# Light-Induced Mirror Symmetry Breaking and Charge Transport

Naoya Arakawa1 E-mail address<EMAIL_ADDRESS>Kenji Yonemitsu1,2
1The Institute of Science and Engineering1The Institute of Science and
Engineering Chuo University Chuo University Bunkyo Bunkyo Tokyo Tokyo 112-8551
112-8551 Japan
2Department of Physics Japan
2Department of Physics Chuo University Chuo University Bunkyo Bunkyo Tokyo
112-8551 Tokyo 112-8551 Japan Japan

###### Abstract

We propose that light can break mirror symmetries and combining symmetries
with a uniform time translation, and their breaking is characterized by an
off-diagonal charge conductivity. Taking periodically driven graphene as an
example, we show that mirror symmetries about the $xz$ and $yz$ planes and the
combining symmetries, the symmetries of combinations of the mirror operations
about these planes and a uniform time translation, can be broken by linearly
or circularly polarized light. We also show that this symmetry breaking
induces the time-averaged off-diagonal symmetric or antisymmetric charge
conductivity in a nonequilibrium steady state with linearly or circularly,
respectively, polarized light. Our results are experimentally testable in
pump-probe measurements. This work will pave the way for controlling mirror
symmetries via light and utilizing the light-induced mirror symmetry breaking.

##  1 Introduction

Light can break symmetries in time and space. For example, circularly
polarized light (CPL) can break the time-reversal symmetry [1, 2, 3]. If CPL
is applied to a non-magnetic material, it can induce the magnetization [4];
the direction of this light-induced magnetization can be reversed by changing
the helicity of CPL [5]. CPL can also induce the anomalous Hall effect (AHE)
[6, 7], in which a charge current perpendicular to an applied electric field
is generated [8, 9, 10]; the magnitude and direction of this current can be
changed by varying the amplitude and helicity of CPL [6, 7, 11, 12]. Then,
bicircularly polarized light [13, 14], which consists of a linear combination
of left-handed and right-handed CPL, can break not only the time-reversal, but
also the inversion symmetry [15]. In fact, it can be used to realize
noncentrosymmetric magnetic topological phases [15] and generate electric
polarization [16]. Since the application of light enables us to engineer
electronic, magnetic, or transport properties without changing materials, it
is crucial to understand which symmetry is broken by light and how its
symmetry breaking affects the properties.

In this paper, we show the mirror symmetry breaking by CPL or by linearly
polarized light (LPL), which results in an off-diagonal antisymmetric or
symmetric, respectively, charge conductivity (i.e.,
$\sigma_{xy}^{\textrm{C}}=-\sigma_{yx}^{\textrm{C}}$ or
$\sigma_{xy}^{\textrm{C}}=\sigma_{yx}^{\textrm{C}}$, respectively). This is
demonstrated for periodically driven graphene. The difference between the
cases with CPL and LPL comes from the difference in the time-reversal
symmetry. The main results are summarized in Table 1. Our results suggest that
a combination of time-reversal symmetry breaking and mirror symmetry breaking
is the origin of the light-induced AHE, and that the off-diagonal symmetric
charge conductivity could be used to detect whether mirror symmetries are
broken or preserved in the presence of the time-reversal symmetry.

Table 1:  Properties of systems driven by CPL or LPL. The difference among LPL1, LPL2, and LPL3 is about the polarization: $A_{x}\neq 0$ and $A_{y}\neq 0$ in LPL1; $A_{x}\neq 0$ and $A_{y}=0$ in LPL2; and $A_{x}=0$ and $A_{y}\neq 0$ in LPL3. $T_{\textrm{rev}}$ represents the time-reversal symmetry, $\sigma_{\textrm{m}}$ or $\sigma_{\textrm{m}}^{\prime}$ represents the mirror symmetry about the $xz$ or $yz$ plane, respectively, and $C_{3}$ represents the $C_{3}$ rotational symmetry around the $z$ axis. $\sigma_{\textrm{m}}T_{t}$, $\sigma_{\textrm{m}}^{\prime}T_{t}$, or $C_{3}T_{t}$ represents the symmetry of a combination of the mirror operation about the $xz$ or $yz$ plane or the $C_{3}$ rotation operation and a uniform time translation $T_{t}$. $\sigma_{yx}^{\textrm{C}}$ represents an off-diagonal charge conductivity. $\sigma_{\textrm{m}}T_{t}$ or $\sigma_{\textrm{m}}^{\prime}T_{t}$ is preserved with LPL3 or LPL2, respectively, if $T_{t}:t\rightarrow t-\frac{\pi}{\Omega}$. $C_{3}T_{t}$ is preserved with CPL if $T_{t}:t\rightarrow t+\frac{2\pi}{3\Omega}$.  | CPL | LPL1 | LPL2 | LPL3
---|---|---|---|---
$T_{\textrm{rev}}$ | Broken | Preserved | Preserved | Preserved
$\sigma_{\textrm{m}}$ | Broken | Broken | Preserved | Broken
$\sigma_{\textrm{m}}T_{t}$ | Broken | Broken | Preserved | Preserved
$\sigma_{\textrm{m}}^{\prime}$ | Broken | Broken | Broken | Preserved
$\sigma_{\textrm{m}}^{\prime}T_{t}$ | Broken | Broken | Preserved | Preserved
$C_{3}$ | Broken | Broken | Broken | Broken
$C_{3}T_{t}$ | Preserved | Broken | Broken | Broken
$\sigma_{yx}^{\textrm{C}}$ | Antisymmetric | Symmetric | Vanishing | Vanishing

##  2 Model

Our periodically driven electron system is described by the Hamiltonian,

$\displaystyle H=H_{\textrm{s}}(t)+H_{\textrm{b}}+H_{\textrm{sb}}.$ (1)

Here $H_{\textrm{s}}(t)$ is the Hamiltonian of the system driven by a light
field $\bm{A}(t)$, the effect of which is treated as the Peierls phase
factors:

$\displaystyle H_{\textrm{s}}(t)$
$\displaystyle=\sum_{\bm{k}}\sum_{a,b=A,B}\sum_{\sigma=\uparrow,\downarrow}\epsilon_{ab}(\bm{k},t)c_{\bm{k}a\sigma}^{\dagger}c_{\bm{k}b\sigma},$
(2)

where
$\epsilon_{AB}(\bm{k},t)=\epsilon_{BA}(\bm{k},t)^{\ast}=t_{\textrm{NN}}\sum_{l=0}^{2}e^{-i[\bm{k}+e\bm{A}(t)]\cdot\bm{R}_{l}}$,
$\epsilon_{aa}(\bm{k},t)=0$, $\bm{R}_{0}={}^{t}(0\ 1)$,
$\bm{R}_{1}={}^{t}(-\frac{\sqrt{3}}{2}\ -\frac{1}{2})$,
$\bm{R}_{2}={}^{t}(\frac{\sqrt{3}}{2}\ -\frac{1}{2})$, $t_{\textrm{NN}}$ is
the hopping integral between nearest neighbor sites on a honeycomb lattice
[17] without $\bm{A}(t)$, and $c_{\bm{k}a\sigma}^{\dagger}$ and
$c_{\bm{k}a\sigma}$ are the creation and annihilation operators of an electron
for momentum $\bm{k}$, sublattice $a$, and spin $\sigma$. Hereafter, we set
$\hbar=c=k_{\textrm{B}}=a_{\textrm{NN}}=1$, where $a_{\textrm{NN}}$ is the
length between nearest neighbor sites. Then, $H_{\textrm{b}}$ is the
Hamiltonian of the Buttiker-type heat bath [18, 19], which is in equilibrium
at temperature $T$:
$H_{\textrm{b}}=\sum_{i}\sum_{p}(\epsilon_{p}-\mu_{\textrm{b}})b_{ip}^{\dagger}b_{ip}$,
where $b_{ip}$ and $b_{ip}^{\dagger}$ are the annihilation and creation
operators of a bath’s fermion at site $i$ for mode $p$, and $\epsilon_{p}$ and
$\mu_{\textrm{b}}$ are the energy and chemical potential of a bath’s fermion;
$\mu_{\textrm{b}}$ is determined from the condition that there is no current
between the system and bath. In addition, $H_{\textrm{sb}}$ is the system-bath
coupling Hamiltonian [20, 11, 12]:
$H_{\textrm{sb}}=\sum_{i}\sum_{p}\sum_{a=A,B}\sum_{\sigma=\uparrow,\downarrow}V_{pa\sigma}(c_{ia\sigma}^{\dagger}b_{ip}+b_{ip}^{\dagger}c_{ia\sigma})$,
where $V_{pa\sigma}$ is the system-bath coupling constant.

We have considered $H_{\textrm{b}}$ and $H_{\textrm{sb}}$, as well as
$H_{\textrm{s}}(t)$, because the damping due to the system-bath coupling makes
the system a nonequilibrium steady state [20, 11, 12]. Such a relaxation
mechanism is necessary for periodically driven systems, in which the heating
due to the driving field exists [21, 22].

