Light-enhanced nonlinear Hall effect

Abstract

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. We transcend this challenge by devising optical driving and quench protocols that enable practical and direct access to large BCD. 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.

Introduction

The nonlinear Hall effect is an exciting response contribution beyond the much-studied quantum anomalous Hall effect^{1,2,3,4,5,6,7,8}, 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)², which generically exists whenever inversion symmetry is broken^2,3,4,5. Venturing beyond the topological paradigm, the BCD leads to an interesting nonlinear anisotropic current response accompanied by higher harmonic generation^{9,10,11,12,13,14,15,16} 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^{8,17,18,19,20,21}, BiTeI²², bilayer graphene²³, twisted bilayer graphene^24,25, twisted double bilayer graphene²⁶, strained twisted bilayer graphene^27,28, two-dimensional monolayer WSe₂²⁹, strained twisted bilayer WSe₂³⁰, two-dimensional monolayer MoS₂^31,32, Weyl semimetal TaIrTe₄³³, three-dimensional Dirac semimetal Cd₃As₂^34,35, ${{\mathbb{Z}}}_{2}$ topological insulator Bi₂Se₃³⁶, the Weyl–Kondo semimetal Ce₃Bi₄Pd₃³⁷, and T_d-MoTe₂³⁸. The dominance of the intrinsic nonlinear Hall effect in antiferromagnets with ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry has been studied^{39,40,41,42,43} and subsequently experimentally probed in thick T_d-MoTe₂ samples⁴⁴. More recently, the quantum metric-induced nonlinear Hall effect in a topological antiferromagnet MnBi₂Te₄ has also been reported^45,46,47.

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 ~ 100 K^5,8), a key challenge is that the BCD is only significant in a narrow window very close to a topological phase transition³. Accessing it hence requires precise control of external parameters such as magnetic field^45,47, electric field^5,8,17,19,44, temperature^8,44,45,47, charge/carrier density^8,45,47, strain^{26,27,28,29,30}, or impurities^{4,24,25,48,49}, 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^{8,17,18,19,20,21,23,26}, intrinsic nonlinear Hall conductivity^25,44,45,47, or both²⁴.

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 an approach 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^50,51,52, 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.

Results

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^{2,3,4,5,8,18,25,42,43,44,45,47,48,49} 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 E is measurable through the electric current density J. We expand the electric current density J_i in increasing orders of the electric field as

$${J}_{i}={\sigma }_{ij}{E}_{j}+{\chi }_{ijk}{E}_{j}{E}_{k}+{\xi }_{ijkl}{E}_{j}{E}_{k}{E}_{l}+\cdots \,,$$

(1)

where {i, j, k, l} ∈ {x, y, z}, and 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 τ, the relaxation time approximation yields the following Boltzmann equation^2,53,54,55

$$-\tau {\partial }_{t}f+\frac{e\tau }{\hslash }{E}_{a}{\partial }_{{k}_{a}}f=f-{f}_{0},$$

(2)

where  − e is the electron charge, ℏ is the reduced Planck’s constant, k_a is the wave vector of electronic wavepackets along the a ∈ {x, y, z} directions, f₀ is the equilibrium occupancy given by the Fermi-Dirac distribution function, and f is the non-equilibrium distribution function. Here, τ sets the timescale for f to relax to its equilibrium distribution f₀. By inverting the derivative operators, we can directly solve for the non-equilibrium distribution function f as follows:

$$f=\frac{{f}_{0}}{1+\tau {\partial }_{t}-\frac{e\tau }{\hslash }{E}_{a}{\partial }_{{k}_{a}}}\approx \sum\limits_{n=0}^{\infty }{\left(-\tau {\partial }_{t}+\frac{e\tau }{\hslash }{E}_{a}{\partial }_{{k}_{a}}\right)}^{n}{f}_{0}.$$

(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

$${{{\bf{J}}}}({{{\bf{E}}}})\approx -e\int\frac{{d}^{2}{{{\bf{k}}}}}{{(2\pi )}^{2}}\left(\frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}+\frac{e}{\hslash }{{{\bf{E}}}}\times {{{{\mathbf{\Omega }}}}}_{{{{\bf{k}}}}}\right){{{\rm{Re}}}}({f}_{0}+{f}_{1}+{f}_{2}),$$

