# Floquet-Weyl semimetals generated by an optically resonant interband-
transition

Runnan Zhang  Doctoral Program in Materials Science, Graduate School of Pure
and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Ken-ichi Hino<EMAIL_ADDRESS>Division of Materials Science, Faculty
of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8573, Japan
Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577,
Japan  Nobuya Maeshima  Center for Computational Sciences, University of
Tsukuba, Tsukuba 305-8577, Japan  Division of Materials Science, Faculty of
Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8573, Japan

###### Abstract

Floquet-Weyl semimetals (FWSMs) generated by irradiation of a continuous-wave
laser with left-hand circular polarization (rotating in counterclockwise sense
with time) on the group II-V narrow gap semiconductor Zn3As2 are theoretically
investigated, where the frequency of the laser is set nearly resonant with a
band gap of the crystal. It is found that the excitation of the crystal by
such a laser induce two types of FWSM phases that differ absolutely in
characters. To be specific, the associated two pairs of Weyl points are stably
formed by band touching between Floquet sidebands ascribable to a valence band
labeled as $J_{z}=\pm 3/2$ and a conduction band labeled as $J_{z}=\pm 1/2$,
where $J_{z}$ represents the $z$-component of total angular momentum quantum
number of $\Gamma$-point and a double sign corresponds. Here, one FWSM state
composed of the up-spin Floquet sidebands relevant to $J_{z}=3/2$ and $1/2$
shows almost quadratic band-touching in the vicinity of the associated pair of
Weyl points, while the other FWSM state composed of the down-spin Floquet
sidebands relevant to $J_{z}=-3/2$ and $-1/2$ shows linear band-touching.
Further, it is revealed that both up-spin and down-spin sidebands host
nontrivial two-dimensional surface states that are pinned to the respective
pairs of the Weyl points. Both surface states also show different energy
dispersions and physical properties. More detailed discussion is made in the
text on the origin of the above findings, chirality of the FWSM phases,
alteration of topological order, laser-induced magnetic properties, and so on.

††preprint: APS/123-QED

##  I Introduction

Topological materials have been studied for more than a decade Kane2005 ;
Bernevig2006 ; Hasan2010 ; Qi2011  and growing interest has been directed
toward the exploration of a class of topological semimetals (SMs)
Murakami2007a ; Murakami2007b ; Wang2012 ; Wang2013 ; Young2012 ; Young2015 ;
Young2015 ; Yang2014 ; Yi2014 ; Xu2015 ; Kargarian2016 ; Park2017 ; Doh2017 ;
Yan2017 ; Armitage2018 ; Yang2018 ; Ramankutty2018 ; Luo2020  — such as Weyl
SMs (WSMs), Dirac SMs (DSMs), and nodal-line SMs (NLSMs) — in addition with
further deepening of the studies of prototypical topological insulators, for
instance, on non-Hermitian topological systems,Kunst2018 ; Yao2018
topological photonics,Ozawa2019  higher-order topological insulators,
Benalcazar2017 ; Langbehn2017 ; Peng2017  topological excitons, Budich2014 ;
Entina2016 ; Chen2017  topological surface plasmon polaritons, Li2014 ;
Deshko2016  and so on. WSMs and DSMs are three dimensional (3D) gapless phases
of materials, in which bands cross linearly at points protected by topology
and symmetry. There is close connection with the chiral anomaly (the Adler-
Bell-Jackiw anomaly) in linearly dispersing fermionic excitations in particle
physics, which gives rise to the nonconservation of an axial current even for
a massless particle. Weyl1929 ; Majorana1937 ; Gynther2011 ; Elliott2015 ;
Adler1969 ; Bell1969 ; Nielsen1983  In accord with the prediction of the
chiral anomaly, large negative magnetic resistance is observed in
magnetotransport in WSMs. Son2013 ; Burkov2013 ; Burkov2014 ; Xiong2015
Further, these SMs exhibit a great number of novel transport properties such
as ultra-high mobility, titanic magnetic resistance, and anomalous Hall
conductivity. Wang2013 ; Jeon2014 ; Feng2015 ; Liang2015 ; Wang2016 ; Son2013
; Arnold2016 ; Huang2015 ; Ali2014 ; Shekhar2015 ; Yan2017  In NLSMs, bands
cross along special lines in the Brillouin zone (BZ) in the shape of a closed
ring or a line. Breaking of either time- reversal (T-) symmetry or spatial-
inversion (I-) symmetry leads DSMs and NLSMs to WSMs.Yang2018  In the
T-breaking WSMs, there are a pair of Weyl nodes with opposite chirality, on
which a surface state is pinned with a characteristic Fermi arc, while the
number of Weyl nodes in the I-breaking WSMs is a multiple of four.
Murakami2007b ; Armitage2018

An interaction of topological SMs with a continuous-wave laser provides the
studies of topological materials with another avenue from the perspective of
the quantum control of underlying topological properties by means of built-in
laser parameters — intensity, frequency $\omega$, and polarization — and the
exploration of topological phases that are in non-equilibrium. Wang2014 ;
Taguchi2016a ; Taguchi2016b ; Hubener2017 ; Zou2016 ; Zhang2016 ; Chen2018 ;
Kumar2019 ; Zhu2020 ; Salerno2020 ; Gao2020 ; Kawaguchi2020 ; Tokman2020 ;
Zhang2021 ; Liang2021 ; Ma2015 ; Juan2017 ; Ghorashi2018 ; Umer2021a ;
Umer2021b  Here, the total Hamiltonian $H(t)$ of concern at time $t$ has
temporal periodicity $H(t)=H(t+T)$ to ensure the Floquet theorem with
$T=2\pi/\omega$. Shirley1965  By the drive with a circularly polarized laser —
in place of the application of a static intrinsic Zeeman field —, the
T-symmetry in DSMs and NLSMs is broken to form WSMs, and these are termed as
Floquet WSMs (FWSMs). This scenario for creating FWSMs is applied to the DSMs
of alkali pnictides NaBi3,Hubener2017  type-II and hybrid LNSMs,Chen2018  and
3D stacked graphene systems.Zou2016  In particular, it is reported that the
drive of a 3D Luttinger semimetal by an elliptically polarized light leads to
the coexistence of WSM phases with double and single Weyl points, which can be
tuned to be type-I or type-II. Ghorashi2018  Further, by virtue of Floquet
engineering due to periodical driving of hybrid multi-WSM phases, the number
of various isolated band touching points can be increased on-demand by tuning
system parameters, where intricate Fermi arc structures are hosted. Umer2021a
In addition, NLSMs are driven to result in FWSMs, revealing a photovoltaic
anomalous Hall effect associated with the Weyl point nodes.Taguchi2016b  Very
recently, frequency-independent magnetization mechanisms in response to
circularly polarized light are studied in WSMs. Gao2020 ; Tokman2020

As regards the I-symmetry, this is also broken by the introduction of an
interaction of electron with the continuous-wave laser. However, the time-
glide I-symmetry holds correct instead to realize the same invariance in
$H(t)$ as the I-symmetry.Zhang2021 ; Morimoto2017  The symmetry operation
associated with this symmetry is represented by the operation of putting time
$t$ ahead by a half period $T/2$, followed by the I-operation.

In most of theoretical studies of Floquet topological materials, Kitagawa2010
; Oka2009 ; Zhenghao2011 ; Lindner2011 ; Rudner2013 ; Rechtsman2013 ;
Claassen2017 ; Kitamura2017 ; Hansen2017 ; Nakagawa2020  an electron-light
interaction is introduced exclusively by employing the Peierls phase transform
— given by the replacement of a Bloch momentum $\bm{k}$ by $\bm{k}+\bm{A}(t)$
— under the off-resonant condition that $\omega\gg E_{g}$. Here, $\bm{A}(t)$
and $E_{g}$ represent a vector potential of the laser at $t$, and a bandgap of
the concerned material, respectively, and the atomic units are used. Further,
the effective Floquet Hamiltonian is constructed by relying on the Floquet-
Magnus expansion with respect to $E_{g}/\omega$.Mananga2011 ; Mananga2016 ;
Haga2019  It is remarked that the convergence of this expansion is not ensured
at the resonant limit of $\omega=E_{g}$.Haga2019  The resulting Floquet bands
are likely modified from original bands before laser irradiation. It is
assumed that under the above off-resonant condition, effects of interband
electric-dipole transitions between a valence band and a conduction band are
negligibly small, compared with those of the above-mentioned Peierls
interaction. Wang2014 ; Hubener2017 ; Hansen2017  Here, interband and
intraband couplings caused by the electron-light interaction due to the
Peierls phase transform are termed as the Peierls interaction to make a
distinction from the interband electric-dipole interaction.

The aim of this paper is to create a FWSM phase by driving Zn3As2 LinChung1969
; Okamoto1992 ; Sieranski1994 ; Botha1999  belonging to the group II-V
semiconductors with a circularly polarized laser which meets an almost on-
resonant condition $\omega\approx E_{g}$,Lindner2011 ; Zhang2021  and to
explore the properties of surface states hosted by this FWSM. It is remarked
that this material is topologically trivial without band inversion. This
differs from a DSM of Cd3As2,LinChung1969 ; Sieranski1994 ; Neupane2014 ;
Liu2014 ; Crassee2018  realized by the band-inversion mechanism due to the
presence of a $n(>2)$-fold uniaxial rotational symmetry along a symmetry line,
hosting edge modes known as double Fermi arcs at the surfaces. Wang2012 ;
Wang2013 ; Yang2014 ; Kargarian2016 ; Armitage2018  The creation of the FWSM
is governed by an almost resonant transition due to the interband electric-
dipole interaction rather than that due to the Peierls interaction, as shown
in more detail later (Sec. III). This is the key issue of this paper.

Below, a more detailed explanation is made on this key issue based on the
four-band model of the semiconductor Zn3As2, where the valence and conduction
bands are labeled as $J_{z}=\pm 3/2$ and $\pm 1/2$, respectively, with $J_{z}$
as the $z$-component of total angular momentum quantum number at the
$\Gamma$-point ($\bm{k}=\bm{0}$). First, given the driving laser with a left-
hand circular polarization (rotating in counterclockwise sense with time), the
coupling of this light with an electric dipole moment induced by the
transition between the down-spin bands with $J_{z}=-3/2$ and $-1/2$ is
dominant over that between the up-spin bands with $J_{z}=3/2$ and
$1/2$.Allen2003  This is maximized when the on-resonant condition is met. It
is remarked that the roles of the up-spin and down-spin bands are exchanged
for a laser with a right-hand circular polarization. A left-hand polarization
is favored throughout this paper unless otherwise stated. Second, as the laser
intensity increases, the ac-Stark effect gives rise to larger energy splitting
of each down-spin band into two leaves with conspicuous modification of the
band profile, Zhang2021 ; Autler1955 ; Knight1980 ; Sie2015  whereas the up-
spin bands are just slightly affected. The ac-Stark effect is also maximized
by the on-resonant condition. Third, the present resonant interband-transition
yields real carrier excitation, differing from virtual carrier excitation due
to the off-resonant one. Thus, it is likely that orbital magnetization results
from the inverse Faraday effect that is a non-linear optical process caused by
a circularly polarized laser field. Gao2020 ; Kawaguchi2020 ; Tokman2020 ;
Liang2021 ; Pershan1966 ; Kimel2005 ; Hertel2006 ; Zhang2009 ; Battiato2014
Consequently, it is expected that the laser drive with a circular polarization
in the almost on-resonant condition provides intriguing physics with FWSMs,
which is sharply distinct from the conventional off-resonant laser
drive.Wang2014 ; Hubener2017 ; Hansen2017

The remainder of this paper is organized as follows. Section II describes the
theoretical framework, Sec. III presents the results and discussion, and Sec.
IV presents the conclusion. Further, three appendices are included. Hereafter,
the atomic units (a.u.) are used throughout, unless otherwise stated.

##  II Theory

###  II.1 Effective Hamiltonian

The crystal of concern, Zn3As2, is a narrow gap semiconductor, the structure
of which is very similar to that of Cd3As2, though in the latter, a band is
inverted to result in a DSM. LinChung1969 ; Sieranski1994 ; Neupane2014 ;
Liu2014 ; Crassee2018  There are many equilibrium phases of Zn3As2 depending
on pressure and temperature, for instance, $\alpha$Zn3As2 (with a body-
centered tetragonal structure I4${}_{1}/$cd) and $\alpha^{\prime}$Zn3As2 (with
a tetragonal structure P4${}_{2}/$nbc). Okamoto1992  Here, one employs the
structure of $\alpha^{\prime}$Zn3As2 having the C4-rotational symmetry along
the $\Gamma-$Z axis in the BZ for constructing an effective Hamiltonian. The
low-energy electronic properties of it are mostly determined by the conduction
band composed of Zn 4$s$-orbitals and the valence band composed of As
4$p$-orbitals.

Here, an effective electronic Hamiltonian for Zn3As2 is constructed by
following the Kane model used in Ref. Wang2013,  for the crystal structure of
Cd3As2 with a tetragonal structure P4${}_{2}/$nbc. To be specific, one
considers the following four states as conduction $s$-states
$|\Gamma_{6},J_{z}=\pm 1/2\rangle$ and heavy-hole $p$-states
$|\Gamma_{7},J_{z}=\pm 3/2\rangle$, and light-hole states and split-off states
are disregarded because of relatively large energy separation from these four
states at the $\Gamma$-point. The effective Hamiltonian is read as the
$4\times 4$-matrix: Kane1957 ; Luttinger1956

$\mathcal{H}(\bm{k})=c(\bm{k})I+\sum_{j=3}^{5}d_{j}(\bm{k})\gamma_{j}$ (1)

with $\bm{k}=(k_{x},k_{y},k_{z})$ as a 3D Bloch momentum. Here, $\gamma_{j}$’s
represent the four-dimensional Dirac matrices for the Clifford algebra,
defined by
$\gamma_{1}=\tau_{x}\otimes\sigma_{x},\>\gamma_{2}=\tau_{x}\otimes\sigma_{y},\>\gamma_{3}=\tau_{x}\otimes\sigma_{z},\>\gamma_{4}=\tau_{z}\otimes
I_{2}$, and $\gamma_{5}=\tau_{y}\otimes I_{2}$, where $I$ and $I_{2}$
represent the $4\times 4$ and $2\times 2$ unit matrices, respectively,
$\tau_{l}$ and $\sigma_{l}$ with $l=x,\>y,\>z$ represent the Pauli matrices
for orbital and spin degrees of freedom, respectively, and the anti-
commutation relation,
$\\{\gamma_{j},\gamma_{j^{\prime}}\\}=2\delta_{jj^{\prime}}$, is ensured.
According to the above definition of $\gamma_{j}$’s, it is understood that the
states of
$|\Gamma_{6},J_{z}=1/2\rangle,\>|\Gamma_{7},J_{z}=3/2\rangle,\>|\Gamma_{6},J_{z}=-1/2\rangle$,
and $|\Gamma_{7},J_{z}=-3/2\rangle$ are labeled as $1,2,3$, and 4,
respectively, for the matrix elements of $\mathcal{H}(\bm{k})$, namely,
$\\{\mathcal{H}_{mn}(\bm{k})\\}$ with $m,n=1\sim 4$. Moreover,
$d_{j}(\bm{k})$’s are given by

$\displaystyle\left\\{\begin{array}[]{l}d_{3}(\bm{k})=t_{sp}\sin{(k_{x}d_{x})}\\\
d_{4}(\bm{k})=\Delta_{g}+\sum_{l=x,y,z}\epsilon_{l}(k_{l})\\\
d_{5}(\bm{k})=t_{sp}\sin{(k_{y}d_{y})}\end{array}\right.,$ (5)

where

$\epsilon_{l}(k_{l})=-2t^{xy}\\{1-\cos{(k_{l}d_{l})}\\}$ (6)

for $l=x,y$, and

$\epsilon_{z}(k_{z})=-2t^{z}\\{1-\cos{(k_{z}d_{z})}\\}.$ (7)

Here, $t^{l}$ represents a hopping matrix between identical bands in the
$l$-direction with $l=x,y,z$, where $t^{l}<0$, $t^{xy}\equiv t^{x}=t^{y}$, and
$t_{sp}$ represents a hopping matrix between different bands due to a spin-
orbit coupling. Further, $d_{l}$ represents a lattice constant in the
$l$-direction, and the bandgap at the $\Gamma$-point $E_{g}$ is given by
$E_{g}=E_{\Gamma_{6}}-E_{\Gamma_{7}}$ with $\Delta_{g}=E_{g}/2$, where the
conduction and valence band energies at $\Gamma_{6}$ and $\Gamma_{7}$ are
represented as $E_{\Gamma_{6}}$ and $E_{\Gamma_{7}}$, respectively. An
additional energy $c(\bm{k})$ is given by

$c(\bm{k})=E_{F}+2\sum_{l=x,y,z}m^{l}\\{1-\cos{(k_{l}d_{l})}\\}$ (8)

with $m^{l}$’s constants, and the Fermi energy $E_{F}$ is set equal to zero:
$E_{F}=0$. Hence, $E_{\Gamma_{6}}=c(\bm{0})+d_{4}(\bm{0})$ and
$E_{\Gamma_{7}}=c(\bm{0})-d_{4}(\bm{0})$.