##  3 Light-induced mirror symmetry breaking

First, we analyze the polarization dependence of the light-induced mirror
symmetry breaking. For our periodically driven electron system, whether a
mirror symmetry is preserved or broken is determined by the symmetry of the
kinetic energy, which is characterized by

$\displaystyle\epsilon_{AB}(\bm{k},t)$
$\displaystyle=t_{AB}^{Z}(t)e^{-ik_{y}}+t_{AB}^{X}(t)e^{i\frac{\sqrt{3}}{2}k_{x}}e^{i\frac{k_{y}}{2}}$
$\displaystyle+t_{AB}^{Y}(t)e^{-i\frac{\sqrt{3}}{2}k_{x}}e^{i\frac{k_{y}}{2}},$
(3) $\displaystyle\epsilon_{BA}(\bm{k},t)$
$\displaystyle=t_{BA}^{Z}(t)e^{ik_{y}}+t_{BA}^{X}(t)e^{-i\frac{\sqrt{3}}{2}k_{x}}e^{-i\frac{k_{y}}{2}}$
$\displaystyle+t_{BA}^{Y}(t)e^{i\frac{\sqrt{3}}{2}k_{x}}e^{-i\frac{k_{y}}{2}},$
(4)

where $t_{AB}^{Z}(t)=t_{BA}^{Z}(t)^{\ast}$,
$t_{AB}^{X}(t)=t_{BA}^{X}(t)^{\ast}$, and
$t_{AB}^{Y}(t)=t_{BA}^{Y}(t)^{\ast}$. Here $Z$, $X$, and $Y$ represent the
three bonds between nearest neighbor sites (see Fig. 1). If the hopping
integrals satisfy

$\displaystyle
t_{AB}^{Z}(t)=t_{BA}^{Z}(t),t_{AB}^{Y}(t)=t_{BA}^{X}(t),t_{AB}^{X}(t)=t_{BA}^{Y}(t),$
(5)

the mirror symmetry about the $xz$ plane [Fig. 1(a)] is preserved; otherwise,
it is broken.

Figure 1:  (Color online) The honeycomb lattice and (a) the $xz$ or (b) the
$yz$ mirror plane. The dashed lines denote the mirror planes. The green arrows
represent the bonds which are connected by the mirror symmetry. $A$ or $B$
represents sublattice $A$ or $B$, respectively. The red, blue, and dark green
bonds represent $Z$, $X$, and $Y$ bonds, respectively. The $x$ and $y$ axes
are also drawn.

We begin with the system driven by LPL. The field of LPL is described by

$\displaystyle\bm{A}_{\textrm{pump}}(t)={}^{t}(A_{0}\alpha_{x}\cos\Omega t\
A_{0}\alpha_{y}\cos\Omega t),$ (6)

where $\Omega=2\pi/T_{\textrm{p}}$ is the light frequency, and
$T_{\textrm{p}}$ is the period of $\bm{A}_{\textrm{pump}}(t)$. In this case,
$t_{AB}^{Z}(t)=t_{\textrm{NN}}e^{-iu\alpha_{y}\cos\Omega t}$,
$t_{AB}^{X}(t)=t_{\textrm{NN}}e^{iu\alpha_{x}\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\alpha_{y}\frac{1}{2}\cos\Omega t}$, and
$t_{AB}^{Y}(t)=t_{\textrm{NN}}e^{-iu\alpha_{x}\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\alpha_{y}\frac{1}{2}\cos\Omega t}$, where $u=eA_{0}$. For the LPL with
$\alpha_{x}\neq 0$, $\alpha_{y}=0$, these hopping integrals satisfy Eq. (5),
which means that the mirror symmetry about the $xz$ plane is preserved.
Meanwhile, for the LPL with $\alpha_{x}=0$, $\alpha_{y}\neq 0$ or with
$\alpha_{x}\neq 0$, $\alpha_{y}\neq 0$, this mirror symmetry is broken.

We turn to the case with CPL. The field of CPL is given by

$\displaystyle\bm{A}_{\textrm{pump}}(t)={}^{t}(A_{0}\cos\Omega t\
A_{0}\sin\Omega t).$ (7)

In a similar way, we can show that the mirror symmetry about the $xz$ plane is
broken.

We make five remarks. First, we can similarly show that the mirror symmetry
about the $yz$ plane [Fig. 1(b)] is broken by LPL with $\alpha_{x}\neq 0$,
$\alpha_{y}=0$, by that with $\alpha_{x}\neq 0$, $\alpha_{y}\neq 0$, and by
CPL, whereas it is preserved by LPL with $\alpha_{x}=0$, $\alpha_{y}\neq 0$.
Second, the same polarization dependence holds for the Floquet Hamiltonian
(see Appendix A). Third, the similar arguments can be used to discuss whether
a mirror symmetry is broken or not in the other periodically driven electron
systems. Fourth, a mirror symmetry of a periodically driven electron system
does not necessarily match that of the trajectory of
$\bm{A}_{\textrm{pump}}(t)$. For example, CPL breaks the mirror symmetry about
the $xz$ plane, whereas the trajectory of its $\bm{A}_{\textrm{pump}}(t)$ has
the mirror symmetry in the $A_{x}$-$A_{y}$ plane [see Fig. 2(a)]. Fifth, our
mirror symmetry, the symmetry about a mirror operation in crystals, is
essentially different from a symmetry discussed in Ref. [23], the symmetry
about the energy spectrum as a function of magnetic flux. Namely, the light-
induced symmetry breaking discussed in Ref. [23] is not about crystal’s mirror
symmetry. To the best of our knowledge, our paper is the first work
demonstrating the light-induced breaking of crystal’s mirror symmetry.

The above arguments show that the mirror symmetries about the $xz$ and $yz$
planes are broken with CPL or LPL1 (i.e., LPL with finite $\alpha_{x}$ and
$\alpha_{y}$). To study a time-averaged quantity in a nonequilibrium steady
state, we need to discuss not only the mirror symmetries, but also its
combining symmetries, the symmetries of combinations of the mirror operations
about the $xz$ and $yz$ planes and a uniform time translation, because that
quantity is not affected by such a translation. (Note that such a combining
symmetry is sometimes called a space-time or dynamical symmetry.) In general,
there is a case that a spatial symmetry is broken, but its combining symmetry
is preserved; in such a case, a time-averaged quantity in a nonequilibrium
steady state behaves as if the spatial symmetry were preserved. As we show in
Appendix B, the combining symmetries for the mirror operations are also broken
with CPL or LPL1. This contrasts with the combining symmetry for the $C_{3}$
rotation in graphene driven by CPL (see Apppendix B): the $C_{3}$ rotational
symmetry is broken, but the symmetry of a combination of the $C_{3}$ rotation
and the uniform time translation $T_{t}:t\rightarrow t+\frac{2\pi}{3\Omega}$
is preserved [11]. This combining symmetry may be called a time-screw
symmetry. In appendix B, we also show that the combining symmetry for the
mirror operation about the $yz$ or $xz$ plane is preserved with LPL2 (i.e.,
LPL with $\alpha_{x}\neq 0$, $\alpha_{y}=0$) or LPL3 (i.e., LPL with
$\alpha_{x}=0$, $\alpha_{y}\neq 0$), respectively, if $T_{t}:t\rightarrow
t-\frac{\pi}{\Omega}$. This combining symmetry may be called a time-glide
symmetry. We do not necessarily call the combining symmetry for a mirror
operation a time-glide one because an analogy with an axial glide symmetry
suggests that a time-glide symmetry consists of a mirror operation and the
uniform time translation with $T_{\textrm{p}}/2$. Then, the $C_{3}$ rotational
symmetry and its combining symmetry are both broken with LPL (see Appendix B).
These results are summarized in Table 1.

Figure 2:  (Color online) (a) The trajectories of $\bm{A}_{\textrm{pump}}(t)$
for CPL, LPL1, LPL2, and LPL3. (b) The setup for the pump-probe measurements
of $\sigma_{yx}^{\textrm{C}}$ in our periodically driven system. For the setup
of $\sigma_{xy}^{\textrm{C}}$, the directions of the probe field and charge
current are interchanged. The arrow of the pump light indicates the direction
of travel, and that of the probe light indicates the direction of the
component of the corresponding electric field. (c), (d)
$\sigma_{yx}^{\textrm{C}}$ and $\sigma_{xy}^{\textrm{C}}$ as functions of
$u=eA_{0}$ for graphene driven by CPL, LPL1, LPL2, and LPL3. The horizontal
dashed lines in (c) and (d) correspond to $-2e^{2}h^{-1}$ and $2e^{2}h^{-1}$,
respectively.

##  4 Charge transport induced by mirror symmetry breaking

Next, we study the effects of the light-induced mirror symmetry breaking on
transport properties. To do this, we use the Floquet linear-response theory
[12, 11, 24] for pump-probe measurements [Fig. 2(b)]. In this theory, we set
$\bm{A}(t)=\bm{A}_{\textrm{pump}}(t)+\bm{A}_{\textrm{prob}}(t)$ and treat the
effects of $\bm{A}_{\textrm{pump}}(t)$ in the Floquet theory [25, 26] and
those of $\bm{A}_{\textrm{prob}}(t)$ in the linear-response theory [27]. The
$\bm{A}_{\textrm{pump}}(t)$ for LPL or CPL is given by Eq. (6) or (7),
respectively. Note that $\bm{A}_{\textrm{pump}}(t)$ is used to periodically
drive the system, whereas $\bm{A}_{\textrm{prob}}(t)$ is used to analyze its
properties [28]. In this theory, we use the Floquet Hamiltonian for
$H_{\textrm{s}}(t)$, which is distinct from the Hamiltonian obtained in a
high-frequency expansion. Using the Floquet linear-response theory, we obtain
a time-averaged charge conductivity $\sigma_{\mu\nu}^{\textrm{C}}$ in the
nonequilibrium steady state [12, 11] (see Appendix C),

