# Light-enhanced nonlinear Hall effect

Fang Qin<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore 117551, Singapore  Rui Chen<EMAIL_ADDRESS>Department
of Physics, Hubei University, Wuhan 430062, China  Ching Hua Lee
<EMAIL_ADDRESS>Department of Physics, National University of Singapore,
Singapore 117551, Singapore

###### Abstract

It is well known that a nontrivial Chern number results in quantized Hall
conductance. What is less known is that, generically, the Hall response can be
dramatically different from its quantized value in materials with broken
inversion symmetry. This stems from the leading Hall contribution beyond the
linear order, known as the Berry curvature dipole (BCD). While the BCD is in
principle always present, it is typically very small outside of a narrow
window close to a topological transition and is thus experimentally elusive
without careful tuning of external fields, temperature, or impurities. In this
work, we transcend this challenge by devising optical driving and quench
protocols that enable practical and direct access to large BCD and nonlinear
Hall responses. Varying the amplitude of an incident circularly polarized
laser drives a topological transition between normal and Chern insulator
phases, and importantly allows the precise unlocking of nonlinear Hall
currents comparable to or larger than the linear Hall contributions. This
strong BCD engineering is even more versatile with our two-parameter quench
protocol, as demonstrated in our experimental proposal. Our predictions are
expected to hold qualitatively across a broad range of Hall materials, thereby
paving the way for the controlled engineering of nonlinear electronic
properties in diverse media.

##  I Introduction

The nonlinear Hall effect is an exciting response contribution beyond the
much-studied quantum anomalous Hall effect Chang  _et al._ (2013); Sodemann
and Fu (2015); Du  _et al._ (2018, 2019, 2021); Ortix (2021); Bandyopadhyay
_et al._ (2024); Ma  _et al._ (2019), in which an applied electric field
results in a quantized transverse current response in an insulator. It stems
from a higher-order correction in the quantum response known as the Berry
curvature dipole (BCD) Sodemann and Fu (2015), which generically exists
whenever inversion symmetry is broken Sodemann and Fu (2015); Du  _et al._
(2018, 2019, 2021). Venturing beyond the topological paradigm, the BCD leads
to an interesting nonlinear anisotropic current response accompanied by higher
harmonic generation Huang  _et al._ (2023a); Eden (2004); Ghimire  _et al._
(2011); Lee  _et al._ (2015); Tai and Lee (2021); Severt  _et al._ (2021); Li
_et al._ (2023) and has motivated a large number of experimental
investigations. Indeed, the BCD has been observed in a variety of materials,
such as two-dimensional bilayer or few-layer WTe2 Ma  _et al._ (2019); Zhang
_et al._ (2018); Xu  _et al._ (2018); Kang  _et al._ (2019); Xiao  _et al._
(2020); Ye  _et al._ (2023); Huang  _et al._ (2023b), BiTeI Facio  _et al._
(2018), bilayer graphene Ho  _et al._ (2021), twisted bilayer graphene Duan
_et al._ (2022); Huang  _et al._ (2023a), twisted double bilayer graphene
Sinha  _et al._ (2022); Chakraborty  _et al._ (2022), strained twisted bilayer
graphene Pantaleón  _et al._ (2021); Zhang  _et al._ (2022), two-dimensional
monolayer WSe2 Qin  _et al._ (2021), strained twisted bilayer WSe2 Hu  _et
al._ (2022), two-dimensional monolayer MoS2 Lee  _et al._ (2017); Son  _et
al._ (2019), Weyl semimetal TaIrTe4 Kumar  _et al._ (2021), three-dimensional
Dirac semimetal Cd3As2 Shvetsov  _et al._ (2019); Zhao  _et al._ (2023),
$\mathbb{Z}_{2}$ topological insulator Bi2Se3 He  _et al._ (2021), the
Weyl–Kondo semimetal Ce3Bi4Pd3 Dzsaber  _et al._ (2021), and Td-MoTe2 Tiwari
_et al._ (2021). The dominance of the intrinsic nonlinear Hall effect in
antiferromagnets with ${\cal P}{\cal T}$ symmetry has been studied Ma  _et
al._ (2023); Das  _et al._ (2023); Liu  _et al._ (2021); Gao  _et al._ (2014);
Wang  _et al._ (2021) and subsequently experimentally probed in thick Td-MoTe2
samples Lai  _et al._ (2021). More recently, the quantum metric-induced
nonlinear Hall effect in a topological antiferromagnet MnBi2Te4 has also been
reported Gao  _et al._ (2023); Wang  _et al._ (2023).

Yet, despite its supposed ubiquity, observing the BCD (or its associated
nonlinear Hall response) is fraught with practical challenges. Besides generic
experimental demands such as maintaining low temperatures ($10\sim 100$ K Ma
_et al._ (2019); Du  _et al._ (2021)), a key challenge is that the BCD is only
significant in a narrow window very close to a topological phase transition Du
_et al._ (2018). Accessing it hence requires precise control of external
parameters such as magnetic field Gao  _et al._ (2023); Wang  _et al._ (2023),
electric field Xu  _et al._ (2018); Xiao  _et al._ (2020); Ma  _et al._
(2019); Du  _et al._ (2021); Lai  _et al._ (2021), temperature Ma  _et al._
(2019); Lai  _et al._ (2021); Gao  _et al._ (2023); Wang  _et al._ (2023),
charge/carrier density Ma  _et al._ (2019); Gao  _et al._ (2023); Wang  _et
al._ (2023), strain Pantaleón  _et al._ (2021); Zhang  _et al._ (2022); Sinha
_et al._ (2022); Qin  _et al._ (2021); Hu  _et al._ (2022), or impurities Du
_et al._ (2019); Duan  _et al._ (2022); Huang  _et al._ (2023a); Chen  _et
al._ (2023); Atencia  _et al._ (2023), which are not easily tuned. As such,
only a few notable experimental observations of the nonlinear Hall effect
exist till date, through measuring the BCD Ma  _et al._ (2019); Xu  _et al._
(2018); Kang  _et al._ (2019); Xiao  _et al._ (2020); Ye  _et al._ (2023);
Huang  _et al._ (2023b); Ho  _et al._ (2021); Sinha  _et al._ (2022),
intrinsic nonlinear Hall conductivity Lai  _et al._ (2021); Gao  _et al._
(2023); Wang  _et al._ (2023); Huang  _et al._ (2023a), or both Duan  _et al._
(2022).

To circumvent these challenges, we devise optical driving and quench protocols
that can precisely tune a system such that the nonlinear Hall response is
dramatically enhanced. Through the periodic driving of polarized light, we not
only obtain a new avenue for driving topological phase transitions that may be
inaccessible in static systems, but more importantly unlock new routes towards
the versatile engineering of the BCD. Obtaining the effective Hamiltonian
through the high-frequency Floquet-Magnus expansion Magnus (1954); Blanes  _et
al._ (2009); Bukov  _et al._ (2015); Eckardt and Anisimovas (2015); Lee  _et
al._ (2018a), we identify a light-enhanced peak in the BCD that can be tuned
via the intensity of an applied circularly polarized laser in a very versatile
manner. This is accompanied by a critical light-induced band inversion, which
demarcates a topological phase boundary between Chern and normal insulator
phases, notably without involving the tuning of any other physical parameter.
Crucially, the precise tunability of our approach allows for further band
engineering via temporal modulation of the laser strength, resulting in
controlled modulation of the BCD strength over a wide range.

##  II Results

###  II.1 General formalism

We first provide the foundational framework for the nonlinear Hall effect,
such as to contrast it with the more commonly known linear Hall effect. When
subjected to external driving forces, the typical dominant response observed
is the linear response, which, for instance, accounts for the well-known
quantum Hall effect with broken time-reversal symmetry. In general, a linear
response encompasses longitudinal and transverse currents. However, nonlinear
response can also manifest strongly in transverse currents Sodemann and Fu
(2015); Kang  _et al._ (2019); Ma  _et al._ (2019); Gao  _et al._ (2014); Wang
_et al._ (2021); Du  _et al._ (2018, 2019, 2021); Chen  _et al._ (2023);
Atencia  _et al._ (2023); Lai  _et al._ (2021); Huang  _et al._ (2023a); Gao
_et al._ (2023); Wang  _et al._ (2023) due to higher-order Berry curvature
effects, with a doubled frequency component in the current signal in the
transverse direction. Consequently, the transverse current is proportional to
the second power in the longitudinal fields.

The response from a driven electric field $\mathbb{E}$ is measurable through
the electric current density $\mathbb{J}$. We expand the electric current
density $J_{i}$ in increasing orders of the electric field as

$\displaystyle
J_{i}=\sigma_{ij}E_{j}+\chi_{ijk}E_{j}E_{k}+\xi_{ijkl}E_{j}E_{k}E_{l}+\cdots,$
(1)

where $\\{i,j,k,l\\}\in\\{x,y,z\\}$, and $\mathbf{E}=(E_{x},E_{y},E_{z})$ is
the external electric field. Here, the first term denotes the linear response,
and its frequency follows the external electric field. The second term is the
leading nonlinear contribution with a doubled frequency.

Below, we shall derive the linear and nonlinear responses through a semi-
classical approach, where the perturbative effects of an external electric
field are treated via the electronic occupation functions. Subsuming the
effects of electron scattering under a phenomenological relaxation time
$\tau$, the relaxation time approximation yields the following Boltzmann
equation Schliemann and Loss (2003); Sinitsyn (2007); Sodemann and Fu (2015);
Mahan (2013); Lee  _et al._ (2018b, 2020)

$-\tau\partial_{t}f+\frac{e\tau}{\hbar}E_{a}\partial_{k_{a}}f=f-f_{0},$ (2)

where $-e$ is the electron charge, $\hbar$ is the reduced Planck’s constant,
$k_{a}$ is the wave vector of electronic wavepackets along the
$a\in\\{x,y,z\\}$ directions, $f_{0}$ is the equilibrium occupancy given by
the Fermi-Dirac distribution function, and $f$ is the non-equilibrium
distribution function. Here, $\tau$ sets the timescale for $f$ to relax to its
equilibrium distribution $f_{0}$. By inverting the derivative operators, we
can directly solve for the non-equilibrium distribution function $f$ as
follows:

$\displaystyle
f\\!=\\!\frac{f_{0}}{1\\!+\\!\tau\partial_{t}\\!-\\!\frac{e\tau}{\hbar}E_{a}\partial_{k_{a}}}\\!\approx\\!\sum_{n=0}^{\infty}\left(\\!-\tau\partial_{t}\\!+\\!\frac{e\tau}{\hbar}E_{a}\partial_{k_{a}}\right)^{n}f_{0}.$
(3)

To understand how the response originates from this mathematical framework, we
expand $f$ into $f={\rm Re}(f_{0}+f_{1}+f_{2}+\cdots)$, where $f_{j}$ contains
terms with the $j$-th power of the electric fields. Below, we shall only
retain up to the quadratic order, such that

${\bf J}({\bf E})\\!\approx\\!-e\int\frac{d^{2}{\bf
k}}{(2\pi)^{2}}\left(\\!\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{\bf
k}\\!+\\!\frac{e}{\hbar}{\bf E}\\!\times\\!{\bf\Omega}_{\bf k}\\!\right){\rm
Re}(f_{0}\\!+\\!f_{1}\\!+\\!f_{2}),$ (4)

with band dispersion $\epsilon_{\bf k}$ and the Berry curvature
${\bf\Omega}_{\bf k}$ Xiao  _et al._ (2010); Shen (2017). To proceed, we
consider a single-frequency driven electric field ${\bf E}(t)=E_{a}(t){\bf
e}_{a}={\rm Re}(\mathcal{E}_{a}e^{i\omega t}){\bf e}_{a}$ along the $a$
direction and write down the coefficients of non-equilibrium distribution to
the first and second orders:

$f_{1}=f_{1}^{(\omega)}e^{i\omega t},\quad
f_{2}=f_{2}^{(0)}+f_{2}^{(2\omega)}e^{2i\omega t}.$ (5)

Since the second-order coefficient $f_{2}$ comprises both
$\mathcal{E}_{a}^{2}$ and $\mathcal{E}_{a}^{2}e^{i2\omega t}$ terms, it
contains both a constant and a frequency-doubled term. Specifically, by
substituting Eq. (5) into Eq. (4), one can have

$\displaystyle J_{b}(E_{a})={\rm Re}(J_{b}^{0}+J_{b}^{\omega}e^{i\omega
t}+J_{b}^{2\omega}e^{i2\omega t}),$ (6)

where
$J_{b}^{0}\\!=\\!\chi_{baa}^{(0)}\mathcal{E}_{a}^{2},~{}J_{b}^{\omega}\\!=\\!\chi_{ba}^{(1)}\mathcal{E}_{a},~{}J_{b}^{2\omega}\\!=\\!\chi_{baa}^{(2)}\mathcal{E}_{a}^{2}$,
with coefficients given at the leading order by

$\displaystyle\chi_{ba}^{(1)}$
$\displaystyle\approx-\varepsilon^{bac}\sigma_{c},$ (7)
$\displaystyle\chi_{baa}^{(0)}$
$\displaystyle\approx\chi_{baa}^{(2)}\approx\frac{e^{3}\tau\varepsilon^{bac}D_{ac}}{2\hbar^{2}(1+i\omega\tau)}.$
(8)

In the above, we have only kept terms to leading order in
$\frac{e\tau}{\hbar}\mathcal{E}_{a}a$ (with the lattice constant $a$), which
is very small in relevant experiments Ma  _et al._ (2019). For the subleading
contributions, see equations (50) to (52) in the Methods Section III.1. In the
above, $\varepsilon^{bac}$ is the Levi-Civita anti-symmetric tensor; $a$, $b$,
and $c$ index the spatial coordinates $x$, $y$, and $z$. The two relevant
physical quantities are the linear Hall conductance $\sigma_{c}$ and the
nonlinear Hall response BCD ($D_{ac}$). Specializing to the 2D $x-y$ plane
such that the index $c=z$, we write the Hall conductance as Kubo (1957); Kubo
_et al._ (1957); Thouless  _et al._ (1982); Shen (2017)

$\displaystyle\sigma_{H}\\!=\\!\frac{e^{2}}{h}\frac{1}{2\pi}\sum_{n}\int
d^{2}{\bf k}~{}\Omega_{{\bf k},z}^{(n)}f_{0},$ (9)

where $\Omega_{{\bf k},z}^{(n)}$ is the Berry curvature corresponding to the
$n$-th eigenstate. However, the main quantity of focus in this work is the
BCD:

$\displaystyle D_{ac}=-\sum_{n}\int\frac{d^{2}{\bf
k}}{(2\pi)^{2}}(\partial_{k_{a}}\epsilon_{\bf k}^{(n)})\Omega_{{\bf
k},c}^{(n)}\frac{\partial f_{0}}{\partial\epsilon_{\bf k}^{(n)}}.$ (10)

Its detailed derivation can be found in Methods Section III.1. Here, the
derivative of the equilibrium distribution function is given by $\partial
f_{0}/\partial\epsilon_{\bf k}^{(n)}=\frac{-e^{\left(\epsilon_{\bf
k}^{(n)}\\!-\\!E_{F}\right)/(k_{B}T)}}{\left[1+e^{\left(\epsilon_{\bf
k}^{(n)}\\!-\\!E_{F}\right)/(k_{B}T)}\right]^{2}k_{B}T}$, where $T$ is the
temperature and $k_{B}$ is the Boltzmann constant. The above integral for the
BCD indicates that the nonlinear Hall response is predominantly governed by
the states near the Fermi energy $E_{F}$ Du  _et al._ (2018, 2019, 2021); Chen
_et al._ (2023). We can interpret $D_{ac}$ as the momentum-space integral over
the Berry curvature $\Omega_{{\bf k},c}^{(n)}$ weighted by the dispersion
$\partial_{k_{a}}\epsilon_{\bf k}^{(n)}$ and occupation gradient $\partial
f_{0}/\partial\epsilon_{\bf k}^{(n)}$, summed over all eigenstates $n$. Since
inversion symmetry breaking is essential for the non-vanishing Berry
curvature, only systems with broken inversion symmetry can exhibit non-zero
BCD and thus nonlinear (transverse) response Du  _et al._ (2018, 2019, 2021);
Chen  _et al._ (2023).