It is assumed that the off-diagonal block matrices of $\mathcal{H}(\bm{k})$
have little contributions to the band structure under the present tetragonal
symmetry, that is, $d_{1}(\bm{k}),d_{2}(\bm{k})\approx 0$, leading to
$[I_{2}\otimes\sigma_{z},\mathcal{H}(\bm{k})]\approx 0$.comment2  Thus,
$\mathcal{H}(\bm{k})$ is cast into the block-diagonal form

$\mathcal{H}(\bm{k})=\left(\begin{array}[]{cc}h(\bm{k})&0\\\
0&h^{*}(-\bm{k})\end{array}\right),$ (9)

where
$h(\bm{k})=d_{3}(\bm{k})\tau_{x}+d_{4}(\bm{k})\tau_{z}+d_{5}(\bm{k})\tau_{y}.$

An interaction of electron with light is introduced into $\mathcal{H}(\bm{k})$
by replacing $\bm{k}$ by $\bm{K}(t)=\bm{k}+\bm{A}(t)$, followed by adding to
$\mathcal{H}(\bm{K}(t))$ an interband electric dipole interaction represented
by $\mathcal{H}^{\prime}(t).$ comment1  Here, the replacement by $\bm{K}(t)$
results from the Peierls phase transform in the lattice representation of the
effective Hamiltonian, and the optical interaction arising from this
replacement is herein termed as the Peierls interaction, as mentioned in Sec.
I. Further, the interband electric-dipole interaction is provided as
$\mathcal{H}^{\prime}(t)=\bm{F}(t)\cdot\bm{M}$, where $\bm{M}$ represents a
matrix of electric-dipole transition between $|\Gamma_{6},J_{z}=\pm
1/2\rangle$ and $|\Gamma_{7},J_{z}=\pm 3/2\rangle$, independent of $\bm{k}$; a
double sign corresponds. The vector potential is given by

$\bm{A}(t)=\left(-{F_{x}\over\omega}\sin\omega t,{F_{y}\over\omega}\cos\omega
t,0\right)$ (10)

with $F_{x}$ and $F_{y}$ constants, and in view of
$\bm{F}(t)=-\dot{\bm{A}}(t)$, the associated electric field becomes

$\bm{F}(t)=\left(F_{x}\cos\omega t,F_{y}\sin\omega t,0\right).$ (11)

The laser is linearly polarized in the $x$-direction when $F_{x}\not=0$ and
$F_{y}=0$, while left-hand circularly polarized in the $x-y$ plane when
$F_{x}$ is set equal to $F_{y}$, namely, $F_{c}\equiv F_{x}=F_{y}$. The time-
dependent effective Hamiltonian of the driven semiconductor is thus read as
comment1

$H(\bm{k},t)=C(\bm{k},t)I+\sum_{j=3}^{5}D_{j}(\bm{k},t)\gamma_{j}+\mathcal{H}^{\prime}(t),$
(12)

where $C(\bm{k},t)\equiv c(\bm{K}(t))$ and $D_{j}(\bm{k},t)\equiv
d_{j}(\bm{K}(t))$.

Obviously, this Hamiltonian ensures the temporal periodicity,
$H(\bm{k},t+T)=H(\bm{k},t)$ with $T=2\pi/\omega$, and the system of concern
follows the Floquet theorem.Shirley1965  Thus, the present time-dependent
problem ends up with the following Floquet eigenvalue problem as:

$L(\bm{k},t)\Psi_{\alpha}(t)=E_{\alpha}(\bm{k})\Psi_{\alpha}(t),$ (13)

where

$L(\bm{k},t)=H(\bm{k},t)-iI{\partial\over\partial t},$ (14)

$E_{\alpha}(\bm{k})$ represents the $\alpha$th eigenvalue termed a quasienergy
or a Floquet energy, and $\Psi_{\alpha}(t)$ represents the associated
eigenvector ensuring the temporal periodicity,
$\Psi_{\alpha}(t+T)=\Psi_{\alpha}(t)$. In actual calculations, a set of
$E_{\alpha}(\bm{k})$’s are obtained by numerically solving Eq. (13) in the
$\omega$-domain, where the Floquet matrix $L(\bm{k},t)$ is recast into a
Fourier-Floquet matrix element $\tilde{L}_{nn^{\prime}}(\bm{k},\omega)$ with
respect to $n$ and $n^{\prime}$ photon states.Shirley1965  This is read as

$\displaystyle\tilde{L}_{nn^{\prime}}(\bm{k},\omega)$ $\displaystyle=$
$\displaystyle\tilde{C}_{nn^{\prime}}(\bm{k},\omega)I+\sum_{j=3}^{5}\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)\gamma_{j}$
(15) $\displaystyle+\tilde{\mathcal{H}^{\prime}}_{nn^{\prime}}(\omega)+n\omega
I\delta_{nn^{\prime}},$

where it is understood that the Fourier transform of matrix $X(t)$ is
represented by

$\tilde{X}_{nn^{\prime}}(\omega)={1\over
T}\int^{T}_{0}dt\>e^{-i(n-n^{\prime})\omega t}X(t).$ (16)

In addition, it is remarked that in fact,
$\tilde{C}_{nn^{\prime}}(\bm{k},\omega)$ is less dependent on the set of
photon numbers, $n$ and $n^{\prime}$, and almost identical to $c(\bm{k})$.
Hence, hereafter, it is understood that $E_{\alpha}(\bm{k})$’s are reckoned
from $c(\bm{k})$; in other words, the effect of $c(\bm{k})$ on
$E_{\alpha}(\bm{k})$’s is neglected. Explicit expressions of
$\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)\;\;(j=3\sim 5)$ are given in
Appendix A.

###  II.2 Electric dipole couplings

It is convenient to describe an explicit form of $\mathcal{H}^{\prime}(t)$.
This is provided as

$\mathcal{H}^{\prime}(t)=(\Omega_{y}\sin{\omega t})\tau_{x}\otimes
I_{2}+(\Omega_{x}\cos{\omega t})\tau_{y}\otimes\sigma_{z},$ (17)

where $\Omega_{x}=F_{x}P/\sqrt{2}$ and $\Omega_{y}=F_{y}P/\sqrt{2}$. Here, $P$
is a dipole matrix element given by $P=\langle S|x|X\rangle=\langle
S|y|Y\rangle$, where $x$ and $y$ represent the $x$ and $y$ components of
electron position $\bm{r}$, respectively, and the states of
$|\Gamma_{6},J_{z}=\pm 1/2\rangle$ and $|\Gamma_{7},J_{z}=\pm 3/2\rangle$ are
represented by $|\Gamma_{6},J_{z}=\pm 1/2\rangle=i|S\rangle$ and
$|\Gamma_{7},J_{z}=\pm 3/2\rangle=\pm(1/\sqrt{2})|X\pm iY\rangle$,
respectively, in terms of $s,p_{x}$, and $p_{y}$ states denoted as
$|S\rangle,|X\rangle$, and $|Y\rangle$, respectively. Similar to Eq. (9),
$\mathcal{H}^{\prime}(t)$ is block-diagonalized as

$\mathcal{H}^{\prime}(t)=\left(\begin{array}[]{cc}V^{(+)}(t)&0\\\
0&V^{(-)}(t)\end{array}\right),$ (18)

where $V^{(+)}(t)$ and $V^{(-)}(t)$ represent the electric dipole couplings
between the up-spin bands, $|\Gamma_{6},J_{z}=1/2\rangle$ and
$|\Gamma_{7},J_{z}=3/2\rangle$, and the down-spin bands,
$|\Gamma_{6},J_{z}=-1/2\rangle$ and $|\Gamma_{7},J_{z}=-3/2\rangle$,
respectively, given by

$V^{(\pm)}(t)=(\Omega_{y}\sin{\omega t})\tau_{x}\pm(\Omega_{x}\cos{\omega
t})\tau_{y},$ (19)

where a double sigh corresponds. The Fourier transform of $V^{(\pm)}(t)$ into
the $\omega$-domein is given in Appendic A. In view of Eqs. (12) and (18),
$H(\bm{k},t)$ is cast into the block-diagonalized form:

$H(\bm{k},t)=C(\bm{k},t)I+\left(\begin{array}[]{cc}H^{(+)}(\bm{k},t)&0\\\
0&H^{(-)}(\bm{k},t)\end{array}\right),$ (20)

that is, $[I_{2}\otimes\sigma_{z},H(\bm{k},t)]=0$. Here, the up-spin
Hamiltonian $H^{(+)}(\bm{k},t)$ and the down-spin Hamiltonian
$H^{(-)}(\bm{k},t)$ are given by

$\displaystyle H^{(\pm)}(\bm{k},t)$ $\displaystyle=$ $\displaystyle\left[\pm
D_{3}(\bm{k},t)+\Omega_{y}\sin{\omega
t}\right]\tau_{x}+D_{4}(\bm{k},t)\tau_{z}$ (21)
$\displaystyle+\left[D_{5}(\bm{k},t)\pm\Omega_{x}\cos{\omega
t}\right]\tau_{y}.$

The expression of Eq. (19) implies that in general, an optical dipole
interaction between up-spin bands is different from that between down-spin
bands. To be specific, for a linearly polarized light,

$V^{(\pm)}(t)=\pm(\Omega_{x}\cos{\omega t})\tau_{y}=\pm\Omega_{x}\cos{\omega
t}\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right)$ (22)

with $\Omega_{y}=0$, and $V^{(+)}(t)$ is identical to $V^{(-)}(t)$ aside from
an unimportant phase factor $\mp i$. On the other hand, for a left-hand
circularly polarized light,

$\displaystyle V^{(\pm)}(t)$ $\displaystyle=$
$\displaystyle\Omega_{c}\left[(\sin{\omega t})\tau_{x}\pm(\cos{\omega
t})\tau_{y}\right]$ (25) $\displaystyle=$ $\displaystyle\mp
i\Omega_{c}\left(\begin{array}[]{cc}0&e^{\pm i\omega t}\\\ -e^{\mp i\omega
t}&0\end{array}\right)$

with $\Omega_{c}\equiv\Omega_{x}=\Omega_{y}$, and $V^{(+)}(t)$ and
$V^{(-)}(t)$ are different from each other. In particular, this distinction
stands out for a linear optical transition, for instance, from the valence
band at $\Gamma_{7}$ to the conduction band at $\Gamma_{6}$. In view of the
matrix element $V^{(\pm)}_{12}(t)$ of Eq. (25), the transition amplitudes of
the photoabsorption between the up-spin bands, represented as $a^{(+)}$, and
that between the down-spin bands, represented as $a^{(-)}$, are given by

$a^{(\pm)}=\mp
i\Omega_{c}\int^{\infty}_{-\infty}dt\>e^{i(E_{\Gamma_{6}}-E_{\Gamma_{7}}\pm\omega)t}\propto\delta(E_{g}\pm\omega)$
(26)

with $E_{g}=E_{\Gamma_{6}}-E_{\Gamma_{7}}>0$. It is evident that the
transition between the up-spin bands is forbidden, namely, $a^{(+)}=0$, while
that between the down-spin bands is allowed, namely, $a^{(-)}\not=0$, because
of the energy conservation $\omega=E_{g}$. Incidentally, as regards the
related photoemission, in view of the matrix element $V^{(\pm)}_{21}(t)$, the
transition amplitudes of it are given by

$b^{(\pm)}=\pm
i\Omega_{c}\int^{\infty}_{-\infty}dt\>e^{i(E_{\Gamma_{7}}-E_{\Gamma_{6}}\mp\omega)t}\propto\delta(E_{g}\pm\omega),$
(27)

and the same discussion as the photoabsorption is applicable; $b^{(+)}=0$ and
$b^{(-)}\not=0$.

As long as $\omega\approx E_{g}$, these results almost hold correct in non-
linear optical processes including strongly photoinduced processes, though the
contribution from up-spin bands does not vanish because the energy
conservation is not required in virtual states. In other words, the effect of
$V^{(-)}(t)$ is dominant over that of $V^{(+)}(t)$. This is one of the key
issues in this paper, as mentioned in Sec. I. In contrast, as regards off-
resonant cases that $\omega\gg E_{g}$ or $\omega\ll E_{g}$, $V^{(+)}(t)$ and
$V^{(-)}(t)$ would have almost equal, however, vanishingly small contributions
to optical processes, as long as $|E_{g}-\omega|\gg 2\Omega_{c}$; for more
detail, see Eq. (28) or (29) to be shown later.

###  II.3 Symmetries

It is evident that both T- and I-symmetries are conserved in
$\mathcal{H}(\bm{k})$, that is,
$\Theta^{-1}\>\mathcal{H}(-\bm{k})\Theta=\mathcal{H}(\bm{k})$, and
$\Pi^{-1}\>\mathcal{H}(-\bm{k})\Pi=\mathcal{H}(\bm{k})$, where $\Theta$ and
$\Pi$ represent the T- and I-operators, defined by
$\Theta=-iI_{2}\otimes\sigma_{y}K$ and $\Pi=\tau_{z}\otimes I_{2}$,
respectively, where $K$ means an operation of taking complex conjugate.
Further, the T-symmetry is still respected in $H(\bm{k},t)$ for a linearly
polarized light, that is, $\Theta^{-1}H(-\bm{k},-t)\Theta=H(\bm{k},t)$, and
thus, a pair of up-spin and down-spin Floquet bands forms Kramers degeneracy.
On the other hand, the T-symmetry is broken for a circularly polarized light,
that is, $\Theta^{-1}H(-\bm{k},-t)\Theta\not=H(\bm{k},t)$.