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}}=\frac{e^{2}}{V}\sum_{\bm{k}}\sum_{a,b,c,d=A,B}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}\int_{-\Omega/2}^{\Omega/2}\frac{d\omega^{\prime}}{2\pi}$
$\displaystyle\times\textrm{tr}\Bigl{[}v_{ab}^{\mu}(\bm{k})\frac{\partial
G_{b\sigma
c\sigma^{\prime}}^{\textrm{R}}(\bm{k},\omega^{\prime})}{\partial\omega^{\prime}}v_{cd}^{\nu}(\bm{k})G_{d\sigma^{\prime}a\sigma}^{<}(\bm{k},\omega^{\prime})$
$\displaystyle\ \ \ -v_{ab}^{\mu}(\bm{k})G_{b\sigma
c\sigma^{\prime}}^{<}(\bm{k},\omega^{\prime})v_{cd}^{\nu}(\bm{k})\frac{\partial
G_{d\sigma^{\prime}a\sigma}^{\textrm{A}}(\bm{k},\omega^{\prime})}{\partial\omega^{\prime}}\Bigr{]},$
(8)

where the trace is taken over the Floquet states [i.e.,
$\textrm{tr}(ABCD)=\sum_{m,l,n,q=-\infty}^{\infty}A_{ml}B_{ln}C_{nq}D_{qm}$
with Floquet indices $m$, $l$, $n$, and $q$],
$V=\frac{N}{2}\frac{3\sqrt{3}}{2}$, $N$ is the number of sites,
$[v_{ab}^{\nu}(\bm{k})]_{mn}$ is the group velocity in the Floquet
representation, and $[G_{a\sigma
b\sigma^{\prime}}^{\textrm{R}}(\bm{k},\omega^{\prime})]_{mn}$, $[G_{a\sigma
b\sigma^{\prime}}^{\textrm{A}}(\bm{k},\omega^{\prime})]_{mn}$, and
$[G_{a\sigma b\sigma^{\prime}}^{<}(\bm{k},\omega^{\prime})]_{mn}$ are the
retarded, advanced, and lesser Green’s functions, respectively, in the Floquet
representation. (For more details, see Appendix C.) These Green’s functions
are determined from Dyson’s equation with the damping $\Gamma$ due to the
second-order perturbation of $H_{\textrm{sb}}$ (see Appendix D). Note that
$\sigma_{\mu\nu}^{\textrm{C}}$ is equivalent to the anomalous Hall
conductivity if and only if it is antisymmetric.

Using Eq. (8), we numerically evaluate $\sigma_{yx}^{\textrm{C}}$ and
$\sigma_{xy}^{\textrm{C}}$ for graphene driven by CPL, LPL1, LPL2, and LPL3.
(For details of the numerical calculations, see Appendix E.) The directions of
the probe field and the observed charge current are fixed: for
$\sigma_{yx}^{\textrm{C}}$ (or $\sigma_{xy}^{\textrm{C}}$), the charge current
along the $y$ (or $x$) axis is generated with the probe field applied along
the $x$ (or $y$) axis. CPL is described by Eq. (7), and LPL1, LPL2, or LPL3 is
described by Eq. (6) with $\alpha_{x}=\alpha_{y}=1$, with $\alpha_{x}=1$ and
$\alpha_{y}=0$, or with $\alpha_{x}=0$ and $\alpha_{y}=1$, respectively [Fig.
2(a)]; as described above, the mirror symmetry about the $xz$ or $yz$ plane is
preserved only for LPL2 or LPL3, respectively. We set
$\Omega=8t_{\textrm{NN}}$ and $t_{\textrm{NN}}=1$; our light is off-resonant,
i.e., $\Omega>W$, where $W(=6t_{\textrm{NN}})$ is the bandwidth without light.
(We have summarized the main results in Table 1.) Except for the results of
the damping or temperature dependence of $\sigma_{yx}^{\textrm{C}}$ with LPL1,
we set $\Gamma=0.004t_{\textrm{NN}}$ and $T=0.006t_{\textrm{NN}}$. When
discussing the damping dependence, we set $T=0.006t_{\textrm{NN}}$ and compare
the results obtained at $\Gamma=0.004t_{\textrm{NN}}$, $0.002t_{\textrm{NN}}$,
and $0.006t_{\textrm{NN}}$; when discussing the temperature dependence, we set
$\Gamma=0.004t_{\textrm{NN}}$ and compare the results obtained at
$T=0.006t_{\textrm{NN}}$, $0.004t_{\textrm{NN}}$, and $0.008t_{\textrm{NN}}$.

Figure 2(c) shows the $u$ dependences of $\sigma_{yx}^{\textrm{C}}$ in
graphene driven by CPL, LPL1, LPL2, and LPL3. $\sigma_{yx}^{\textrm{C}}$ for
$u\neq 0$ is finite for CPL and LPL1, whereas it vanishes for LPL2 and LPL3.
The similar results are obtained also for $\sigma_{xy}^{\textrm{C}}$ [Fig.
2(d)]. These results are consistent with the properties of the mirror
symmetries about the $xz$ and $yz$ planes and their combining symmetries (see
Table 1). Therefore, the mirror symmetries and their combining symmetries play
a vital role in discussing the off-diagonal charge conductivities. Note that
since $u=eA_{0}=eE_{0}/\Omega$ is dimensionless, the $u$ dependence of
$\sigma_{yx}^{\textrm{C}}$ at fixed $\Omega$ gives its dependence on $E_{0}$,
the amplitude of the light field.

One of the main differences between the cases of CPL and LPL1 is the relation
between $\sigma_{yx}^{\textrm{C}}$ and $\sigma_{xy}^{\textrm{C}}$. Figure 2(d)
shows the $u$ dependences of $\sigma_{xy}^{\textrm{C}}$ in graphene driven by
CPL, LPL1, LPL2, and LPL3. Comparing this figure with Fig. 2(c), we see that
$\sigma_{xy}^{\textrm{C}}=-\sigma_{yx}^{\textrm{C}}$ for CPL, whereas
$\sigma_{xy}^{\textrm{C}}=\sigma_{yx}^{\textrm{C}}$ for LPL1. They are the
Onsager reciprocal relations [29, 30], and their difference comes from the
difference in the time-reversal symmetry. We should note that
$\sigma_{xy}^{\textrm{C}}=\sigma_{yx}^{\textrm{C}}$ for LPL1 does not
contradict the properties of the $C_{3}$ rotational symmetry (see Appendix B).
Since the anomalous Hall conductivity is off-diagonal and antisymmetric, our
results indicate that the light-induced AHE comes from a combination of mirror
symmetry breaking and time-reversal symmetry breaking. This is consistent with
the AHE in nondriven systems [31, 32]. Our results also suggest that the off-
diagonal symmetric charge conductivity can be regarded as an indicator for
mirror symmetry breaking in the presence of the time-reversal symmetry. This
might be used to detect helical higher-order topological insulators [33],
which are protected by the mirror symmetry and time-reversal symmetry, because
that conductivity vanishes with the mirror symmetry or its combining symmetry,
as shown above.

Another difference is about the quantization of $\sigma_{yx}^{\textrm{C}}$.
$\sigma_{yx}^{\textrm{C}}$ is quantized only with CPL. This quantization can
be understood using a high-frequency expansion [34, 35], as shown in previous
studies [36, 11]: the term proportional to $\Omega^{-1}$ gives a pure-
imaginary hopping integral between next-nearest neighbors on the honeycomb
lattice, which is similar to the term vital for the quantum Hall effect [37].
Similarly, we can understand the non-quantized $\sigma_{yx}^{\textrm{C}}$ with
LPL: the $\Omega^{-1}$ term becomes zero. This is consistent with the property
that LPL does not break the time-reversal symmetry. Note that except the above
interpretations, we do not use the high-frequency expansion.

The other difference is about the $\Gamma$ dependence of
$\sigma_{yx}^{\textrm{C}}$. Figure 3(a) shows the $\Gamma$ dependence of
$\sigma_{yx}^{\textrm{C}}$ for graphene driven by LPL1.
$\sigma_{yx}^{\textrm{C}}$ is roughly proportional to $\Gamma^{-1}$. This
contrasts the $\Gamma$ dependence of the off-diagonal charge conductivity for
graphene driven by CPL because it is almost independent of $\Gamma$ [e.g.,
compare the red curve in Fig. 2(d) of this paper and the brown one in Fig. 10
of Ref. [11]]. Note that $\sigma_{yx}^{\textrm{C}}$ with LPL1, as well as that
with CPL, is little dependent on the bath temperature $T$ [Fig. 3(b)]. This is
because the bath temperature may play a similar role to the temperature
appearing in the distribution function. We should note that $\Gamma$ is
independent of temperature in our theory.

Figure 3:  (Color online) (a), (b) The dependences of
$\sigma_{yx}^{\textrm{C}}$ on the damping induced by the system-bath coupling,
$\Gamma$, and the temperature of the bath, $T$, for graphene driven by LPL1.

The sign of the off-diagonal symmetric charge conductivity can be reversed by
replacing LPL1 by the LPL for $\alpha_{x}=-\alpha_{y}=1$ or
$-\alpha_{x}=\alpha_{y}=1$, a counterpart connected by the mirror operation
about the $xz$ or $yz$ plane, respectively. Furthermore, it remains unchanged
by replacing LPL1 by the LPL for $\alpha_{x}=\alpha_{y}=-1$. The similar
properties hold for arbitrary $\theta$ when $\alpha_{x}=\cos\theta$ and
$\alpha_{y}=\sin\theta$. These three additional results are shown in Appendix
F. They also suggest the vital role of the mirror symmetries and their
combining symmetries in the off-diagonal charge conductivity.

Figure 4:  (Color online) (a), (b) Situations with and without the mirror
symmetry (or its combining symmetry with a uniform time translation) under the
probe field applied along the $x$ axis.

##  5 Discussion

The importance of the mirror symmetry breaking is a general concept. Let us
consider a situation where the probe field is applied along the $x$ axis of a
material. If the mirror symmetry about the $xz$ plane (or its combining
symmetry with a uniform time translation) exists, any currents along the $y$
axis are prohibited [Fig. 4(a)]. Meanwhile, if it is broken, the charge
current along the $y$ axis can be induced [Fig. 4(b)]. This current is finite
(i.e., $\sigma_{yx}^{\textrm{C}}\neq 0$) if the mirror symmetry about the $yz$
plane (or its combining symmetry), as well as that about the $xz$ plane, is
broken. Therefore, the mirror symmetry breaking by light plays the key role in
the light-induced off-diagonal charge transport. Although mirror symmetry
breaking about the $xy$ plane is important in several systems with the Rashba
spin-orbit coupling [38, 39], it is not essential for obtaining
$\sigma_{yx}^{\textrm{C}}$ and $\sigma_{xy}^{\textrm{C}}$; such off-site spin-
orbit coupling is absent in our system. Note that the importance of the mirror
symmetry breaking can been seen from Eq. (8) and the expression using the
Berry curvature because both contain the momentum summation of the product of
the $x$ and $y$ components of the group velocity, which can be finite without
the mirror symmetries about the $xz$ and $yz$ planes and their combining
symmetries.