####  II.1.1 Two-component models

The simplest models with nontrivial linear Hall and nonlinear BCD responses
require at least two bands for nontrivial Berry curvature. A generic
2-component model is expressed as the ansatz

$\displaystyle{\cal H}({\bf
k})=h_{0}\sigma_{0}+\sum_{i=x,y,z}h_{i}\sigma_{i},$ (11)

where $\sigma_{x,y,z}$ are the Pauli matrices, and $\sigma_{0}$ is the
$2\times 2$ identity matrix. The eigenenergies for its upper ($+$) and lower
($-$) bands are

$\displaystyle\epsilon_{\mathbf{k}}^{(\pm)}\\!=\\!h_{0}\\!\pm\\!\sqrt{h_{x}^{2}\\!+\\!h_{y}^{2}\\!+\\!h_{z}^{2}},$
(12)

with corresponding eigenvectors $\left|\psi^{(\pm)}\right\rangle$
($\mathbb{k}$ subscript index omitted for brevity). The $z$ component of Berry
curvature, which contributes to the Hall response in two-dimensional systems,
is given by Mahan (2013)

$\displaystyle\Omega^{(\pm)}_{\mathbf{k},z}\\!=\\!-2\varepsilon_{zxy}\\!\frac{{\rm
Im}\\!\left\langle\\!\psi^{(\pm)}\\!\left|\\!\frac{\partial{\cal
H}^{(F)}}{\partial
k_{x}}\right|\\!\psi^{(\mp)}\\!\right\rangle\\!\\!\left\langle\\!\psi^{(\mp)}\\!\left|\\!\frac{\partial\\!{\cal
H}^{(F)}}{\partial
k_{y}}\\!\right|\\!\psi^{(\pm)}\\!\right\rangle}{\left[\epsilon_{\mathbf{k}}^{(\pm)}-\epsilon_{\mathbf{k}}^{(\mp)}\right]^{2}}.$
(13)

####  II.1.2 Effective Floquet Hamiltonian from optical driving

The paradigmatic model for describing nonlinear Hall materials, for instance,
monolayer WTe2 Kang  _et al._ (2019); Ma  _et al._ (2019); Ye  _et al._
(2023); Zhang  _et al._ (2018); Xiao  _et al._ (2020), is the two-dimensional
tilted massive Dirac model Sodemann and Fu (2015); Du  _et al._ (2018); Chen
_et al._ (2023); Muechler  _et al._ (2016), which form the theoretical basis
in nonlinear Hall effect experiments Du  _et al._ (2018); Ma  _et al._ (2019):

$\displaystyle{\cal H}({\bf
k})\\!=\\!t_{0}k_{x}\sigma_{0}\\!+\\!vk_{y}\sigma_{x}\\!+\\!\eta
vk_{x}\sigma_{y}\\!+\\!(m\\!-\\!\alpha k^{2})\sigma_{z},$ (14)

where $\eta$ is allowed to take values of $\pm 1$ in principle (in our
numerics, we set $\eta=-1$), and $t_{0}$, $v$, $m$, and $\alpha$ are
parameters that can be empirically fitted. The tilted term
$t_{0}k_{x}\sigma_{0}$ breaks the inversion symmetry and is key to triggering
the nonlinear Hall effect.

We next discuss how optical driving can modify the effective Hamiltonian and
consequently significantly and precisely enhance the BCD. In a generic
setting, the optical electric field propagating along the $z$ direction can be
expressed as ${\bf E}(t)=\partial{\bf A}(t)/\partial
t=E_{0}(\cos(\tilde{\omega}t),\cos(\tilde{\omega}t+\varphi))$, where $E_{0}$
is the amplitude of the electric field and $\tilde{\omega}$ is the angular
frequency of the light. The phase $\varphi$ controls the polarization:
$\varphi=0$ introduces linear polarization, while $\varphi=\mp\pi/2$
introduces left- or right-handed circular polarization. Integrating, we can
have ${\bf A}(t)={\bf
A}(t+T)=\tilde{\omega}^{-1}E_{0}(\sin(\tilde{\omega}t),\sin(\tilde{\omega}t+\varphi))$
that is of period $T=2\pi/\tilde{\omega}$. Notice that the light frequency
$\tilde{\omega}$ is much higher than the ultralow frequency $\omega$ (17.77 Hz
Ma  _et al._ (2019)) of an longitudinal alternating current (a.c.)
$I^{\omega}_{x}$, which is used to induce the transverse nonlinear Hall effect
in experiments Ma  _et al._ (2019); Du  _et al._ (2021).

Under optical driving, the motion of lattice electrons is governed by minimal
substitution of the lattice momentum with the electromagnetic gauge field
${\bf A}(t)$. Hence, the photon-dressed effective Hamiltonian is given by

$\displaystyle{\cal H}({\bf k},t)={\cal H}\left({\bf k}-\frac{e}{\hbar}{\bf
A}(t)\right).$ (15)

In our Floquet band engineering proposal, we are interested in the off-
resonant regime where the central Floquet band is far away from other
replicas, such that the high-frequency expansion is applicable Magnus (1954);
Blanes  _et al._ (2009); Bukov  _et al._ (2015); Eckardt and Anisimovas
(2015); Lee  _et al._ (2018a). As such, we set the driving optical frequency
to the representative value $\hbar\tilde{\omega}=1$ eV ($\tilde{\omega}\sim
1.519\times 10^{15}$ Hz), which is much larger than the bandwidth Qin  _et
al._ (2023a, 2022a, 2022b). Under periodic driving through $\mathbb{A}(t)$,
the effective Floquet Hamiltonian Oka and Aoki (2009); Lee  _et al._ (2018a);
Qin  _et al._ (2023a, 2022a, 2022b) ${\cal H}^{(F)}({\bf
k})=\frac{i}{T}\ln\left[\mathcal{T}e^{-i\int^{T}_{0}{\cal H}({\bf
k},t)dt}\right]$ is the effective static Hamiltonian with the effects of the
periodic driving “averaged” over one period. In the high-frequency regime, a
closed-form solution exists via the Magnus expansion Magnus (1954); Blanes
_et al._ (2009); Bukov  _et al._ (2015); Eckardt and Anisimovas (2015); Lee
_et al._ (2018a)

$\displaystyle{\cal H}^{(F)}({\bf k})={\cal
H}_{0}+\sum_{n=1}^{\infty}\frac{[{\cal H}_{-n},{\cal
H}_{n}]}{n\hbar\tilde{\omega}}+{\cal O}(\tilde{\omega}^{-2}),$ (16)

where ${\cal H}_{n}=\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{in\tilde{\omega}t}dt$ is the $n$-th time Fourier component of
$\mathcal{H}(\mathbb{k},t)$. For our tilted Dirac model [Eq. (14)], all but
the $n=1$ commutators vanish, as shown in Methods Section III.2, and the
Floquet Hamiltonian takes the form

$\displaystyle{\cal H}^{(F)}({\bf
k})=h_{0}\sigma_{0}+\sum_{i=x,y,z}h_{i}^{(F)}\sigma_{i},$ (17)

where $h_{0}=t_{0}k_{x}$ as before,

$\displaystyle h_{x}^{(F)}\\!=$ $\displaystyle vk_{y}\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (18) $\displaystyle
h_{y}^{(F)}\\!=$ $\displaystyle\eta vk_{x}\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (19) $\displaystyle
h_{z}^{(F)}\\!=$ $\displaystyle
m-A_{0}^{2}\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)-\alpha
k^{2},$ (20)

and $A_{0}=eE_{0}/(\hbar\tilde{\omega})$. The detailed derivations for Eqs.
(17) to (20) can be found in Methods Section III.2. Explicitly, we see that
the optical driving has introduced new contributions proportional to
$A_{0}^{2}$, which acts as a rescaling of the effective mass $m$ (up to an
overall rescaling of the Hamiltonian) and hence ultimately its nonlinear
response properties. To have non-vanishing BCD, note that $t_{0}\neq 0$ is
still required for inversion symmetry breaking (Methods Section III.3), even
though time-reversal symmetry is already broken for any values of $t_{0}$ and
$A_{0}$ (Methods Section III.4). Note that in the later numerical
calculations, the Hamiltonian (17) would be regularized into its corresponding
tight-binding lattice Hamiltonian. The tight-binding lattice model for the
Floquet Hamiltonian is shown in Supplementary Materials SI.1.

Figure 1: Light-induced topological band inversion and its large BCD
contribution. Top row (a1-e1): The $k_{y}=0$ slice of the Floquet band
structure [Eq. (12)] for our nonlinear Hall medium [Eq. (17)], which exhibits
a band inversion at laser amplitude $A_{0}\to A_{0c}\approx
1.0541~{}\text{nm}^{-1}$. Second row (a2)-(e2): Berry curvature
$\Omega^{(-)}_{\mathbb{k},z}$ [Eq. (13)] of the lower band, which integrates
to $+1$ for $A_{0}<A_{0c}$ and $0$ for $A_{0}>A_{0c}$. Third row (a3)-(e3):
The resultant linear Hall conductance [Eq. (9)] as a function of the Fermi
energy $E_{F}$, which is quantized at $+e^{2}/h$ in the gap (pale yellow) for
$A_{0}<A_{0c}$ and $0$ for $A_{0}>A_{0c}$, corresponding to the Chern
insulator (CI) and normal insulator (NI) phases, respectively. Fourth row
(a4)-(e4): The nonlinear Berry curvature dipole (BCD) $D_{xz}$ [Eq. (10)],
which vanishes when $E_{F}$ is in the gap, but which peaks when $E_{F}$ is
near the band edge. Near the transition at $A_{0}\approx A_{0c}$, the peak
becomes drastically higher and narrower. The other parameters are
$\varphi=\pi/2$ (right-handed circularly polarized light),
$\hbar\tilde{\omega}=1$ eV, $t_{0}=0.05$ eV$\cdot$nm, $v=0.1$ eV$\cdot$nm,
$\alpha=0.1$ eV$\cdot$nm2, $m=0.1$ eV, $\eta=-1$, and $k_{B}T=0.003$ eV, i.e.,
$T\approx 34.8136$ K, which are of the same order as those in Refs. Ma  _et
al._ (2019); Muechler  _et al._ (2016); Liu (2021).

To estimate the validity of the high-frequency expansion quantitatively, we
evaluate the maximum instantaneous energy of the time-dependent Hamiltonian
${\cal H}\left({\bf k}-\frac{e}{\hbar}{\bf A}(t)\right)$ averaged over a
Floquet period
$\frac{1}{T}\int_{0}^{T}dt~{}\text{max}\left\\{\big{|}\big{|}{\cal H}({\bf
k},t)\big{|}\big{|}\right\\}<\hbar\tilde{\omega}$ at the $\Gamma$ point
($k_{x}=k_{y}=0$), which gives the following constraint on the optical field
parameters: ${\rm Max}\left(vA_{0},2\alpha
A_{0}^{2}\right)<\hbar\tilde{\omega}$ i.e. $A_{0}<{\rm
Min}\left(\frac{\hbar\tilde{\omega}}{v},\sqrt{\frac{\hbar\tilde{\omega}}{2\alpha}}\right)$.
In the high-frequency regime $\tilde{\omega}\sim 1.519\times 10^{15}$ Hz
($\hbar\tilde{\omega}=1$ eV) with parameters set to $v=0.1$ eV$\cdot$nm and
$\alpha=0.1$ eV$\cdot$nm2, the same order as those in representative 2D Dirac
materials such as WTe2 Kang  _et al._ (2019); Ma  _et al._ (2019); Ye  _et
al._ (2023); Zhang  _et al._ (2018); Xu  _et al._ (2018); Xiao  _et al._
(2020), MoTe2 Muechler  _et al._ (2016), and other WTe2-type materials Liu
(2021), one can obtain $A_{0}=eE_{0}/(\hbar\tilde{\omega})<2.236$ nm-1
($E_{0}=A_{0}\hbar\tilde{\omega}/e<2.236\times 10^{9}$ V/m), which corresponds
to an incident light intensity Paschotta (2016) of
$I=\frac{1}{2}nc\varepsilon_{0}|E_{0}|^{2}<9.954\times 10^{15}$ W/m2, where
$n$ is the refractive index, $c$ is the speed of light in vacuum, and
$\varepsilon_{0}$ is the vacuum permittivity. The refractive index
$n\sim\mathcal{O}(1)$, with $n\approx 1.5$ Buchkov  _et al._ (2021) observed
in monolayer WTe2 in the deep-ultraviolet region.

###  II.2 Divergent nonlinear Hall response near a topological transition

####  II.2.1 Light-induced topological transition

Before showing how a large nonlinear Hall response can be achieved, we first
elaborate on the topological phase transition induced by optical driving.
Shown in Fig. 1 are the energy band structure, Berry curvature, Hall
conductance, and BCD under right-handed circularly polarized light (i.e.,
$\varphi=\pi/2$) at different intensities $A_{0}$.

As can be theoretically predicted from Eqs. (18) to (20), the energy gap [pale
yellow in Figs. 1(a1)-1(e1)] closes at the critical intensity
$A_{0c}=\sqrt{m/\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)}\approx
1.0541$ nm-1 [Fig. 1(c1)]. Tuning the intensity across $A_{0c}$ not only
closes and then opens the energy gap, but also gives rise to band inversion
[Figs. 1(a1)-1(e1)], as evident from the change in sign of the Berry curvature
of the lower band as shown in Figs. 1(a2)-1(e2), as analytically derived in
Supplementary Materials SI.2.

The fact that this band inversion corresponds to a topological transition can
be seen in the change of the quantized value of the Hall conductance
$\sigma_{H}$ for $E_{F}$ in the gapped region (yellow) [Fig. 1(a3)-1(e3)].
When the light intensity $A_{0}$ is below the critical value $A_{0c}$, the
energy spectrum is gapped [Figs. 1(a1) and 1(b1)] and the corresponding Hall
conductance is quantized at $e^{2}/h$ within the gap as shown in Figs. 1(a3)
and 1(b3). This corresponds to the Chern insulator (CI) phase. When the light
intensity $A_{0}>A_{0c}$, the spectrum is also gapped, as shown in Figs. 1(d1)
and 1(e1), but the corresponding Hall conductance equals zero as shown in
Figs. 1(d3) and 1(e3). This is the normal insulator (NI) phase.