(4)

with band dispersion ϵ_k and the Berry curvature Ω_k^56,57. 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 f₂ 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

$${J}_{b}({{{{\mathcal{E}}}}}_{a})={{{\rm{Re}}}}\left({J}_{b}^{0}+{J}_{b}^{\omega }{e}^{i\omega t}+{J}_{b}^{2\omega }{e}^{i2\omega t}\right),$$

(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

$${\chi }_{ba}^{(1)}\approx -{\varepsilon }^{bac}{\sigma }_{c},$$

(7)

$${\chi }_{baa}^{(0)}\approx {\chi }_{baa}^{(2)}\approx \frac{{e}^{3}\tau {\varepsilon }^{bac}{D}_{ac}}{2{\hslash }^{2}(1+i\omega \tau )}.$$

(8)

In the above, we have only kept terms to leading order in $\frac{e\tau }{\hslash }{{{{\mathcal{E}}}}}_{a}a$ (with the lattice constant a), which is very small in relevant experiments⁸. For the subleading contributions, see equations (50) to (52) in the Methods Section VIA. In the above, ε^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 σ_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^57,58,59

$${\sigma }_{H}=\frac{{e}^{2}}{h}\frac{1}{2\pi } {\sum}_{n}\int\,{d}^{2}{{{\bf{k}}}}\,{\Omega }_{{{{\bf{k}}}},z}^{(n)}{f}_{0},$$

(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:

$${D}_{ac}=-{\sum}_{n}\int\frac{{d}^{2}{{{\bf{k}}}}}{{(2\pi )}^{2}}\left({\partial }_{{k}_{a}}{\epsilon }_{{{{\bf{k}}}}}^{(n)}\right){\Omega }_{{{{\bf{k}}}},c}^{(n)}\frac{\partial {f}_{0}}{\partial {\epsilon }_{{{{\bf{k}}}}}^{(n)}}.$$

(10)

Its detailed derivation can be found in Methods Section VIA. 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^3,4,5,48. 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^3,4,5,48.

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

$${{{\mathcal{H}}}}({{{\bf{k}}}})={h}_{0}{\sigma }_{0}+{\sum}_{i=x,y,z}{h}_{i}{\sigma }_{i},$$

(11)

where σ_x,y,z are the Pauli matrices, and σ₀ is the 2 × 2 identity matrix. The eigenenergies for its upper (+) and lower (−) bands are

$${\epsilon }_{{{{\bf{k}}}}}^{(\!\pm\! )}={h}_{0}\pm \sqrt{{h}_{x}^{2}+{h}_{y}^{2}+{h}_{z}^{2}},$$

(12)

with corresponding eigenvectors $\left\vert {\psi }^{(\!\pm\! )}\right\rangle$ (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^{57,58,59,60,61,62}

$${\Omega }_{{{{\bf{k}}}},z}^{(\!\pm \!)}=-2{\varepsilon }_{zxy}\frac{{{{\rm{Im}}}}\left\langle {\psi }^{(\!\pm\! )}\left\vert \frac{\partial {{{{\mathcal{H}}}}}^{(F)}}{\partial {k}_{x}}\right\vert {\psi }^{(\mp )}\right\rangle \left\langle {\psi }^{(\mp )}\left\vert \frac{\partial {{{{\mathcal{H}}}}}^{(F)}}{\partial {k}_{y}}\right\vert {\psi }^{(\!\pm\! )}\right\rangle }{{\left[{\epsilon }_{{{{\bf{k}}}}}^{(\!\pm\! )}-{\epsilon }_{{{{\bf{k}}}}}^{(\mp )}\right]}^{2}}.$$

(13)

Notice that, compared to more realistic lattice models under circularly polarized light, a simple linearized model (for example, the type-I Weyl semimetal) with additional quadratic k² terms can lead to different physics¹¹. For a type-I Weyl semimetal, the population at one Weyl node induced by the left circularly polarized light is the same as that induced at the other Weyl node by the right circularly polarized light¹¹. However, once the non-linearity (near the Weyl nodes) of the band structure is taken into account, the excitations induced near one Weyl node by the left circularly polarized light will no longer overlap perfectly with that induced near the other Weyl node by right circularly polarized light, even though the key Hall contributions remain similar.