As regards the I-symmetry, this is broken, that is,
$\Pi^{-1}H(-\bm{k},t)\Pi\not=H(\bm{k},t)$, because
$D_{j}(-\bm{k},t)\not=-D_{j}(\bm{k},t)$ for $j=$3 and 5,
$D_{4}(-\bm{k},t)\not=D_{4}(\bm{k},t)$, and
$\Pi^{-1}\mathcal{H}^{\prime}(t)\Pi\not=\mathcal{H}^{\prime}(t)$. In fact, it
is shown that in terms of an operator defined as
$\tilde{\Pi}=\Pi\mathcal{T}_{1/2}$, the symmetry
$\tilde{\Pi}^{-1}H(-\bm{k},t+T/2)\tilde{\Pi}=H(\bm{k},t)$ is retrieved, where
$\mathcal{T}_{1/2}$ represents the operation of putting $t$ ahead by a half
period $T/2$, namely, the replacement of $t\rightarrow t+T/2$.Zhang2021 ;
Morimoto2017  This is the time-glide $I$-operator mentioned in Sec. I.
Therefore, despite the breaking of the I-symmetry, a Floquet band disperses in
a symmetric manner with respect to $\bm{k}$, namely,
$E_{\alpha}(\bm{k})=E_{\alpha}(-\bm{k})$. For a linearly polarized light, it
is still probable that a four-fold band crossing occurs at the high-symmetry
points, namely, the time-reversal invariant momenta.

##  III Results and Discussion

In the actual calculations, the following material parametersWang2012 ;
Wang2013  and laser parameters are employed as:
$E_{g}=0.0169\>(0.46\>\text{eV}),\omega=0.0147\>(0.4\>\text{eV}),d_{x}=d_{y}=5.67\>(3\text{\AA}),d_{z}=9.44\>(5\text{\AA}),t^{xy}=-0.0018,t^{z}=-0.0074,t_{sp}=0.0037,F_{x}=F_{c}=0.0003\>(1.54\>\text{MV/cm})$,
and $P=25.9$. Further, the maximum number of photons $(N_{p})$ incorporated in
the calculations is set to be three to reach numerical convergence, that is,
$n,n^{\prime}=-N_{p}\sim N_{p}$ for the Fourier-Floquet matrix
$\tilde{L}_{nn^{\prime}}(\bm{k},\omega)$.

Based on these numerical parameters, one evaluates the degree of magnitude of
effects due to the Peierls interaction. These contributions are determined in
terms of factors given by the $n$th-order Bessel function of the first kind
$J_{n}(z_{l})$ that is included in
$\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)\;(j=3\sim 5)$ of Eqs. (59)-(74) with
$z_{l}\equiv F_{l}d_{l}/\omega\;(l=x,y)$. For $z_{l}=0.116$, one obtains that
$J_{0}(z_{l})-1=-0.0034,J_{1}(z_{l})=0.056$, and $J_{2}(z_{l})=0.0017$,
implying that the hopping matrices of $t^{l}$ and $t_{sp}$ are modified just
by the order of $10^{-2}\sim 10^{-3}$ by the Peierls interaction. Thus, it is
stated that in the system of concern, this interaction plays a less
significant role than the interband electric-dipole interaction.

###  III.1 Qualitative Understanding of Band Structures

It is preferable to show an overall Floquet band structure in the present
system in a qualitative manner prior to the discussion of rather complicated
numerical results. Here, a Floquet state $\alpha$ attributed to a $b$-band
dressed with $n$ photons is denoted as $b(n)$ with $b=e,hh$, where the bands
$e$ and $hh$ represent the $s$ and heavy-hole $p$ orbitals, respectively.
Below, one seeks approximate Floquet bands represented in an analytic closed
form in a two-step manner. First, based on a two-band model incorporating
$e(n-1)$ and $hh(n)$ for each spin state, one seeks expressions of hybridized
bands of states $e(n-1)$ for up- and down-spins, represented as
$\mathcal{E}^{(+)}_{e(n-1)}$ and $\mathcal{E}^{(-)}_{e(n-1)}$, respectively.
Here, the ac-Stark effect with a Rabi frequency $\Omega_{l}$ is incorporated
by employing the rotational-wave approximation. Similarly, the coupling
between $e(n)$ and $hh(n+1)$ for each spin state results in expressions of
hybridized bands of states $hh(n+1)$ for up- and down-spins, represented as
$\mathcal{E}^{(+)}_{hh(n+1)}$ and $\mathcal{E}^{(-)}_{hh(n+1)}$, respectively.
Second, by introducing the residual spin-orbit interaction between
$\mathcal{E}^{(\pm)}_{e(n-1)}$ and $\mathcal{E}^{(\pm)}_{hh(n+1)}$ bands, one
obtains the desired expressions of hybridized Floquet bands represented by
$E^{(\pm)}_{e(n-1)}(\bm{k})$ and $E^{(\pm)}_{hh(n+1)}(\bm{k})$: a double sign
corresponds. For more detail of the derivation, consult Appendices B.1-B.3.

In the case that the band $hh(n+1)$ is located above the band $e(h-1)$, the
resulting energy, represented by $E^{(\pm)}(\bm{k})$, is cast into
$E^{(\pm)}(\bm{k})=E^{(\pm)}_{hh(n+1)}(\bm{k})\geq n\omega$ and
$E^{(\pm)}(\bm{k})=E^{(\pm)}_{e(n-1)}(\bm{k})\leq n\omega$, the expressions of
which are given by Eqs. (122) and (123), respectively. To avoid unnecessary
complication in these expressions, the approximations that
$J_{0}(z_{l})\approx 1$ and $J_{n(\not=0)}(z_{l})\approx 0$ are made. Thus,
these are read as

$\displaystyle E^{(\pm)}_{e(n-1)}(\bm{k})$ $\displaystyle\approx$
$\displaystyle-\left[\left\\{\omega/2-\left(\eta(\bm{k})^{2}+|\mathcal{W}_{0}^{(\pm)}|^{2}\right)^{1/2}\right\\}^{2}\right.$
(28) $\displaystyle\left.+|\mathcal{V}_{0}^{(\pm)}|^{2}\right]^{1/2}+n\omega,$

and

$\displaystyle E^{(\pm)}_{hh(n+1)}(\bm{k})$ $\displaystyle\approx$
$\displaystyle\left[\left\\{\omega/2-\left(\eta(\bm{k})^{2}+|\mathcal{W}_{0}^{(\pm)}|^{2}\right)^{1/2}\right\\}^{2}\right.$
(29) $\displaystyle\left.+|\mathcal{V}_{0}^{(\pm)}|^{2}\right]^{1/2}+n\omega,$

where

$\eta(\bm{k})=d_{4}(\bm{k})-{\omega\over 2},$ (30)

and $d_{4}(\bm{k})$ is given in Eq. (5). Further,
$\mathcal{W}_{0}^{(+)}=[\mathcal{W}_{0}^{(-)}]^{*}=-i\Omega_{x}/2$ for the
linearly polarized light, while $\mathcal{W}_{0}^{(+)}=0$ and
$\mathcal{W}_{0}^{(-)}=i\Omega_{c}$ for the circularly polarized light.
Defining $\mathcal{V}_{0}^{\pm)}$ as an approximation expression of
$\mathcal{V}^{(\pm)}$ of Eq. (119) in view of the above approximations, one
has

$|\mathcal{V}_{0}^{(\pm)}|=t_{sp}\sigma^{(\pm)}\sqrt{\sin^{2}{(k_{x}d_{x})}+\sin^{2}{(k_{y}d_{y})}},$
(31)

where the prefactor $\sigma^{(\pm)}$ depending on the polarization of light is
given by Eq. (141). In Eqs. (28) and (29), $\mathcal{V}_{0}^{(\pm)}$ is
attributed to $\mathcal{D}_{3}(\bm{k},t)$ and $\mathcal{D}_{5}(\bm{k},t)$ in
Eq. (12), while $\mathcal{W}_{0}^{(\pm)}$ is attributed to
$\mathcal{H}^{\prime}(t)$. Hereafter, it is understood that in the opposite
case that $hh(n+1)$ is located below $e(h-1)$, the subscript of $e(n-1)$ is
replaced by that of $hh(n+1)$ in the above equations, that is,
$E^{(\pm)}(\bm{k})=E^{(\pm)}_{hh(n+1)}(\bm{k})\leq n\omega$ and
$E^{(\pm)}(\bm{k})=E^{(\pm)}_{e(n-1)}(\bm{k})\geq n\omega$.

Figure 1: The scheme of the surface state formation. (a) In the FDSM arising
from the drive of linearly polarized laser, this phase hosts non-trivial
surface states with up-spin (red) and down-spin (blue) that are pinned to a
pair of surface Dirac nodes at $k_{z}=\pm k_{z}^{D(\pm)}$. (Here,
$k_{z}^{D(\pm)}$ is replaced by the notation of $k_{z}^{D}$ just for the sake
of simplicity.) (b) In the FWSM arising from the drive of left-hand circularly
polarized laser, due to the breaking of the T-symmetry, the Dirac nodes of the
above FDSM are split into a pair of Weyl nodes, and each FWSM phase hosts a
non-trivial surface state. One surface state (red) is characteristic of an up-
spin band and is pinned to a pair of surface Dirac nodes at $k_{z}=\pm
k_{z}^{(+)}$. The other state (blue) is characteristic of a down-spin band and
is pinned to a pair of surface Dirac nodes at $k_{z}=\pm k_{z}^{(-)}$. (Here,
$k_{z}^{W(\pm)}$ is replaced by the notation of $k_{z}^{(\pm)}$ just for the
sake of simplicity.) For more detail, consult the text.

Now, one examines the possibility of creating Dirac nodal points on the
$k_{z}$-axis that result from band inversion for the irradiation of the
linearly polarized light. It is likely that the band $e(n-1)$ crosses the band
$hh(n^{\prime}+1)$ for $n=n^{\prime}$ at $\bm{k}=(0,0,k_{z})$ when
$E_{e(n-1)}(\bm{k})=E_{hh(n+1)}(\bm{k})$. Here, one takes account of the pair
of Floquet bands of $e(-1)$ and $hh(1)$. These bands are enabled to be
inverted to form a pair of Dirac nodes at the positions
$\pm\bm{k}^{D(\pm)}\equiv\pm(0,0,k_{z}^{D(\pm)})$ subject to the equation

$\epsilon_{z}(k_{z}^{D(\pm)})={1\over 2}\left[\omega-
E_{g}+\sqrt{\omega^{2}-\Omega_{x}^{2}}\right]$ (32)

in terms of $\epsilon_{z}(k_{z})$ defined in Eq. (7) under the condition that

$\omega_{1}^{D}<\omega<\omega_{2}^{D},$ (33)

where

$\omega_{1}^{D}=\text{Max}\left(\Omega_{x},\frac{\Delta^{2}_{g}+(\Omega_{x}/2)^{2}}{\Delta_{g}}\right),$
(34)

and

$\omega_{2}^{D}=\frac{(\Delta_{g}-4t^{z})^{2}+(\Omega_{x}/2)^{2}}{\Delta_{g}-4t^{z}}.$
(35)

For more accurate expressions than Eqs. (32) and (33), consult Eqs. (124) and
(129), respectively.

The existence of these nodes exhibits the manifestation of the Floquet DSM
(FDSM) phases in the original crystal of Zn3As2 that is in a topologically
trivial phase. Due to the T-symmetry in addition with the time-glide
I-symmetry, the up-spin and down-spin bands for the states $e(-1)$ and $hh(1)$
are doubly degenerate, that is,
$E^{(+)}_{e(-1)}(\bm{k})=E^{(-)}_{e(-1)}(\bm{k})$ and
$E^{(+)}_{hh(1)}(\bm{k})=E^{(-)}_{hh(1)}(\bm{k})$, and the above Dirac nodes
are four-fold degenerate, namely, $k_{z}^{D(+)}=k_{z}^{D(-)}$. Thus, it is
considered that the FDSM carries Chern number zero and is not topologically
protected.Armitage2018  In addition, as shown in Eq. (143),
$E^{(\pm)}(\bm{k})\geq 0$ forms an upper part of the Dirac cone (linear
dispersion) in the vicinity of $\bm{k}=\bm{k}^{D(\pm)}$, that is,

$\displaystyle E^{(\pm)}(\bm{k})$
$\displaystyle\approx\left[\sum_{l=x,y,}(\xi^{D}_{l})^{2}(k_{l}d_{l})^{2}+(\xi^{D}_{z})^{2}(\Delta
k_{z}d_{z})^{2}\right]^{1/2},$ (36)

where $\Delta k_{z}=k_{z}-k_{z}^{(D(\pm)}$, and the constants of
$\xi^{D}_{x}=\xi^{D}_{y}$ and $\xi^{D}_{z}$ are given right below Eq. (143).

Further, in the similar manner to DSMs created by the band inversion mechanism
in stationary systems such as Cd3As2 and Na3Bi,Wang2012 ; Wang2013 ; Yang2014
; Kargarian2016 ; Armitage2018  two-dimensional (2D) nontrivial surface states
are also expected in the FDSM phase. As shown schematically in Fig. 1(a),
these surface states are composed of up-spin and down-spin states forming a
Kramers pair, each energy band of which is attached to the same pair of the
Dirac nodes that are projected to the surface 2D-BZ; hereafter, these
projected Dirac nodes are termed as surface Dirac nodes or surface Dirac
points. The intersection of the Fermi energy with these two leaves of surface
bands would result in the formation of double Fermi arcs, supposing that the
whole of carriers are occupied just below $E_{F}$ in disregard of the non-
equilibrium system of concern.

Next, as regards the circularly polarized laser drive, the T-symmetry is
broken to lift the two-fold degeneracy between up-spin and down-spin bands.
Thus, the four-fold degeneracy at the Dirac nodes (at $\pm\bm{k}^{D(\pm)}$)
are also lifted to be split into two pairs of Weyl nodes residing at
$\pm\bm{k}^{W(-)}\equiv\pm(0,0,k_{z}^{W(-)})$ and
$\pm\bm{k}^{W(+)}\equiv\pm(0,0,k_{z}^{W(+)})$; a double sign corresponds. The
nodal momentum $\pm\bm{k}^{W(-)}$ is attributed to the down-spin Floquet band,
and its location is subject to the similar equation as Eq. (32), aside from
the replacement of $\Omega_{x}$ by $2\Omega_{c}$,

$\epsilon_{z}(k_{z}^{W(-)})={1\over 2}\left[\omega-
E_{g}+\sqrt{\omega^{2}-4\Omega_{c}^{2}}\right]$ (37)

under the condition that

$\omega_{1}^{W(-)}<\omega<\omega_{2}^{W(-)},$ (38)

where

$\omega_{1}^{W(-)}=\text{Max}\left(2\Omega_{c},\frac{\Delta^{2}_{g}+\Omega_{c}^{2}}{\Delta_{g}}\right),$
(39)

and

$\omega_{2}^{W(-)}=\frac{(\Delta_{g}-4t^{z})^{2}+\Omega_{c}^{2}}{\Delta_{g}-4t^{z}}.$
(40)

Here the ac-Stark effect plays a key role. On the other hand, the nodal
momentum $\pm\bm{k}^{W(+)}$ is attributed to the up-spin Floquet band, and its
location is subject to the equation

$\epsilon_{z}(k_{z}^{W(+)})=\omega-\Delta_{g}$ (41)

under the condition that

$\omega_{1}^{W(+)}<\omega<\omega_{2}^{W(+)},$ (42)

where $\omega_{1}^{W(+)}=\Delta_{g}$ and
$\omega_{2}^{W(+)}=\Delta_{g}-4t^{z}.$ Here, in contrast, the ac-Stark effect
is less significant because the optical interaction given by $V^{(+)}(t)$ of
Eq. (25) has negligibly small contributions in the case of $\omega\approx
E_{g}$; consult Sec. II.2. Thus, it is obvious that
$k_{z}^{W(+)}>k_{z}^{D(\pm)}>k_{z}^{W(-)}$; based on Eqs. (32), (37) and (41),
approximate values of $k_{z}^{W(+)}$, $k_{z}^{D(\pm)}$, and $k_{z}^{W(-)}$ are
estimated as $k_{z}^{W(+)}=0.956/d_{z},k_{z}^{D(\pm)}=0.910/d_{z}$, and
$k_{z}^{W(-)}=0.732/d_{z}$, respectively.