Most saliently, at the topological phase transition $A_{0c}$, the BCD
($D_{xz}$) diverges for $E_{F}$ near the band edge, just outside of the pale
yellow gap region. Very close to the transition, as shown in Fig. 1(c4), it is
orders of magnitude larger than that away from the transition, i.e., Figs.
1(a4), 1(b4), 1(d4), and 1(e4). Since the BCD is proportional to the nonlinear
Hall conductance, which can be directly measured, this divergence would have
profound physical consequences, as explored in the following subsection.
Analogous results under left-handed circularly polarized light
($\varphi=-\pi/2$) are qualitatively similar and can be found in Supplementary
Materials SI.3.

####  II.2.2 Non-quantized current from nonlinear light-enhanced BCD

Figure 2:  Light enhanced BCD and significant departure from linear Hall
response. (a) BCD [Eq. (10)] peak max$(|D_{xz}|)$ (black) and band gap
$\Delta={\rm min}[\epsilon_{\bf k}^{(+)}]-{\rm max}[\epsilon_{\bf k}^{(-)}]$
(blue) as a function of light amplitude $A_{0}$. While the gap $\Delta$
vanishes as $A_{0}\to A_{0c}\approx 1.0541$ nm-1, leading to a transition
between the CI and NI phases, the BCD is dramatically enhanced. The BCD peak
values correspond to the maximum of $D_{xz}$ over the broad window
$E_{F}\in[-0.2,0.2]$ eV. (b) The root-mean-square nonlinear Hall current
density $\sqrt{\langle J_{y}^{2}\rangle}$ [Eq. (21)] as a function of $A_{0}$,
from metallic ($E_{F}=0.1$ eV) to purely insulating ($E_{F}\to 0$) cases.
While $\sqrt{\langle J_{y}^{2}\rangle}$ is understandably not quantized along
the dashed line when the system is not insulating (purple, yellow), it is
still not completely quantized even when $E_{F}$ is well within the gap (red).
Instead, it exhibits a pronounced spike when $A_{0}$ is very close to $A_{0c}$
due to the large BCD contribution. Despite its narrowness, this spike is
experimentally accessible due to the experimental ease of tuning $A_{0}$.
Other than the electric field amplitude $\mathcal{E}_{x}$=0.1 V/m,
$\tau\approx 4.12434\times 10^{-14}$ s Ma  _et al._ (2019), a.c. frequency
$\omega=17.777$ Hz Ma  _et al._ (2019), and the other parameters used are
identical to those in Fig. 1.

While it is commonly expected for the Hall current to be quantized according
to the Chern number, in the presence of a large BCD, the Hall current should
actually deviate considerably from its quantized value. In principle, this
should be observable since a large BCD generally exists near a topological
Chern transition with inversion symmetry breaking. However, in most realistic
experimental settings, it is usually extremely difficult to tune the system to
the BCD peak, which is extremely narrow, as plotted in black in Fig. 2(a) for
our system with $\varphi=\pi/2$ (right-handed circularly polarized light). A
key advantage of our Floquet-induced BCD approach is that, by adjusting the
laser intensity $A_{0}$, one is able to very precisely tune the system across
the topological transition value $A_{0c}$, where the BCD (black) peaks and its
corresponding bandgap $\Delta$ (blue) vanishes.

This precision in tuning $A_{0}$ allows one to observe the Hall current
deviating greatly from its quantized value. Plotted in Fig. 2(b) is the root-
mean-square of the Hall current density $J_{y}$ [Eq. (6)] averaged over a
period $\tilde{T}=2\pi/\omega$ as the driving light intensity $A_{0}$ is
varied for an applied field $E_{x}^{\omega}={\rm Re}(\mathcal{E}_{x}e^{i\omega
t})$. Explicitly, from Eq. (6) to (8),

$\displaystyle\langle J_{y}^{2}(\mathcal{E}_{x})\rangle\\!=\\!$
$\displaystyle\int_{0}^{\tilde{T}}\\!\frac{dt}{\tilde{T}}\\!\left[J_{y}^{0}+J_{y}^{\omega}\cos(\omega
t)+J_{y}^{2\omega}\cos(2\omega t)\right]^{2}$ $\displaystyle\\!=\\!$
$\displaystyle\left(J_{y}^{0}\right)^{2}+\frac{1}{2}\left(J_{y}^{\omega}\right)^{2}+\frac{1}{2}\left(J_{y}^{2\omega}\right)^{2}$
$\displaystyle\\!=\\!$
$\displaystyle\left[\chi_{yxx}^{(0)}\mathcal{E}_{x}^{2}\right]^{2}\\!+\\!\frac{1}{2}\left[\chi_{yx}^{(1)}\mathcal{E}_{x}\right]^{2}\\!+\\!\frac{1}{2}\left[\chi_{yxx}^{(2)}\mathcal{E}_{x}^{2}\right]^{2}$
$\displaystyle\approx$
$\displaystyle\,\frac{\mathcal{E}_{x}^{2}}{2}\left[\sigma_{H}^{2}+\frac{3e^{6}\tau^{2}D_{xz}^{2}}{4\hbar^{4}}\mathcal{E}_{x}^{2}\right],$
(21)

where $\sigma_{H}$ gives the linear Hall conductance contribution and the BCD
($D_{xz}$) provides the nonlinear contribution. When $E_{F}$ is outside of the
bandgap (see Fig. 1), $\sqrt{\langle J_{y}^{2}(\mathcal{E}_{x})\rangle}$ is
not quantized as expected. However, for smaller Fermi energy, i.e.,
$E_{F}=0.001$ eV (red) where the system is clearly insulating, $\sqrt{\langle
J_{y}^{2}(\mathcal{E}_{x})\rangle}$ does not remain quantized all the time;
when $A_{0}$ is tuned very close to $A_{0c}$ where a Chern transition occurs,
$\sqrt{\langle J_{y}^{2}(\mathcal{E}_{x})\rangle}$ exhibits a sharp spike due
to the BCD ($D_{xz}$) peak. For a very narrow window of light intensity
$A_{0}$, this nonlinear contribution can in fact be much larger than the usual
linear contribution. Importantly, this window, albeit narrow, is readily
experimentally accessible due to the ease of accurately tuning $A_{0}$ Zhou
_et al._ (2023); Kobayashi  _et al._ (2023). Further discussion on the
competition between the linear and nonlinear current contributions can be
found in Methods Section III.5.

###  II.3 Enhanced BCD through Floquet quench

An interesting extension of our above-mentioned approach involves performing a
Floquet quench on the normal and Chern insulator phases. Since from Fig. 1,
the Chern insulator phase exists without optical driving and the normal
insulator phase is generated by a strong polarized light, we shall
periodically quench the polarized light to alternate rapidly between light-off
and light-on, as can be achieved in experiments on attosecond pulses of light
Castelvecchi and Sanderson (2023); Paul  _et al._ (2001); Hentschel  _et al._
(2001); Schultze  _et al._ (2010); Isinger  _et al._ (2017). Without loss of
generality, we employ right-handed circularly polarized light with
$\varphi=\pi/2$ in the following discussions.

Figure 3: Quench-induced topological band inversion and BCD peaks. (a1-c1)
Floquet energy bands [Eq. (22)] for the $k_{y}=0$ slice. In (a1) where
$T_{1}=0.1T_{2}$ and (a3) where $T_{2}=0.1T_{1}$, the ${\cal H}_{2}({\bf k})$
with light amplitude $A_{0}=\sqrt{2}A_{0c}$ and ${\cal H}_{1}({\bf k})$ with
$A_{0}=0$ are respectively dominant. They respectively correspond to the NI
and CI phases, and are both gapped (pale yellow). But in (b1) where
$T_{1}=T_{2}$, nontrivial contributions from ${\cal H}_{1}({\bf k})$ and
${\cal H}_{2}({\bf k})$ cancel each other out, leaving a gapless band
structure. (a2-c2) Berry curvature [Eq. (13)] of the lower band, whose
integral over the $k_{x}$-$k_{y}$ plane interpolates between the quantized
values of $0$ and $+1$ as $T_{1}/T_{2}$ increases. (a3-c3) The corresponding
Hall conductance [Eq. (9)], which exhibits these quantized values when $E_{F}$
is within the band gap (pale yellow). (a4-c4) The BCD ($D_{xz}$) [Eq. (10)],
which vanishes for $E_{F}$ within the band gap but which peaks at the band
edges, particularly when $T_{1}=T_{2}$. The other parameters are identical to
those in Fig. 1. Here, $T_{2}$ is fixed at $0.1\hbar/$eV$\approx 6.58212\times
10^{-17}$ s.

We consider a periodic two-step quench with a total period of $T_{1}+T_{2}$,
such that each period consists of an odd (even) step governed by the
Hamiltonian under light-off (light-on) polarized light described by ${\cal
H}_{1}({\bf k})$ [${\cal H}_{2}({\bf k})$], for a duration of $T_{1}$
[$T_{2}$]. ${\cal H}_{1}({\bf k})$ denotes the Hamiltonian without light,
i.e., $A_{0}=0$, and ${\cal H}_{2}({\bf k})$ denotes the Hamiltonian under
right-handed circularly polarized light with $\varphi=\pi/2$ and
$A_{0}=\sqrt{2}A_{0c}=\sqrt{2m/\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)}\approx
1.49071$ nm-1, which from Fig. 1 is well within the normal insulating phase.
The effective Floquet Hamiltonian is given by Qin  _et al._ (2023a); Xiong
_et al._ (2016); Liu  _et al._ (2019); Li  _et al._ (2018); Lee and Song
(2021); Ghosh  _et al._ (2022a, b)

$\displaystyle{\cal H}_{\text{eff}}({\bf
k})\equiv\frac{i}{T_{1}+T_{2}}\ln[e^{-i{\cal H}_{2}({\bf k})T_{2}}e^{-i{\cal
H}_{1}({\bf k})T_{1}}],$ (22)

whose gap (pale yellow in Fig. 3) depends intimately on the values of both
$T_{1}$ and $T_{2}$. For our choice of light amplitudes $A_{0}=0$ for ${\cal
H}_{1}({\bf k})$ and $A_{0}=\sqrt{2}A_{0c}$ for ${\cal H}_{2}({\bf k})$, the
gap closes at $T_{1}=T_{2}$ [Fig. 3(b1)], which also induces a band inversion
that separates the CI phase (with $T_{1}>T_{2}$) from the NI phase (with
$T_{1}<T_{2}$). This is shown through the Berry curvature of the lower band
$\Omega_{{\bf k},z}^{(-)}$ in Fig. 3(a2-c2) and its corresponding linear Hall
conductance in Fig. 3(a3-c3). Similar to that in Fig. 1, the BCD ($D_{xz}$)
also exhibits a sharp peak near the band edge when the band gap vanishes,
although this behavior is now precisely tunable through the ratio
$T_{1}/T_{2}$ (with $T_{2}$ fixed at $0.1\hbar/eV\approx 6.58212\times
10^{-17}$s), rather than the laser amplitude $A_{0}$. Analogous results for
left-handed circularly polarized optical driving ($\varphi=-\pi/2$) can be
found in the Supplementary Materials SI.3.

Figure 4: Tunable band gap and BCD peaks from Floquet quench. Our quenching
protocol offers versatility in the tuning of the nonlinear Hall response
through the two independent quench durations $T_{1}$ and $T_{2}$. (a) The band
gap $\Delta$ [Eq. (22)] in the $(T_{1},T_{2})$ parameter space. (b) The
quench-enhanced BCD peak [Eq. (10)], which saliently peaks around the
$T_{1}=T_{2}$ line where the gap closes. The other parameters are identical to
those in Fig. 1.

With independent control over both $T_{1}$ and $T_{2}$ step durations, this
Floquet quench approach offers even greater versatility in tuning the
topological phase transition and approaching the BCD peak. Shown in Fig. 4 are
the band gap $\Delta$ and BCD peak $\text{max}|D_{xz}|$ in the parameter
spaces of $T_{1}$ and $T_{2}$. Evidently, the phase boundary occurs along the
$T_{1}=T_{2}$ line, which also corresponds to the peaks in the BCD. By
simultaneously adjusting $T_{1}$ and $T_{2}$, a maximal nonlinear Hall
response from the BCD term can be obtained.

###  II.4 Measurement of the nonlinear BCD response

The nonlinear response due to light-enhanced BCD can be experimentally
measured in the schematic setup shown in Fig. 5(a). Floquet driving is
achieved through high-frequency laser pumping (purple), whose amplitude
$A_{0}$ can be tuned to sensitively adjust the BCD strength. The nonlinear
response current density $J_{y}^{2\omega}$, which is perpendicular to the
longitudinal electric field $E_{x}^{\omega}={\rm Re}(\mathcal{E}_{x}e^{i\omega
t})$ induced by an alternating current (a.c.) $I_{x}^{\omega}$ at ultralow
frequency $\omega$, can be measured with a lock-in amplifier on its
corresponding potential difference $V_{y}^{2\omega}$.

Figure 5: Proposed experimental setup and signatures of nonlinear Hall
response. (a) In our nonlinear Hall insulator, a lock-in amplifier can be used
to measure the nonlinear Hall voltage $V_{y}^{2\omega}$ resulting from a
longitudinal electrical field $E_{x}^{\omega}={\rm
Re}(\mathcal{E}_{x}e^{i\omega t})$ induced by an a.c. $I^{\omega}_{x}$ of
ultralow frequency $\omega$. High-frequency irradiated light (purple) of
amplitude $A_{0}\approx A_{0c}$ produces the requisite Floquet-enhanced BCD.
(b) The BCD ($D_{xz}$) [Eq. (10)] as a function of the Fermi energy $E_{F}$
for various light amplitudes $A_{0}$ close to the critical value
$A_{0c}\approx 1.0541$ nm-1 where a topological transition occurs, as labeled
in (c). The BCD is dramatically enhanced nearest to $A_{0c}$, peaking at $\sim
3$ nm that is comparable to experimentally measured values in Ma  _et al._
(2019). (c) The peak of the second-harmonic Hall current density
$J_{y}^{2\omega}=\chi_{yxx}^{(2)}\mathcal{E}_{x}^{2}$ [Eq. (6)], which varies
nonlinearly with the amplitude of the applied perpendicular electric field
$\mathcal{E}_{x}$, and is greatly enhanced for $A_{0}$ nearest to $A_{0c}$.
The other parameters are $\varphi=\pi/2$ (right-handed circularly polarized
light), $\hbar\tilde{\omega}=1$ eV, $t_{0}=0.05$ eV$\cdot$nm, $v=0.1$
eV$\cdot$nm, $\alpha=0.1$ eV$\cdot$nm2, $m=0.1$ eV [Eq. (14)], $k_{B}T=0.003$
eV, i.e., $T\approx 34.8$ K, $\tau\approx 4.124\times 10^{-14}$ s Ma  _et al._
(2019), and $\omega=17.77$ Hz Ma  _et al._ (2019).