Effective Floquet Hamiltonian from optical driving

The paradigmatic model for describing nonlinear Hall materials, for instance, monolayer WTe₂^8,18,19,20, is the two-dimensional tilted massive Dirac model^2,3,48,63, which form the theoretical basis in nonlinear Hall effect experiments^3,8:

$${{{\mathcal{H}}}}({{{\bf{k}}}})={t}_{0}{k}_{x}{\sigma }_{0}+v{k}_{y}{\sigma }_{x}+\eta v{k}_{x}{\sigma }_{y}+(m-\alpha {k}^{2}){\sigma }_{z},$$

(14)

where η is allowed to take values of  ± 1 in principle (in our numerics, we set η = − 1), and t₀, v, m, and α are parameters that can be empirically fitted. The tilted term t₀k_xσ₀ 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 E₀ is the amplitude of the electric field and $\tilde{\omega }$ is the angular frequency of the light. The phase φ controls the polarization: φ = 0 introduces linear polarization, while φ = ∓ π/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 ω (17.77 Hz⁸) of an longitudinal alternating current (a.c.) ${I}_{x}^{\omega }$, which is used to induce the transverse nonlinear Hall effect in experiments^5,8.

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

$${{{\mathcal{H}}}}({{{\bf{k}}}},t)={{{\mathcal{H}}}}\left({{{\bf{k}}}}-\frac{e}{\hslash }{{{\bf{A}}}}(t)\right).$$

(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^50,51,52. As such, we set the driving optical frequency to the representative value $\hslash \tilde{\omega }=1$ eV⁶⁴ ($\tilde{\omega } \sim 1.519\times 1{0}^{15}$ Hz), which is much larger than the bandwidth^61,65,66. Under periodic driving through A(t), the effective Floquet Hamiltonian^{52,61,65,66,67}${{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}})=\frac{i}{T}\ln \left[{{{\mathcal{T}}}}{e}^{-i\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t)dt}\right]$ 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^50,51,52

$${{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}})={{{{\mathcal{H}}}}}_{0}+\sum\limits_{n=1}^{\infty }\frac{[{{{{\mathcal{H}}}}}_{-n},{{{{\mathcal{H}}}}}_{n}]}{n\hslash \tilde{\omega }}+{{{\mathcal{O}}}}({\tilde{\omega }}^{-2}),$$

(16)

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

$${{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}})={h}_{0}{\sigma }_{0}+{\sum}_{i=x,y,z}{h}_{i}^{(F)}{\sigma }_{i},$$

(17)

where h₀ = t₀k_x as before,

$${h}_{x}^{(F)}=v{k}_{y}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right),$$

(18)

$${h}_{y}^{(F)}=\eta v{k}_{x}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right),$$

(19)

$${h}_{z}^{(F)}=m-{A}_{0}^{2}\left(\alpha +\eta \frac{{v}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right)-\alpha {k}^{2},$$

(20)

and ${A}_{0}=e{E}_{0}/(\hslash \tilde{\omega })$. Specifically, the intensity of the employed light is below the damage threshold. The maximum value of light amplitude used in our work is A₀ ~ 1.49 nm⁻¹, which corresponds to the amplitude of the electric field ${E}_{0}=\hslash \tilde{\omega }{A}_{0}/e \sim 1.49\times 1{0}^{9}$ V/m. This maximum value of E₀ is smaller than 3.0 × 10⁹ V/m in refs. ^68,69, and it is also smaller than the maximum value 1.06 × 10¹⁰ V/m in ref. ⁷⁰. The detailed derivations for Eqs. (17) to (20) can be found in Methods Section VIB. 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 t₀ ≠ 0 is still required for inversion symmetry breaking (Methods Section VIC), even though time-reversal symmetry is already broken for any values of t₀ and A₀ (Methods Section VID). 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 Note 1.

