\\
V_{2}\\\ V_{3}\\\ V_{4}\\\ V_{5}\\\ V_{6}\end{pmatrix},$ (127)

where $T_{11}=T_{22}=T_{33}=T_{44}=T_{55}=T_{66}=0$. If the six electrodes are
identical, we have $T_{j+1,j}=T_{1,6}=n_{R}$ and $T_{j,j+1}=T_{6,1}=n_{L}$
with $j=1,2,\cdots,5$; $T_{j+2,j}=T_{2,6}=T_{1,5}$ and
$T_{j,j+2}=T_{6,2}=T_{5,1}$ with $j=1,2,3,4$; $T_{j+3,j}=T_{3,6}$ and
$T_{j,j+3}=T_{6,3}$, i.e., $T_{j+3,j}=T_{j,j+3}$ with $j=1,2,3$. For the non-
neighboring terminals, we set that $T_{ij}=0$. Therefore, the corresponding
transmission matrix has the form

$\displaystyle T=$
$\displaystyle\\!\begin{pmatrix}n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0&0&0&\\!-\\!n_{R}\\\
\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0&0&0\\\
0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0&0\\\
0&0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0\\\
0&0&0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}\\\
\\!-\\!n_{L}&0&0&0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}\end{pmatrix}.\\!$ (128)

As shown in Fig. 5(g)-(i), we can apply a bias voltage leads on electrodes 1
and 4, and the currents which leads on electrodes 2, 3, 5, and 6 are set to
zero. Owing to the current conservation, one finds that $I_{1}=-I_{4}\equiv
I_{14}$. We set $V_{5}=0$, which allows us to truncate the fifth row and the
fifth column of the matrix and write

$\displaystyle\begin{pmatrix}V_{1}\\\ V_{2}\\\ V_{3}\\\ V_{4}\\\
V_{6}\end{pmatrix}=\frac{h}{e^{2}}T_{d}^{-1}\begin{pmatrix}I_{14}\\\ 0\\\ 0\\\
-I_{14}\\\ 0\end{pmatrix},$ (129)

where

$\displaystyle T_{d}=$
$\displaystyle\\!\begin{pmatrix}n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0&0&\\!-\\!n_{R}\\\
\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0&0\\\
0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&\\!-\\!n_{L}&0\\\
0&0&\\!-\\!n_{R}&n_{L}\\!+\\!n_{R}&0\\\
\\!-\\!n_{L}&0&0&0&n_{L}\\!+\\!n_{R}\end{pmatrix}\\!.$ (130)

Then the voltages of the six terminals are given by Eq. (129) as

$\displaystyle V_{1}$
$\displaystyle=\frac{hI_{14}}{e^{2}}\frac{n_{R}}{n_{L}^{2}+n_{R}^{2}-n_{L}n_{R}},$
(131) $\displaystyle V_{2}$
$\displaystyle=\frac{hI_{14}}{e^{2}}\frac{n_{R}^{2}+n_{L}n_{R}-n_{L}^{2}}{n_{L}^{3}+n_{R}^{3}},$
(132) $\displaystyle V_{3}$
$\displaystyle=\frac{hI_{14}}{e^{2}}\frac{n_{R}-n_{L}}{n_{L}^{2}+n_{R}^{2}-n_{L}n_{R}},$
(133) $\displaystyle V_{4}$
$\displaystyle=-\frac{hI_{14}}{e^{2}}\frac{n_{L}^{2}}{n_{L}^{3}+n_{R}^{3}},$
(134) $\displaystyle V_{5}$ $\displaystyle=0,$ (135) $\displaystyle V_{6}$
$\displaystyle=\frac{hI_{14}}{e^{2}}\frac{n_{L}n_{R}}{n_{L}^{3}+n_{R}^{3}},$
(136)

where $n_{L}$ and $n_{R}$ are the coefficients of the transmission probability
matrix. They determine the number of the clockwise and anti-clockwise pseudo-
spin currents as shown in the Fig. 5(f). For the QAHI with $C=1$, we have
$n_{L}=1$ and $n_{R}=0$. For the QAHI with $C=-1$, we have $n_{L}=0$ and
$n_{R}=1$. For the QSHI, we have $n_{L}=n_{R}=1$. For the NI, we have
$n_{L}=n_{R}=0$. As shown in Fig. 5(g)-(i), the nonlocal resistances can be
analytically found as