Our results can be tested experimentally. In our system, the nonequilibrium
steady state can be achieved due to $\Gamma$ at times larger than
$\tau=\frac{\hbar}{2\Gamma}=O(10\textrm{fs})$. Then, the off-diagonal charge
conductivity in graphene driven by LPL1 could be observed experimentally in
pump-probe measurements. Note that
$u=\frac{eE_{0}a_{\textrm{NN}}}{\Omega}=0.1$ at
$\Omega=8t_{\textrm{NN}}\approx 24$ eV corresponds to $E_{0}\approx 171$ MV
cm-1. In this estimate, we have used $a_{\textrm{NN}}\approx 0.14$ nm [40]. We
have also set $t_{\textrm{NN}}\approx 3$ eV because, according to the first-
principles calculations without $\bm{A}(t)$ [41], the energy difference
between the two bands in our model at $\bm{k}=\bm{0}$ corresponds to about
$19$ eV, i.e., $6t_{\textrm{NN}}\approx 19$ eV. Because of
$t_{\textrm{NN}}\approx 3$ eV, $k_{\textrm{B}}T=0.006t_{\textrm{NN}}(\approx
0.018$ eV$)$ corresponds to about $209$K, where $k_{\textrm{B}}\approx
8.6\times 10^{-5}$ eV K-1 is used. Since $\sigma_{yx}^{\textrm{C}}$ and
$\sigma_{xy}^{\textrm{C}}$ become finite at nonzero $u$’s [Figs. 2(c) and
2(d)], the off-diagonal charge transport induced by LPL1 is testable.

##  6 Conclusion

We have studied the polarization dependence of the light-induced mirror
symmetry breaking and its effects on charge transport in periodically driven
graphene. We showed that the mirror symmetries about the $xz$ and $yz$ planes
and their combining symmetries are broken by CPL and by the LPL whose $A_{x}$
and $A_{y}$ are both nonzero. This mirror symmetry breaking leads to the
light-induced AHE in the absence of the time-reversal symmetry. This indicates
that the origin of the light-induced AHE is a combination of the time-reversal
symmetry breaking and the mirror symmetry breaking. In the presence of time-
reversal symmetry, the mirror symmetry breaking results in the off-diagonal
symmetric charge conductivity. This conductivity could be used to detect the
mirror symmetry breaking with the time-reversal symmetry. Our results
highlight the overlooked role of the light-induced mirror symmetry breaking in
the light-induced AHE and reveal the emergence of the off-diagonal symmetric
charge transport induced by LPL.

###### Acknowledgements.

This work was supported by JST CREST Grant No. JPMJCR1901, JSPS KAKENHI Grant
No. JP22K03532, and MEXT Q-LEAP Grant No. JP-MXS0118067426.

##  Appendix A Light-induced mirror symmetry breaking for the Floquet
Hamiltonian

We analyze the polarization dependence of the light-induced mirror symmetry
breaking for the Floquet Hamiltonian. As we will show below, this polarization
dependence is the same as that for the time-dependent Hamiltonian, which has
been shown in the main text. (Note that the Floquet Hamiltonian is distinct
from that obtained in a high-frequency expansion.) The momentum dependence of
the Floquet Hamiltonian is characterized by

$\displaystyle[\epsilon_{AB}(\bm{k})]_{mn}$
$\displaystyle=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}\epsilon_{AB}(\bm{k},t)$
$\displaystyle=[t_{AB}^{Z}]_{mn}e^{-ik_{y}}+[t_{AB}^{X}]_{mn}e^{i\frac{\sqrt{3}}{2}k_{x}}e^{i\frac{k_{y}}{2}}$
$\displaystyle+[t_{AB}^{Y}]_{mn}e^{-i\frac{\sqrt{3}}{2}k_{x}}e^{i\frac{k_{y}}{2}},$
(9) $\displaystyle[\epsilon_{BA}(\bm{k})]_{mn}$
$\displaystyle=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}\epsilon_{BA}(\bm{k},t)$
$\displaystyle=[t_{BA}^{Z}]_{mn}e^{ik_{y}}+[t_{BA}^{X}]_{mn}e^{-i\frac{\sqrt{3}}{2}k_{x}}e^{-i\frac{k_{y}}{2}}$
$\displaystyle+[t_{BA}^{Y}]_{mn}e^{i\frac{\sqrt{3}}{2}k_{x}}e^{-i\frac{k_{y}}{2}},$
(10)

where

$\displaystyle[t_{ab}^{Z}]_{mn}=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}t_{ab}^{Z}(t),$ (11)
$\displaystyle[t_{ab}^{X}]_{mn}=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}t_{ab}^{X}(t),$ (12)
$\displaystyle[t_{ab}^{Y}]_{mn}=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}t_{ab}^{Y}(t).$ (13)

Note that $\epsilon_{AB}(\bm{k},t)$ and $\epsilon_{BA}(\bm{k},t)$ have been
given by Eqs. (3) and (4), respectively. (We neglect the energy shifts due to
the light frequency in the Floquet Hamiltonian because such momentum-
independent terms do not affect mirror symmetries.) Note that we have
considered $[\epsilon_{AB}(\bm{k})]_{mn}$ and $[\epsilon_{BA}(\bm{k})]_{mn}$
because they give the finite components of $H_{m,n}$, which is part of the
matrix used to obtain the quasienergy in the Floquet state [11]. As we have
explained in the main text, the mirror symmetry about the $xz$ plane is
preserved if the hopping integrals as a function of time satisfy Eq. (5).
Therefore, for the Floquet Hamiltonian, this mirror symmetry is preserved if

$\displaystyle[t_{AB}^{Z}]_{mn}=[t_{BA}^{Z}]_{mn},\
[t_{AB}^{Y}]_{mn}=[t_{BA}^{X}]_{mn},\ [t_{AB}^{X}]_{mn}=[t_{BA}^{Y}]_{mn}.$
(14)

In the case of the system driven by LPL described by Eq. (6), the hopping
integrals are given by

$\displaystyle[t_{AB}^{Z}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{m-n}(\alpha_{y}u),$
(15)
$\displaystyle[t_{AB}^{X}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{n-m}(\frac{\sqrt{3}\alpha_{x}+\alpha_{y}}{2}u),$
(16)
$\displaystyle[t_{AB}^{Y}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{m-n}(\frac{\sqrt{3}\alpha_{x}-\alpha_{y}}{2}u),$
(17)
$\displaystyle[t_{BA}^{Z}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{n-m}(\alpha_{y}u),$
(18)
$\displaystyle[t_{BA}^{X}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{m-n}(\frac{\sqrt{3}\alpha_{x}+\alpha_{y}}{2}u),$
(19)
$\displaystyle[t_{BA}^{Y}]_{mn}=t_{\textrm{NN}}i^{n-m}\mathcal{J}_{n-m}(\frac{\sqrt{3}\alpha_{x}-\alpha_{y}}{2}u),$
(20)

where $\mathcal{J}_{l}(x)$ is the Bessel function of the first kind with the
order $l$. Therefore, the mirror symmetry about the $xz$ plane is preserved
for the LPL with $\alpha_{x}\neq 0$, $\alpha_{y}=0$, whereas it is broken by
the LPL with $\alpha_{x}=0$, $\alpha_{y}\neq 0$ or $\alpha_{x}\neq 0$,
$\alpha_{y}\neq 0$. Similarly, we can show that this mirror symmetry is broken
by CPL described by Eq. (7). These results are the same as the polarization
dependence shown in the main text.

##  Appendix B $C_{3}$ rotational symmetry and mirror symmetries without or
with a uniform time translation

We discuss the $C_{3}$ rotational symmetry, the mirror symmetries, and the
combining symmetries with a uniform time translation. First, we show that the
$C_{3}$ rotational symmetry is broken with CPL or LPL, that the combining
symmetry, the symmetry of a combination of the $C_{3}$ rotation and a uniform
time translation, is preserved with CPL if $T_{t}:t\rightarrow
t+\frac{2\pi}{3\Omega}$, and that the combining symmetry is broken with LPL.
Next, we review the mirror symmetry about the $xz$ or $yz$ plane with CPL or
LPL, which has been discussed in Sect. III. Then, we show that, as well as the
mirror symmetries, the combining symmetries, the symmetries of combinations of
the mirror operations about the $xz$ and $yz$ planes and a uniform time
translation, are broken with CPL or LPL1. We also show that the symmetry of a
combination of the mirror operation about the $yz$ or $xz$ plane and the
uniform time translation $T_{t}:t\rightarrow t-\frac{\pi}{\Omega}$ is
preserved with LPL2 or LPL3, respectively. As shown in Sect. III, the mirror
symmetry about the $yz$ or $xz$ plane is broken with LPL2 or LPL3,
respectively. Note that for LPL1 $\alpha_{x}\neq 0$ and $\alpha_{y}\neq 0$ in
Eq. (6), for LPL2 $\alpha_{x}\neq 0$ and $\alpha_{y}=0$, and for LPL3
$\alpha_{x}=0$ and $\alpha_{y}\neq 0$.