To estimate the validity of the high-frequency expansion quantitatively, we evaluate the maximum instantaneous energy of the time-dependent Hamiltonian ${{{\mathcal{H}}}}\left({{{\bf{k}}}}-\frac{e}{\hslash }{{{\bf{A}}}}(t)\right)$ averaged over a Floquet period $\frac{1}{T}\int_{0}^{T}dt\,\max \left\{\left\vert \right.\left\vert \right.{{{\mathcal{H}}}}({{{\bf{k}}}},t)\left\vert \right.\left\vert \right.\right\} < \hslash \tilde{\omega }$ at the Γ point (k_x = k_y = 0), which gives the following constraint on the optical field parameters: ${{{\rm{Max}}}}\left(v{A}_{0},2\alpha {A}_{0}^{2}\right) < \hslash \tilde{\omega }$ i.e. ${A}_{0} < {{{\rm{Min}}}}\left(\frac{\hslash \tilde{\omega }}{v},\sqrt{\frac{\hslash \tilde{\omega }}{2\alpha }}\right)$. In the high-frequency regime $\tilde{\omega } \sim 1.519\times 1{0}^{15}$ Hz ($\hslash \tilde{\omega }=1$ eV) with parameters set to v = 0.1 eV ⋅ nm and α = 0.1 eV ⋅ nm², the same order as those in representative 2D Dirac materials such as WTe₂^{8,17,18,19,20}, MoTe₂⁶³, and other WTe₂-type materials⁷¹, one can obtain ${A}_{0}=e{E}_{0}/(\hslash \tilde{\omega }) \, < \, 2.236$ nm⁻¹ (${E}_{0}={A}_{0}\hslash \tilde{\omega }/e \, < \, 2.236\times 1{0}^{9}$ V/m), which corresponds to an incident light intensity⁷² of $I=\frac{1}{2}nc{\varepsilon }_{0}| {E}_{0}{| }^{2} < 9.954\times 1{0}^{15}$ W/m², where n is the refractive index, c is the speed of light in vacuum, and ε₀ is the vacuum permittivity. The refractive index $n \sim {{{\mathcal{O}}}}(1)$, with n ≈ 1.5⁷³ observed in monolayer WTe₂ in the deep-ultraviolet region.

Divergent nonlinear Hall response near a topological transition

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., φ = π/2) at different intensities A₀.

Fig. 1: Light-induced topological band inversion and its large Berry curvature dipole (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 A₀ → A_0c ≈ 1.0541nm⁻¹. Second row a2–e2: Berry curvature ${\Omega }_{{{{\bf{k}}}},z}^{(-)}$ (Eq. (13)) of the lower band, which integrates to  + 1 for A₀ < A_0c and 0 for A₀ > 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  + e²/h in the gap (pale yellow) for A₀ < A_0c and 0 for A₀ > A_0c, corresponding to the Chern insulator (CI) and normal insulator (NI) phases, respectively. Fourth row a4–e4: The nonlinear 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 A₀ ≈ A_0c, the peak becomes drastically higher and narrower. The other parameters are φ = π/2 (right-handed circularly polarized light), $\hslash \tilde{\omega }=1$ eV, t₀ = 0.05 eV ⋅ nm, v = 0.1 eV ⋅ nm, α = 0.1 eV ⋅ nm², m = 0.1 eV, η = − 1, and k_BT = 0.003 eV, i.e., T ≈ 34.8136 K, which are of the same order as those in refs. ^8,63,71.