$\displaystyle R_{14,14}$
$\displaystyle=\frac{V_{1}-V_{4}}{I_{14}}=\frac{h}{e^{2}}\frac{n_{L}^{2}+n_{R}^{2}+n_{R}n_{L}}{n_{L}^{3}+n_{R}^{3}},$
(137) $\displaystyle R_{14,23}$
$\displaystyle=\frac{V_{2}-V_{3}}{I_{14}}=\frac{h}{e^{2}}\frac{n_{L}n_{R}}{n_{L}^{3}+n_{R}^{3}},$
(138) $\displaystyle R_{14,26}$
$\displaystyle=\frac{V_{2}-V_{6}}{I_{14}}=\frac{h}{e^{2}}\frac{n_{R}-n_{L}}{n_{L}^{2}+n_{R}^{2}-n_{L}n_{R}}.$
(139)

For the QAHI with $C=1$, we have $R_{14,14}=\frac{h}{e^{2}}$, $R_{14,23}=0$,
and $R_{14,26}=-\frac{h}{e^{2}}$. For the QAHI with $C=-1$, we have
$R_{14,14}=\frac{h}{e^{2}}$, $R_{14,23}=0$, and $R_{14,26}=\frac{h}{e^{2}}$.
For the QSHI, we obtain $R_{14,14}=\frac{3h}{2e^{2}}$,
$R_{14,23}=\frac{h}{2e^{2}}$, and $R_{14,26}=0$. For the NI, we have
$R_{14,14}=R_{14,23}=R_{14,26}=0$.

##  IX Summary

We investigate the impact of high-frequency pumping on a ferromagnetic
topological insulator with the circularly polarized optical field. It is found
that the intensity of the circularly polarized light can be used as a knob to
drive a topological transition. With modulating the strength of the polarized
optical field in an experimentally accessible range, there are four different
regions: the NI phase, the time-reversal-symmetry-broken QSHI phase, and two
different QAHI phases. This is different from the situation in the absent of
optical field.

We propose an experimental scheme to manipulate the topological phases in Cr-
doped Bi2Se3 with high-frequency pumping light. Our proposal can be realized
in an experimentally accessible range. Particularly, to realize the light
driven topological phases, the frequency and intensity of the light are both
within the experimental accessibility lightexp2013 ; lightexp2016 . In most of
the recent experiments XueExp2013 ; CheckelskyExp2014 ; QAHIexp2014 ;
QAHIexp2015 ; XueExp2015 ; YoshimiExp2015 ; ChangExp2016 ; GrauerExp2017 ;
XueExp2018 ; ZhangExp2020 ; QAHIexp2020 ; ZhaoExp2020 ; MogiExp2021 , people
focus on using the magnetic fields to manipulate the topological phases of the
system. However, the topological phases obtained in this way may be confused
with the quantum Hall effect. Luckily, our proposal avoids this. Therefore,
the theoretically investigations we put forward will be helpful to the future
experiments.

Note Added. When finishing this paper, we became aware of another preprint,
Ref. Liu2021arxive , which also did the Floquet engineering of magnetism in
topological insulator thin films.

###### Acknowledgements.

Fang Qin and Rui Chen contribute equally to this work. We acknowledge helpful
discussions with Dong-Hui Xu, Hai-Peng Sun, Xiao-Bin Qiang, De-Huan Cai, and
Hui Xiong. This work was supported by the National Natural Science Foundation
of China (Grants No. 11534001, No. 11925402, and No. 11404106), the Strategic
Priority Research Program of Chinese Academy of Sciences (Grant No.
XDB28000000), Guangdong province (Grants No. 2016ZT06D348 and No.
2020KCXTD001), the National Key R & D Program (Grant No. 2016YFA0301700),
Shenzhen High-level Special Fund (Grants No. G02206304 and No. G02206404), and
the Science, Technology and Innovation Commission of Shenzhen Municipality
(No. ZDSYS20190902092905285, No. ZDSYS20170303165926217, No.
JCYJ20170412152620376, and No. KYTDPT20181011104202253). F.Q. acknowledges
support from the project funded by the China Postdoctoral Science Foundation
(Grant No. 2019M662150 and No. 2020T130635) and the SUSTech Presidential
Postdoctoral Fellowship. R.C. acknowledges support from the project funded by
the China Postdoctoral Science Foundation (Grant No. 2019M661678) and the
SUSTech Presidential Postdoctoral Fellowship.