To obtain the BCD ($D_{xz}$) explicitly, we invoke the nonlinear Hall response
relation $J_{y}^{2\omega}\\!=\\!\chi_{yxx}^{(2)}\mathcal{E}_{x}^{2}$ in Eq.
(8):

$\displaystyle
D_{xz}=\frac{2\hbar^{2}(1+i\omega\tau)J_{y}^{2\omega}}{e^{3}\tau\varepsilon^{yxx}\mathcal{E}_{x}^{2}},$
(23)

in which $\mathcal{E}_{x}$ and $J_{y}^{2\omega}$ can be obtained from the
excitation current $I_{x}^{\omega}$ and measured perpendicular
$V_{y}^{2\omega}$ via Ohm’s law as
$\mathcal{E}_{x}=\frac{I_{x}^{\omega}}{\sigma W}$ and
$J_{y}^{2\omega}=\frac{\sigma V_{y}^{2\omega}}{W}$. As shown in Fig. 5(a), $W$
is the effective sample width, and the longitudinal conductance is of the
order of $\sigma\sim 10^{-5}$ S Ma  _et al._ (2019). From the Drude formula
$\sigma=\frac{n_{0}e^{2}\tau}{\tilde{m}}$, one can also estimate the
relaxation time $\tau$ to be $\sim 10^{-14}$ s with the effective mass being
$\tilde{m}=0.3m_{e}$, $m_{e}$ the electron mass Ma  _et al._ (2019), and
$n_{0}$ the charge density. Tuning this charge density $n_{0}$ also modifies
the Fermi energy $E_{F}$, as demonstrated by first-principle calculations in a
related system Ma  _et al._ (2019).

Experimentally, it suffices to use an ultralow a.c. frequency $\omega$ of the
order of $10-10^{2}$ Hz Ma  _et al._ (2019), such that $\omega\tau\to 0$ and
we obtain

$\displaystyle
D_{xz}=\frac{2\hbar^{2}\sigma^{3}V_{y}^{2\omega}W}{e^{3}\tau(I_{x}^{\omega})^{2}}.$
(24)

Based on these physical parameters, the BCD ($D_{xz}$) is plotted in Fig. 5(b)
for various optical driving amplitudes $A_{0}$ near the critical value
$A_{0c}\approx 1.0541~{}\text{nm}^{-1}$. It exhibits peaks of up to $3$ nm for
$A_{0}=1.0540~{}\text{nm}^{-1}$ [blue curve in Fig. 5(b)], the same order as
in the experimentally measured BCD in bilayer WTe2 Ma  _et al._ (2019), even
though our system is monolayer. The peak width increases with temperature and
is about $E_{F}\sim 10^{-2}$ eV when calculated with the temperature of
$T\approx 34.8$ K taken from the experiment in Ma  _et al._ (2019).

As evident, $D_{xz}$ exhibits high sensitivity to the value of $A_{0}$ near
the critical value $A_{0c}$. This also translates to the high sensitivity of
the peak of the nonlinear Hall current density $J_{y}^{2\omega}$, as plotted
in Fig. 5(c). The nonlinearity of the current response is evident from the
curvature of max($J_{y}^{2\omega}$) plotted against the perpendicular electric
field amplitude $\mathcal{E}_{x}$, particularly for values of $A_{0}$ closest
to $A_{0c}$ (blue, pink).

###  II.5 Conclusion

We discovered that nonlinear Hall materials can exhibit a strong light-
enhanced Berry curvature dipole (BCD) and hence a nonlinear Hall response when
excited by circularly polarized lasers. This was established using the two-
dimensional tilted massive Dirac Hamiltonian that accurately models known
nonlinear Hall materials such as WTe2 Kang  _et al._ (2019); Ma  _et al._
(2019); Ye  _et al._ (2023); Zhang  _et al._ (2018); Xu  _et al._ (2018); Xiao
_et al._ (2020), MoTe2 Lai  _et al._ (2021); Muechler  _et al._ (2016), and
other WTe2-type materials Liu (2021). More generally, however, the dramatic
light-induced BCD enhancement is expected to occur in all media with
simultaneously broken time-reversal and inversion symmetries, albeit only in
close proximity to a topological transition.

The generically very narrow window of large BCD places significant challenges
on its experimental observation. This challenge is crucially addressed through
our Floquet approach, which enables convenient, precise access to the
topological transition by varying the laser amplitude or quench duration. We
find a significant light-enhanced peak for the BCD at a specific critical
intensity of incident light, at which there is a light-induced topological
band inversion: when the light intensity is subcritical, the system is a Chern
insulator, whereas exceeding the critical intensity results in a transition to
a normal insulator phase. Conversely, by measuring the BCD peak as a function
of light intensity, the location of the topological phase transition point can
also be accurately determined. In all, our approach not only transcends
experimental challenges in precisely assessing a topological transition and
its accompanying BCD peak, but also enhances the sensitivity and
reproducibility of nonlinear Hall measurements, opening up avenues for novel
discoveries in the larger realm of nonlinear physics Russomanno  _et al._
(2017); Knüppel  _et al._ (2019); He  _et al._ (2019); Tuloup  _et al._
(2020); Pizzi  _et al._ (2021).

###### Acknowledgements.

We acknowledge helpful discussions with Hao-Jie Lin and Xiao-Bin Qiang. F.Q.
is supported by the QEP2.0 Grant from the Singapore National Research
Foundation (Grant No. NRF2021-QEP2-02-P09) and the MOE Tier-II Grant (Proposal
ID: T2EP50222-0008). R.C. acknowledges the support from the National Natural
Science Foundation of China (Grant No. 12304195) and the Chutian Scholars
Program in Hubei Province.

##  III METHODS

###  III.1 Nonlinear Hall effect

In this subsection, we will introduce the nonlinear Hall effect, where the
transverse electric current contains both linear and quadratic (or even
higher) contributions from the external electric field due to higher-order
Berry curvature corrections Gao  _et al._ (2014); Sodemann and Fu (2015); Du
_et al._ (2018, 2019, 2021); Chen  _et al._ (2023); Ma  _et al._ (2019); Lai
_et al._ (2021).

In the quantum Hall (or quantum anomalous Hall) effect, under broken time-
reversal symmetry, longitudinal current does not flow due to the band gap.
Instead, due to topological in-gap states, a transverse linear Hall current
density $J_{y}=\sigma_{xy}E_{x}$ exists at the same a.c. frequency. It is well
known that at zero temperature, a perturbative expansion in linear response
theory gives the Hall conductance
$\sigma_{xy}=-\frac{e^{2}2\pi}{h}\int\frac{d^{2}{\bf
k}}{(2\pi)^{2}}\varepsilon^{xyz}\Omega_{{\bf k},z}$, where $\varepsilon^{xyz}$
is the Levi-Civita anti-symmetric tensor. But the nonlinear Hall response
tells of another story: Instead of the usual Ohm’s law, we have a quadratic
$I$-$V$ relation, i.e., the transverse current density is proportional to the
second power of the longitudinal electric field, due to high-order Berry
curvature-like contributions. Generically, we can write the response to the
electric current density as

$\displaystyle
J_{i}=\sigma_{ij}E_{j}+\chi_{ijk}E_{j}E_{k}+\xi_{ijkl}E_{j}E_{k}E_{l}+\cdots,$
(25)

where $\\{i,j,k,l\\}\in\\{x,y,z\\}$, $\mathbf{E}=(E_{x},E_{y},E_{z})$ is the
external electric field; the first term is the linear term with frequency as
the external electric field; and the second term is the leading-order
nonlinear transverse response with a doubled a.c. frequency that can be
measured with a lock-in amplifier. In the following subsubsections, we derive
and study the coefficient $\chi_{ijk}$ for the nonlinear Hall effect.

####  III.1.1 Equations of motion

Under an electric field ${\bf E}$, the equations of motion of semi-classical
electronic wavepackets are given by Chang and Niu (1995, 1996); Sundaram and
Niu (1999); Xiao  _et al._ (2010); Shen (2017)

$\displaystyle\dot{\bf r}$ $\displaystyle=\frac{1}{\hbar}\nabla_{\bf
k}\epsilon_{\bf k}-\dot{\bf k}\times{\bf\Omega}_{\bf k},$ (26)
$\displaystyle\dot{\bf k}$ $\displaystyle=-\frac{e}{\hbar}{\bf E},$ (27)

where both the position ${\bf r}$ and wave vector ${\bf k}$ simultaneously
characterize the center-of-mass and momentum of a wavepacket, $\dot{\bf r}$
and $\dot{\bf k}$ are their time derivatives, ${\bf E}$ is the external
longitudinal electric field, $\epsilon_{\bf k}$ is the energy band dispersion,
and ${\bf\Omega}_{\bf k}$ is the Berry curvature defined in ${\bf k}$ space
Xiao  _et al._ (2010); Shen (2017).

Substituting Eq. (27) into Eq. (26), the formula of $\dot{\bf r}$ becomes

$\displaystyle\tilde{\bf v}_{\bf k}=\dot{\bf r}={\bf v}_{\bf
k}+\frac{e}{\hbar}{\bf E}\times{\bf\Omega}_{\bf k},$ (28)

where we have defined the wavepacket velocity as Xiao  _et al._ (2010)

$\displaystyle{\bf v}_{\bf k}=\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{\bf k}.$
(29)

####  III.1.2 Boltzmann equation within the relaxation time approximation

In the case of thermal equilibrium without an external field, the distribution
function is the Fermi-Dirac distribution function Pathria (2016)

$\displaystyle f_{0}(\epsilon_{\bf k})=\frac{1}{e^{\beta(\epsilon_{\bf
k}-\mu)}+1},$ (30)

where $\beta=1/(k_{B}T)$ with Boltzmann constant $k_{B}$ and temperature $T$,
and $\mu$ is the chemical potential.

When collisions exist, we consider the non-equilibrium distribution function
$f({\bf r},{\bf k},t)$ Haug  _et al._ (2008)

$\displaystyle\frac{df({\bf r},{\bf k},t)}{dt}=\left(\frac{\partial
f}{\partial t}\right)_{\rm coll}.$ (31)

Further, we expand the non-equilibrium distribution as

$\displaystyle f({\bf r},{\bf k},t)=f({\bf r}-d{\bf r},{\bf k}-d{\bf
k},t-dt)=f({\bf r}-\dot{\bf r}dt,{\bf k}-\dot{\bf
k}dt,t-dt)+\left(\frac{\partial f}{\partial t}\right)_{\rm coll}dt,$ (32)

and obtain

$\displaystyle f({\bf r}-\dot{\bf r}dt,{\bf k}-\dot{\bf k}dt,t-dt)=f({\bf
r},{\bf k},t)-\dot{\bf r}\cdot\frac{\partial f}{\partial{\bf r}}dt-\dot{\bf
k}\cdot\frac{\partial f}{\partial{\bf k}}dt-\frac{\partial f}{\partial
t}dt+{\cal O}(dt^{2}).$ (33)

Thus, the Boltzmann equation acts as

$\displaystyle\dot{\bf r}\cdot\frac{\partial f}{\partial{\bf r}}+\dot{\bf
k}\cdot\frac{\partial f}{\partial{\bf k}}+\frac{\partial f}{\partial
t}=\left(\frac{\partial f}{\partial t}\right)_{\rm coll},$ (34)

where the left side is a drift term and the right side is a collision term.

Under an external electric field ${\bf E}$, the drift term of the Boltzmann
equation [Eq. (34)] becomes

$\displaystyle\frac{\partial f}{\partial t}+\dot{\bf r}\cdot\frac{\partial
f}{\partial{\bf r}}+\dot{\bf k}\cdot\frac{\partial f}{\partial{\bf
k}}=\frac{\partial f}{\partial t}-\frac{e}{\hbar}{\bf E}\cdot\frac{\partial
f}{\partial{\bf k}},$ (35)

where we have used $\frac{\partial f}{\partial{\bf r}}=0$ since the field is
spatially homogeneous on the length scale of a wavepacket. We employ a simple
relaxation time approximation

$\displaystyle\left(\frac{\partial f}{\partial t}\right)_{\rm
coll}\approx-\frac{f-f_{0}}{\tau}.$ (36)

Then, the Boltzmann equation (34) becomes

$\displaystyle f-f_{0}$ $\displaystyle=-\tau\frac{\partial f}{\partial
t}+\frac{e\tau}{\hbar}{\bf E}\cdot\frac{\partial f}{\partial{\bf k}}$
$\displaystyle=-\tau\frac{\partial f}{\partial t}+e\tau{\bf
E}\cdot\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{\bf k}\frac{\partial
f}{\partial\epsilon_{\bf k}}$ $\displaystyle=-\tau\frac{\partial f}{\partial
t}+e\tau{\bf E}\cdot{\bf v}_{\bf k}\frac{\partial f}{\partial\epsilon_{\bf
k}},$ (37)

where ${\bf v}_{\bf k}=\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{\bf k}$.

Furthermore, the Boltzmann equation (III.1.2) becomes

$\displaystyle f-f_{0}=-\tau\frac{\partial f}{\partial t}+e\tau{\bf v}_{\bf
k}\frac{\partial f}{\partial\epsilon_{\bf k}}\cdot{\bf E}=-\tau\frac{\partial
f}{\partial t}+\frac{e\tau}{\hbar}{\bf E}\cdot\frac{\partial f}{\partial{\bf
k}}=-\tau\frac{\partial f}{\partial t}+\frac{e\tau}{\hbar}E_{a}\frac{\partial
f}{\partial k_{a}}.$ (38)

Importantly, this allows for a controlled approximation of the distribution
function as

$\displaystyle
f=\frac{f_{0}}{1+\tau\partial_{t}-\frac{e\tau}{\hbar}E_{a}\partial_{k_{a}}}\approx\sum_{n=0}^{\infty}\left(-\tau\partial_{t}+\frac{e\tau}{\hbar}E_{a}\partial_{k_{a}}\right)^{n}f_{0}=f_{0}+f_{1}+f_{2}+\cdots\approx
f_{0}+f_{1}+f_{2},$ (39)

where $f_{n}$ refers to the contribution to $f$ that is of the $n$-order in
the field $\mathbb{E}$ (and also of $e\tau/\hbar$). Here, we noted that
$\partial_{t}f_{0}=0$.