In addition, as shown in Eq. (146), $E^{(-)}(\bm{k})$ forms an upper part of
the Weyl cone (linear dispersion) in the vicinity of $\bm{k}=\bm{k}^{W(-)}$,
that is,

$\displaystyle E^{(-)}(\bm{k})$
$\displaystyle\approx\left[\sum_{l=x,y,}(\xi^{W}_{l})^{2}(k_{l}d_{l})^{2}+(\xi^{W}_{z})^{2}(\Delta
k_{z}d_{z})^{2}\right]^{1/2},$ (43)

where $\Delta k_{z}=k_{z}-k_{z}^{(W(-)}$, and the constants of
$\xi^{W}_{x}=\xi^{W}_{y}$ and $\xi^{W}_{z}$ are given right below Eq. (146).
On the other hand, as shown in Eq. (145), in contrast wth $E^{(-)}(\bm{k})$,
$E^{(+)}(\bm{k})$ forms a quadratic dispersion with respect to $k_{x}$ and
$k_{y}$ and a linear dispersion with respect to $\Delta k_{z}$ in the vicinity
of $\bm{k}=\bm{k}^{W(+)}$, that is,

$\displaystyle
E^{(+)}(\bm{k})\approx\left|(-t^{xy})\sum_{l=x,y}(k_{l}d_{l})^{2}+\eta_{z}^{(1)}(\Delta
k_{z}d_{z})\right|,$ (44)

where $\Delta k_{z}=k_{z}-k_{z}^{W(+)}$, and
$\eta^{(1)}_{z}=-2t^{z}\sin{(k_{z}^{W(+)}d_{z})}$. Here, a term of linear
dispersion represented by $\sum_{l=x,y}\nu_{l}^{(+)}(k_{l}d_{l})$ is
considered negligibly small, because the expansion coefficient is given by
$|\nu_{l}^{(+)}|=t_{sp}[t_{sp}J_{1}(z_{c})/(2\eta(\bm{k}^{W(+)}))]^{2}$ due to
the reduction of the spin-orbit interaction by the Peierls interaction by a
factor of the order of $J_{1}(z_{c})$; see also the explanation below Eq.
(145).

Further, the FWSM band structure $E^{(+)}(\bm{k})$ in the vicinity of the
$k_{x}-k_{y}$ plane $(k_{z}=0)$ is examined. Following Eq. (29) for
$d_{4}(\bm{k})>\omega/2$, this is represented simply as

$E^{(+)}(\bm{k})\approx\omega-d_{4}(\bm{k})$ (45)

within the order of $t_{sp}J_{1}(z_{c})\approx 0$ due again to the reduction
of $t_{sp}$ by the Peierls interaction. Thus, there is a closed ring in the
$k_{x}-k_{y}$ plane $(k_{z}=0)$ on which $E^{(+)}(\bm{k})\approx 0$; the locus
of this ring is given by

$\displaystyle-2t^{xy}\sum_{l=x,y}[1-\cos{(k_{l}d_{l})}]=\omega-\Delta_{g}>0.$
(46)

As regards the FWSM of the down-spin state, an effect of $\Omega_{c}$ causes a
gap to open between $E^{(-)}_{hh(1)}(\bm{k})$ and $E^{(-)}_{e(-1)}(\bm{k})$ in
this plane. Consult Eq. (149) and the related discussion in Appendix B.3 on
the closed ring formation in the $k_{x}-k_{y}$ plane $(k_{z}\not=0)$ and the
origin of the difference between the up- and down-spin states.

Given the relation between Dirac points and Weyl points in stationary systems,
the surface Dirac point in the FDSM is regarded as the stable merger of two
Weyl points in the FWSM that have different handedness and are projected to
the same surface momentum. Hereafter, these Weyl points are termed as surface
Weyl nodes or surface Weyl points. Due to the breaking of the T-symmetry and
the resulting splitting of the Dirac node into of the pair of Weyl nodes, the
associated energy bands of the two surfaces with different spin states are
pinned to different surface Weyl points, as shown schematically in Fig. 1(b).
That is, the surface band characteristic of up-spin/down-spin state is pinned
to the surface Weyl point projected from the bulk Weyl points at
$\pm\bm{k}^{W(+)}/\pm\bm{k}^{W(-)}$. Further, it is considered that the energy
gap $\mathcal{E}_{g}^{(+)}$ arising from the hybridization between the up-spin
Floquet bands $e(-1)$ and $hh(1)$ are largely different from the energy gap
$\mathcal{E}_{g}^{(-)}$ attributed to the down-spin bands mostly due to the
difference of magnitude between $V^{(+)}(t)$ and $V^{(-)}(t)$; to be more
specific, $\mathcal{E}_{g}^{(+)}\ll\mathcal{E}_{g}^{(-)}$. Such difference is
straightforward reflected on the band gaps projected to the surface BZ; see
Fig. 1(b). Therefore, it is speculated that the most parts of down-spin
surface band are energetically separated from the up-spin surface band; for
more detail, consult Sec. III.3.

###  III.2 Floquet Band Structures of FDSM and FWSM

Figure 2: Band structures of FDSM and FWSM. (a) The calculated band structure
of FDSM with the drive of a linearly polarized laser. Inset: Expanded view of
the band structure in the vicinity of the anticrossing along the $\Gamma$-X
line. (b) The calculated band structure of FWSM with the drive of a left-hand
circularly polarized laser. (c) Expanded view of panel (b) in the vicinity of
the band crossing along the $\Gamma$-Z line with specification of the Weyl
nodes at $k_{z}^{(+)}$ and $k_{z}^{(-)}$; these are the abbreviation of
$k_{z}^{W(+)}$ and $k_{z}^{W(-)}$, respectively. In panels (a), (b), and (c),
bands dominated rather by the $s/p$-orbital component are denoted by a
red/blue solid line. (d) The bulk BZ of the crystal Zn3As2.

Figures 2(a) and 2(b) show the calculated Floquet band structures of FDSM and
FWSM, respectively, for the crystal structure of Zn3As2 given in Fig. 2(d). In
Fig. 2(a), it is found that there is a Dirac node along the $\Gamma$-Z line at
$k_{z}^{D(\pm)}$ in addition with anticrossings along the lines of $\Gamma$-X
and $\Gamma$-M with energy differences of approximately 3 and 30 meV,
respectively. As shown in Fig. 2(b), the two-fold degeneracy confirmed in
panel (a) is lifted to result in energy splitting between the up-spin and
down-spin bands. It is noted that a pair of Weyl nodes emerges along the
$\Gamma$-Z line at different $k_{z}$’s following $k_{z}^{W(-)}<k_{z}^{W(+)}$,
as shown in the enlarged view of Fig. 2(c). As regards the up-spin bands, the
anticrossing along the line of $\Gamma$-M is largely reduced from that in
panel (a) to approximately 0.4 meV, while the energy difference along the line
of $\Gamma$-X is almost the same as 2 meV. In contrast, as regards the down-
spin bands, the energy differences of anticrossings along the lines of
$\Gamma$-X is largely enhanced from that in panel (a) to approximately 23 meV,
while that along the line of $\Gamma$-M is slightly changed to approximately
20 meV.

Figure 3: Energy dispersions $E^{(\pm)}(\bm{k})$ of up-spin and down-spin
bands at $k_{z}=0$ and in the vicinity of the Weyl points. The axis of
abscissa $k_{l}$ is gauged in the unit of $1/d_{l}$ with $l=x,y,z$. Here,
$E^{(\pm)}(\bm{k})$ is simply represented as $E(\bm{k})$, and bands dominated
rather by the $s/p$-orbital component are denoted by a red/blue solid line.
(a) $E(\bm{k})$ in the $k_{x}-k_{y}$ plane at $k_{z}=0$ for the up-spin bands.
(b) The same as panel (a) but for the down-spin bands. (c) $E(\bm{k})$ as a
function of $k_{x}$ with $k_{y}=0$ in the vicinity of the Weyl point
$\bm{k}^{W(+)}$ for the up-spin bands. (d) The same as panel (c) but in the
vicinity of the Weyl point $\bm{k}^{W(-)}$ for the down-spin bands. (e)
$E(\bm{k})$ as a function of $k_{z}$ with $k_{x}=k_{y}=0$ for the up-spin
bands with the Weyl points $\pm k_{z}^{W(+)}=\pm 0.887/d_{z}$. (f) The same as
panel (e) but for the down-spin bands with the Weyl points $\pm
k_{z}^{W(-)}=\pm 0.775/d_{z}$.

Figures 3(a) and 3(b) show the energy dispersions of the up-spin and down-spin
bands at $k_{z}=0$. In Fig. 3(a), the up-spin band structure is reminiscent of
a NLSM phase with a nodal ring on the $k_{x}-k_{y}$ plane; see also the
enlarged figure of it given in Appendix C. According to the analytic model
developed in Sec. III.1, the locus of the ring is approximately represented as
Eq. (46). In fact, this ring is slightly blurred at most by
$\mathcal{E}_{g}^{(+)}\approx 2$ meV around $E=0$ that corresponds to the
above-mentioned energy difference along the line of $\Gamma$-X. On the other
hand, it is obviously seen in Fig. 3(b) that the down-spin band is gapped by
the order of $\mathcal{E}_{g}^{(-)}\approx 20$ meV due to the relatively
strong anticrossing between $e(-1)$ and $hh(1)$; see also the enlarged figure
of it given in Appendix C.

The definite contrast in the energy dispersions between up-spin and down-spin
bands seen in Figs. 3(a) and 3(b) is caused by the different manner of
couplings between the Floquet states of $e(-1)$ and $hh(1)$. It is noted that
the leading contribution arises from a two-photon coupling between $e(-1)$ and
$hh(1)$, because the difference of the photon number of these Floquet bands
equals two. This coupling is given by a successive interaction composed of the
coupling due to $\tilde{D}_{4,nn^{\prime}}(\bm{k},\omega)$ and one of the
three terms, $\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega),(j=3,5)$ and
$\tilde{\mathcal{H}}^{\prime}_{nn^{\prime}}(\omega)$, in Eq. (15) with
$|n-n^{\prime}|=1$. For example, for down-spin bands, it is likely that
$hh(1)$ is mediated by a two-photon interaction,
$\tilde{\mathcal{H}}^{\prime}_{01}(\omega)$ followed by
$\tilde{D}_{4,-10}(\bm{k},\omega)$, to be coupled with $e(-1)$. As regards up-
spin bands, because the effect of $\tilde{\mathcal{H}}^{\prime}_{01}(\omega)$
is negligibly small, $hh(1)$ is couplied with $e(-1)$ by a two-photon
interaction, $\tilde{D}_{j,01}(\bm{k},\omega),(j=3,5)$ followed by
$\tilde{D}_{4,-10}(\bm{k},\omega)$. The magnitudes of interactions
$\tilde{\mathcal{H}}^{\prime}_{01}(\omega)$ and
$\tilde{D}_{j,01}(\bm{k},\omega),(j=3,5)$ are roughly evaluated to be
$\Omega_{c}$ and $J_{1}(z_{c})t_{sp}$, respectively; consult Eqs. (25), (59),
and (74). Thus, it is stated that the inverted band gap
$\mathcal{E}_{g}^{(-)}$ in the down-spin state is mostly caused by a strong
resonant electric dipole coupling, while $\mathcal{E}_{g}^{(+)}$ in the up-
spin state is just attributed to an optical spin-orbit coupling, namely, a
spin-orbit coupling reduced by a factor of $J_{1}(z_{c})$; actually,
$\Omega_{c}=5.49\times 10^{-3}\gg t_{sp}J_{1}(z_{c})=2.07\times 10^{-4}$. The
resulting FWSM phase for the up-spin state is considered as a Floquet NLSM
(FNLSM) phase that is generated by the drive of the present circularly
polarized light.

Such sharp distinction between the up-spin and down-spin bands is also seen in
the energy dispersions in the $k_{x}$-direction in the vicinity of the Weyl
points at $\bm{k}^{W(+)}$ and $\bm{k}^{W(-)}$, respectively, as shown in Figs.
3(c) and 3(d). It is noted that the up-spin band crosses with quadratic band
touching, following

$E^{(+)}(\bm{k})\approx(-t^{xy})(k_{x}d_{x})^{2}\geq 0,$ (47)

as given in Eq. (44), while the down-spin band crosses with linear band
touching, as often happens, following

$E^{(-)}(\bm{k})\approx\xi^{W}_{x}|k_{x}d_{x}|\geq 0,$ (48)

as given in Eq. (43), where $\xi^{W}_{x}$ is of the order of $t_{sp}$. In
fact, there is a contribution from the linear dispersion of the form of
$\nu_{x}^{(+)}(k_{x}d_{x})$ in Eq. (47), however, this is neglected because of
$\nu_{x}^{(+)}\ll 1$, as mentioned before.

Further, it is seen in Figs. 3(e) and 3(f) that the two bands of $e(-1)$ and
$hh(1)$ are inverted to form a pair of Weyl nodes along the $k_{z}$-axis at
$k_{z}^{W(\pm)}$ and $-k_{z}^{W(\pm)}$. These dispersions in the vicinity of
$\bm{k}^{W(\pm)}$ are given by

$E^{(+)}(\bm{k})\approx\eta_{z}^{(1)}|\Delta k_{z}d_{z}|\geq 0$ (49)

for the up-spin state, and

$E^{(-)}(\bm{k})\approx\xi^{W}_{z}|\Delta k_{z}d_{z}|\geq 0$ (50)

for the down-spin state, following Eqs. (44) and (43), respectively. Here, the
obtained numerical value of $k_{z}^{W(+)}(=0.887/d_{z})$ is greater than that
of $k_{z}^{W(-)}(=0.775/d_{z})$, which is in harmony with the qualitative
discussion based on the approximated expressions of Eqs. (37) and (41);
consult the values thus obtained for $k_{z}^{W(\pm)}$ below Eq. (42). It is
speculated that the difference of the former numerical values from the latter
approximate ones is attributed to the non-resonant contributions of interband
couplings beyond the rotational-wave approximation in the nearly resonant two-
band model adopted in Sec. III.1. Actually, the intense laser field is applied
to the system of concern with the order of $\Omega_{c}/\omega\approx 0.37$,
and hence, for instance, the Floquet band $hh(1)$ is somewhat coupled with
other non-resonant bands of $e(n\not=0)$ in addition with the nearly resonant
band $e(0)$.