##  Appendix A Derivations of Eq. (67)

###  A.1 Semi-infinite boundary conditions

The surface states have a finite distribution near the boundary. For a film
thick enough that the states at opposite surfaces barely couple to each other,
we can focus on just one surface. Without loss of generality, we study a
system from $z=0$ to $+\infty$. The boundary condition is given as

$\displaystyle\Psi_{\uparrow}(z=0)=0,~{}~{}~{}~{}~{}~{}~{}\Psi_{\uparrow}(z=+\infty)=0.$
(140)

The condition of $\Psi_{\uparrow}(z=+\infty)=0$ requires that
$\Psi_{\uparrow}$ contains only the four terms in which $\beta$ is negative
and that the real part of $\lambda_{\alpha}$ be positive.

With Eq. (V), we have

$\displaystyle\Psi_{\uparrow}(z)$
$\displaystyle=C_{1-}\left(\begin{array}[]{cc}D_{+}\lambda_{1}^{2}-L_{-}+E_{0}\\\
iA_{1}\lambda_{1}\end{array}\right)e^{-\lambda_{1}z}$ (143)
$\displaystyle~{}~{}+C_{2-}\left(\begin{array}[]{cc}D_{+}\lambda_{2}^{2}-L_{-}+E_{0}\\\
iA_{1}\lambda_{2}\end{array}\right)e^{-\lambda_{2}z}.$ (146)

Applying the boundary conditions of equation (140) to the general solution of
equation (143), the secular equation of the nontrivial solution to the
coefficients $C_{\alpha\beta}$ leads to

$\displaystyle(E_{0}\\!-\\!L_{-})(C_{1-}\\!+\\!C_{2-})+D_{+}(C_{1-}\lambda_{1}^{2}\\!+\\!C_{2-}\lambda_{2}^{2})$
$\displaystyle\\!=\\!0,$ (147) $\displaystyle
C_{1-}\lambda_{1}\\!+\\!C_{2-}\lambda_{2}$ $\displaystyle\\!=\\!0.$ (148)

Substituting $C_{2-}=-C_{1-}\lambda_{1}/\lambda_{2}$ into Eq. (147), we have

$\displaystyle E_{0}-L_{-}$
$\displaystyle=D_{+}\lambda_{1}\lambda_{2}=\frac{\sqrt{F^{2}-R}}{2D_{-}}$
$\displaystyle=\frac{\sqrt{D_{+}D_{-}(E_{0}-L_{+})(E_{0}-L_{-})}}{D_{-}}.$
(149)

Therefore, we have

$\displaystyle E_{\pm}^{0}$ $\displaystyle\equiv
E_{0}=\frac{D_{+}L_{+}-D_{-}L_{-}}{D_{+}-D_{-}}=\frac{D_{1}M_{0}+B_{1}C_{0}}{B_{1}}$
$\displaystyle=C_{0}+\frac{D_{1}M_{0}}{B_{1}}.$ (150)

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

In a lattice, one makes the following replacements Shen2012

$\displaystyle k_{j}\rightarrow\frac{1}{a_{j}}\sin(k_{j}a_{j}),$ (151)
$\displaystyle k_{j}^{2}\rightarrow\frac{2}{a_{j}^{2}}[1-\cos(k_{j}a_{j})],$
(152)

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

$\displaystyle
k_{\perp}^{2}=k_{x}^{2}+k_{y}^{2}\rightarrow\frac{2}{a_{x}^{2}}[1-\cos(k_{x}a_{x})]+\frac{2}{a_{y}^{2}}[1-\cos(k_{y}a_{y})].$
(153)