We specialize in an oscillating (i.e., a.c.) electric field, ${\bf
E}(t)=E_{a}^{\omega}(t){\bf e}_{a}=\frac{1}{2}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t}+\mathcal{E}^{*}_{a}e^{-i\omega t}){\bf
e}_{a}={\rm Re}(\mathcal{E}_{a}e^{i\omega t}){\bf e}_{a}$, where the driving
field oscillates harmonically in time, but it is uniform in space. With the
amplitude vector $\mathcal{E}_{a}$ and frequency $\omega$, we have the linear
contribution $f_{1}$ Du  _et al._ (2019)

$\displaystyle f_{1}$
$\displaystyle=\sum_{n=0}^{\infty}(-\tau\partial_{t})^{n}\frac{e\tau}{\hbar}{\rm
Re}(\mathcal{E}_{a}e^{i\omega
t})\partial_{k_{a}}f_{0}=\frac{e\tau}{\hbar}\sum_{n=0}^{\infty}(-\tau\partial_{t})^{n}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t})\partial_{k_{a}}f_{0}$
$\displaystyle=\frac{e\tau}{\hbar}\sum_{n=0}^{\infty}{\rm
Re}[\mathcal{E}_{a}e^{i\omega
t}(-i\omega\tau)^{n}]\partial_{k_{a}}f_{0}=\frac{e\tau}{\hbar}{\rm
Re}\left(\frac{\mathcal{E}_{a}e^{i\omega
t}}{1+i\omega\tau}\right)\partial_{k_{a}}f_{0},$ (40)

where we noted that

$\displaystyle\sum_{n=1}^{\infty}(-\tau\partial_{t})^{n-1}\frac{e\tau}{\hbar}E_{a}^{\omega}\partial_{k_{a}}=\frac{e\tau}{\hbar}\sum_{n^{\prime}=0}^{\infty}(-\tau\partial_{t})^{n^{\prime}}E_{a}^{\omega}\partial_{k_{a}}\propto
E_{a}^{\omega}$ (41)

is still linear in the electric field $E_{a}^{\omega}$. Furthermore, the
leading-order nonlinear response contribution is given by

$\displaystyle f_{2}$
$\displaystyle=\sum_{n=0}^{\infty}(-\tau\partial_{t})^{n}\frac{e\tau}{2\hbar}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t}+\mathcal{E}^{*}_{a}e^{-i\omega
t})\partial_{k_{a}}\left[\sum_{m=0}^{\infty}(-\tau\partial_{t})^{m}\frac{e\tau}{2\hbar}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t}+\mathcal{E}^{*}_{a}e^{-i\omega
t})\partial_{k_{a}}\right]f_{0}$
$\displaystyle=\frac{e^{2}\tau^{2}}{4\hbar^{2}}\sum_{n=0}^{\infty}(-\tau\partial_{t})^{n}\left\\{{\rm
Re}(\mathcal{E}_{a}e^{i\omega t}+\mathcal{E}^{*}_{a}e^{-i\omega
t})\partial_{k_{a}}\left[{\rm Re}\left(\frac{\mathcal{E}_{a}e^{i\omega
t}}{1+i\omega\tau}+\frac{\mathcal{E}^{*}_{a}e^{-i\omega
t}}{1-i\omega\tau}\right)\partial_{k_{a}}\right]f_{0}\right\\}$
$\displaystyle=\frac{e^{2}\tau^{2}}{4\hbar^{2}}\sum_{n=0}^{\infty}(-\tau\partial_{t})^{n}{\rm
Re}\left[\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}^{*}e^{-i2\omega
t}}{1-i\omega\tau}\right)+\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1-i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}\right)\right]\partial_{k_{a}}\partial_{k_{a}}f_{0}$
$\displaystyle=\frac{e^{2}\tau^{2}}{4\hbar^{2}}\sum_{n=0}^{\infty}{\rm
Re}\left[\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}(-i2\omega\tau)^{n}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}^{*}e^{-i2\omega
t}(i2\omega\tau)^{n}}{1-i\omega\tau}\right)+\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1-i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}\right)\right]\partial_{k_{a}}\partial_{k_{a}}f_{0}$
$\displaystyle=\frac{e^{2}\tau^{2}}{4\hbar^{2}}{\rm
Re}\left[\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}}{(1+i\omega\tau)(1+i2\omega\tau)}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}^{*}e^{-i2\omega
t}}{(1-i\omega\tau)(1-i2\omega\tau)}\right)+\left(\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1-i\omega\tau}\right)\right]\partial_{k_{a}}\partial_{k_{a}}f_{0}$
$\displaystyle=\frac{e^{2}\tau^{2}}{2\hbar^{2}}{\rm
Re}\left[\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}}{(1+i\omega\tau)(1+i2\omega\tau)}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}\right]\partial_{k_{a}}\partial_{k_{a}}f_{0},$
(42)

where $\tau$ is taken to be spatially constant, i.e.,
$\partial_{k_{a}}\tau=0$. Note that $f_{2}$ contains two distinct types of
nonlinear contributions, one oscillating at the doubled frequency $2\omega$
and the other static.

####  III.1.3 Electric current density

Explicitly, from the definition of the non-equilibrium distribution function,
we have the electric current density

$\displaystyle{\bf J}({\bf E})=\frac{-e}{V}\sum_{\bf k}\dot{\bf r}f_{\bf
k}\,\approx\,\frac{-e}{V}\sum_{\bf k}\dot{\bf r}(f_{0}+f_{1}+f_{2}),$ (43)

where $V=L^{D}$ is the volume in $D$ dimensions. As $\mathbb{k}$ lives in the
lattice Brillouin zone, we should replace the sum over ${\bf k}$ by the
integral $\sum_{\bf k}\rightarrow V\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}$.
Substituting Eqs. (28) and (38) into (43), we obtain, to the leading nonlinear
order,

$\displaystyle{\bf J}({\bf E})\approx\frac{-e}{V}\sum_{\bf k}\dot{\bf
r}(f_{0}+f_{1}+f_{2})=\frac{-e}{V}\sum_{\bf k}\left({\bf v}_{\bf
k}+\frac{e}{\hbar}{\bf E}\times{\bf\Omega}_{\bf k}\right)(f_{0}+f_{1}+f_{2}).$
(44)

Further, we have the $b$-component in the electric current density

$\displaystyle J_{b}$ $\displaystyle=\frac{-e}{V}\sum_{\bf
k}\left(v_{k_{b}}+\frac{e}{\hbar}\varepsilon^{bac}E_{a}^{\omega}\Omega_{{\bf
k},c}\right)(f_{0}+f_{1}+f_{2})$ $\displaystyle=\frac{-e}{V}\sum_{\bf
k}\left[v_{k_{b}}+\frac{e}{2\hbar}\varepsilon^{bac}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t}+\mathcal{E}_{a}^{*}e^{-i\omega
t})\Omega_{{\bf k},c}\right]$
$\displaystyle~{}~{}\times\left\\{f_{0}+\frac{e\tau}{2\hbar}{\rm
Re}\left(\frac{\mathcal{E}_{a}e^{i\omega
t}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}e^{-i\omega
t}}{1-i\omega\tau}\right)\partial_{k_{a}}f_{0}+\frac{e^{2}\tau^{2}}{4\hbar^{2}}{\rm
Re}\left(\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1-i\omega\tau}\right)\partial_{k_{a}}\partial_{k_{a}}f_{0}\right.$
$\displaystyle\left.~{}~{}+\frac{e^{2}\tau^{2}}{4\hbar^{2}}{\rm
Re}\left[\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}}{(1+i\omega\tau)(1+i2\omega\tau)}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}^{*}e^{-i2\omega
t}}{(1-i\omega\tau)(1-i2\omega\tau)}\right]\partial_{k_{a}}\partial_{k_{a}}f_{0}\right\\}$
$\displaystyle=\frac{-e}{V}\sum_{\bf
k}\left[v_{k_{b}}+\frac{e}{\hbar}\varepsilon^{bac}{\rm
Re}(\mathcal{E}_{a}e^{i\omega t})\Omega_{{\bf k},c}\right]$
$\displaystyle~{}~{}\times\left\\{f_{0}+\frac{e\tau}{\hbar}{\rm
Re}\left(\frac{\mathcal{E}_{a}e^{i\omega
t}}{1+i\omega\tau}\right)\partial_{k_{a}}f_{0}+\frac{e^{2}\tau^{2}}{2\hbar^{2}}{\rm
Re}\left(\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1+i\omega\tau}\right)\partial_{k_{a}}\partial_{k_{a}}f_{0}+\frac{e^{2}\tau^{2}}{2\hbar^{2}}{\rm
Re}\left[\frac{\mathcal{E}_{a}\mathcal{E}_{a}e^{i2\omega
t}}{(1+i\omega\tau)(1+i2\omega\tau)}\right]\partial_{k_{a}}\partial_{k_{a}}f_{0}\right\\}$
$\displaystyle\equiv{\rm Re}(J_{b}^{0}+J_{b}^{\omega}e^{i\omega
t}+J_{b}^{2\omega}e^{i2\omega t}),$ (45)

where $\varepsilon^{abc}$ is the Levi-Civita anti-symmetric tensor; $a$, $b$,
and $c$ stand for the spatial coordinates $x$, $y$, and $z$,

$\displaystyle{\rm Re}(J_{b}^{0})$
$\displaystyle=\frac{e^{2}\tau}{4\hbar^{2}}\frac{(-e)}{V}\sum_{\bf
k}\left[\varepsilon^{bac}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1-i\omega\tau}\right)\Omega_{{\bf
k},c}\partial_{k_{a}}+\tau v_{k_{b}}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1-i\omega\tau}\right)\partial_{k_{a}}\partial_{k_{a}}\right]f_{0}$
$\displaystyle=\frac{e^{2}\tau}{4\hbar^{2}}\frac{(-e)}{V}\sum_{\bf
k}\left[\varepsilon^{bac}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1-i\omega\tau}\right)\Omega_{{\bf
k},c}\partial_{k_{a}}+\tau v_{k_{b}}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1-i\omega\tau}\right)\partial_{k_{a}}\partial_{k_{a}}\right]f_{0}$
$\displaystyle=\frac{e^{2}\tau}{4\hbar^{2}}\frac{(-e)}{V}\sum_{\bf k}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}+\frac{\mathcal{E}_{a}^{*}\mathcal{E}_{a}}{1-i\omega\tau}\right)\left(\varepsilon^{bac}\Omega_{{\bf
k},c}\partial_{k_{a}}+\tau
v_{k_{b}}\partial_{k_{a}}\partial_{k_{a}}\right)f_{0}$
$\displaystyle=\frac{e^{2}\tau}{2\hbar^{2}}\frac{(-e)}{V}\sum_{\bf k}{\rm
Re}\left(\frac{\mathcal{E}_{a}\mathcal{E}_{a}^{*}}{1+i\omega\tau}\right)\left(\varepsilon^{bac}\Omega_{{\bf
k},c}\partial_{k_{a}}+\tau
v_{k_{b}}\partial_{k_{a}}\partial_{k_{a}}\right)f_{0},$ (46)
$\displaystyle{\rm Re}(J_{b}^{\omega})$
$\displaystyle=\frac{e}{\hbar}\frac{(-e)}{V}\sum_{\bf
k}\left[\varepsilon^{bac}\mathcal{E}_{a}\Omega_{{\bf k},c}+{\rm
Re}\left(\frac{\tau
v_{k_{b}}\mathcal{E}_{a}}{1+i\omega\tau}\right)\partial_{k_{a}}\right]f_{0}$
$\displaystyle=\frac{e\mathcal{E}_{a}}{\hbar}\frac{(-e)}{V}\sum_{\bf
k}\left[\varepsilon^{bac}\Omega_{{\bf k},c}+{\rm Re}\left(\frac{\tau
v_{k_{b}}}{1+i\omega\tau}\right)\partial_{k_{a}}\right]f_{0},$ (47)
$\displaystyle{\rm Re}(J_{b}^{2\omega})$
$\displaystyle=\frac{e^{2}\tau}{2\hbar^{2}}\frac{(-e)}{V}\sum_{\bf
k}\left\\{{\rm
Re}\left(\frac{\varepsilon^{bac}\mathcal{E}_{a}\mathcal{E}_{a}}{1+i\omega\tau}\right)\Omega_{{\bf
k},c}\partial_{k_{a}}+{\rm Re}\left[\frac{\tau
v_{k_{b}}\mathcal{E}_{a}\mathcal{E}_{a}}{(1+i\omega\tau)(1+i2\omega\tau)}\right]\partial_{k_{a}}\partial_{k_{a}}\right\\}f_{0}$
$\displaystyle={\rm
Re}\left[\frac{e^{2}\tau}{2\hbar^{2}(1+i\omega\tau)}\right]\frac{(-e)}{V}\sum_{\bf
k}\left[{\rm Re}(\varepsilon^{bac}\mathcal{E}_{a}\mathcal{E}_{a})\Omega_{{\bf
k},c}\partial_{k_{a}}+{\rm Re}\left(\frac{\tau
v_{k_{b}}\mathcal{E}_{a}\mathcal{E}_{a}}{1+i2\omega\tau}\right)\partial_{k_{a}}\partial_{k_{a}}\right]f_{0}$
$\displaystyle={\rm
Re}\left[\frac{e^{2}\tau\mathcal{E}_{a}\mathcal{E}_{a}}{2\hbar^{2}(1+i\omega\tau)}\frac{(-e)}{V}\sum_{\bf
k}\left(\varepsilon^{bac}\Omega_{{\bf k},c}\partial_{k_{a}}+\frac{\tau
v_{k_{b}}}{1+i2\omega\tau}\partial_{k_{a}}\partial_{k_{a}}\right)f_{0}\right].$
(48)

By defining

$\displaystyle
J_{b}^{0}=\chi_{baa}^{(0)}\mathcal{E}_{a}\mathcal{E}_{a}^{*},~{}~{}J_{b}^{\omega}=\chi_{ba}^{(1)}\mathcal{E}_{a},~{}~{}J_{b}^{2\omega}=\chi_{baa}^{(2)}\mathcal{E}_{a}\mathcal{E}_{a},$
(49)

where we identify the linear Hall coefficient $\chi_{ba}^{(1)}$ and nonlinear
Hall coefficients $\chi_{ba}^{(0)},\chi_{ba}^{(2)}$ as

$\displaystyle\chi_{baa}^{(0)}$
$\displaystyle=\frac{e^{3}\tau}{2\hbar^{2}(1+i\omega\tau)}\left(\varepsilon^{bac}D_{ac}+\tau
U_{baa}\right)\approx\frac{e^{3}\tau\varepsilon^{bac}D_{ac}}{2\hbar^{2}(1+i\omega\tau)},$
(50) $\displaystyle\chi_{ba}^{(1)}$
$\displaystyle=-\frac{e^{2}}{\hbar}\left[\varepsilon^{bac}A_{c}+\frac{\tau}{(1+i\omega\tau)}C_{ba}\right]\approx-\frac{e^{2}}{\hbar}\varepsilon^{bac}A_{c}=-\varepsilon^{bac}\sigma_{c},$
(51) $\displaystyle\chi_{baa}^{(2)}$
$\displaystyle=\frac{e^{3}\tau}{2\hbar^{2}(1+i\omega\tau)}\left[\varepsilon^{bac}D_{ac}+\frac{\tau}{(1+i2\omega\tau)}U_{baa}\right]\approx\frac{e^{3}\tau\varepsilon^{bac}D_{ac}}{2\hbar^{2}(1+i\omega\tau)},$
(52)

where ${\bf v}_{\bf k}=\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{\bf k}$,
$v_{a}=\frac{1}{\hbar}\partial_{k_{a}}\epsilon_{\bf k}$, and

$\displaystyle A_{c}$ $\displaystyle\\!=\\!\sum_{n}\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}\Omega_{{\bf k},c}^{(n)}f_{0},$ (53) $\displaystyle\sigma_{c}$
$\displaystyle\\!=\\!\frac{e^{2}2\pi}{h}A_{c}\\!=\\!\frac{e^{2}2\pi}{h}\sum_{n}\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}\Omega_{{\bf k},c}^{(n)}f_{0},$ (54) $\displaystyle C_{ba}$
$\displaystyle\\!=\\!\frac{2\pi}{h}\sum_{n}\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}(\partial_{k_{b}}\epsilon_{\bf
k}^{(n)})(\partial_{k_{a}}f_{0}),$ (55) $\displaystyle D_{ac}$
$\displaystyle\\!=\\!-\sum_{n}\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}(\partial_{k_{a}}\epsilon_{\bf k}^{(n)})\Omega_{{\bf
k},c}^{(n)}\frac{\partial f_{0}}{\partial\epsilon_{\bf k}^{(n)}},$ (56)
$\displaystyle U_{baa}$
$\displaystyle\\!=\\!-\frac{2\pi}{h}\sum_{n}\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}(\partial_{k_{b}}\epsilon_{\bf
k}^{(n)})(\partial_{k_{a}}\partial_{k_{a}}f_{0}).$ (57)

Here, we can use

$\displaystyle\frac{\partial f_{0}}{\partial\epsilon_{\bf
k}^{(n)}}=\frac{-e^{\left(\epsilon_{\bf
k}^{(n)}\\!-\\!E_{F}\right)/(k_{B}T)}}{\left[1+e^{\left(\epsilon_{\bf
k}^{(n)}\\!-\\!E_{F}\right)/(k_{B}T)}\right]^{2}k_{B}T}=\frac{-1}{2k_{B}T\left\\{1+\cosh\left[\left(\epsilon_{\bf
k}^{(n)}-E_{F}\right)/(k_{B}T)\right]\right\\}}.$ (58)

In the above, we have approximated the expressions to the leading order in a
small dimensionless parameter $\frac{e\tau}{\hbar}\mathcal{E}_{a}a$ (with the
lattice constant $a$) that is proportional to $\tau$. Taking the Hamiltonian
(14) as an example, the factor $\partial_{k_{y}}\epsilon_{\bf k}^{(n)}$ in
both $C_{yx}$ and $U_{yxx}$ is an odd function of $k_{y}$, and the factor
$\partial_{k_{x}}f_{0}$ in $C_{yx}$ is an even function of $k_{y}$. Therefore,
the integrand $(\partial_{k_{y}}\epsilon_{\bf
k}^{(n)})(\partial_{k_{x}}f_{0})$ in $C_{yx}$ is an odd function of $k_{y}$,
which contributes zero with the integral of $k_{y}$. Similarly, the integrand
$(\partial_{k_{y}}\epsilon_{\bf
k}^{(n)})(\partial_{k_{x}}\partial_{k_{x}}f_{0})$ in $U_{yxx}$ is an odd
function of $k_{y}$, which also contributes zero with the integral of $k_{y}$.