We begin with the properties about the $C_{3}$ rotational symmetry for
graphene driven by CPL. This periodically driven system has the $C_{3}$
rotational symmetry if the $H_{\textrm{s}}(t)$ remains unchanged after a
counterclockwise rotation of $120$ degrees around the $z$ axis. This condition
can be expressed as the following equation:

$\displaystyle C_{3}^{-1}H_{\textrm{s}}(t)C_{3}=H_{\textrm{s}}(t).$ (21)

In the case of graphene, this equation can be rewritten as follows:

$\displaystyle C_{3}^{-1}t_{AB}^{Z}(t)C_{3}=t_{AB}^{Z}(t),$ (22)
$\displaystyle C_{3}^{-1}t_{AB}^{X}(t)C_{3}=t_{AB}^{X}(t),$ (23)
$\displaystyle C_{3}^{-1}t_{AB}^{Y}(t)C_{3}=t_{AB}^{Y}(t),$ (24)

where $t_{AB}^{Z}(t)=t_{\textrm{NN}}e^{-ie\bm{A}(t)\cdot\bm{R}_{0}}$,
$t_{AB}^{X}(t)=t_{\textrm{NN}}e^{-ie\bm{A}(t)\cdot\bm{R}_{1}}$, and
$t_{AB}^{Y}(t)=t_{\textrm{NN}}e^{-ie\bm{A}(t)\cdot\bm{R}_{2}}$. Since the
$C_{3}$ rotation transforms the $Z$, $X$, and $Y$ bonds on the honeycomb
lattice into the $X$, $Y$, and $Z$ bonds, respectively [see Fig. 1(a) or
1(b)], the left-hand sides of Eqs. (22)–(24) become

$\displaystyle C_{3}^{-1}t_{AB}^{Z}(t)C_{3}=t_{AB}^{X}(t),$ (25)
$\displaystyle C_{3}^{-1}t_{AB}^{X}(t)C_{3}=t_{AB}^{Y}(t),$ (26)
$\displaystyle C_{3}^{-1}t_{AB}^{Y}(t)C_{3}=t_{AB}^{Z}(t).$ (27)

Therefore, if the hopping integrals satisfy

$\displaystyle C_{3}^{-1}t_{AB}^{Z}(t)C_{3}=t_{AB}^{X}(t)=t_{AB}^{Z}(t),$ (28)
$\displaystyle C_{3}^{-1}t_{AB}^{X}(t)C_{3}=t_{AB}^{Y}(t)=t_{AB}^{X}(t),$ (29)
$\displaystyle C_{3}^{-1}t_{AB}^{Y}(t)C_{3}=t_{AB}^{Z}(t)=t_{AB}^{Y}(t),$ (30)

the $C_{3}$ rotational symmetry is preserved; otherwise, it is broken.
Combining Eqs. (28)–(30) with the equations of the hopping integrals for
graphene driven by CPL,

$\displaystyle t_{AB}^{Z}(t)=t_{\textrm{NN}}e^{-iu\sin(\Omega t)},$ (31)
$\displaystyle t_{AB}^{X}(t)=t_{\textrm{NN}}e^{-iu\sin(\Omega
t-\frac{2\pi}{3})},$ (32) $\displaystyle
t_{AB}^{Y}(t)=t_{\textrm{NN}}e^{-iu\sin(\Omega t-\frac{4\pi}{3})},$ (33)

we find that these hopping integrals do not satisfy Eqs. (28)–(30), which
means that the $C_{3}$ rotational symmetry is broken for graphene driven by
CPL. Meanwhile, the periodically driven system has the combining symmetry, the
symmetry of a combination of the $C_{3}$ rotation and a uniform time
translation $T_{t}$, if

$\displaystyle
T_{t}^{-1}C_{3}^{-1}H_{\textrm{s}}(t)C_{3}T_{t}=H_{\textrm{s}}(t),$ (34)

which is reduced in the case of graphene to

$\displaystyle T_{t}^{-1}C_{3}^{-1}t_{AB}^{Z}(t)C_{3}T_{t}=t_{AB}^{Z}(t),$
(35) $\displaystyle
T_{t}^{-1}C_{3}^{-1}t_{AB}^{X}(t)C_{3}T_{t}=t_{AB}^{X}(t),$ (36)
$\displaystyle T_{t}^{-1}C_{3}^{-1}t_{AB}^{Y}(t)C_{3}T_{t}=t_{AB}^{Y}(t).$
(37)

If we consider the uniform time translation,

$\displaystyle T_{t}:t\rightarrow t+\frac{2\pi}{3\Omega},$ (38)

the left-hand sides of Eqs. (35)–(37) are written in graphene driven by CPL as
follows:

$\displaystyle
T_{t}^{-1}C_{3}^{-1}t_{AB}^{Z}(t)C_{3}T_{t}=T_{t}^{-1}t_{AB}^{X}(t)T_{t}=t_{AB}^{Z}(t),$
(39) $\displaystyle
T_{t}^{-1}C_{3}^{-1}t_{AB}^{X}(t)C_{3}T_{t}=T_{t}^{-1}t_{AB}^{Y}(t)T_{t}=t_{AB}^{X}(t),$
(40) $\displaystyle
T_{t}^{-1}C_{3}^{-1}t_{AB}^{Y}(t)C_{3}T_{t}=T_{t}^{-1}t_{AB}^{Z}(t)T_{t}=t_{AB}^{Y}(t),$
(41)

where we have used Eqs. (25)–(27) and (31)–(33). Therefore, the combining
symmetry, which may be called a time-screw symmetry, is preserved for graphene
driven by CPL. Because of this property, the time-averaged off-diagonal charge
conductivities in the nonequilibrium steady state with CPL satisfy
$\sigma_{xy}^{\textrm{C}}=-\sigma_{yx}^{\textrm{C}}$ [11] even without using
the Onsager reciprocal relation.

In a similar way, we can show for graphene driven by LPL that the $C_{3}$
rotational symmetry and its combining symmetry are both broken. This result
holds for arbitrary $\alpha_{x}$ and $\alpha_{y}$ of Eq. (6). Because of the
breaking of these symmetries, there is no need to satisfy
$\sigma_{xy}^{\textrm{C}}=-\sigma_{yx}^{\textrm{C}}$ in the case of LPL.
Namely, $\sigma_{xy}^{\textrm{C}}=\sigma_{yx}^{\textrm{C}}$ with LPL1 is
symmetrically reasonable. Note that for graphene driven by LPL the hopping
integrals are given by

$\displaystyle t_{AB}^{Z}(t)=t_{\textrm{NN}}e^{-iu\alpha_{y}\cos(\Omega t)},$
(42) $\displaystyle
t_{AB}^{X}(t)=t_{\textrm{NN}}e^{iu\frac{\sqrt{3}}{2}\alpha_{x}\cos(\Omega
t)}e^{iu\frac{\alpha_{y}}{2}\cos(\Omega t)},$ (43) $\displaystyle
t_{AB}^{Y}(t)=t_{\textrm{NN}}e^{-iu\frac{\sqrt{3}}{2}\alpha_{x}\cos(\Omega
t)}e^{iu\frac{\alpha_{y}}{2}\cos(\Omega t)}.$ (44)

We turn to the properties of the mirror symmetry about the $xz$ or $yz$ plane.
The periodically driven system is symmetric with respect to the $xz$ mirror
plane [Fig. 1(a)] if

$\displaystyle\sigma_{\textrm{m}}^{-1}H_{\textrm{s}}(t)\sigma_{\textrm{m}}=H_{\textrm{s}}(t),$
(45)

which is written in the case of graphene as

$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}=t_{AB}^{Z}(t),$
(46)
$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}=t_{AB}^{X}(t),$
(47)
$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}=t_{AB}^{Y}(t).$
(48)

As we can see from Fig. 1(a), the left-hand sides of these equations become

$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}=t_{BA}^{Z}(t),$
(49)
$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}=t_{BA}^{Y}(t),$
(50)
$\displaystyle\sigma_{\textrm{m}}^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}=t_{BA}^{X}(t).$
(51)

A combination of Eqs. (46)–(48) and Eqs. (49)–(51) gives Eq. (5). Therefore,
as we have shown in Sect. III, the mirror symmetry about the $xz$ plane is
broken for graphene driven by CPL and LPL with $\alpha_{y}\neq 0$, whereas it
is preserved for graphene driven by LPL with $\alpha_{y}=0$ (see Table 1). For
the mirror symmetry about the $yz$ plane, Eqs. (45) and (46)–(48) are replaced
by

$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}H_{\textrm{s}}(t)\sigma_{\textrm{m}}^{\prime}=H_{\textrm{s}}(t),$
(52)

and

$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{Z}(t),$
(53)
$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{X}(t),$
(54)
$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{Y}(t).$
(55)

Moreover, as we can see from Fig. 1(b), the left-hand sides of Eqs. (53)–(55)
are expressed as

$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{Z}(t),$
(56)
$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{Y}(t),$
(57)
$\displaystyle(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}^{\prime}=t_{AB}^{X}(t).$
(58)

Therefore, as we have remarked in Sect. III, the mirror symmetry about the
$yz$ plane is broken for graphene driven by CPL or LPL with $\alpha_{x}\neq
0$, whereas it is preserved for graphene driven by LPL with $\alpha_{x}=0$
(see Table 1). Then, for the combining symmetry for the $xz$ mirror plane,
Eqs. (45) and (52) are replaced by