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 for topological thin film with high-frequency pumping in the
basis $(c_{{\bf k},+,\uparrow},c_{{\bf k},-,\uparrow},c_{{\bf
k},+,\downarrow},c_{{\bf k},-,\downarrow})^{T}$ with ${\bf k}=(k_{x},k_{y})$
as

$\displaystyle\\!H_{\text{film}}^{(F)}\\!=\\!\left(\begin{array}[]{cccc}h_{f1}(k_{x},k_{y})&0\\\
0&h_{f2}(k_{x},k_{y})\end{array}\right),$ (156)

where

$\displaystyle~{}~{}\\!h_{f1}(k_{x},k_{y})$
$\displaystyle\\!=\\!\left(\tilde{E}_{1}^{0}\\!-\\!D\left\\{\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\right)\sigma_{0}$
$\displaystyle\\!+\\!\left(\frac{\tilde{\Delta}}{2}\\!-\\!B\left\\{\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\\!+\\!\tilde{m}\right)\sigma_{z}$
$\displaystyle\\!+\\!i\gamma_{1}\left(\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right)\sigma_{+}$
$\displaystyle\\!-\\!i\gamma_{1}\left(\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right)\sigma_{-},$
(157) $\displaystyle~{}~{}\\!h_{f2}(k_{x},k_{y})$
$\displaystyle\\!=\\!\left(\tilde{E}_{2}^{0}\\!-\\!D\left\\{\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\right)\sigma_{0}$
$\displaystyle\\!+\\!\left(-\frac{\tilde{\Delta}}{2}\\!+\\!B\left\\{\frac{2}{a_{x}^{2}}[1\\!-\\!\cos(k_{x}a_{x})]\\!+\\!\frac{2}{a_{y}^{2}}[1\\!-\\!\cos(k_{y}a_{y})]\right\\}\\!+\\!\tilde{m}\right)\sigma_{z}$
$\displaystyle\\!+\\!i\gamma_{2}\left(\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!-\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right)\sigma_{+}$
$\displaystyle\\!-\\!i\gamma_{2}\left(\frac{1}{a_{x}}\sin(k_{x}a_{x})\\!+\\!\frac{i}{a_{y}}\sin(k_{y}a_{y})\right)\sigma_{-}.$
(158)

Performing the Fourier transformation, one obtains

$\displaystyle c_{{\bf k},s,\sigma}$
$\displaystyle=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{-ik_{y}j_{y}a_{y}}c_{k_{x},j_{y},s,\sigma},$
(159) $\displaystyle c_{{\bf k},s,\sigma}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{ik_{y}j_{y}a_{y}}c_{k_{x},j_{y},s,\sigma}^{\dagger}.$
(160) $\displaystyle C_{{\bf k},\sigma}$ $\displaystyle=\begin{pmatrix}c_{{\bf
k},+,\sigma}\\\ c_{{\bf
k},-,\sigma}\end{pmatrix}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{-ik_{y}j_{y}a_{y}}C_{k_{x},j_{y},\sigma},$
(161) $\displaystyle C_{{\bf k},\sigma}^{\dagger}$
$\displaystyle=\begin{pmatrix}c_{{\bf k},+,\sigma}^{\dagger}&c_{{\bf
k},-,\sigma}^{\dagger}\end{pmatrix}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{x},j_{y}}^{N_{y}}e^{ik_{y}j_{y}a_{y}}C_{k_{x},j_{y},\sigma}^{\dagger}.$
(162)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\uparrow}^{\dagger}\left[h_{f1}(k_{x},k_{y})\right]C_{{\bf k},\uparrow}$
$\displaystyle\\!=\\!\sum_{k_{x},j_{y}}\\!\left[\\!\tilde{E}_{1}^{0}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y},\uparrow}$
$\displaystyle\\!+\\!\frac{D}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{k_{x},j_{y}}\\!\left[\\!\frac{\tilde{\Delta}}{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\tilde{m}\\!\right]\\!$
$\displaystyle\\!\times\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y},\uparrow}$
$\displaystyle\\!+\\!\frac{B}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\\!\sigma_{z}\\!C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\frac{i\gamma_{1}}{a_{x}}\sum_{k_{x},j_{y}}\sin(k_{x}a_{x})\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}-\sigma_{-})C_{k_{x},j_{y},\uparrow}\\!\right]\\!$
$\displaystyle\\!-\\!\frac{\gamma_{1}}{2a_{y}}\sum_{k_{x},j_{y}}\\!\left[\\!iC_{k_{x},j_{y},\uparrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{k_{x},j_{y}+1,\uparrow}\\!+\\!{\rm
h.c.}\\!\right].$ (163)