###  III.3 Surface States

Here, it is considered that a vanishing boundary condition in the
$y$-direction is imposed on the Floquet eigenvalue problem given by Eq. (13)
in place of a periodic boundary condition. To be specific, an electron is
confined in the finite range of $y$ from $L_{1}=0$ to $L_{2}=40$ a.u., while
it moves freely in the $x-z$ plane. Such confinement results in energy
dispersions $\mathcal{E}(\bar{\bm{k}})$ that are the projection of bulk bands
$E(\bm{k})$ on the $k_{x}-k_{z}$ plane where $\bar{\bm{k}}=(k_{x},k_{z})$.
Further, it is likely that surface states are hosted by the projected bands.
For the sake of the later convenience, the positions of surface Weyl nodes for
the up-spin and down-spin bands are represented as
$\pm\bar{\bm{k}}^{W(+)}=\pm(0,k_{z}^{W(+)})$ and
$\pm\bar{\bm{k}}^{W(-)}=\pm(0,k_{z}^{W(-)})$, respectively.

Figure 4: Projected energy dispersions $\mathcal{E}(\bar{\bm{k}})$ with a
surface state of down-spin. The axis of abscissa $k_{x}$ is gauged in the unit
of $1/d_{x}$. Here, bands dominated rather by the $s/p$-orbital component are
denoted by a red/blue solid line. (a) $\mathcal{E}(\bar{\bm{k}})$ at
$k_{z}=0.7$ slightly smaller than $k_{z}^{W(-)}$. (b) The same as pannel (a)
but at $k_{z}=0.35$. (c) The same as pannel (a) but at $k_{z}=0$. (d) Enlarged
view of pannel (c).

Figure 4 shows the projected energy dispersions of down-spin bands with
surface states at three different $k_{z}$’s. As shown in Fig. 4(a), at $k_{z}$
close to $k_{z}^{W(-)}$, the inverted bands of $e(-1)$ and $hh(1)$ form a
definite energy gap $\mathcal{E}_{g}^{(-)}$, hosting a pair of surface states
just in a small range of $k_{x}$. It is evident that as $k_{z}$ becomes closer
to $k_{z}^{(-)}$, the range of $k_{x}$ becomes more reduced, and eventually,
the pair of surface states are embedded in the surface Weyl point at
$\bar{\bm{k}}^{W(-)}$. Meanwhile, it is remarked that the appearance of such a
pair is due to a numerical artifact ascribable to the above-mentioned
confinement of electron in the finite range in place of a semi-infinite
confinement corresponding to $L_{2}=\infty$. Here, it is understood that in
all of the figures in Fig. 4, just the surface states with a positive gradient
are taken account of. As shown in Figs. 4(b)-4(d), with the further decrease
of $k_{z}$, the range of $k_{x}$ in which the surface state is supported
becomes larger, and is maximized at $k_{z}=0$, where this range extends over a
half of the BZ in the $k_{x}$ direction. Moreover, as $k_{z}$ changes from
$k_{z}=0$ to the negative $k_{z}$-direction, the range of $k_{x}$ turns to a
decrease, and eventually, at $k_{z}=-k_{z}^{W(-)}$, the surface state is
incorporated with another surface Weyl point at $-\bar{\bm{k}}^{W(-)}$; though
not shown here. These nontrivial surface states sliced in the interval
$-k_{z}^{W(-)}\leq k_{z}\leq k_{z}^{W(-)}$ are unified to form a tilted
surface band in the $k_{x}-k_{z}$ plane. Both edges of it are pinned to the
respective surface Weyl points at $\pm\bar{\bm{k}}^{W(-)}$. This surface band
is schematically depicted as the tilted surface that is colored blue in the
right figure of Fig. 1(b).

Figure 5: Projected energy dispersions $\mathcal{E}(\bar{\bm{k}})$ with a
surface state of up-spin. The axis of abscissa $k_{x}$ is gauged in the unit
of $1/d_{x}$. Here, bands dominated rather by the $s/p$-orbital component are
denoted by a red/blue solid line. (a) $\mathcal{E}(\bar{\bm{k}})$ at
$k_{z}=0.8$ slightly smaller than $k_{z}^{W(+)}$. (b) The same as pannel (a)
but at $k_{z}=0.4$. (c) The same as pannel (a) but at $k_{z}=0$. (d) Enlarged
view of pannel (c).

Figures 5(a)-5(c) show the projected energy dispersions of up-spin bands with
surface states at three different $k_{z}$’s. It is seen that the pattern of
variance of the surface states formed here follows that shown in the down-spin
bands of Fig. 4. However, the energy gap $\mathcal{E}_{g}^{(+)}$ is extremely
smaller than $\mathcal{E}_{g}^{(-)}$, and as seen in Fig. 5(d), the surface
band is slightly tilted with overall negative gradient and undulation. The
pattern of variance in the negative $k_{z}$-direction is also subject to that
seen in the down-spin bands; though not shown here. As a result, the
nontrivial surface states sliced in the interval $-k_{z}^{W(+)}\leq k_{z}\leq
k_{z}^{W(+)}$ form a slightly tilted and undulated surface band in the
$k_{x}-k_{z}$ plane. Both edges of it are pinned to the respective surface
Weyl points at $\pm\bar{\bm{k}}^{W(+)}$. This surface band is schematically
depicted as the tilted surface that is colored red in the left figure of Fig.
1(b).

###  III.4 Physical Properties

First, discussion is made on the chirality of the FWSM phases and the related
topological phase transitions. It is considered that based on the qualitative
discussion in Sec. III.1, the conditions of generating the Weyl nodes for the
up-spin and down-spin states are approximately evaluated as Eqs. (42) and
(38), respectively. According to these, when $\omega$ is made greater from
$\omega_{1}^{W(\pm)}$ and eventually identical to $\omega_{2}^{W(\pm)}$, the
Weyl nodes at $\bm{k}^{W(\pm)}$ move along $k_{z}$ axis from the $\Gamma$
point $k_{z}=0$ toward the boundary of the BZ at $k_{z}=\pi/d_{z}$ to
annihilate with the other pair of the Weyl nodes at $-\bm{k}^{W(\pm)}$ that
move in the opposite direction toward the boundary at $k_{z}=-\pi/d_{z}$; a
double sign corresponds. This implies that the Weyl nodes at $\bm{k}^{W(\pm)}$
possess opposite handedness from that at the other Weyl nodes at
$-\bm{k}^{W(\pm)}$. In other words, there should be the relations that

$h_{+}^{(+)}h_{-}^{(+)}=-1,\;h_{+}^{(-)}h_{-}^{(-)}=-1,$ (51)

where $h_{\pm}^{(+)}$ and $h_{\pm}^{(-)}$, which are either 1 or -1, represent
helicities of the Weyl cones at $\pm\bm{k}^{W(+)}$ for the up-spin and
$\pm\bm{k}^{W(-)}$ for the down-spin, respectively. Further, it is noted that
the handedness of the Weyl node for the up-spin state at
$\bm{k}^{W(+)}(-\bm{k}^{W(+)})$ is opposite from that for the down-spin state
at $\bm{k}^{W(-)}(-\bm{k}^{W(-)})$, because a pair of Weyl nodes for the up-
spin and down-spin states at $\bm{k}^{W(+)}$ and $\bm{k}^{W(-)}$,
respectively, are generated by splitting of the Dirac node at
$\bm{k}^{D}(-\bm{k}^{D})$ due to the breaking of the T-symmetry. That is,
there should be the relations that

$h_{+}^{(+)}h_{+}^{(-)}=-1,\;h_{-}^{(+)}h_{-}^{(-)}=-1.$ (52)

Actually, the above relations of Eqs. (51) and (52) are confirmed by defining
these helicities as Eqs. (160) and (171), followed by mathematical evaluation,
as developed in Appendix B.4. Here, these expressions of helicities are
extracted from effective Fourier-Weyl Hamiltonians of Eqs. (154) and (165),
which are reduced from the original Floquet Hamiltonian of Eq. (15). Here, one
mentions that recently, dynamical characterization of Floquet-Weyl nodes is
discussed in Ref. Umer2021b, .

In passing, when $\omega$ exceeds $\omega_{2}^{W(\pm)}$, the topological order
is changed from the FWSM phase to a phase of Floquet topological insulator due
to the gap opening. Further, the reduction of $\omega$ below
$\omega_{1}^{W(\pm)}$ in the other direction brings the FWSM phase just back
to a trivial insulator phase.

Figure 6: Frequency dependence of energy dispersions $E^{(\pm)}(\bm{k})$ of
up-spin and down-spin bands as a function of $k_{z}$ with $k_{x}=k_{y}=0$ for
$\omega$ equal to (a) 0.2 eV (7.35$\times 10^{-3}$ a.u.), (b) 0.3 eV
(1.10$\times 10^{-2}$ a.u.), (c) 0.35 eV (1.29$\times 10^{-2}$ a.u.), (d) 0.4
eV (1.47$\times 10^{-2}$ a.u.), (e) 0.5 eV (1.84$\times 10^{-2}$ a.u.), and
(f) 0.7 eV (2.57$\times 10^{-2}$ a.u.). The abscissa $k_{z}$ is gauged in the
unit of $1/d_{z}$. The panel (d) is the same as Figs. 3(e) and 3(f). Here, the
legend of ordinate $E^{(\pm)}(\bm{k})$ is simply represented as $E$. The up-
spin bands $e(-1)$ and $hh(1)$ are depicted by blue solid lines, and the down-
spin bands $e(-1)$ and $hh(1)$ are depicted by red solid lines. The parent
bands are also labeled as $e(0)$ and $hh(0)$. In all panels, red solid lines
are partially superimposed on blue solid lines.

Figure 6 shows the energy dispersions $E^{(\pm)}(\bm{k})$ of up-spin and down-
spin bands as a function of $k_{z}$ with $k_{x}=k_{y}=0$ for $\omega=0.2\sim
0.7$ eV. Below, discussion is made on the alteration of just $e(-1)$ and
$hh(1)$ bands of an up-spin state (depicted by blue solid lines) and a down-
spin state (depicted by red solid lines) with respect to $\omega$. In the
panel (a) for $\omega=0.2$ eV, both spin bands are not inverted, and in the
panel (b) for $\omega=0.3$ eV, the up-spin band is inverted to form a FWSM
phase with a pair of Floquet-Weyl nodes while the down-spin band is left open.
In the panel (c) for $\omega=0.35$ eV, both bands become inverted to form FWSM
phases with two pairs of Floquet-Weyl nodes, and similarly, in the panel (d)
for $\omega=0.4$ eV $(<E_{g}$), both bands remain inverted. Such band
inversion is still retained in the panels (e) and (f) even for $\omega=$0.5
and 0.7 eV $(>E_{g}$). Incidentally, the discontinuities of the down-spin band
seen in the panels (e) and (f) are due to an anticrossing between $e(-1)$ and
$hh(0)$ and that between $e(0)$ and $hh(1)$. It is worth comparing these
numerical results with the results estimated by Eqs. (42) and (38) based on
the analytic model in Sec. III.1, where
$\omega_{1}^{W(+)}=0.23\>{\text{e}V},\omega_{2}^{W(+)}=0.64\>{\text{e}V},\omega_{1}^{W(-)}=0.33\>{\text{e}V}$,
and $\omega_{2}^{W(-)}=0.67\>{\text{e}V}$. It is found that aside from the
panel (f), the above-stated changes of topological order with respect to
$\omega$ are well consistent with these estimated existence conditions of FWSM
phases. The variance seen in the panel (f) is due to the breaking of the
rotational-wave approximation adopted in this model. Actually, this
approximation is considered accurate under the situation that $\omega\approx
E_{g}(=0.46$ eV).

Second, discussion is made on a magnetic property induced by the irradiation
of the intense laser with a left-hand circular polarization. As far as the
nearly resonant optical transition is concerned, down-spin electrons that are
situated in a valence band before the irradiation are selectively excited to a
conduction band, and some fractions of the excited electrons are deexcited
back to the valence band due to the Rabi oscillation, whereas up-spin
electrons remain almost in the valence band; consult Sec. II.2. In terms of
the Floquet picture, these excitation and deexcitation processes in a series
of the non-equilibrium dynamics are interpreted as couplings between one pair
of down-spin bands $hh(1)$ and $e(0)$, and between another pair of down-spin
bands $hh(0)$ and $e(-1)$, respectively. Thus, carriers are likely distributed
to both bands of $e(-1)$ and $hh(1)$, which are further coupled by the two-
photon interaction mentioned in Sec.III.2 to form the FWSM phase through the
ac-Stark splitting. On the contrary, it is considered that the up-spin bands
of $e(-1)$ and $hh(1)$ are almost unoccupied. Therefore, the down-spin
electrons are exclusively distributed over the surface, while these coexist
with the up-spin electrons in the bulk though both electronic states are
energetically separated by the amount of $\mathcal{E}_{g}^{(-)}$.

This implies that the system of concern exhibits transient surface
magnetization with down spins that survives for as long as the associated
population relaxation time, besides bulk magnetization that is expected to be
induced as well. In addition with such an effect of spin magnetization, it is
likely that the circularly polarized laser induces the inverse Faraday effect,
which is a sort of a generation mechanism of orbital magnetization.
Pershan1966 ; Kimel2005 ; Hertel2006 ; Zhang2009 ; Battiato2014  This effect
is expected to contribute the above surface magnetization to a certain extent.
Indeed, the surface magnetization seems faint and transient, but the magnitude
of it can be somewhat enhanced by increasing the strength of the circularly
polarized laser. Moreover, the measurement of such an intriguing phenomenon
would be feasible by means of the longitudinal magneto-optic Kerr effect that
can detect the degree of strength of magnetization manifested just in the
surface. Kerr1877 ; Weinberg2008 ; Haider2017  To do this, a pump-probe
measurement is expected to be effective, in which a linear polarized laser
causing the magneto-optic Kerr effect is incorporated as a probe in addition
with the pump laser with the left-hand circular polarization.

Below, additional comments on the results described in Secs III.1-III.3 are
enumerated.

(1) The up-spin FWSM band is considered as a FNLSM phase, because the band gap
of $\mathcal{E}_{g}^{(+)}$ is of the order of 2 meV; consult Sec III.2.
Actually, such small energy separation and the concomitant surface state would
be possibly smeared with homogeneous broadening due to an electron correlation
effect and inhomogeneous broadening due to finite temporal width of a laser
pulse — in place of the ideal continuous-wave laser —, which is of the order
of a couple of meV for a pico-second pulse. Although the up-spin bands are
almost unoccupied as stated above, these would be detectable by reconstructing
the optical system of concern as follows: the up-spin bands of $e(-1)$ and
$hh(1)$ are excited in advance by an intense ultrashort pulse laser with
linear polarization, followed by the irradiation of the pico-second pulse (the
continuous-wave laser) with the left-hand circular polarization.