###  III.2 Expression for the Floquet Hamiltonian (17)

In this subsection, we derive the concrete analytical expressions for the time
Fourier components ${\cal H}_{0}$, ${\cal H}_{-n}$, and ${\cal H}_{n}$ that
enter the Floquet Hamiltonian (16) of the main text. With ${\bf
A}(t)=\tilde{\omega}^{-1}E_{0}(\sin(\tilde{\omega}t),\sin(\tilde{\omega}t+\varphi))$
and $A_{0}=eE_{0}/(\hbar\tilde{\omega})$, we have the photon-dressed effective
Hamiltonian as

$\displaystyle{\cal H}({\bf k},t)=$ $\displaystyle{\cal H}\left({\bf
k}-\frac{e}{\hbar}{\bf A}(t)\right)$ $\displaystyle=$ $\displaystyle
t_{0}[k_{x}-A_{0}\sin(\tilde{\omega}t)]\sigma_{0}\\!+\\!v[k_{y}-A_{0}\sin(\tilde{\omega}t+\varphi)]\sigma_{x}\\!+\\!\eta
v[k_{x}-A_{0}\sin(\tilde{\omega}t)]\sigma_{y}$
$\displaystyle\\!+\\!\left\\{m-\alpha\left\\{[k_{x}-A_{0}\sin(\tilde{\omega}t)]^{2}\\!+\\![k_{y}-A_{0}\sin(\tilde{\omega}t+\varphi)]^{2}\right\\}\right\\}\sigma_{z}$
$\displaystyle=$ $\displaystyle
t_{0}k_{x}\sigma_{0}\\!+\\!vk_{y}\sigma_{x}\\!+\\!\eta
vk_{x}\sigma_{y}\\!+\\!m\sigma_{z}\\!-\\!A_{0}\frac{t_{0}}{2i}\left(e^{i\tilde{\omega}t}-e^{-i\tilde{\omega}t}\right)\sigma_{0}$
$\displaystyle\\!-\\!A_{0}\frac{v}{2i}\left[e^{i(\tilde{\omega}t+\varphi)}-e^{-i(\tilde{\omega}t+\varphi)}\right]\sigma_{x}\\!-\\!A_{0}\eta\frac{v}{2i}\left(e^{i\tilde{\omega}t}-e^{-i\tilde{\omega}t}\right)\sigma_{y}$
$\displaystyle\\!-\\!\alpha\left\\{\left[k_{x}-\frac{A_{0}}{2i}\left(e^{i\tilde{\omega}t}-e^{-i\tilde{\omega}t}\right)\right]^{2}+\left[k_{y}-\frac{A_{0}}{2i}\left(e^{i(\tilde{\omega}t+\varphi)}-e^{-i(\tilde{\omega}t+\varphi)}\right)\right]^{2}\right\\}\sigma_{z}.$
(59)

We can extract the time Fourier components in ${\cal H}({\bf k},t)$ to give
analytical expressions for ${\cal H}_{0}$, ${\cal H}_{-n}$, and ${\cal H}_{n}$
in the Floquet Hamiltonian (16):

$\displaystyle{\cal H}_{0}$ $\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal
H}({\bf k},t)dt\\!=\\!t_{0}k_{x}\sigma_{0}\\!+\\!vk_{y}\sigma_{x}\\!+\\!\eta
vk_{x}\sigma_{y}\\!+\\!\left(m-\alpha A_{0}^{2}-\alpha
k^{2}\right)\sigma_{z},$ (60) $\displaystyle\\!{\cal H}_{-1}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{-i\tilde{\omega}t}dt\\!=\\!i\frac{A_{0}t_{0}}{2}\sigma_{0}\\!+\\!i\frac{A_{0}v}{2}e^{i\varphi}\sigma_{x}\\!+\\!\eta
i\frac{A_{0}v}{2}\sigma_{y}\\!-\\!iA_{0}\alpha\left(k_{x}+k_{y}e^{i\varphi}\right)\sigma_{z},$
(61) $\displaystyle\\!{\cal H}_{1}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{i\tilde{\omega}t}dt\\!=\\!-i\frac{A_{0}t_{0}}{2}\sigma_{0}\\!-\\!i\frac{A_{0}v}{2}e^{-i\varphi}\sigma_{x}\\!-\\!\eta
i\frac{A_{0}v}{2}\sigma_{y}\\!+\\!iA_{0}\alpha\left(k_{x}+k_{y}e^{-i\varphi}\right)\sigma_{z},$
(62) $\displaystyle\\!{\cal H}_{-2}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{-i2\tilde{\omega}t}dt\\!=\\!\frac{1}{4}A_{0}^{2}\alpha(1+e^{2i\varphi})\sigma_{z},$
(63) $\displaystyle\\!{\cal H}_{2}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{i2\tilde{\omega}t}dt\\!=\\!\frac{1}{4}A_{0}^{2}\alpha(1+e^{-2i\varphi})\sigma_{z},$
(64) $\displaystyle\\!{\cal H}_{-n<-2}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{-in\tilde{\omega}t}dt\\!=\\!0,$ (65) $\displaystyle\\!{\cal H}_{n>2}$
$\displaystyle\\!=\\!\frac{1}{T}\int_{0}^{T}{\cal H}({\bf
k},t)e^{in\tilde{\omega}t}dt\\!=\\!0.$ (66)

Importantly, only the $n=1$ commutator in the Floquet-expanded effective
Hamiltonian evaluates to nonzero:

$\displaystyle\frac{[{\cal H}_{-1},{\cal H}_{1}]}{\hbar\tilde{\omega}}=$
$\displaystyle\frac{A_{0}^{2}}{\hbar\tilde{\omega}}\left\\{\eta\frac{v^{2}}{4}e^{i\varphi}[\sigma_{x},\sigma_{y}]+\eta\frac{v^{2}}{4}e^{-i\varphi}[\sigma_{y},\sigma_{x}]\right.$
$\displaystyle\left.-\frac{v\alpha}{2}e^{i\varphi}\left(k_{x}+k_{y}e^{-i\varphi}\right)[\sigma_{x},\sigma_{z}]-\frac{v\alpha}{2}e^{-i\varphi}\left(k_{x}+k_{y}e^{i\varphi}\right)[\sigma_{z},\sigma_{x}]\right.$
$\displaystyle\left.-\eta\frac{v\alpha}{2}\left(k_{x}+k_{y}e^{-i\varphi}\right)[\sigma_{y},\sigma_{z}]-\eta\frac{v\alpha}{2}\left(k_{x}+k_{y}e^{i\varphi}\right)[\sigma_{z},\sigma_{y}]\right\\}$
$\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{\hbar\tilde{\omega}}\left\\{\eta\frac{v^{2}}{4}e^{i\varphi}[\sigma_{x},\sigma_{y}]+\eta\frac{v^{2}}{4}e^{-i\varphi}[\sigma_{y},\sigma_{x}]\right.$
$\displaystyle\left.-\frac{v\alpha}{2}\left(k_{x}e^{i\varphi}+k_{y}\right)[\sigma_{x},\sigma_{z}]-\frac{v\alpha}{2}\left(k_{x}e^{-i\varphi}+k_{y}\right)[\sigma_{z},\sigma_{x}]\right.$
$\displaystyle\left.-\eta\frac{v\alpha}{2}\left(k_{x}+k_{y}e^{-i\varphi}\right)[\sigma_{y},\sigma_{z}]-\eta\frac{v\alpha}{2}\left(k_{x}+k_{y}e^{i\varphi}\right)[\sigma_{z},\sigma_{y}]\right\\}$
$\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{\hbar\tilde{\omega}}\left\\{i\eta\frac{v^{2}}{2}(e^{i\varphi}-e^{-i\varphi})\sigma_{z}+iv\alpha(e^{i\varphi}-e^{-i\varphi})k_{x}\sigma_{y}+i\eta
v\alpha(e^{i\varphi}-e^{-i\varphi})k_{y}\sigma_{x}\right\\}$ $\displaystyle=$
$\displaystyle-\frac{A_{0}^{2}(\sin\varphi)}{\hbar\tilde{\omega}}\left(\eta
v^{2}\sigma_{z}+2v\alpha k_{x}\sigma_{y}+2\eta v\alpha
k_{y}\sigma_{x}\right),$ (67) $\displaystyle\frac{[{\cal H}_{-2},{\cal
H}_{2}]}{2\hbar\tilde{\omega}}=$
$\displaystyle\frac{A_{0}^{2}\alpha}{8\hbar\tilde{\omega}}\left\\{(1+e^{2i\varphi})(1+e^{-2i\varphi})[\sigma_{z},\sigma_{z}]\right\\}=0,$
(68) $\displaystyle\frac{[{\cal H}_{-n<-2},{\cal
H}_{n>2}]}{n\hbar\tilde{\omega}}=$ $\displaystyle 0,$ (69)

where we have used $[\sigma_{a},\sigma_{b}]=2i\varepsilon_{abc}\sigma_{c}$.

Therefore, the Floquet Hamiltonian can be written as

$\displaystyle{\cal H}^{(F)}({\bf
k})\\!=\\!t_{0}k_{x}\sigma_{0}\\!+\\!vk_{y}\left(1\\!-\\!\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{x}\\!+\\!\eta
vk_{x}\left(1\\!-\\!\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{y}\\!+\\!\left(m\\!-\\!\alpha
A_{0}^{2}\\!-\\!\eta\frac{v^{2}A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\\!-\\!\alpha
k^{2}\right)\sigma_{z}.$ (70)

###  III.3 Inversion symmetry

We show that the Floquet Hamiltonian (70) (or Eq. (17) in the main text) is
inversion-symmetric only when $t_{0}=0$. As such, a nonzero $t_{0}$ is
necessary for the BCD.

The Floquet Hamiltonian (70) under inversion transformation becomes

$\displaystyle~{}~{}{\cal I}{\cal H}^{(F)}({\bf k}){\cal I}^{-1}$
$\displaystyle\\!=\\!t_{0}k_{x}\sigma_{0}\\!-\\!vk_{y}\left(1\\!-\\!\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{x}\\!-\\!\eta
vk_{x}\left(1\\!-\\!\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{y}\\!+\\!\left(m\\!-\\!\alpha
A_{0}^{2}\\!-\\!\eta\frac{v^{2}A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\\!-\\!\alpha
k^{2}\right)\sigma_{z}$ $\displaystyle\\!=\\!{\cal H}^{(F)}(-{\bf
k})\\!+\\!2t_{0}k_{x}\sigma_{0},$ (71)

where ${\cal I}=\sigma_{z}$ Chen  _et al._ (2015) is the inversion operator,
and we have used

$\displaystyle(\sigma_{z})(\sigma_{0})(\sigma_{z})^{-1}=\sigma_{0},$ (72)
$\displaystyle(\sigma_{z})(\sigma_{x})(\sigma_{z})^{-1}=-\sigma_{x},$ (73)
$\displaystyle(\sigma_{z})(\sigma_{y})(\sigma_{z})^{-1}=-\sigma_{y},$ (74)
$\displaystyle(\sigma_{z})(\sigma_{z})(\sigma_{z})^{-1}=\sigma_{z}.$ (75)

As a result of Eq. (III.3), if $t_{0}=0$, we have ${\cal I}{\cal H}^{(F)}({\bf
k}){\cal I}^{-1}={\cal H}^{(F)}(-{\bf k})$ which satisfies inversion symmetry.
However, for $t_{0}\neq 0$, Eq. (III.3), the extra $2t_{0}k_{x}\sigma_{0}$
term appears due to broken inversion symmetry.

###  III.4 Time-reversal symmetry is broken

Here we show that the Floquet Hamiltonian (70) (or Eq. (17) in the main text)
does not satisfy the time-reversal symmetry regardless of the values of
$t_{0}$ and $A_{0}$.

The Floquet Hamiltonian (70) under time-reversal transformation becomes

$\displaystyle~{}~{}{\cal T}{\cal H}^{(F)}({\bf k}){\cal T}^{-1}$
$\displaystyle\\!=\\!t_{0}k_{x}\sigma_{0}\\!-\\!vk_{y}\left(1\\!-\\!\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{x}\\!-\\!\eta
vk_{x}\left(1\\!-\\!\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)\sigma_{y}\\!-\\!\left(m\\!-\\!\alpha
A_{0}^{2}\\!-\\!\eta\frac{v^{2}A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\\!-\\!\alpha
k^{2}\right)\sigma_{z}$ $\displaystyle\\!=\\!{\cal H}^{(F)}(-{\bf
k})\\!+\\!2t_{0}k_{x}\sigma_{0}\\!-\\!2\left(m\\!-\\!\alpha
A_{0}^{2}\\!-\\!\eta\frac{v^{2}A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\\!-\\!\alpha
k^{2}\right)\sigma_{z},$ (76)

where ${\cal T}=i\sigma_{y}{\cal K}$ Chen  _et al._ (2015) is the time-
reversal operator with the complex conjugate operator ${\cal K}$ such that
${\cal K}{\cal H}^{(F)}({\bf k}){\cal K}^{-1}={\cal H}^{(F)^{*}}({\bf k})$,
and

$\displaystyle(i\sigma_{y})(\sigma_{0})(i\sigma_{y})^{-1}=\sigma_{0},$ (77)
$\displaystyle(i\sigma_{y})(\sigma_{x})(i\sigma_{y})^{-1}=-\sigma_{x},$ (78)
$\displaystyle(i\sigma_{y})(K\sigma_{y}K^{-1})(i\sigma_{y})^{-1}=-\sigma_{y},$
(79) $\displaystyle(i\sigma_{y})(\sigma_{z})(i\sigma_{y})^{-1}=-\sigma_{z}.$
(80)

Due to the presence of the $\sigma_{0}$ and $\sigma_{z}$ terms, Eq. (III.4)
shows that the time-reversal symmetry is broken, i.e., ${\cal T}{\cal
H}^{(F)}({\bf k}){\cal T}^{-1}\neq{\cal H}^{(F)}(-{\bf k})$ whether $t_{0}$
and $A_{0}$ equal zero or not. In the same way, we can show that the original
static tilted Dirac Hamiltonian (Eq. (14) in the main text) also does not
satisfy the time-reversal symmetry, regardless of whether $t_{0}$ vanishes.