$\displaystyle~{}~{}\sum_{{\bf k},s}C_{{\bf
k},\downarrow}^{\dagger}\left[h_{f2}(k_{x},k_{y})\right]C_{{\bf
k},\downarrow}$
$\displaystyle\\!=\\!\sum_{k_{x},j_{y}}\\!\left[\\!\tilde{E}_{2}^{0}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{x}^{2}}\cos(k_{x}a_{x})\\!\right]\\!C_{k_{x},j_{y},\downarrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y},\downarrow}$
$\displaystyle\\!+\\!\frac{D}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\uparrow}^{\dagger}\sigma_{0}C_{k_{x},j_{y}+1,\downarrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\sum_{k_{x},j_{y}}\\!\left[\\!-\\!\frac{\tilde{\Delta}}{2}\\!+\\!\frac{2B}{a_{x}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\\!-\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\tilde{m}\\!\right]\\!$
$\displaystyle\\!\times\\!C_{k_{x},j_{y},\downarrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y},\downarrow}$
$\displaystyle\\!-\\!\frac{B}{a_{y}^{2}}\sum_{k_{x},j_{y}}\\!\left[\\!C_{k_{x},j_{y},\downarrow}^{\dagger}\sigma_{z}C_{k_{x},j_{y}+1,\downarrow}\\!+\\!{\rm
h.c.}\\!\right]\\!$
$\displaystyle\\!+\\!\frac{i\gamma_{2}}{a_{x}}\sum_{k_{x},j_{y}}\sin(k_{x}a_{x})\\!\left[\\!C_{k_{x},j_{y},\downarrow}^{\dagger}(\sigma_{+}-\sigma_{-})C_{k_{x},j_{y},\downarrow}\\!\right]\\!$
$\displaystyle\\!-\\!\frac{\gamma_{2}}{2a_{y}}\sum_{k_{x},j_{y}}\\!\left[\\!iC_{k_{x},j_{y},\downarrow}^{\dagger}(\sigma_{+}+\sigma_{-})C_{k_{x},j_{y}+1,\downarrow}\\!+\\!{\rm
h.c.}\\!\right].$ (164)

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

$\displaystyle H_{y-OBC}=\begin{pmatrix}h&T&0&\cdots&0\\\
T^{\dagger}&h&T&\cdots&0\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&0&T^{\dagger}&h&T\\\ 0&0&0&\ddots&\ddots\end{pmatrix}_{4N_{y}\times
4N_{y}},$ (165)