(2) In view of the above comment (1), the surface states hosted by the down-
spin band are entirely embedded in the continuum (bulk) of the FNLSM phase of
the up-spin band; consult Figs. 4 and 5. When a spin flip interaction
attributed to the spin-orbit coupling is tuned on, the surface states become
somewhat unstable due to the effect of Fano resonance, namely, the collapse of
the discrete levels of the surface states into the continuum states which is
caused by the interaction between both of these states.Fano1961  The spin flip
interaction becomes effective when either $d_{1}(\bm{k})$ or $d_{2}(\bm{k})$
has a non-negligible contribution to the effective Hamiltonian given in Eq.
(9).

(3) The crystal Zn3As2 has a bulk rotational symmetry around the $z$-axis, and
this leads to the formation of FWSMs under the conditions of Eqs. (38) and
(42). In fact, there remains internal compression normal to this axis within
the crystal, and this symmetry is considered partially broken. Such breaking
will open up a slight gap to make the Floquet system of concern
insulating.Wang2012

##  IV Conclusions

It is found that the narrow gap semiconductor Zn3As2 is driven by a left-hand
circularly-polarized continuous-wave laser with frequency nearly resonant with
the band gap $E_{g}$ to produce the two types of FWSM phases simultaneously in
the crystal, which are sharply distinguished by their spins. The bulk
rotational symmetry around the $z$-axis protects a pair of Weyl nodes with
opposite chirality along the $k_{z}$-axis in the respective FWSM phases under
the condition of either Eq. (38) or Eq. (42). In the down-spin FWSM phase, the
Floquet bands of $e(-1)$ and $hh(1)$ touch in a linear manner in the vicinity
of the Weyl nodes situated at $\pm\bm{k}^{W(-)}$, hosting the nontrivial
surface states pinned to both nodes. Since the above-mentioned laser makes
electrons excited exclusively in the down-spin Floquet bands, it is considered
that the surface states are selectively occupied by such spin-polarized
electrons, showing transient magnetization with partial modification by the
inverse Faraday effect. This surface magnetization would be measured by virtue
of the magneto-optic Kerr effect. On the other hand, in the up-spin FWSM
phase, the Floquet bands of $e(-1)$ and $hh(1)$ touch in the vicinity of the
Weyl nodes situated at $\pm\bm{k}^{W(+)}$ almost in a quadratic manner in the
$k_{x}$\- and $k_{y}$-directions and in a linear manner in the
$k_{z}$-direction. Because of the negligibly small band gap, this up-spin FWSM
phase is rather considered as the FNLSM phase. To detect this phase somehow or
other, it would be necessary to make excited electrons occupied in the up-spin
bands in advance prior to the irradiation of the circularly polarized laser.
The exploration of the transient non-equilibrium dynamics of the concerned
system is inevitable in addition with Floquet band structures to deepen the
understandings of the underlying physics of the FWSMs.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant No. JP19K03695. The authors are
grateful to Prof. J. Fujioka for fruitful comments and discussion.

##  Appendix A Explicit expressions of
$\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)\;\;(j=3\sim 5)$ and
$\tilde{V}^{(\pm)}_{nn^{\prime}}$

The Floquet matrix element $\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)$ seen in
Eq. (15) is given by

$\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)={1\over
T}\int_{0}^{T}dt\>e^{-i\Delta n\omega t}D_{j}(\bm{k},t)$ (53)

with $\Delta n=n-n^{\prime}$. This is expressed in terms of the $N$th-order
Bessel function of the first kind

$J_{N}(z_{l})={1\over
2\pi}\int_{0}^{2\pi}d\theta\>e^{-iN\theta}e^{iz_{l}\sin{\theta}}$ (54)

with $z_{l}=F_{l}d_{l}/\omega\;\;(l=x,y)$ as follows:

$\displaystyle\tilde{D}_{3,nn^{\prime}}(\bm{k},\omega)=\left\\{\begin{array}[]{l}t_{sp}J_{\Delta
n}(z_{x})\sin{(k_{x}d_{x})}\\\ \hskip 25.60747pt{\rm for}\;\;\Delta n=0,\pm
2,\pm 4,\cdots\\\ it_{sp}J_{\Delta n}(z_{x})\cos{(k_{x}d_{x})}\\\ \hskip
25.60747pt{\rm for}\;\;\Delta n=\pm 1,\pm 3,\cdots\end{array}\right.,$ (59)
$\displaystyle\tilde{D}_{4,nn^{\prime}}(\bm{k},\omega)$
$\displaystyle=\left\\{\begin{array}[]{l}\Delta_{g}-\Sigma_{l=x,y}2t^{xy}\left[1-J_{0}(z_{l})\cos{(k_{l}d_{l})}\right]\\\
\hskip 25.60747pt-2t^{z}\left[1-\cos{(k_{z}d_{z})}\right]\\\ \hskip
85.35826pt{\rm for}\;\;\Delta n=0\\\ 2t^{xy}\left[J_{\Delta
n}(z_{x})\cos{\left(k_{x}d_{x}\right)}\right.\\\ \hskip
25.60747pt\left.+J_{\Delta n}(z_{y})\cos{\left(k_{y}d_{y}+{\Delta n\over
2}\pi\right)}\right]\\\ \hskip 85.35826pt{\rm for}\;\;\Delta n=\pm 2,\pm
4,\cdots\\\ 2t^{xy}\left[-iJ_{\Delta
n}(z_{x})\sin{\left(k_{x}d_{x}\right)}\right.\\\ \hskip
25.60747pt\left.+J_{\Delta n}(z_{y})\cos{\left(k_{y}d_{y}+{\Delta n\over
2}\pi\right)}\right]\\\ \hskip 85.35826pt{\rm for}\;\;\Delta n=\pm 1,\pm
3,\cdots\\\ \end{array}\right.,$ (69)

and

$\displaystyle\tilde{D}_{5,nn^{\prime}}(\bm{k},\omega)=\left\\{\begin{array}[]{l}i^{\Delta
n}t_{sp}J_{\Delta n}(z_{y})\sin{(k_{y}d_{y})}\\\ \hskip 25.60747pt{\rm
for}\;\;\Delta n=0,\pm 2,\pm 4,\cdots\\\ i^{(\Delta n-1)}t_{sp}J_{\Delta
n}(z_{y})\cos{(k_{y}d_{y})}\\\ \hskip 25.60747pt{\rm for}\;\;\Delta n=\pm
1,\pm 3,\cdots\end{array}\right..$ (74)

Further, the Fourier transform of the electric-dipole interaction
$V^{(\pm)}(t)$ of Eq. (19), given by

$\tilde{V}^{(\pm)}_{nn^{\prime}}={1\over T}\int_{0}^{T}dt\>e^{-i\Delta n\omega
t}V^{(\pm)}(t),$ (75)

is cast into

$\tilde{V}^{(\pm)}_{nn^{\prime}}=\left(\begin{array}[]{cc}0&{1\over
2i}(\Omega_{y}\pm\Omega_{x})\delta_{n,n^{\prime}+1}\\\ &-{1\over
2i}(\Omega_{y}\mp\Omega_{x})\delta_{n,n^{\prime}-1}\\\ {1\over
2i}(\Omega_{y}\mp\Omega_{x})\delta_{n,n^{\prime}+1}\\\ -{1\over
2i}(\Omega_{y}\pm\Omega_{x})\delta_{n,n^{\prime}-1}&0\end{array}\right).$ (76)

##  Appendix B Analytic expressions of energy dispersion $E(\bm{k})$

###  B.1 Floquet Hamiltonian and approximate eigenvalues

The eigenvalue problem of the following Floquet Hamiltonian for up- and down-
spins

$L^{(\pm)}(\bm{k},t)=H^{(\pm)}(\bm{k},t)+V^{(\pm)}(t)-iI_{2}{\partial\over\partial
t}$ (77)

is solved approximately to obtain analytic expressions of eigenvalue
$E(\bm{k})$, where $H^{(\pm)}(\bm{k},t)$ and $V^{(\pm)}(t)$ are given in Eqs.
(21) and (19), respectively. The associated Fourier-Floquet matrix
$\tilde{L}^{(\pm)}(\bm{k},\omega)$ is represented as

$\tilde{L}^{(\pm)}(\bm{k},\omega)=\left(\begin{array}[]{@{\,}cccc@{\,}}\ddots&\vdots&\vdots&\reflectbox{$\ddots$}\\\
\ldots&\mathcal{L}^{(\pm)}_{n,n}&\mathcal{X}^{(\pm)}_{n,n-1}&\ldots\\\
\ldots&\mathcal{X}^{(\pm)}_{n-1,n}&\mathcal{L}^{(\pm)}_{n-1,n-1}&\ldots\\\
\reflectbox{$\ddots$}&\vdots&\vdots&\ddots\end{array}\right)$ (78)

where $\mathcal{L}^{(\pm)}_{n,n}$ and $\mathcal{X}^{(\pm)}_{n,n-1}$ are
$2\times 2$-block-matrices defined by

$\mathcal{L}^{(\pm)}_{n,n}=\left(\begin{array}[]{@{\,}cccc@{\,}}\tilde{L}^{(\pm)}_{1(n)1(n)}&\tilde{L}^{(\pm)}_{1(n)2(n+1)}\\\
\tilde{L}^{(\pm)}_{2(n+1)1(n)}&\tilde{L}^{(\pm)}_{2(n+1)2(n+1)}\end{array}\right)$
(79)

and

$\mathcal{X}^{(\pm)}_{n,n-1}=\left(\begin{array}[]{@{\,}cccc@{\,}}\tilde{L}^{(\pm)}_{1(n)1(n-1)}&\tilde{L}^{(\pm)}_{1(n)2(n)}\\\
\tilde{L}^{(\pm)}_{2(n+1)1(n-1)}&\tilde{L}^{(\pm)}_{2(n+1)2(n)}\end{array}\right)$
(80)

with $\mathcal{X}^{(\pm)}_{n-1,n}=[\mathcal{X}^{(\pm)}_{n,n-1}]^{\dagger},$
respectively. Here, $\tilde{L}^{(\pm)}_{b(n)b^{\prime}(n^{\prime})}$ is a
Fourier-transform of the $(b,b^{\prime})$-matrix element of
$L^{(\pm)}(\bm{k},t)$, that is,

$\tilde{L}^{(\pm)}_{b(n)b^{\prime}(n^{\prime})}={1\over
T}\int_{0}^{T}dt\>e^{-i\Delta n\omega t}L^{(\pm)}_{bb^{\prime}}(\bm{k},t),$
(81)

where it is understood that the index $b=1(2)$ represents the conduction
(valence) band.

Now, an approximation is made that $\tilde{L}^{(\pm)}(\bm{k},\omega)$ is
replaced by a sequence of $4\times 4$-block matrices $\\{L^{(\pm)}_{n}\\}$,

$\tilde{L}^{(\pm)}(\bm{k},\omega)\approx\left(\begin{array}[]{@{\,}ccccc@{\,}}\ddots&&&&\\\
&L^{(\pm)}_{n+1}&&O&\\\ &&L^{(\pm)}_{n}&&\\\ &{}^{t}O&&L^{(\pm)}_{n-1}&\\\
&&&&\ddots\end{array}\right),$ (82)

where the off-diagonal elements are represented in terms of a null upper-
triangular block $O$ and its transpose ${}^{t}O$, and the diagonal block
matrix is given by

$L_{n}^{(\pm)}=\left(\begin{array}[]{@{\,}cc@{\,}}\mathcal{L}^{(\pm)}_{n,n}&\mathcal{X}^{(\pm)}_{n,n-1}\\\
\mathcal{X}^{(\pm)}_{n-1,n}&\mathcal{L}^{(\pm)}_{n-1,n-1}\end{array}\right).$
(83)

It is obvious that this corresponds to the rotational-wave approximation, in
which just almost resonant coupling terms of $\mathcal{X}^{(\pm)}_{n,n-1}$ and
$\mathcal{X}^{(\pm)}_{n-1,n}$ are sustained, and the rest terms are
disregarded. In Eq. (79), the diagonal elements of
$\tilde{L}^{(\pm)}_{1(n)1(n)}$ and $\tilde{L}^{(\pm)}_{2(n+1)2(n+1)}$ of
$\mathcal{L}^{(\pm)}_{n,n}$ are strongly coupled by the off-diagonal element,
$\tilde{L}^{(\pm)}_{1(n)2(n+1)}$ and $\tilde{L}^{(\pm)}_{2(n+1)1(n)}$
representing interband interactions, when an almost on-resonant condition is
met. Setting a set of eigenvalues and the associated eigenvectors of
$\mathcal{L}^{(\pm)}_{n,n}$ as $\mathcal{E}^{(\pm)}_{n,\alpha}$ and
$u^{(\pm)}_{n,\alpha}$ with $\alpha=1,2$, respectively, that is,

$\mathcal{L}^{(\pm)}_{n,n}u^{(\pm)}_{n,\alpha}=\mathcal{E}^{(\pm)}_{n,\alpha}u^{(\pm)}_{n,\alpha},$
(84)

with $U^{(\pm)}_{n}$ as a $2\times 2$-unitary matrix given by

$U^{(\pm)}_{n}=\left(u^{(\pm)}_{n,1}u^{(\pm)}_{n,2}\right),$ (85)

$\bar{L}^{(\pm)}_{n}$ defined as

$\bar{L}^{(\pm)}_{n}=U^{(\pm)\dagger}_{n}L^{(\pm)}_{n}U^{(\pm)}_{n}$ (86)

becomes of the form

$\displaystyle\bar{L}^{(\pm)}_{n}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{@{\,}cc@{\,}}\mathcal{E}^{(\pm)}_{n}&\bar{\mathcal{X}}^{(\pm)}_{n,n-1}\\\
\bar{\mathcal{X}}^{(\pm)}_{n-1,n}&\mathcal{E}^{(\pm)}_{n-1}\end{array}\right)$
(89) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{@{\,}c|cc|c@{\,}}\mathcal{E}^{(\pm)}_{n,1}&0&\times&\times\\\
\hline\cr 0&\mathcal{E}^{(\pm)}_{n,2}&\mathcal{V}^{(\pm)}&\times\\\
\times&\mathcal{V}^{(\pm)*}&\mathcal{E}^{(\pm)}_{n-1,1}&0\\\
\hline\cr\times&\times&0&\mathcal{E}^{(\pm)}_{n-1,2}\end{array}\right),$ (94)

where

$\mathcal{E}^{(\pm)}_{n}=\left(\begin{array}[]{@{\,}cc@{\,}}\mathcal{E}^{(\pm)}_{n,1}&0\\\
0&\mathcal{E}^{(\pm)}_{n,2}\end{array}\right)$ (95)

and

$\bar{\mathcal{X}}^{(\pm)}_{n.n-1}=U^{(\pm)\dagger}_{n}\mathcal{X}^{(\pm)}_{n,n-1}U^{(\pm)}_{n-1}\equiv\left(\begin{array}[]{@{\,}cc@{\,}}\times&\times\\\
\mathcal{V}^{(\pm)}&\times\end{array}\right).$ (96)

In the second equality of the above equation, $\mathcal{V}^{(\pm)}$ represents
the (2,1)-components of $\bar{\mathcal{X}}^{(\pm)}_{n.n-1}$ and all other
components are expressed just as a symbol $\times$; $\mathcal{V}^{(\pm)}$ is
independent of $n$, as shown later. Here, it is supposed that just the
component $\mathcal{V}^{(\pm)}$ is retained with neglecting the components
denoted as $\times$.