###  III.5 Competition between the linear and nonlinear current densities

In this subsection, we present more data on the competition between the linear
and nonlinear current densities as a function of the Fermi energy at different
light intensities.

As shown in Fig. 6, the root-mean-square total current density [Eq. (21) as
defined in the main text] along the $y$ direction is plotted as a function of
the Fermi energy $E_{F}$ at different light intensities, for fixed electronic
field intensity or amplitude $\mathcal{E}_{x}$=0.1 V/m.

When $A_{0}$ is sufficiently small such that the system is in the Chern
insulator phase, i.e., $A_{0}=0.5$ nm-1, there is an obvious nonzero flat
$\sqrt{\langle J_{y}^{2}\rangle}$ region as shown in Fig. 6(a), which is
approximately at the value of the quantized linear Hall current density. The
two broad peaks at larger $E_{F}$ (which are in the bulk bands) are not from
the nonlinear Hall response, but rather result from the non-quantized Hall
response from incomplete band filling.

But when $A_{0}$ tends to the critical value $A_{0c}\approx 1.0541$ nm-1, the
gap and hence the flat region disappear, with two very pronounced sharp peaks
near the point $E_{F}=0$ as shown in Figs. 6(b) and 6(c). While they are also
within the bulk bands, teetering at their edges, the peak values of the root-
mean-square current density are far higher. That is due mostly to the
nonlinear Hall contribution from divergently large BCD. We emphasize that
although such a large nonlinear Hall response seemingly requires fine-tuning
to observe, the light amplitude $A_{0}$ is exactly such a very tunable
parameter. When $A_{0}$ becomes even larger such that the system is in the
topologically trivial phase with vanishing Chern number, i.e., $A_{0}=1.5$
nm-1, there is a flat region at zero as shown in Fig. 6(d), where both the
linear and nonlinear Hall effects essentially vanish.

Figure 6: Competition between the linear and nonlinear current densities. The
root-mean-squrare total current density [Eq. (21)] along the $y$ direction,
averaged over an oscillation period as a function of the Fermi energy $E_{F}$
at different light intensities from (a) in the Chern insulator phase, to (b,c)
just before and after the topological phase transition, and (d) in the trivial
phase. Very sharp and divergently large current density peaks occur very close
to the phase transition. Other parameters are the electronic field intensity
$\mathcal{E}_{x}$=0.1 V/m, $\varphi=\pi/2$ (right-handed circularly polarized
light), $\hbar\tilde{\omega}=1$ eV, $t_{0}=0.05$ eV$\cdot$nm, $v=0.1$
eV$\cdot$nm, $\alpha=0.1$ eV$\cdot$nm2, $m=0.1$ eV, $\eta=-1$, $k_{B}T=0.003$
eV, i.e., $T\approx 34.8136$ K, $\tau\approx 4.12434\times 10^{-14}$ s Ma  _et
al._ (2019), and $\omega=17.777$ Hz Ma  _et al._ (2019).

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##  SI Supplementary Materials

###  SI.1 Tight-binding model

In this subsection, we derive the lattice tight-binding model for the Floquet
Hamiltonian (70). The Floquet Hamiltonian is written as

$\displaystyle{\cal H}^{(F)}({\bf
k})=t_{0}k_{x}\sigma_{0}+\sum_{i=x,y,z}h_{i}^{(F)}\sigma_{i},$ (S1)

where

$\displaystyle h_{x}^{(F)}\\!=$ $\displaystyle vk_{y}\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S2) $\displaystyle
h_{y}^{(F)}\\!=$ $\displaystyle\eta vk_{x}\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S3) $\displaystyle
h_{z}^{(F)}\\!=$ $\displaystyle m-\alpha
A_{0}^{2}-\eta\frac{v^{2}A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}-\alpha
k^{2}=m-A_{0}^{2}\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)-\alpha
k^{2}.$ (S4)

Here $A_{0}=eE_{0}/(\hbar\tilde{\omega})$ is the electromagnetic potential
amplitude.

To regularize the model onto a lattice, one makes the following replacements
Shen (2017):

$\displaystyle k_{j}\rightarrow\frac{1}{a_{j}}\sin(k_{j}a_{j}),$ (S5)
$\displaystyle k_{j}^{2}\rightarrow\frac{2}{a_{j}^{2}}[1-\cos(k_{j}a_{j})],$
(S6)

where $j=x,y$ and $a_{j}$ is the lattice constant along the $j$ direction.
With the mappings (S5) and (S6) and $a_{x}=a_{y}=a$, one obtains

$\displaystyle h_{0}$ $\displaystyle\rightarrow\frac{t_{0}}{a}\sin(k_{x}a),$
(S7) $\displaystyle h_{x}^{(F)}$
$\displaystyle\rightarrow\frac{v}{a}\sin(k_{y}a)\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S8) $\displaystyle
h_{y}^{(F)}$
$\displaystyle\rightarrow\eta\frac{v}{a}\sin(k_{x}a)\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S9) $\displaystyle
h_{z}^{(F)}$ $\displaystyle\rightarrow
m-A_{0}^{2}\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)-\frac{2\alpha}{a^{2}}[2-\cos(k_{x}a)-\cos(k_{y}a)].$
(S10)

This gives the lattice eigenenergies for the upper ($+$) and lower ($-$) bands
as follows:

$\displaystyle\epsilon_{\mathbf{k}}^{(\pm)}\\!=\\!h_{0}\\!\pm\\!\sqrt{[h_{x}^{(F)}]^{2}\\!+\\![h_{y}^{(F)}]^{2}\\!+\\![h_{z}^{(F)}]^{2}},$
(S11)

which gives Floquet band dispersion velocities according to

$\displaystyle v_{k_{x}}^{(\pm)}$
$\displaystyle=\frac{1}{\hbar}\frac{\partial\epsilon_{\bf k}^{(\pm)}}{\partial
k_{x}}=\frac{\partial h_{0}}{\partial
k_{x}}\pm\frac{\left(h_{x}^{(F)}\frac{\partial h_{x}^{(F)}}{\partial
k_{x}}+h_{y}^{(F)}\frac{\partial h_{y}}{\partial
k_{x}}+h_{z}^{(F)}\frac{\partial h_{z}^{(F)}}{\partial
k_{x}}\right)}{\hbar\sqrt{[h_{x}^{(F)}]^{2}\\!+\\![h_{y}^{(F)}]^{2}\\!+\\![h_{z}^{(F)}]^{2}}},$
(S12) $\displaystyle v_{k_{y}}^{(\pm)}$
$\displaystyle=\frac{1}{\hbar}\frac{\partial\epsilon_{\bf k}^{(\pm)}}{\partial
k_{y}}=\pm\frac{\left(h_{x}^{(F)}\frac{\partial h_{x}^{(F)}}{\partial
k_{y}}+h_{y}\frac{\partial h_{y}^{(F)}}{\partial
k_{y}}+h_{z}^{(F)}\frac{\partial h_{z}^{(F)}}{\partial
k_{y}}\right)}{\hbar\sqrt{[h_{x}^{(F)}]^{2}\\!+\\![h_{y}^{(F)}]^{2}\\!+\\![h_{z}^{(F)}]^{2}}},$
(S13)

where

$\displaystyle\frac{\partial h_{0}}{\partial k_{x}}$
$\displaystyle=t_{0}\cos(k_{x}a),$ (S14) $\displaystyle\frac{\partial
h_{x}^{(F)}}{\partial k_{x}}$ $\displaystyle=0,$ (S15)
$\displaystyle\frac{\partial h_{y}^{(F)}}{\partial k_{x}}$ $\displaystyle=\eta
v\cos(k_{x}a)\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S16)
$\displaystyle\frac{\partial h_{z}^{(F)}}{\partial k_{x}}$
$\displaystyle=-\frac{2\alpha}{a}\sin(k_{x}a),$ (S17)
$\displaystyle\frac{\partial h_{x}^{(F)}}{\partial k_{y}}$
$\displaystyle=v\cos(k_{y}a)\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$ (S18)
$\displaystyle\frac{\partial h_{y}^{(F)}}{\partial k_{y}}$ $\displaystyle=0,$
(S19) $\displaystyle\frac{\partial h_{z}^{(F)}}{\partial k_{y}}$
$\displaystyle=-\frac{2\alpha}{a}\sin(k_{y}a).$ (S20)

###  SI.2 Berry curvature

For pedagogical purposes, we shall derive the analytical expression for the
Berry curvature of our Floquet lattice tight-binding model Hamiltonian (S1)
through two different methods. Both are generically applicable to any two-band
model, although the computational complexities differ.

####  SI.2.1 Method one

We first show that the traceless part of any two-band Hamiltonian, such as our
Hamiltonian in Eq. (S1), can be parametrized in terms of quantities $R$ and
$D$. Explicitly, we start by decomposing the model as ${\cal
H}^{(F)}=h_{0}\sigma_{0}+h_{+}^{(F)}\sigma_{+}+h_{-}^{(F)}\sigma_{-}+h_{z}^{(F)}\sigma_{z}=h_{0}\sigma_{0}+[(h_{+}^{(F)}+h_{-}^{(F)})/2]\sigma_{x}+[i(h_{+}^{(F)}-h_{-}^{(F)})/2]\sigma_{y}+h_{z}^{(F)}\sigma_{z}$
with $h_{\pm}^{(F)}=h_{x}^{(F)}\mp ih_{y}^{(F)}$ and
$\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, such that we can perform the
following parametrization on the Bloch sphere Qin  _et al._ (2023b):

$\displaystyle h_{x}^{(F)}$
$\displaystyle=h^{(F)}\sin\theta\cos\phi=h^{(F)}\sqrt{1-D^{2}}\cos\phi,$ (S21)
$\displaystyle h_{y}^{(F)}$
$\displaystyle=h^{(F)}\sin\theta\sin\phi=h^{(F)}\sqrt{1-D^{2}}\sin\phi,$ (S22)
$\displaystyle h_{z}^{(F)}$ $\displaystyle=h^{(F)}\cos\theta=h^{(F)}D,$ (S23)
$\displaystyle h^{(F)}$
$\displaystyle=\sqrt{[h_{x}^{(F)}]^{2}+[h_{y}^{(F)}]^{2}+[h_{z}^{(F)}]^{2}},$
(S24) $\displaystyle D$
$\displaystyle=\frac{h_{z}^{(F)}}{\sqrt{[h_{x}^{(F)}]^{2}+[h_{y}^{(F)}]^{2}+[h_{z}^{(F)}]^{2}}},$
(S25) $\displaystyle R$
$\displaystyle=\sqrt{\frac{h_{+}^{(F)}}{h_{-}^{(F)}}}=\sqrt{\frac{h_{x}^{(F)}-ih_{y}^{(F)}}{h_{x}^{(F)}+ih_{y}^{(F)}}}=\sqrt{\frac{e^{-i\phi}}{e^{i\phi}}}=e^{-i\phi},$
(S26)

where we use
$h_{x}^{(F)}+ih_{y}^{(F)}=h_{\perp}^{(F)}(\cos\phi+i\sin\phi)=h_{\perp}^{(F)}e^{i\phi}$
and
$h_{x}^{(F)}-ih_{y}^{(F)}=h_{\perp}^{(F)}(\cos\phi-i\sin\phi)=h_{\perp}^{(F)}e^{-i\phi}$
with $h_{\perp}^{(F)}=\sqrt{[h_{x}^{(F)}]^{2}+[h_{y}^{(F)}]^{2}}$. Hence, our
Hamiltonian can be expressed as

$\displaystyle{\cal H}^{(F)}$
$\displaystyle=h_{0}\sigma_{0}+h_{x}^{(F)}\sigma_{x}+h_{y}^{(F)}\sigma_{y}+h_{z}^{(F)}\sigma_{z}=h_{0}\sigma_{0}+\begin{pmatrix}h_{z}^{(F)}&h_{x}^{(F)}-ih_{y}^{(F)}\\\
h_{x}^{(F)}+ih_{y}^{(F)}&-h_{z}^{(F)}\end{pmatrix}$
$\displaystyle=h_{0}\sigma_{0}+\begin{pmatrix}h^{(F)}D&h^{(F)}\sqrt{1-D^{2}}(\cos\phi-i\sin\phi)\\\
h^{(F)}\sqrt{1-D^{2}}(\cos\phi+i\sin\phi)&-h^{(F)}D\end{pmatrix}$
$\displaystyle=h_{0}\sigma_{0}+h^{(F)}\begin{pmatrix}D&\sqrt{1-D^{2}}e^{-i\phi}\\\
\sqrt{1-D^{2}}e^{i\phi}&-D\end{pmatrix}=h_{0}\sigma_{0}+h^{(F)}\begin{pmatrix}D&\sqrt{1-D^{2}}R\\\
\sqrt{1-D^{2}}R^{-1}&-D\end{pmatrix}.$ (S27)

Explicitly, its right and left eigenstates are expressible entirely in terms
of $R$ and $D$:

$\displaystyle|\psi^{(\pm)}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm
R\sqrt{1\pm D}\\\ \sqrt{1\mp D}\end{pmatrix},$ (S28)
$\displaystyle\langle\psi^{(\pm)}|=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm\sqrt{1\pm
D}/R&\sqrt{1\mp D}\end{pmatrix}.$ (S29)