where

$\displaystyle
h\\!=\\!\begin{pmatrix}\\!\bar{E}_{1}^{0}\sigma_{0}\\!+\\!\bar{\Delta}_{1}\sigma_{z}\\!+\\!i\bar{\gamma}_{1}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!)&0\\\
0&\bar{E}_{2}^{0}\sigma_{0}\\!+\\!\bar{\Delta}_{2}\sigma_{z}\\!+\\!i\bar{\gamma}_{2}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!)\\!\end{pmatrix},$
(166)
$\displaystyle\bar{E}_{1}^{0}=\tilde{E}_{1}^{0}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{x}^{2}}\cos(k_{x}a_{x}),$
(167)
$\displaystyle\bar{E}_{2}^{0}=\tilde{E}_{2}^{0}\\!-\\!\frac{2D}{a_{x}^{2}}\\!-\\!\frac{2D}{a_{y}^{2}}\\!+\\!\frac{2D}{a_{x}^{2}}\cos(k_{x}a_{x}),$
(168)
$\displaystyle\bar{\Delta}_{1}=\frac{\tilde{\Delta}}{2}\\!-\\!\frac{2B}{a_{x}^{2}}\\!-\\!\frac{2B}{a_{y}^{2}}\\!+\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\tilde{m},$
(169)
$\displaystyle\bar{\Delta}_{2}=-\frac{\tilde{\Delta}}{2}\\!+\\!\frac{2B}{a_{x}^{2}}\\!+\\!\frac{2B}{a_{y}^{2}}\\!-\\!\frac{2B}{a_{x}^{2}}\cos(k_{x}a_{x})\\!+\\!\tilde{m},$
(170)
$\displaystyle\bar{\gamma}_{1}=\frac{\gamma_{1}}{a_{x}}\sin(k_{x}a_{x}),$
(171)
$\displaystyle\bar{\gamma}_{2}=\frac{\gamma_{2}}{a_{x}}\sin(k_{x}a_{x}),$
(172) $\displaystyle
T\\!=\\!\begin{pmatrix}\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\frac{i\gamma_{1}}{2a_{y}}\sigma_{x}&0\\\
0&\frac{D}{a_{y}^{2}}\sigma_{0}\\!-\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\frac{i\gamma_{2}}{2a_{y}}\sigma_{x}\\!\end{pmatrix}\\!,$
(173) $\displaystyle
T^{\dagger}\\!=\\!\begin{pmatrix}\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\frac{i\gamma_{1}}{2a_{y}}\sigma_{x}&0\\\
0&\frac{D}{a_{y}^{2}}\sigma_{0}\\!-\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\frac{i\gamma_{2}}{2a_{y}}\sigma_{x}\\!\end{pmatrix}\\!,$
(174)

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

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

$\displaystyle
H_{y-OBC}^{\uparrow}=\begin{pmatrix}h_{\uparrow}&T_{\uparrow}&0&\cdots&0\\\
T^{\dagger}_{\uparrow}&h_{\uparrow}&T_{\uparrow}&\cdots&0\\\
0&\ddots&\ddots&\ddots&0\\\
\vdots&0&T^{\dagger}_{\uparrow}&h_{\uparrow}&T_{\uparrow}\\\
0&0&0&\ddots&\ddots\end{pmatrix}_{2N_{y}\times 2N_{y}},$ (175)

where

$\displaystyle
h_{\uparrow}\\!=\\!\bar{E}_{1}^{0}\sigma_{0}\\!+\\!\bar{\Delta}_{1}\sigma_{z}\\!+\\!i\bar{\gamma}_{1}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!),$
(176) $\displaystyle
T_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\frac{i\gamma_{1}}{2a_{y}}\sigma_{x},$
(177) $\displaystyle
T^{\dagger}_{\uparrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!+\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\frac{i\gamma_{1}}{2a_{y}}\sigma_{x}.$
(178)

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

$\displaystyle
H_{y-OBC}^{\downarrow}=\begin{pmatrix}h_{\downarrow}&T_{\downarrow}&0&\cdots&0\\\
T^{\dagger}_{\downarrow}&h_{\downarrow}&T_{\downarrow}&\cdots&0\\\
0&\ddots&\ddots&\ddots&0\\\
\vdots&0&T^{\dagger}_{\downarrow}&h_{\downarrow}&T_{\downarrow}\\\
0&0&0&\ddots&\ddots\end{pmatrix}_{2N_{y}\times 2N_{y}},$ (179)

where

$\displaystyle
h_{\downarrow}\\!=\\!\bar{E}_{2}^{0}\sigma_{0}\\!+\\!\bar{\Delta}_{2}\sigma_{z}\\!+\\!i\bar{\gamma}_{2}(\\!\sigma_{+}\\!-\\!\sigma_{-}\\!),$
(180) $\displaystyle
T_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!-\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!-\\!\frac{i\gamma_{2}}{2a_{y}}\sigma_{x},$
(181) $\displaystyle
T^{\dagger}_{\downarrow}\\!=\\!\frac{D}{a_{y}^{2}}\sigma_{0}\\!-\\!\frac{B}{a_{y}^{2}}\sigma_{z}\\!+\\!\frac{i\gamma_{2}}{2a_{y}}\sigma_{x}.$
(182)

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is well-defined, definite, and nondegenerate.