Thus, one obtains two kinds of eigenenergies from Eq. (94), — denoted as
$E^{(\pm)}_{n,\beta}(\bm{k})$ with $\beta=1,2$ —, which are the eigenvalues of
the $2\times 2$-block matrix

$\bar{L}^{(\pm)}_{D,n}\equiv\left(\begin{array}[]{@{\,}cc@{\,}}\mathcal{E}^{(\pm)}_{n,2}&\mathcal{V}^{(\pm)}\\\
\mathcal{V}^{(\pm)*}&\mathcal{E}^{(\pm)}_{n-1,1}\end{array}\right).$ (97)

Explicit expressions of $E^{(\pm)}_{n,\beta}$ are provided as

$\displaystyle E^{(\pm)}_{n,\beta}(\bm{k})$ $\displaystyle=$
$\displaystyle{1\over
2}\left[\mathcal{E}^{(\pm)}_{n,2}+\mathcal{E}^{(\pm)}_{n-1,1}-(-1)^{\beta}\right.$
(98)
$\displaystyle\left.\times\sqrt{\left(\mathcal{E}^{(\pm)}_{n,2}-\mathcal{E}^{(\pm)}_{n-1,1}\right)^{2}+4|\mathcal{V}^{(\pm)}|^{2}}\right].$

Therefore, the approximate eigenvalues of $\tilde{L}^{(\pm)}(\bm{k},\omega)$
of Eq. (82) are given in terms of a series of the set of eigenvalues
$\\{E^{(\pm)}_{n,1}(\bm{k}),E^{(\pm)}_{n,2}(\bm{k})\\}$.

###  B.2 Energy dispersions of FWSM states

The more detailed expressions of $E^{(\pm)}_{n,\beta}(\bm{k})$ are sought. To
do this, $\mathcal{E}^{(\pm)}_{n,\alpha}$ is represented in terms of the
matrix elements of Eq. (79) as

$\mathcal{E}^{(\pm)}_{n,\alpha}={1\over
2}\left[\epsilon^{0(\pm)}_{n}-(-1)^{\alpha}\sqrt{\left(\Delta\epsilon^{(\pm)}_{n}\right)^{2}+4|\mathcal{W}^{(\pm)}|^{2}}\right],$
(99)

where

$\epsilon^{0(\pm)}_{n}=\tilde{L}^{(\pm)}_{1(n)1(n)}+\tilde{L}^{(\pm)}_{2(n+1)2(n+1)},$
(100)
$\Delta\epsilon^{(\pm)}_{n}=\tilde{L}^{(\pm)}_{1(n)1(n)}-\tilde{L}^{(\pm)}_{2(n+1)2(n+1)},$
(101)

and

$\mathcal{W}^{(\pm)}=\tilde{L}^{(\pm)}_{1(n)2(n+1)}.$ (102)

According to the explicit expressions of
$\tilde{D}_{j,nn^{\prime}}(\bm{k},\omega)$ obtained in Appendix A, it is shown
that

$\epsilon^{0(\pm)}_{n}=(2n+1)\omega$ (103)

and

$\Delta\epsilon^{(\pm)}_{n}\equiv 2\eta(\bm{k})=2D(\bm{k})-\omega,$ (104)

where

$\displaystyle D(\bm{k})$ $\displaystyle=$
$\displaystyle\Delta_{g}+\epsilon_{z}(k_{z})$ (105)
$\displaystyle-\sum_{l=x,y}2t^{xy}\left[1-J_{0}(z_{l})\cos{(k_{l}d_{l})}\right],$

with

$\epsilon_{z}(k_{z})=-2t^{z}\left[1-\cos{(k_{z}d_{z})}\right].$ (106)

Similarly, in view of $\tilde{V}^{(\pm)}_{nn^{\prime}}$ given in Appendix A,
one has

$\displaystyle\left\\{\begin{array}[]{l}\mathcal{W}^{(+)}=-{i\over
2}\Omega_{x}-it_{sp}J_{1}(z_{x})\cos{(k_{x}d_{x})}\\\
\mathcal{W}^{(-)}=(\mathcal{W}^{(+)})^{*}\end{array}\right.$ (109)

for the linearly polarized light, and

$\displaystyle\left\\{\begin{array}[]{lll}\mathcal{W}^{(+)}&=&-it_{sp}J_{1}(z_{c})\left[\cos{(k_{x}d_{x})}+\cos{(k_{y}d_{y})}\right]\\\
\mathcal{W}^{(-)}&=&it_{sp}J_{1}(z_{c})\left[\cos{(k_{x}d_{x})}-\cos{(k_{y}d_{y})}\right]\\\
&&+i\Omega_{c}\end{array}\right.$ (113)

for the circularly polarized light, with
$\Omega_{c}\equiv\Omega_{x}=\Omega_{y}$.

Further, $U^{(\pm)}_{n}$ of Eq. (85) is given by

$u^{(\pm)}_{n,1}=\left(\begin{array}[]{l}\cos{\Theta^{(\pm)}}\\\
\sin{\Theta^{(\pm)}}e^{-i\Phi^{(\pm)}}\end{array}\right)$ (114)

and

$u^{(\pm)}_{n,2}=\left(\begin{array}[]{l}\sin{\Theta^{\prime(\pm)}}\>e^{i\Phi^{(\pm)}}\\\
\cos{\Theta^{\prime(\pm)}}\end{array}\right),$ (115)

where

$\tan{\Theta^{(\pm)}}={|\mathcal{W}^{(\pm)}|\over\eta(\bm{k})+\sqrt{\eta(\bm{k})^{2}+|\mathcal{W}^{(\pm)}|^{2}}},$
(116) $\tan{\Theta^{\prime(\pm)}}=-\tan{\Theta^{(\pm)}},$ (117)

and

$e^{i\Phi^{(\pm)}}={\mathcal{W}^{(\pm)}\over|\mathcal{W}^{(\pm)}|}.$ (118)

Thus, $\mathcal{V}^{(\pm)}$ of Eq. (96) is approximately given by

$\mathcal{V}^{(\pm)}\approx\tilde{L}^{(\pm)}_{1(n)2(n)}\sin{\Theta^{(\pm)}}\sin{\Theta^{\prime(\pm)}}e^{-2i\Phi^{(\pm)}},$
(119)

where just the most dominant component $\tilde{L}^{(\pm)}_{1(n)2(n)}$ in
$\mathcal{X}^{(\pm)}_{n,n-1}$ of Eq. (80) is kept under the condition that
$J_{0}(z_{l})\gg J_{n(\geq 1)}(z_{l})$ in the range of $z_{l}$ concerned here.
To be more specific, $\tilde{L}^{(\pm)}_{1(n)2(n)}$ is represented as

$\tilde{L}^{(\pm)}_{1(n)2(n)}=\pm
t_{sp}\left[J_{0}(z_{x})\sin{(k_{x}d_{x})}\mp i\sin{(k_{y}d_{y})}\right]$
(120)

for the the linearly polarized light, and

$\tilde{L}^{(\pm)}_{1(n)2(n)}=\pm
t_{sp}J_{0}(z_{c})\left[\sin{(k_{x}d_{x})}\mp i\sin{(k_{y}d_{y})}\right]$
(121)

for the the circularly polarized light.

According to the above results, $E^{(\pm)}_{n,\beta}(\bm{k})$ of Eq. (98) is
cast into

$E^{(\pm)}_{n,\beta}(\bm{k})=n\omega+E^{(\pm)}_{0,\beta}(\bm{k}),$ (122)

where

$\displaystyle E^{(\pm)}_{0,\beta}(\bm{k})$ $\displaystyle=$
$\displaystyle(-1)^{\beta-1}\left[\left\\{\omega/2-\left(\eta(\bm{k})^{2}+|\mathcal{W}^{(\pm)}|^{2}\right)^{1/2}\right\\}^{2}\right.$
(123) $\displaystyle\left.+|\mathcal{V}^{(\pm)}|^{2}\right]^{1/2}.$

Obviously, it is likely that these two photon sidebands,
$E^{(\pm)}_{n,1}(\bm{k})$ and $E^{(\pm)}_{n^{\prime},2}(\bm{k})$, touch each
other when the following conditions are met: $n=n^{\prime}$ and the expression
inside the square brackets of Eq. (123) vanishes. The second condition is
ensured only when $k_{x}=k_{y}=0$, that is, $\mathcal{V}^{(\pm)}=0$. Thus,
this becomes the existence condition of a touching point at $k_{z}^{D/W(\pm)}$
satisfying the relation

$\epsilon_{z}(k_{z}^{D/W(\pm)})=\omega/2-\Delta^{\prime}_{g}+\sqrt{(\omega/2)^{2}-|\mathcal{W}_{0}^{(\pm)}|^{2}},$
(124)

where

$\Delta^{\prime}_{g}=\Delta_{g}-\sum_{l=x,y}2t^{xy}[1-J_{0}(z_{l})]$ (125)

and

$\mathcal{W}_{0}^{(\pm)}=\mathcal{W}^{(\pm)}\left|{}_{k_{x}=k_{y}=0}\right..$
(126)

Here, $k_{z}^{D(\pm)}$ represents a solution of Eq. (124) for the drive of
linearly polarized laser, while $k_{z}^{W(\pm)}$ for the drive of circularly
polarized laser. In view of Eq. (106), a certain pair of values,
$k_{z}^{D/W(\pm)}$ and $-k_{z}^{D/W(\pm)}$, exists under the condition that

$\displaystyle-\omega/2+\Delta^{\prime}_{g}$ $\displaystyle<$
$\displaystyle\sqrt{(\omega/2)^{2}-|\mathcal{W}_{0}^{(\pm)}|^{2}}$ (127)
$\displaystyle<$ $\displaystyle-\omega/2+\Delta^{\prime}_{g}-4t^{z}$

with $t^{z}<0$, and

$\omega/2>|\mathcal{W}_{0}^{(\pm)}|.$ (128)

The above condition is recast into

Max $\displaystyle\left(2|\mathcal{W}_{0}^{(\pm)}|,\frac{\Delta^{\prime
2}_{g}+|\mathcal{W}_{0}^{(\pm)}|^{2}}{\Delta^{\prime}_{g}}\right)<\omega$
(129)
$\displaystyle<\frac{(\Delta^{\prime}_{g}-4t^{z})^{2}+|\mathcal{W}_{0}^{(\pm)}|^{2}}{\Delta^{\prime}_{g}-4t^{z}}.$

It is evident that these pairs of values $k_{z}^{D/W(\pm)}$ and
$-k_{z}^{D/W(\pm)}$ correspond to nodal points lying on the $\Gamma-Z$ axis in
the BZ for FDSM/FWSM states.

###  B.3 Band structures of $E^{(\pm)}_{hh(1)}(\bm{k},\omega)$ and
$E^{(\pm)}_{e(-1)}(\bm{k},\omega)$

Here, the label $b(n)$ (with $b=e,hh$) is introduced to represent a Floquet
state attributed to $b$-band with $n$ photon dressing; $e$ and $hh$ mean
electron and heavy-hole bands, respectively. Following this, a Floquet energy
$E^{(\pm)}(\bm{k})$ is given by

$E^{(\pm)}_{hh(n+1)}(\bm{k})\equiv E^{(\pm)}_{n,1}(\bm{k})\geq n\omega$ (130)

and

$E^{(\pm)}_{e(n-1)}(\bm{k})\equiv E^{(\pm)}_{n,2}(\bm{k})\leq n\omega$ (131)

in the case that a Floquet band $hh(n+1)$ is located above a Floquet band
$e(n-1)$. In the opposite case that $hh(n+1)$ is located below $e(n-1)$, it is
understood that the above definition of $E^{(\pm)}_{hh(n+1)}(\bm{k})$ is
replaced by that of $E^{(\pm)}_{e(n-1)}(\bm{k})$.

Below, an energy dispersion of
$E^{(\pm)}(\bm{k})=E^{(\pm)}_{hh(1)}(\bm{k})\geq 0$ in the vicinity of
$\bm{k}^{D/W(\pm)}\equiv(0,0,k_{z}^{D/W(\pm)})$ is examined, where
$E^{(\pm)}_{e(-1)}(\bm{k})=-E^{(\pm)}_{hh(1)}(\bm{k})$. To do this,
$\eta(\bm{k})$, $\mathcal{W}^{(\pm)}$, and $\mathcal{V}^{(\pm)}$ are expanded
around this point as follows:

$\displaystyle\eta(\bm{k})$ $\displaystyle\approx$
$\displaystyle\eta_{0}+\eta_{z}^{(1)}(\Delta k_{z}d_{z})$ (132)
$\displaystyle+\sum_{l=x,y}\eta_{l}(k_{l}d_{l})^{2}+\eta_{z}(\Delta
k_{z}d_{z})^{2},$
$\mathcal{W}^{(\pm)}\approx\mathcal{W}^{(\pm)}_{0}+\sum_{l=x,y}\omega^{(\pm)}_{l}(k_{l}d_{l})^{2},$
(133)

and

$\mathcal{V}^{(\pm)}\approx\sum_{l=x,y}\nu^{(\pm)}_{l}(k_{l}d_{l}),$ (134)

where $\Delta k_{z}=k_{z}-k_{z}^{D/W(\pm)}$, and the expansion coefficients
$\eta_{0},\eta_{z}^{(1)},\eta_{l},\eta_{z},\mathcal{W}^{(\pm)}_{0},\omega_{l}^{(\pm)}$,
and $\nu_{l}^{(\pm)}$ are represented by

$\displaystyle\eta_{0}$ $\displaystyle=$
$\displaystyle\Delta^{\prime}_{g}+\epsilon_{z}(k_{z}^{D/W(\pm)})-\omega/2,\eta_{z}^{(1)}=-2t^{z}\sin{(k_{z}^{D/W(\pm)}d_{z})},$
$\displaystyle\eta_{l}$ $\displaystyle\approx$
$\displaystyle-t^{xy},\eta_{z}=-t^{z}\cos{(k_{z}^{D/W(\pm)}d_{z})},$ (135)
$\displaystyle\left\\{\begin{array}[]{lll}\mathcal{W}^{(+)}_{0}=-{i\over
2}\Omega_{x},&\mathcal{W}^{(-)}_{0}={i\over 2}\Omega_{x}&\text{(linear)}\\\
\mathcal{W}^{(+)}_{0}=0,&\mathcal{W}^{(-)}_{0}=i\Omega_{c}&\text{(circular)}\end{array}\right.,$
(138) $\omega_{l}^{(\pm)}\approx 0$ (139)

and

$\nu_{x}^{(\pm)}=\pm\sigma^{(\pm)}t_{sp},\;\;\nu_{y}^{(\pm)}=-i\sigma^{(\pm)}t_{sp},$
(140)

respectively. It is considered that $J_{0}(z_{l})\approx 1$ in the range of
$z_{l}$ concerned here in Eq. (135), terms including $t_{sp}J_{1}(z_{l})$ are
neglected due to $|J_{1}(z_{l})|\ll 1$ in Eqs. (138) and (139), and the
prefactor $\sigma^{(\pm)}$ is given by