$\displaystyle
T_{t}^{-1}\sigma_{\textrm{m}}^{-1}H_{\textrm{s}}(t)\sigma_{\textrm{m}}T_{t}=H_{\textrm{s}}(t),$
(59)

and

$\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}H_{\textrm{s}}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=H_{\textrm{s}}(t),$
(60)

respectively. In the case of graphene, the former is decomposed into

$\displaystyle
T_{t}^{-1}\sigma_{\textrm{m}}^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}T_{t}=T_{t}^{-1}t_{BA}^{Z}(t)T_{t}=t_{AB}^{Z}(t),$
(61) $\displaystyle
T_{t}^{-1}\sigma_{\textrm{m}}^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}T_{t}=T_{t}^{-1}t_{BA}^{Y}(t)T_{t}=t_{AB}^{X}(t),$
(62) $\displaystyle
T_{t}^{-1}\sigma_{\textrm{m}}^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}T_{t}=T_{t}^{-1}t_{BA}^{X}(t)T_{t}=t_{AB}^{Y}(t),$
(63)

and the latter is decomposed into

$\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{Z}(t)T_{t}=t_{AB}^{Z}(t),$
(64) $\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{Y}(t)T_{t}=t_{AB}^{X}(t),$
(65) $\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{X}(t)T_{t}=t_{AB}^{Y}(t).$
(66)

In deriving the first and the last three equations, we have used Eqs.
(49)–(51) and Eqs. (56)–(58), respectively. Using Eqs. (31)–(33) or (42)–(44),
we can show that the combining symmetries for the $xz$ and $yz$ mirror planes
are broken for graphene driven by CPL or LPL1. This is because in the case of
graphene driven by CPL or LPL1, there is no uniform time translation which
makes the system after the mirror operation the same as that before it. For
example, in the case of the mirror symmetry about the $yz$ plane with CPL, if
we choose $T_{t}$ in Eqs. (64)–(66) as Eq. (38), we get

$\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Z}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{Z}(t)T_{t}$
$\displaystyle=t_{\textrm{NN}}e^{-iu\sin(\Omega
t+\frac{2\pi}{3})}=t_{AB}^{Y}(t)\neq t_{AB}^{Z}(t),$ (67) $\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{X}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{Y}(t)T_{t}$
$\displaystyle=t_{\textrm{NN}}e^{-iu\sin(\Omega
t-\frac{2\pi}{3})}=t_{AB}^{X}(t),$ (68) $\displaystyle
T_{t}^{-1}(\sigma_{\textrm{m}}^{\prime})^{-1}t_{AB}^{Y}(t)\sigma_{\textrm{m}}^{\prime}T_{t}=T_{t}^{-1}t_{AB}^{X}(t)T_{t}$
$\displaystyle=t_{\textrm{NN}}e^{-iu\sin(\Omega t)}=t_{AB}^{Z}(t)\neq
t_{AB}^{Y}(t).$ (69)

There is no uniform time translation that the three conditions for the hopping
integrals are satisfied simultaneously. In contrast, the combining symmetry
for the $yz$ or $xz$ mirror plane is preserved in graphene driven by LPL2 or
LPL3, respectively. As we have shown in Sect. III, the mirror symmetry about
the $xz$ plane is preserved for graphene driven by LPL2, whereas it is broken
for graphene driven by LPL3. Meanwhile, the hopping integrals for graphene
driven by LPL3 satisfy Eqs. (61)–(63) if the $T_{t}$ is chosen as

$\displaystyle T_{t}:t\rightarrow t-\frac{\pi}{\Omega}.$ (70)

By using the same $T_{t}$, we can show that the hopping integrals for graphene
driven by LPL2 satisfy Eqs. (64)–(66). Therefore, the symmetry of a
combination of the mirror operation about the $yz$ or $xz$ plane and the
uniform time translation of Eq. (70), which may be called a time-glide
symmetry, is preserved for graphene driven by LPL2 or LPL3, respectively.

##  Appendix C Derivation of Eq. (8)

We derive Eq. (8). Since this derivation has been explained, for example, in
Ref. [12], we explain the main points below. Treating
$\bm{A}_{\textrm{prob}}(t)$ in the linear-response theory, we express a charge
conductivity as

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}}(t,t^{\prime})$
$\displaystyle=\frac{1}{i\omega}\frac{\delta\langle
j_{\textrm{C}}^{\mu}(t)\rangle}{\delta A_{\textrm{prob}}^{\nu}(t^{\prime})},$
(71)

where $\langle j_{\textrm{C}}^{\mu}(t)\rangle$ is the expectation value of the
operator of the charge current density
$j_{\textrm{C}}^{\mu}(t)=J_{\textrm{C}}^{\mu}(t)/V$, $J_{\textrm{C}}^{\mu}(t)$
is the charge current operator,

$\displaystyle
J_{\textrm{C}}^{\mu}(t)=(-e)\sum_{\bm{k}}\sum_{a,b}\sum_{\sigma=\uparrow,\downarrow}v_{ab}^{\mu}(\bm{k},t)c_{\bm{k}a\sigma}^{\dagger}(t)c_{\bm{k}b\sigma}(t),$
(72)

and $v_{ab}^{\mu}(\bm{k},t)=\frac{\partial\epsilon_{ab}(\bm{k},t)}{\partial
k_{\mu}}$. By substituting Eq. (72) into Eq. (71) and doing some calculations
[12], we get

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}}(t,t^{\prime})$
$\displaystyle=\sigma_{\mu\nu}^{\textrm{C}(1)}(t,t^{\prime})+\sigma_{\mu\nu}^{\textrm{C}(2)}(t,t^{\prime}),$
(73)

where

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}(1)}(t,t^{\prime})$
$\displaystyle=\frac{e}{\omega
V}\sum_{\bm{k}}\sum_{a,b}\sum_{\sigma=\uparrow,\downarrow}\frac{\delta
v_{ab}^{\mu}(\bm{k},t)}{\delta A_{\textrm{prob}}^{\nu}(t^{\prime})}G_{b\sigma
a\sigma}^{<}(\bm{k};t,t),$ (74)
$\displaystyle\sigma_{\mu\nu}^{\textrm{C}(2)}(t,t^{\prime})$
$\displaystyle=\frac{(-e)^{2}}{\omega
V}\sum_{\bm{k}}\sum_{a,b,c,d}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}v_{ab}^{\mu}(\bm{k},t)v_{cd}^{\nu}(\bm{k},t^{\prime})$
$\displaystyle\times\Bigl{[}G_{b\sigma
c\sigma^{\prime}}^{\textrm{R}}(\bm{k};t,t^{\prime})G_{d\sigma^{\prime}a\sigma}^{<}(\bm{k};t^{\prime},t)$
$\displaystyle+G_{b\sigma
c\sigma^{\prime}}^{<}(\bm{k};t,t^{\prime})G_{d\sigma^{\prime}a\sigma}^{\textrm{A}}(\bm{k};t^{\prime},t)\Bigr{]},$
(75)

and the lesser, retarded, and advanced Green’s functions are defined as
follows:

$\displaystyle G_{b\sigma^{\prime}a\sigma}^{<}(\bm{k};t,t^{\prime})$
$\displaystyle=i\langle
c_{\bm{k}a\sigma}^{\dagger}(t^{\prime})c_{\bm{k}b\sigma^{\prime}}(t)\rangle,$
(76) $\displaystyle G_{a\sigma
b\sigma^{\prime}}^{\textrm{R}}(\bm{k};t,t^{\prime})$
$\displaystyle=-i\theta(t-t^{\prime})\langle\\{c_{\bm{k}a\sigma}(t),c_{\bm{k}b\sigma^{\prime}}^{\dagger}(t^{\prime})\\}\rangle,$
(77) $\displaystyle G_{a\sigma
b\sigma^{\prime}}^{\textrm{A}}(\bm{k};t,t^{\prime})$
$\displaystyle=i\theta(t^{\prime}-t)\langle\\{c_{\bm{k}a\sigma}(t),c_{\bm{k}b\sigma^{\prime}}^{\dagger}(t^{\prime})\\}\rangle.$
(78)

Since we consider charge transport in the nonequilibrium steady state, we
introduce the time-averaged charge conductivity,

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}}$
$\displaystyle=\lim_{\omega\rightarrow
0}\textrm{Re}\int_{0}^{T_{\textrm{p}}}\frac{dt_{\textrm{av}}}{T_{\textrm{p}}}\int_{-\infty}^{\infty}dt_{\textrm{rel}}e^{i\omega
t_{\textrm{rel}}}\sigma_{\mu\nu}^{\textrm{C}}(t,t^{\prime}),$ (79)

where $t_{\textrm{rel}}=t-t^{\prime}$ and $t_{\textrm{av}}=(t+t^{\prime})/2$.
By combining Eq. (79) with Eqs. (73)–(75) and performing some calculations
[12], we obtain

$\displaystyle\sigma_{\mu\nu}^{\textrm{C}}=\frac{(-e)^{2}}{V}\sum_{\bm{k}}\sum_{a,b,c,d}\sum_{\sigma,\sigma^{\prime}=\uparrow,\downarrow}\int_{-\Omega/2}^{\Omega/2}\frac{d\omega^{\prime}}{2\pi}\sum_{m,l,n,q=-\infty}^{\infty}$
$\displaystyle\times\Bigl{\\{}[v_{ab}^{\mu}(\bm{k})]_{ml}\frac{\partial[G_{b\sigma
c\sigma^{\prime}}^{\textrm{R}}(\bm{k},\omega^{\prime})]_{ln}}{\partial\omega^{\prime}}[v_{cd}^{\nu}(\bm{k})]_{nq}[G_{d\sigma^{\prime}a\sigma}^{<}(\bm{k},\omega^{\prime})]_{qm}$
$\displaystyle-[v_{ab}^{\mu}(\bm{k})]_{ml}[G_{b\sigma
c\sigma^{\prime}}^{<}(\bm{k},\omega^{\prime})]_{ln}[v_{cd}^{\nu}(\bm{k})]_{nq}\frac{\partial[G_{d\sigma^{\prime}a\sigma}^{\textrm{A}}(\bm{k},\omega^{\prime})]_{qm}}{\partial\omega^{\prime}}\Bigr{\\}},$
(80)

where the group velocity and Green’s functions in the Floquet representation
are defined as

$\displaystyle[v_{ab}^{\mu}(\bm{k})]_{mn}$
$\displaystyle=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}v_{ab}^{\mu}(\bm{k},t),$ (81) $\displaystyle[G_{a\sigma
b\sigma^{\prime}}^{r}(\bm{k},\omega^{\prime})]_{mn}$
$\displaystyle=\int_{-\infty}^{\infty}dt_{\textrm{rel}}e^{i(\omega^{\prime}+\frac{m+n}{2}\Omega)t_{\textrm{rel}}}\int_{0}^{T_{\textrm{p}}}\frac{dt_{\textrm{av}}}{T_{\textrm{p}}}$
$\displaystyle\times e^{i(m-n)\Omega t_{\textrm{av}}}G_{a\sigma
b\sigma^{\prime}}^{r}(\bm{k};t,t^{\prime}).$ (82)

Equation (80) is equivalent to Eq. (8).