###### Proof.

To simplify notation throughout this proof, we identify the spaces $W_{r}$ and
$\\{0\\}\times W_{r}$ and write

$\displaystyle\widetilde{W}:=T_{[1,0,0]}j(W_{r})\subseteq
T_{[1,0,0]}Z\qquad\text{and}\qquad\widetilde{U}:=\mathrm{d}\Psi\big{(}\widetilde{W}\big{)}\subset
T_{m}M.$

We begin with three observations. First, since the map $\Psi$ pulls back the
momentum map on $M$ to the one on $Z$, we get that

$\displaystyle\mathrm{d}\Psi\big{(}\ker(\mathrm{d}\Phi_{Z})_{[1,0,0]}\big{)}=\ker\mathrm{d}\Phi_{m}.$
(5.4)

Second, from the $G$-equivariance of the diffeomorphism $\Psi$ we get that

$\displaystyle\mathrm{d}\Psi\big{(}T_{[1,0,0]}(G\cdot[1,0,0])\big{)}=T_{m}(G\cdot
m).$ (5.5)

Third, note that the linear subspace $\widetilde{W}$ is $K$-invariant under
the $K$-representation it inherits from the action of $G$ on the manifold $M$.
Also recall that $W_{r}$ is a neighborhood in the symplectic slice $W$. Thus,
in particular, the linear subspace $\widetilde{W}$ is such that

$\displaystyle\ker(\mathrm{d}\Phi_{Z})_{[1,0,0]}=T_{[1,0,0]}(G\cdot[1,0,0])\oplus\widetilde{W}.$
(5.6)

Using (5.4), (5.5), and (5.6) we get that $\widetilde{U}$ is a $K$-invariant
subspace of $T_{m}M$ such that

$\displaystyle\ker\mathrm{d}\Phi_{m}=T_{m}(G\cdot m)\oplus\widetilde{U}.$
(5.7)

Using (5.7), the assumptions on the space $U$, and Lemma 5.2, we obtain that
the Hessian $\mathrm{d}^{2}h^{\xi}(m)|_{\widetilde{U}}$ is definite and
nondegenerate.

Now observe that the augmented Hamiltonian $h^{\xi}_{Z}$ of the pullback
$h_{Z}$ is the pullback of the augmented Hamiltonian $h^{\xi}$:

$\displaystyle
h^{\xi}_{Z}=h_{Z}-\langle\Phi_{Z},\xi\rangle=\Psi^{*}h-\langle\Psi^{*}\Phi,\xi\rangle=\Psi^{*}h^{\xi}.$

Thus, since the point $m$ is a critical point of the augmented Hamiltonian
$h^{\xi}$ we have that

$\displaystyle\mathrm{d}h^{\xi}_{Z}([1,0,0])=\mathrm{d}\big{(}\Psi^{*}h^{\xi}\big{)}([1,0,0])=\Psi^{*}\big{(}\mathrm{d}h^{\xi}\big{)}(m)=\Psi^{*}(0)=0.$

Hence, the Hessian $\mathrm{d}^{2}h^{\xi}_{Z}([1,0,0])$ is well-defined.

Finally, by the pullback properties of Hessians (see Remark 4.7), we have that

$\displaystyle\mathrm{d}^{2}h^{\xi}_{Z}([1,0,0])|_{\widetilde{W}}=\mathrm{d}^{2}\big{(}\Psi^{*}h^{\xi}\big{)}([1,0,0])|_{\widetilde{W}}=\Psi^{*}\big{(}\mathrm{d}^{2}h^{\xi}(m)\big{)}|_{\widetilde{U}}.$

Since the map $\Psi$ is a diffeomorphism, this implies that the Hessian
$\mathrm{d}^{2}h^{\xi}_{Z}([1,0,0])|_{\widetilde{W}}$ is definite and
nondegenerate because the Hessian $\mathrm{d}^{2}h^{\xi}(m)|_{\widetilde{U}}$
is definite and nondegenerate. ∎

### Acknowledgements

The author would like to thank Eugene Lerman for providing guidance throughout
this project, for his enduring patience with my many questions, and for the
many interesting conversations on the subject. The author would also like to
express their gratitude to the anonymous referee for their helpful comments
and careful review of the preprint of this paper.

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