$\sigma^{(\pm)}=\sin{\Theta^{(\pm)}_{0}}\sin{\Theta^{\prime(\pm)}_{0}}e^{-2i\Phi^{(\pm)}_{0}}$
(141)

in Eq. (140), where

$\displaystyle\Theta^{(\pm)}_{0}=\Theta^{(\pm)}|_{\bm{k}=\bm{k}^{D/W(\pm)}},\;\;\Theta^{\prime(\pm)}_{0}=\Theta^{\prime(\pm)}|_{\bm{k}=\bm{k}^{D/W(\pm)}},$
$\displaystyle\Phi^{(\pm)}_{0}=\Phi^{(\pm)}|_{\bm{k}=\bm{k}^{D/W(\pm)}}.$
(142)

For the FDSM driven by the linearly polarized light,

$\displaystyle E^{(\pm)}(\bm{k})$
$\displaystyle\approx\left[\left|\sum_{l=x,y}\nu_{l}^{(\pm)}(k_{l}d_{l})\right|^{2}+\frac{(\eta_{0}\eta_{z}^{(1)})^{2}}{(\omega/2)^{2}}(\Delta
k_{z}d_{z})^{2}\right]^{1/2}$
$\displaystyle=\left[\sum_{l=x,y}(\xi^{D}_{l})^{2}(k_{l}d_{l})^{2}+(\xi^{D}_{z})^{2}(\Delta
k_{z}d_{z})^{2}\right]^{1/2},$ (143)

where $\Delta k_{z}=k_{z}-k_{z}^{D(\pm)}$,
$\xi^{D}_{x}=\xi^{D}_{y}=|\sigma^{(\pm)}|t_{sp}$ with

$\displaystyle|\sigma^{(\pm)}|=\frac{(\Omega_{x}/2)^{2}}{(\Omega_{x}/2)^{2}+\left(\eta_{0}+\sqrt{\eta_{0}^{2}+(\Omega_{x}/2)^{2}}\right)^{2}},$
(144)

and $\xi^{D}_{z}=|\eta_{0}\eta_{z}^{(1)}|/(\omega/2)$. Hence, it is seen that
the Floquet bands of $E^{(\pm)}_{hh(1)}(\bm{k})$ and
$E^{(\pm)}_{e(-1)}(\bm{k})$ cross linearly at the Dirac points of
$\bm{k}^{D(\pm)}$ and $-\bm{k}^{D(\pm)}$. On the other hand, for the FWSM
driven by the circularly polarized light,

$\displaystyle E^{(+)}(\bm{k})$
$\displaystyle\approx\left|\sum_{l=x,y}\eta_{l}(k_{l}d_{l})^{2}+\eta_{z}^{(1)}(\Delta
k_{z}d_{z})+\eta_{z}(\Delta k_{z}d_{z})^{2}\right|$
$\displaystyle=\left|(-t^{xy})\sum_{l=x,y}(k_{l}d_{l})^{2}+\eta_{z}^{(1)}(\Delta
k_{z}d_{z})+\eta_{z}(\Delta k_{z}d_{z})^{2}\right|,$ (145)

due to $\mathcal{W}^{(+)}_{0}=0$ and
$\nu_{x}^{(+)}=i\nu_{y}{(+)}=\sigma^{(+)}t_{sp}\approx 0$ with
$|\sigma^{(+)}|=[t_{sp}J_{1}(z_{c})/(2\eta_{0})]^{2}$, where $\Delta
k_{z}=k_{z}-k_{z}^{W(+)}$, while

$\displaystyle E^{(-)}(\bm{k})$
$\displaystyle\approx\left[\left|\sum_{l=x,y}\nu_{l}^{(-)}(k_{l}d_{l})\right|^{2}+\frac{(\eta_{0}\eta_{z}^{(1)})^{2}}{(\omega/2)^{2}}(\Delta
k_{z}d_{z})^{2}\right]^{1/2}$
$\displaystyle=\left[\sum_{l=x,y}(\xi^{W}_{l})^{2}(k_{l}d_{l})^{2}+(\xi^{W}_{z})^{2}(\Delta
k_{z}d_{z})^{2}\right]^{1/2},$ (146)

where $\Delta k_{z}=k_{z}-k_{z}^{W(-)}$,
$\xi^{W}_{x}=\xi^{W}_{y}=|\sigma^{(-)}|t_{sp}$ with

$\displaystyle|\sigma^{(-)}|=\frac{(\Omega_{c})^{2}}{(\Omega_{c})^{2}+\left(\eta_{0}+\sqrt{\eta_{0}^{2}+(\Omega_{c})^{2}}\right)^{2}},$
(147)

and $\xi^{W}_{z}=|\eta_{0}\eta_{z}^{(1)}|/(\omega/2)$. Hence, it is seen that
the Floquet bands of $E^{(+)}_{hh(1)}(\bm{k})$ and $E^{(+)}_{e(-1)}(\bm{k})$
cross in a quadratic manner in the $k_{x}$\- and $k_{y}$-directions and in a
linear manner in the $k_{z}$-direction at the Weyl points of $\bm{k}^{W(+)}$
and $-\bm{k}^{W(+)}$. On the other hand, those of $E^{(-)}_{hh(1)}(\bm{k})$
and $E^{(-)}_{e(-1)}(\bm{k})$ cross in a linear manner at the Weyl points of
$\bm{k}^{W(-)}$ and $-\bm{k}^{W(-)}$, similar to the FDSMs.

Finally, band structures in the vicinity of the $k_{x}-k_{y}$ plane with
$k_{z}$ fixed are examined. For the FWSM of the up-spin state,
$E^{(+)}(\bm{k})$ is represented simply as

$E^{(+)}(\bm{k})\approx|\omega/2-D(\bm{k})|$ (148)

due to Eq. (123), where $\mathcal{W}^{(+)}$ and $\mathcal{V}^{(+)}$ are
neglected within the order of $t_{sp}J_{1}(z_{c})\approx 0$. Thus,
$E^{(+)}_{hh(1)}(\bm{k})\approx 0$ for $\bm{k}$’s situated on the closed
surface

$\displaystyle-2t^{xy}\sum_{l=x,y}[1-J_{0}(z_{l})\cos{(k_{l}d_{l})}]-2t^{z}[1-\cos{(k_{z}d_{z})}]$
$\displaystyle=\omega-\Delta_{g}>0.$ (149)

This shows that there exists a nodal ring in the $k_{x}-k_{y}$ plane, which is
reminiscent of a NLSM phase in the FWSM of concern. As regards the FWSM of the
down-spin state, $E^{(-)}_{hh(1)}(\bm{k})$ and $E^{(-)}_{e(-1)}(\bm{k})$ are
gapped out in the $k_{x}-k_{y}$ plane due to $\mathcal{V}^{(-)}$ that is not
negligible. It is remarked that an accidental band crossing occurs between
these two bands at high-symmenty points of
$\bm{k}^{*}=(0,0,0),(0,\pm\pi/d_{y},0),(\pm\pi/d_{x},0,0)$, and
$(\pm\pi/d_{x},\pm\pi/d_{y},0)$ at which $\mathcal{V}^{(-)}=0$, only when the
relation

$\omega/2=\sqrt{\eta(\bm{k}^{*})^{2}+\Omega_{c}^{2}}$ (150)

is ensured. In passing, the similar result with that of the FWSM of the down-
spin state is applied for the FDSMs.

###  B.4 Floquet-Weyl Hamiltonian and Chirality

Here, effective Floquet-Weyl Hamiltonians the eigenvalues of which are given
by Eqs. (145) and (146) are derived from Eq. (97) in the vicinity of the Weyl
points $\bm{k}^{W(+)}$ and $\bm{k}^{W(-)}$. The Hamiltonian for the up-spin
band, represented as $L_{n}^{W(+)}\equiv\bar{L}_{D,n}^{(+)}$, is cast into

$\displaystyle L_{n}^{W(+)}$ $\displaystyle=$ $\displaystyle
n\omega+\left(\begin{array}[]{cc}-\eta_{z}^{(1)}(\Delta
k_{z}d_{z})&\sum_{l=x,y}\nu^{(+)}_{l}(k_{l}d_{l})\\\
\sum_{l=x,y}\nu_{l}^{(+)*}(k_{l}d_{l})&\eta_{z}^{(1)}(\Delta
k_{z}d_{z})\end{array}\right)$ (153) $\displaystyle=$ $\displaystyle
n\omega+h_{+}^{(+)}\bm{p}^{(+)}\cdot\bm{\sigma},$ (154)

where Eqs. (99), (132), (133), and (134) are used. Further, the effective
momentum $\bm{p}^{(+)}$, defined in the right-handed system, is given by

$\displaystyle p_{l}^{(+)}$ $\displaystyle=$ $\displaystyle
h_{+}^{(+)}\sum_{l^{\prime}=x,y,z}v_{ll^{\prime}}^{(+)}(k_{l^{\prime}}-k_{l^{\prime}}^{W(+)})$
(155) $\displaystyle\equiv$ $\displaystyle
h_{+}^{(+)}\bm{v}_{l}^{(+)}\cdot(\bm{k}-\bm{k}^{W(+)}),$

where

$\displaystyle\left\\{\begin{array}[]{l}\bm{v}_{x}^{(+)}=(\nu^{(+)}_{x}d_{x},0,0)\\\
\bm{v}_{y}^{(+)}=(0,\nu^{(+)}_{x}d_{y},0)\\\
\bm{v}_{z}^{(+)}=(0,0,-\eta_{z}^{(1)}d_{z})\end{array}\right.$ (159)

and $h_{+}^{(+)}$ is a helicity of particle at the Weyl point $\bm{k}^{W(+)}$,
defined by

$h_{+}^{(+)}=\text{sgn}(\bm{v}_{x}^{(+)}\times\bm{v}_{y}^{(+)}\cdot\bm{v}_{z}^{(+)}),$
(160)

which is either +1 or -1. Thus, the positive eigenvalue of Eq. (154) for $n=0$
is given by

$\displaystyle E^{W(+)}(\bm{k})$ $\displaystyle=$
$\displaystyle\left|(\eta_{z}^{(1)})^{2}(\Delta
k_{z}d_{z})^{2}+\nu_{x}^{(+)2}\sum_{l=x,y}(k_{l}d_{l})^{2}\right|^{1/2}$ (161)
$\displaystyle\approx$ $\displaystyle\left|\eta_{z}^{(1)}(\Delta
k_{z}d_{z})\right|,$

where in the second equality the fact that $\nu^{(+)}_{x}\approx 0$ is
considered. This is identical to Eq. (145) within the first order with respect
to $k_{x},k_{y},$ and $\Delta k_{z}$.

Similarly, the Hamiltonian for the down-spin band, represented as
$L_{n}^{W(-)}\equiv\bar{L}_{D,n}^{(-)}$, is cast into

$\displaystyle L_{n}^{W(-)}$ $\displaystyle=$ $\displaystyle
n\omega+\left(\begin{array}[]{cc}-{\eta_{0}\eta_{z}^{(1)}\over(\omega/2)}(\Delta
k_{z}d_{z})&\sum_{l=x,y}\nu^{(-)}_{l}(k_{l}d_{l})\\\
\sum_{l=x,y}\nu^{(-)*}_{l}(k_{l}d_{l})&{\eta_{0}\eta_{z}^{(1)}\over(\omega/2)}(\Delta
k_{z}d_{z})\end{array}\right)$ (164) $\displaystyle=$ $\displaystyle
n\omega+h_{+}^{(-)}\bm{p}^{(-)}\cdot\bm{\sigma}.$ (165)

Here, the effective momentum $\bm{p}^{(-)}$, defined in the right-handed
system, is given by

$\displaystyle p_{l}^{(-)}$ $\displaystyle=$ $\displaystyle
h_{+}^{(-)}\sum_{l^{\prime}=x,y,z}v_{ll^{\prime}}^{(-)}(k_{l^{\prime}}-k_{l^{\prime}}^{W(-)})$
(166) $\displaystyle\equiv$ $\displaystyle
h_{+}^{(-)}\bm{v}_{l}^{(-)}\cdot(\bm{k}-\bm{k}^{W(-)}),$

where

$\displaystyle\left\\{\begin{array}[]{l}\bm{v}_{x}^{(-)}=(\nu^{(-)}_{x}d_{x},0,0)\\\
\bm{v}_{y}^{(-)}=(0,-\nu^{(-)}_{x}d_{y},0)\\\
\bm{v}_{z}^{(-)}=(0,0,-{\eta_{0}\eta_{z}^{(1)}\over(\omega/2)}d_{z})\end{array}\right.$
(170)

and $h_{+}^{(-)}$ is a helicity of particle at the Weyl point $\bm{k}^{W(-)}$,
defined by

$h_{+}^{(-)}=\text{sgn}(\bm{v}_{x}^{(-)}\times\bm{v}_{y}^{(-)}\cdot\bm{v}_{z}^{(-)}),$
(171)

which is either +1 or -1. It is obvious that the positive eigenvalue of Eq.
(165) for $n=0$, $E^{W(-)}(\bm{k})$, is identical to Eq. (146).

Finally, the chiralities of FWSMs for both up- and down-spins are examined.
Let the helicities of the Weyl cones at $-\bm{k}^{W(+)}$ and $-\bm{k}^{W(-)}$
be represented as $h_{-}^{(+)}$ and $h_{-}^{(-)}$, respectively. It is evident
that $h_{+}^{(+)}h_{-}^{(+)}=-1$ and $h_{+}^{(-)}h_{-}^{(-)}=-1$, since
according to Eqs (135) and (140), the replacement of the nodal position at
$\bm{k}^{W(\pm)}$ by that at $-\bm{k}^{W(\pm)}$ still keeps $\nu_{x}^{(\pm)}$
unaltered, whereas $\eta_{z}^{(1)}$ changes its sign; a double sign
corresponds. Further, it is also seen that $h_{+}^{(+)}h_{+}^{(-)}=-1$ and
$h_{-}^{(+)}h_{-}^{(-)}=-1$, since the sign of $\eta_{z}^{(1)}$ remains
unaltered, whereas the sign of $\nu_{\pm}^{(+)}$ is different from that of
$\nu_{\pm}^{(-)}$; a double sign corresponds. Therefore, it is verified that
each of four pairs of the Weyl cones at $\pm\bm{k}^{W(+)}$,
$\pm\bm{k}^{W(-)}$, $\bm{k}^{W(\pm)}$, and $-\bm{k}^{W(\pm)}$ possesses
opposite chiralities.

##  Appendix C Enlarged view of Figs. 3(a) and 3(b)

Energy dispersions shown in Figs. 3(a) and 3(b) are enlarged in Fig. 7 to make
clearer the difference of band gaps between the up-spin and down-spin bands
around $E=0$.

Figure 7: Energy dispersions $E(\bm{k})$ of up-spin and down-spin bands at
$k_{z}=0$ which are enlarged around $E=0$. Here, bands dominated rather by the
$s/p$-orbital component are denoted by a red/blue solid line. The axis of
abscissa $k_{l}$ is gauged in the unit of $1/d_{l}$ with $l=x,y,z$. (a)
$E(\bm{k})$ in the $k_{x}-k_{y}$ plane at $k_{z}=0$ for the up-spin bands. (b)
The same as panel (a) but for the down-spin bands.

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