##  Appendix D Dyson’s equation for the Green’s functions

The Green’s functions appearing in Eq. (8) are determined from Dyson’s
equation in a matrix form [12],

$\displaystyle G=G_{0}+G_{0}\Sigma G,$ (83)

where

$\displaystyle
G=\left(\begin{array}[]{@{\,}cc@{\,}}G^{\textrm{R}}&G^{\textrm{K}}\\\\[3.0pt]
0&G^{\textrm{A}}\end{array}\right),\
G_{0}=\left(\begin{array}[]{@{\,}cc@{\,}}G^{\textrm{R}}_{0}&G^{\textrm{K}}_{0}\\\\[3.0pt]
0&G^{\textrm{A}}_{0}\end{array}\right),\
\Sigma=\left(\begin{array}[]{@{\,}cc@{\,}}\Sigma^{\textrm{R}}&\Sigma^{\textrm{K}}\\\\[3.0pt]
0&\Sigma^{\textrm{A}}\end{array}\right).$ (90)

Here $G^{\textrm{R}}$, $G^{\textrm{A}}$, and $G^{\textrm{K}}$ are the
retarded, advanced, and Keldysh Green’s functions with $H_{\textrm{sb}}$,
$G^{\textrm{R}}_{0}$, $G^{\textrm{A}}_{0}$, and $G^{\textrm{K}}_{0}$ are those
without $H_{\textrm{sb}}$, and $\Sigma^{\textrm{R}}$, $\Sigma^{\textrm{A}}$,
and $\Sigma^{\textrm{K}}$ are the retarded, advanced, and Keldysh self-
energies due to the second-order perturbation of $H_{\textrm{sb}}$; the matrix
$G^{\textrm{R}}$ is, for instance, given by
$G^{\textrm{R}}=([G^{\textrm{R}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn})$, where $a$, $b=A$, $B$, $\sigma$,
$\sigma^{\prime}=\uparrow$, $\downarrow$, and $m$,
$n=-\infty,\cdots,0,1,\cdots,\infty$. The retarded, advanced, and Keldysh
components are related to the lesser component via the relation, such as

$\displaystyle
G^{<}=\frac{1}{2}(G^{\textrm{K}}-G^{\textrm{R}}+G^{\textrm{A}}).$ (91)

In the second-order perturbation theory, in which $H_{\textrm{sb}}$ is treated
as perturbation, $\Sigma^{\textrm{R}}$, $\Sigma^{\textrm{A}}$, and
$\Sigma^{\textrm{K}}$ are given by [12]

$\displaystyle[\Sigma^{\textrm{R}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn}$
$\displaystyle=-i\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}\Gamma,$
(92) $\displaystyle[\Sigma^{\textrm{A}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn}$
$\displaystyle=+i\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}\Gamma,$
(93) $\displaystyle[\Sigma^{\textrm{K}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn}$
$\displaystyle=-2i\Gamma\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}\tanh\frac{\omega+m\Omega}{2T},$
(94)

where $\Gamma$ is the damping. Then, the matrices $G^{\textrm{R}}$,
$G^{\textrm{A}}$, and $G^{\textrm{K}}$ can be determined from the following
relations [12]:

$\displaystyle(G^{\textrm{R}})^{-1}=(G^{-1})^{\textrm{R}},$ (95)
$\displaystyle(G^{\textrm{A}})^{-1}=(G^{-1})^{\textrm{A}},$ (96)
$\displaystyle
G^{\textrm{K}}=-G^{\textrm{R}}(G^{-1})^{\textrm{K}}G^{\textrm{A}},$ (97)

where

$\displaystyle
G^{-1}=\left(\begin{array}[]{@{\,}cc@{\,}}(G^{-1})^{\textrm{R}}&(G^{-1})^{\textrm{K}}\\\\[3.0pt]
0&(G^{-1})^{\textrm{A}}\end{array}\right).$ (100)

Therefore, we obtain the retarded and advanced Green’s functions with
$H_{\textrm{sb}}$ using Eqs. (95) and (96) with the equations,

$\displaystyle[(G^{-1})^{\textrm{R}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn}$
$\displaystyle=(\omega+m\Omega+i\Gamma)\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}$
$\displaystyle-[\epsilon_{ab}(\bm{k})]_{mn}\delta_{\sigma,\sigma^{\prime}},$
(101) $\displaystyle[(G^{-1})^{\textrm{A}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)]_{mn}$
$\displaystyle=(\omega+m\Omega-i\Gamma)\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}$
$\displaystyle-[\epsilon_{ab}(\bm{k})]_{mn}\delta_{\sigma,\sigma^{\prime}},$
(102)

where

$\displaystyle[\epsilon_{ab}(\bm{k})]_{mn}=\int_{0}^{T_{\textrm{p}}}\frac{dt}{T_{\textrm{p}}}e^{i(m-n)\Omega
t}\epsilon_{ab}(\bm{k},t).$ (103)

We also get the Keldysh Green’s function with $H_{\textrm{sb}}$ using these
Green’s functions, Eq. (97), and

$\displaystyle[(G^{-1})^{\textrm{K}}_{a\sigma
b\sigma^{\prime}}(\bm{k},\omega)$
$\displaystyle]_{mn}=2i\Gamma\delta_{m,n}\delta_{a,b}\delta_{\sigma,\sigma^{\prime}}\tanh\frac{\omega+m\Omega}{2T}.$
(104)

Then, using these three Green’s functions and Eq. (91), we obtain the lesser
Green’s function with $H_{\textrm{sb}}$.

##  Appendix E Details of the numerical calculations

We numerically calculate $\sigma_{\mu\nu}^{\textrm{C}}$ of Eq. (8) in the
following procedure. To calculate the momentum summation, we set
$\bm{k}=\frac{m_{1}}{N_{1}}\bm{b}_{1}+\frac{m_{2}}{N_{2}}\bm{b}_{2}$ and
$N_{1}=N_{2}=360$, where $0\leq m_{1}<N_{1}$, $0\leq m_{2}<N_{2}$,
$\bm{b}_{1}={}^{t}(\frac{2\pi}{\sqrt{3}}\ \frac{2\pi}{3})$,
$\bm{b}_{2}={}^{t}(\frac{2\pi}{\sqrt{3}}\ -\frac{2\pi}{3})$, and
$N_{1}N_{2}=\frac{N}{2}$. We calculated the frequency integral using
$\int_{-\Omega/2}^{\Omega/2}d\omega^{\prime}F(\omega^{\prime})\approx\sum_{s=0}^{W-1}\Delta\omega^{\prime}F(\omega^{\prime}_{s})$,
where $\omega^{\prime}_{s}=-\Omega/2+s\Delta\omega^{\prime}$,
$\omega^{\prime}_{W}=\Omega/2$, and
$\Delta\omega^{\prime}=0.001t_{\textrm{NN}}$. Then, to calculate the frequency
derivatives of the Green’s functions, we used $\frac{\partial
F(\omega^{\prime})}{\partial\omega^{\prime}}\approx\frac{F(\omega^{\prime}+\Delta\omega^{\prime})-F(\omega^{\prime}-\Delta\omega^{\prime})}{2\Delta\omega^{\prime}}$.
We took the trace over the Floquet states [i.e.,
$\textrm{tr}(ABCD)=\sum_{m,l,n,q=-\infty}^{\infty}A_{ml}B_{ln}C_{nq}D_{qm}$],
replaced the summation over the Floquet indices,
$\sum_{m,l,n,q=-\infty}^{\infty}$, by
$\sum_{m,l,n,q=-n_{\textrm{max}}}^{n_{\textrm{max}}}$, and set
$n_{\textrm{max}}=2$.

##  Appendix F Additional numerical results

Figure 5:  (Color online) (a)–(f) The polarization dependences of
$\sigma_{yx}^{\textrm{C}}$ and $\sigma_{xy}^{\textrm{C}}$ as functions of
$u=eA_{0}$ for graphene driven by LPL. The blue, green, light blue, and yellow
lines correspond to the cases with the LPL for $\alpha_{x}=\alpha_{y}=1$, for
$\alpha_{x}=-\alpha_{y}=1$, for $-\alpha_{x}=\alpha_{y}=1$, and for
$\alpha_{x}=\alpha_{y}=-1$, respectively.  Figure 6:  (Color online) The
$\theta$ dependences of $\sigma_{yx}^{\textrm{C}}$ and
$\sigma_{xy}^{\textrm{C}}$ at (a) $u=0.4$ and (b) $u=0.8$ in graphene driven
by LPL for $\alpha_{x}=\cos\theta$ and $\alpha_{y}=\sin\theta$. In these
panels, $\Omega=8t_{\textrm{NN}}$, $\Gamma=0.004t_{\textrm{NN}}$, and
$T=0.006t_{\textrm{NN}}$.