From this, we obtain the Berry curvature as

$\displaystyle\Omega_{{\bf k},z}^{(+)}$
$\displaystyle=i\left(\langle\partial_{k_{x}}\psi^{(+)}|\partial_{k_{y}}\psi^{(+)}\rangle-\langle\partial_{k_{y}}\psi^{(+)}|\partial_{k_{x}}\psi^{(+)}\rangle\right)$
$\displaystyle=\frac{i}{2}\left[\begin{pmatrix}\partial_{k_{x}}(\sqrt{1+D}/R)&\partial_{k_{x}}\sqrt{1-D}\end{pmatrix}\begin{pmatrix}\partial_{k_{y}}(R\sqrt{1+D})\\\
\partial_{k_{y}}\sqrt{1-D}\end{pmatrix}-\begin{pmatrix}\partial_{k_{y}}(\sqrt{1+D}/R)&\partial_{k_{y}}\sqrt{1-D}\end{pmatrix}\begin{pmatrix}\partial_{k_{x}}(R\sqrt{1+D})\\\
\partial_{k_{x}}\sqrt{1-D}\end{pmatrix}\right]$
$\displaystyle=\frac{i}{2}\left[\partial_{k_{x}}(\sqrt{1\\!+\\!D}/R)\partial_{k_{y}}(R\sqrt{1\\!+\\!D})\\!+\\!\partial_{k_{x}}\sqrt{1\\!-\\!D}\partial_{k_{y}}\sqrt{1\\!-\\!D}\\!-\\!\partial_{k_{y}}(\sqrt{1\\!+\\!D}/R)\partial_{k_{x}}(R\sqrt{1\\!+\\!D})\\!-\\!\partial_{k_{y}}\sqrt{1\\!-\\!D}\partial_{k_{x}}\sqrt{1\\!-\\!D}\right]$
$\displaystyle=\frac{i}{2}\left[\partial_{k_{x}}(\sqrt{1\\!+\\!D}/R)\partial_{k_{y}}(R\sqrt{1\\!+\\!D})\\!-\\!\partial_{k_{y}}(\sqrt{1\\!+\\!D}/R)\partial_{k_{x}}(R\sqrt{1\\!+\\!D})\right]$
$\displaystyle=\frac{i}{2}\left[\left(\frac{1}{2R\sqrt{1\\!+\\!D}}\partial_{k_{x}}D-\frac{\sqrt{1\\!+\\!D}}{R^{2}}\partial_{k_{x}}R\right)\left(\frac{R}{2\sqrt{1\\!+\\!D}}\partial_{k_{y}}D+\sqrt{1\\!+\\!D}\partial_{k_{y}}R\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}\left.\\!-\\!\left(\frac{1}{2R\sqrt{1\\!+\\!D}}\partial_{k_{y}}D-\frac{\sqrt{1\\!+\\!D}}{R^{2}}\partial_{k_{y}}R\right)\left(\frac{R}{2\sqrt{1\\!+\\!D}}\partial_{k_{x}}D+\sqrt{1\\!+\\!D}\partial_{k_{x}}R\right)\right]$
$\displaystyle=\frac{i}{2R}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}R)-(\partial_{k_{y}}D)(\partial_{k_{x}}R)\right]=\frac{i}{2}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}\ln
R)-(\partial_{k_{y}}D)(\partial_{k_{x}}\ln R)\right],$ (S30)
$\displaystyle\Omega_{{\bf k},z}^{(-)}$
$\displaystyle=i\left(\langle\partial_{k_{x}}\psi^{(-)}|\partial_{k_{y}}\psi^{(-)}\rangle-\langle\partial_{k_{y}}\psi^{(-)}|\partial_{k_{x}}\psi^{(-)}\rangle\right)$
$\displaystyle=\frac{i}{2}\left[\begin{pmatrix}\partial_{k_{x}}(-\sqrt{1-D}/R)&\partial_{k_{x}}\sqrt{1+D}\end{pmatrix}\begin{pmatrix}\partial_{k_{y}}(-R\sqrt{1-D})\\\
\partial_{k_{y}}\sqrt{1+D}\end{pmatrix}-\begin{pmatrix}\partial_{k_{y}}(-\sqrt{1-D}/R)&\partial_{k_{y}}\sqrt{1+D}\end{pmatrix}\begin{pmatrix}\partial_{k_{x}}(-R\sqrt{1-D})\\\
\partial_{k_{x}}\sqrt{1+D}\end{pmatrix}\right]$
$\displaystyle=\frac{i}{2}\left[\partial_{k_{x}}(\sqrt{1\\!-\\!D}/R)\partial_{k_{y}}(R\sqrt{1\\!-\\!D})\\!+\\!\partial_{k_{x}}\sqrt{1\\!+\\!D}\partial_{k_{y}}\sqrt{1\\!+\\!D}\\!-\\!\partial_{k_{y}}(\sqrt{1\\!-\\!D}/R)\partial_{k_{x}}(R\sqrt{1\\!-\\!D})\\!-\\!\partial_{k_{y}}\sqrt{1\\!+\\!D}\partial_{k_{x}}\sqrt{1\\!+\\!D}\right]$
$\displaystyle=\frac{i}{2}\left[\partial_{k_{x}}(\sqrt{1\\!-\\!D}/R)\partial_{k_{y}}(R\sqrt{1\\!-\\!D})\\!-\\!\partial_{k_{y}}(\sqrt{1\\!-\\!D}/R)\partial_{k_{x}}(R\sqrt{1\\!-\\!D})\right]$
$\displaystyle=\frac{i}{2}\left[\left(\frac{1}{2R\sqrt{1\\!-\\!D}}\partial_{k_{x}}(-D)-\frac{\sqrt{1\\!-\\!D}}{R^{2}}\partial_{k_{x}}R\right)\left(\frac{R}{2\sqrt{1\\!-\\!D}}\partial_{k_{y}}(-D)+\sqrt{1\\!-\\!D}\partial_{k_{y}}R\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}\left.\\!-\\!\left(\frac{1}{2R\sqrt{1\\!-\\!D}}\partial_{k_{y}}(-D)-\frac{\sqrt{1\\!-\\!D}}{R^{2}}\partial_{k_{y}}R\right)\left(\frac{R}{2\sqrt{1\\!-\\!D}}\partial_{k_{x}}(-D)+\sqrt{1\\!-\\!D}\partial_{k_{x}}R\right)\right]$
$\displaystyle=-\frac{i}{2R}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}R)-(\partial_{k_{y}}D)(\partial_{k_{x}}R)\right]=-\frac{i}{2}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}\ln
R)-(\partial_{k_{y}}D)(\partial_{k_{x}}\ln R)\right].$ (S31)

Finally, substituting in our specific model (S1), the analytical expression
for the Berry curvature is

$\displaystyle\Omega_{{\bf k},z}^{(\pm)}=$
$\displaystyle\pm\frac{i}{2}\left[(\partial_{k_{x}}D)(\partial_{k_{y}}\ln
R)-(\partial_{k_{y}}D)(\partial_{k_{x}}\ln R)\right]$ $\displaystyle=$
$\displaystyle\pm\frac{\eta\tilde{v}^{2}\left\\{\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)\cos(k_{x}a)\cos(k_{y}a)+\frac{2\alpha}{a^{2}}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]\right\\}}{2\left\\{\tilde{M}^{2}({\bf
k})+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}^{1/2}}$
$\displaystyle\times\frac{1}{\left\\{\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)^{2}+\left(\frac{2\alpha}{a^{2}}\right)^{2}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]^{2}+\frac{4\alpha}{a^{2}}\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)\left[\cos(k_{x}a)+\cos(k_{y}a)\right]+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}}$
$\displaystyle=$
$\displaystyle\pm\frac{\eta\tilde{v}^{2}\left\\{\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)\cos(k_{x}a)\cos(k_{y}a)+\frac{2\alpha}{a^{2}}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]\right\\}}{2\left\\{\tilde{M}^{2}({\bf
k})+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}^{1/2}}$
$\displaystyle\times\frac{1}{\left\\{\left[\tilde{m}-\frac{4\alpha}{a^{2}}+\frac{2\alpha}{a^{2}}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]\right]^{2}+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}}$
$\displaystyle=$
$\displaystyle\pm\frac{\eta\tilde{v}^{2}\left\\{\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)\cos(k_{x}a)\cos(k_{y}a)+\frac{2\alpha}{a^{2}}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]\right\\}}{2\left\\{\tilde{M}^{2}({\bf
k})+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}^{3/2}},$
(S32)

where

$\displaystyle\tilde{m}$
$\displaystyle=m-A_{0}^{2}\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right),$
(S33) $\displaystyle\tilde{M}({\bf k})$
$\displaystyle=\tilde{m}-\frac{2\alpha}{a^{2}}\left[2-\cos(k_{x}a)-\cos(k_{y}a)\right],$
(S34) $\displaystyle\tilde{v}$ $\displaystyle=v\left(1-\eta\frac{2\alpha
A_{0}^{2}\sin\varphi}{\hbar\tilde{\omega}}\right).$ (S35)

####  SI.2.2 Method two

From the expression of the Chern number Saha (2016)

$\displaystyle\mathcal{C}=\frac{1}{4\pi}\int_{\text{BZ}}d^{2}\mathbf{k}~{}\frac{\mathbf{h}^{(F)}}{|\mathbf{h}^{(F)}|^{3}}\cdot(\partial_{k_{x}}\mathbf{h}^{(F)}\times\partial_{k_{y}}\mathbf{h}^{(F)}),$
(S36)

where BZ stands for the Brillouin zone, we can compute the Berry curvature
directly in terms of ${\mathbb{h}}^{(F)}$. We have

$\displaystyle\partial_{k_{x}}\mathbf{h}^{(F)}\times\partial_{k_{y}}\mathbf{h}^{(F)}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijk}\partial_{k_{x}}h_{j}^{(F)}\partial_{k_{y}}h_{k}^{(F)}=\left|\begin{array}[]{ccc}\mathbf{e}_{x}&\mathbf{e}_{y}&\mathbf{e}_{z}\\\
\partial_{k_{x}}h_{x}^{(F)}&\partial_{k_{x}}h_{y}^{(F)}&\partial_{k_{x}}h_{z}^{(F)}\\\
\partial_{k_{y}}h_{x}^{(F)}&\partial_{k_{y}}h_{y}^{(F)}&\partial_{k_{y}}h_{z}^{(F)}\end{array}\right|$
(S40) $\displaystyle=$
$\displaystyle(\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}}h_{z}^{(F)}-\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{y}^{(F)})\mathbf{e}_{x}+(\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{x}^{(F)}-\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{z}^{(F)})\mathbf{e}_{y}$
$\displaystyle+(\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{y}^{(F)}-\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}}h_{x}^{(F)})\mathbf{e}_{z}$
$\displaystyle=$
$\displaystyle[(\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}}h_{z}^{(F)}-\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{y}^{(F)}),(\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{x}^{(F)}-\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{z}^{(F)}),(\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{y}^{(F)}-\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}}h_{x}^{(F)})],$
(S41)
$\displaystyle\mathbf{h}^{(F)}\cdot(\partial_{k_{x}}\mathbf{h}^{(F)}\times\partial_{k_{y}}\mathbf{h}^{(F)})$
$\displaystyle=$ $\displaystyle
h_{i}^{(F)}\varepsilon_{ijk}\partial_{k_{x}}h_{j}^{(F)}\partial_{k_{y}}h_{k}^{(F)}$
$\displaystyle=$ $\displaystyle
h_{x}^{(F)}(\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}}h_{z}^{(F)}-\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{y}^{(F)})+h_{y}^{(F)}(\partial_{k_{x}}h_{z}^{(F)}\partial_{k_{y}}h_{x}^{(F)}-\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{z}^{(F)})$
$\displaystyle+h_{z}^{(F)}(\partial_{k_{x}}h_{x}^{(F)}\partial_{k_{y}}h_{y}^{(F)}-\partial_{k_{x}}h_{y}^{(F)}\partial_{k_{y}^{(F)}}h_{x}^{(F)}).$
(S42)

Such that the Berry curvature for our specific model (S1) is given by Ning
_et al._ (2023)

$\displaystyle\Omega_{{\bf k},z}^{(\pm)}=$
$\displaystyle\mp\frac{1}{2}\frac{\mathbf{h}^{(F)}}{|\mathbf{h}^{(F)}|^{3}}\cdot(\partial_{k_{x}}\mathbf{h}^{(F)}\times\partial_{k_{y}}\mathbf{h}^{(F)})$
$\displaystyle=$
$\displaystyle\pm\frac{\eta\tilde{v}^{2}\left\\{\left(\tilde{m}-\frac{4\alpha}{a^{2}}\right)\cos(k_{x}a)\cos(k_{y}a)+\frac{2\alpha}{a^{2}}\left[\cos(k_{x}a)+\cos(k_{y}a)\right]\right\\}}{2\left\\{\tilde{M}^{2}({\bf
k})+\frac{\tilde{v}^{2}}{a^{2}}\left[\sin^{2}(k_{x}a)+\sin^{2}(k_{y}a)\right]\right\\}^{3/2}}.$
(S43)

###  SI.3 Energy spectrum, Berry curvature, Hall conductance, and BCD under
left-handed circularly polarized light

Here, we present the Floquet driving and quench results when left-handed
($\varphi=-\pi/2$) instead of right-handed ($\varphi=\pi/2$) polarized light
is used. The results are qualitatively similar to those in the main text,
apart from some subtle differences. The main purpose of this subsection is to
show that there is negligible difference in the results between the right-
handed and the left-handed circularly polarized lights, except for the
critical intensity of light $A_{0c}$.

Figure S1: Floquet band structure [Eq. (12)], Berry curvature [Eq. (13)], Hall
conductance [Eq. (9)], and BCD [Eq. (10)] of the nonlinear Hall materials from
the Chern insulator phase to the normal insulator phase. First row (a1)-(e1):
Energy spectrum as a function of $k_{x}$ with $k_{y}=0$ under different light
intensities: (a1) $A_{0}=0$ nm-1, (b1) $A_{0}=0.8$ nm-1, (c1) $A_{0}\approx
A_{0c}\approx 0.9535$ nm-1, (d1) $A_{0}=1.1$ nm-1, and (e1) $A_{0}=1.2$ nm-1.
Second row (a2)-(e2): Berry curvature of the lower band as a function of
$k_{x}$ and $k_{y}$ under different light intensities. Third row (a3)-(e3):
Hall conductance as a function of the Fermi energy $E_{F}$ under different
light intensities. Fourth row (a4)-(e4): BCD as a function of the Fermi energy
$E_{F}$ under different light intensities. The other parameters are
$\varphi=-\pi/2$ (left-handed circularly polarized light),
$\hbar\tilde{\omega}=1$ eV, $t_{0}=0.05$ eV$\cdot$nm, $v=0.1$ eV$\cdot$nm,
$\alpha=0.1$ eV$\cdot$nm2, $m=0.1$ eV, $\eta=-1$, and $k_{B}T=0.003$ eV, i.e.,
$T\approx 34.8136$ K. These parameters are of the same order as those in two-
dimensional massive Dirac models Ma  _et al._ (2019); Muechler  _et al._
(2016); Liu (2021). Figure S2: Floquet quench band structure [Eq. (22)], Berry
curvature [Eq. (13)], Hall conductance [Eq. (9)], and BCD [Eq. (10)] are
quenched from the normal insulator phase to the Chern insulator phase. (a1-c1)
Floquet energy bands with $k_{y}=0$. In (a1) $T_{1}=0.1T_{2}$. The ${\cal
H}_{2}({\bf k})$ is dominant, gapping the energy band (yellow). In (b1),
$T_{1}=T_{2}$. The contributions from ${\cal H}_{1}({\bf k})$ and ${\cal
H}_{2}({\bf k})$ cancel each other out, so that the energy band is gapless. In
(c1), $T_{1}=10T_{2}$. The ${\cal H}_{1}({\bf k})$ is dominant, gapping the
energy band (yellow). Here ${\cal H}_{1}({\bf k})$ denotes the Hamiltonian
without light, i.e., $A_{0}=0$, and ${\cal H}_{2}({\bf k})$ denotes the
Hamiltonian under right-handed circularly polarized light with
$\varphi=-\pi/2$ and
$A_{0}=\sqrt{2}A_{0c}=\sqrt{2m/\left(\alpha+\eta\frac{v^{2}\sin\varphi}{\hbar\tilde{\omega}}\right)}\approx
1.3484$ nm-1. (a2-c2) Berry curvature of the lower band as a function of
$k_{x}$ and $k_{y}$. (a3-c3) Hall conductance as a function of the Fermi
energy $E_{F}$. (a4-c4) BCD as a function of the Fermi energy $E_{F}$. Here,
$T_{2}=0.1\hbar/$eV$\approx 6.58212\times 10^{-17}$ s and the other parameters
are the same as those in Fig. 3.