We show additional numerical results. Figure 5(a) compares the $u$ dependences
of $\sigma_{yx}^{\textrm{C}}$ in graphene driven by the LPL for
$\alpha_{x}=\alpha_{y}=1$, which has been considered as the case of LPL1 in
the main text, and by the LPL for $\alpha_{x}=-\alpha_{y}=1$. (Note that in
both cases the mirror symmetries about the $xz$ and $yz$ planes and their
combining symmetries are both broken, which means that the latter LPL also
belongs to LPL1.) The $\sigma_{yx}^{\textrm{C}}$’s in these two cases are the
same in magnitude and opposite in sign. In addition,
$\sigma_{yx}^{\textrm{C}}=\sigma_{xy}^{\textrm{C}}$ holds in both cases, as
shown in Fig. 5(b). We should note that the systems driven by the LPL for
$\alpha_{x}=\alpha_{y}=1$ and by the LPL for $\alpha_{x}=-\alpha_{y}=1$ are
connected by a mirror operation with respect to the $xz$ plane, which
interchanges parts of the system above and below the $xz$ mirror plane of Fig.
1(a). This is because this mirror operation replaces $t_{AB}^{Z}(t)$,
$t_{AB}^{Y}(t)$, and $t_{AB}^{X}(t)$ by $t_{BA}^{Z}(t)$, $t_{BA}^{X}(t)$, and
$t_{BA}^{Y}(t)$, respectively [see Eq. (5)] and $t_{BA}^{Z}(t)$,
$t_{BA}^{X}(t)$, and $t_{BA}^{Y}(t)$ in the former case are the same as
$t_{AB}^{Z}(t)$, $t_{AB}^{Y}(t)$, and $t_{AB}^{X}(t)$ in the latter case. Note
that these hopping integrals with the LPL for $\alpha_{x}=\alpha_{y}=1$ are
given by

$\displaystyle t_{AB}^{Z}(t)$ $\displaystyle=t_{\textrm{NN}}e^{-iu\cos\Omega
t},$ $\displaystyle t_{AB}^{X}(t)$
$\displaystyle=t_{\textrm{NN}}e^{iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\frac{1}{2}\cos\Omega t},$ $\displaystyle t_{AB}^{Y}(t)$
$\displaystyle=t_{\textrm{NN}}e^{-iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\frac{1}{2}\cos\Omega t},$ (105) $\displaystyle t_{BA}^{Z}(t)$
$\displaystyle=t_{\textrm{NN}}e^{iu\cos\Omega t},$ $\displaystyle
t_{BA}^{X}(t)$
$\displaystyle=t_{\textrm{NN}}e^{-iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{-iu\frac{1}{2}\cos\Omega t},$ $\displaystyle t_{BA}^{Y}(t)$
$\displaystyle=t_{\textrm{NN}}e^{iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{-iu\frac{1}{2}\cos\Omega t},$ (106)

whereas those with the LPL for $\alpha_{x}=-\alpha_{y}=1$ are given by

$\displaystyle t_{AB}^{Z}(t)$ $\displaystyle=t_{\textrm{NN}}e^{iu\cos\Omega
t},$ $\displaystyle t_{AB}^{X}(t)$
$\displaystyle=t_{\textrm{NN}}e^{iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{-iu\frac{1}{2}\cos\Omega t},$ $\displaystyle t_{AB}^{Y}(t)$
$\displaystyle=t_{\textrm{NN}}e^{-iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{-iu\frac{1}{2}\cos\Omega t},$ (107) $\displaystyle t_{BA}^{Z}(t)$
$\displaystyle=t_{\textrm{NN}}e^{-iu\cos\Omega t},$ $\displaystyle
t_{BA}^{X}(t)$
$\displaystyle=t_{\textrm{NN}}e^{-iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\frac{1}{2}\cos\Omega t},$ $\displaystyle t_{BA}^{Y}(t)$
$\displaystyle=t_{\textrm{NN}}e^{iu\frac{\sqrt{3}}{2}\cos\Omega
t}e^{iu\frac{1}{2}\cos\Omega t}.$ (108)

Moreover, Figs. 5(c) and 5(d) show that the sign of
$\sigma_{yx}^{\textrm{C}}(=\sigma_{xy}^{\textrm{C}})$ can be reversed by
changing from the LPL for $\alpha_{x}=\alpha_{y}=1$ to that for
$-\alpha_{x}=\alpha_{y}=1$. This result can be similarly understood because
the system driven by the LPL for $-\alpha_{x}=\alpha_{y}=1$ is a counterpart
connected by the mirror operation about the $yz$ plane. Then,
$\sigma_{yx}^{\textrm{C}}(=\sigma_{xy}^{\textrm{C}})$ driven by the LPL for
$\alpha_{x}=\alpha_{y}=-1$ becomes the same as that for
$\alpha_{x}=\alpha_{y}=1$, as shown in Figs. 5(e) and 5(f). This is because
the system driven by the LPL for $\alpha_{x}=\alpha_{y}=-1$ is connected to
that driven by the LPL for $\alpha_{x}=-\alpha_{y}=1$ by the mirror operation
about the $yz$ plane (or to that driven by the LPL for
$-\alpha_{x}=\alpha_{y}=1$ by the mirror operation about the $xz$ plane). The
similar properties hold in more general cases in which $\alpha_{x}$ and
$\alpha_{y}$ are written as $\alpha_{x}=\cos\theta$ and
$\alpha_{y}=\sin\theta$, as shown in Figs. 6(a) and 6(b). Namely,
$\sigma_{yx}^{\textrm{C}}(=\sigma_{xy}^{\textrm{C}})$’s driven by the LPL for
$\theta=\theta_{0}$ and $\theta_{0}+180^{\circ}$ (e.g., $\theta=30^{\circ}$
and $210^{\circ}$) are the same in magnitude and sign, whereas those for
$\theta=360^{\circ}-\theta_{0}$ and $180^{\circ}-\theta_{0}$ (e.g.,
$\theta=330^{\circ}$ and $150^{\circ}$) have the opposite sign to
$\theta=\theta_{0}$ and $\theta_{0}+180^{\circ}$ and the same magnitude; these
properties can be understood in a similar way. Therefore, these results
indicate that the sign of the off-diagonal symmetric charge conductivity can
be changed by switching LPL1 to a counterpart connected by the mirror
operation about the $xz$ or $yz$ plane. Note that the relation between the LPL
for $\alpha_{x}=\alpha_{y}=1$ and for $\alpha_{x}=-\alpha_{y}=1$ is similar to
that between left- and right-handed circularly polarized light because the
systems driven by left- and right-handed circularly polarized light are
connected by the mirror operation (and also by a time-reversal operation
[12]).

## References

  * [1] Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Science 342, 453-457 (2013).
  * [2] M. Claassen, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Nat. Commun. 8, 1192 (2017).
  * [3] S. Kitamura, T. Oka, and H. Aoki, Phys. Rev. B 96, 014406 (2017).
  * [4] J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, Phys. Rev. Lett. 15, 190 (1965).
  * [5] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and Th. Rasing, Nature 435, 655–657 (2005).
  * [6] T. Oka and H. Aoki, Phys. Rev. B 79, 081406(R) (2009).
  * [7] J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri, Nat. Phys. 16, 38–41 (2020).
  * [8] E. H. Hall, Philos. Mag. 12, 157 (1881).
  * [9] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954).
  * [10] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * [11] T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. B 93, 144307 (2016).
  * [12] N. Arakawa and K. Yonemitsu, Comms. Phys. 6, 43 (2023).
  * [13] O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, Nat. Photonics 9, 99 (2015).
  * [14] T. Nag, R.-J. Slager, T. Higuchi, and T. Oka, Phys. Rev. B 100, 134301 (2019).
  * [15] T. V. Trevisan, P. V. Arribi, O. Heinonen, R.-J. Slager, and P. P. Orth, Phys. Rev. Lett. 128, 066602 (2022).
  * [16] Y. Ikeda, S. Kitamura, and T. Morimoto, Prog. Theor. Exp. Phys. 2022, 04A101 (2022).
  * [17] D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, Advances in Physics, 59, 261-482 (2010).
  * [18] M. Büttiker, Phys. Rev. B 32, 1846(R) (1985).
  * [19] M. Büttiker, Phys. Rev. B 33, 3020 (1986).
  * [20] N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. Lett. 103, 047403 (2009).
  * [21] L. D’Alessio and M. Rigol, Phys. Rev. X 4, 041048 (2014).
  * [22] A. Lazarides, A. Das, and R. Moessner, Phys. Rev. E 90, 012110 (2014).
  * [23] L. Du, Q. Chen, A. D. Barr, A. R. Barr, and G. A. Fiete, Phys. Rev. B 98, 245145 (2018).
  * [24] M. Eckstein and M. Kollar, Phys. Rev. B 78, 205119 (2008).
  * [25] J. H. Shirley, Phys. Rev. 138, B979 (1965).
  * [26] H. Sambe, Phys. Rev. A 7, 2203 (1973).
  * [27] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
  * [28] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010).
  * [29] L. Onsager, Phys. Rev. 37, 405 (1931).
  * [30] L. Onsager, Phys. Rev. 38, 2265 (1931).
  * [31] H. Kontani, T. Tanaka, and K. Yamada, Phys. Rev. B 75, 184416 (2007).
  * [32] T. Tomizawa and H. Kontani, Phys. Rev. B 80, 100401(R) (2009).
  * [33] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. 4, eaat0346 (2018).
  * [34] A. Eckardt and E. Anisimovas, New J. Phys. 17, 093039 (2015).
  * [35] M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys. 64, 139 (2015).
  * [36] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev. B 84, 235108 (2011).
  * [37] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
  * [38] V. M. Edelstein, Phys. Rev. Lett. 95, 156602 (2005).
  * [39] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * [40] G. Yang, L. Li, W. B. Lee, and M. C. Ng, Sci. Tech. Adv. Mat. 19, 613–648 (2018).
  * [41] E. Kogan, V. U. Nazarov, V. M. Silkin, and M. Kaveh, Phys. Rev. B 89, 165430 (2014).