As can be theoretically predicted from Eqs. (18) to (20), the energy gap (pale yellow in Fig. 1a1–e1 closes at the critical intensity ${A}_{0c}=\sqrt{m/\left(\alpha +\eta \frac{{v}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right)}\approx 1.0541$ nm⁻¹ Fig. 1c1. Tuning the intensity across A_0c not only closes and then opens the energy gap, but also gives rise to band inversion (Fig. 1a1–e1, as evident from the change in sign of the Berry curvature of the lower band as shown in Fig. 1a2–e2, as analytically derived in Supplementary Note 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 σ_H for E_F in the gapped region (yellow) (Fig. 1a3–e3). When the light intensity A₀ is below the critical value A_0c, the energy spectrum is gapped (Fig. 1a1 and b1) and the corresponding Hall conductance is quantized at e²/h within the gap as shown in Fig. 1a3 and b3. This corresponds to the Chern insulator (CI) phase. When the light intensity A₀ > A_0c, the spectrum is also gapped, as shown in Fig. 1d1 and e1, but the corresponding Hall conductance equals zero as shown in Fig. 1d3 and 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. 1c4, it is orders of magnitude larger than that away from the transition, i.e., Fig. 1a4, b4, d4, and 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 (φ = − π/2) are qualitatively similar and can be found in Supplementary Note 3.

Notice that the circularly polarized light (φ = ± π/2) can be used to break the time-reversal symmetry of the nonlinear Hall material (for example, WTe₂^3,8) and realize a topological transition between normal and Chern insulator phases, but the linearly polarized light cannot break the time-reversal symmetry. Therefore, 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, notably without involving the tuning of any other physical parameter. Moreover, the high-harmonic generation is also a nonlinear process¹⁰. As pointed out by ref. ¹⁰, time-reversal symmetry-broken systems can lead to anomalous odd harmonics even based on a linearly polarized driving pulse. This is an alternate way to see the nonlinear Hall effect using a polarized driving light.

Non-quantized current from nonlinear light-enhanced BCD

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. 2a for our system with φ = π/2 (right-handed circularly polarized light). A key advantage of our Floquet-induced BCD approach is that, by adjusting the laser intensity A₀, 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 Δ (blue) vanishes.

Fig. 2: Light enhanced Berry curvature dipole (BCD) and significant departure from linear Hall response.

(a) BCD (Eq. (10)) peak max(∣D_xz∣) (black) and band gap $\Delta =\min [{\epsilon }_{{{{\bf{k}}}}}^{(+)}]-max[{\epsilon }_{{{{\bf{k}}}}}^{(-)}]$ (blue) as a function of light amplitude A₀. While the gap Δ vanishes as A₀ → A_0c ≈ 1.0541 nm⁻¹, leading to a transition between the Chern insulator (CI) and normal insulator (NI) phases, the BCD is dramatically enhanced. The BCD peak values correspond to the maximum of D_xz over the broad window E_F ∈ ( − 0.2, 0.2) eV. (b) The root-mean-square Hall current density $\sqrt{\langle {J}_{y}^{2}\rangle }$ (Eq. (21)) as a function of A₀, from metallic (E_F = 0.1 eV) to purely insulating (E_F → 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 A₀ 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 A₀. Other than the electric field amplitude ${{{{\mathcal{E}}}}}_{x}$ = 0.1 V/m, τ ≈ 4.12434 × 10⁻¹⁴ s⁸, a.c. frequency ω = 17.777 Hz⁸, and the other parameters used are identical to those in Fig. 1.

The tuning of the light amplitude to a precision of A₀ ~ 0.1 nm⁻¹ at least or better has been shown to be experimentally feasible. With fixed $\hslash \tilde{\omega }=1$ eV and the relation ${A}_{0}=e{E}_{0}/(\hslash \tilde{\omega })$, we can get the precision of light amplitude A₀ ~ 0.025 nm⁻¹ with E₀ = 2.5 × 10⁷ V/m⁷⁴, A₀ ~ 0.116 nm⁻¹ with E₀ = 11.6 × 10⁷ V/m⁷⁵, and A₀ ~ 0.04 nm⁻¹ with E₀ = 4.0 × 10⁷ V/m⁷⁶. Here E₀ is the amplitude of the electric field for the light. This precision in tuning A₀ allows one to observe the Hall current deviating greatly from its quantized value. Plotted in Fig. 2b 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 A₀ is varied for an applied field ${E}_{x}^{\omega }={{{\rm{Re}}}}({{{{\mathcal{E}}}}}_{x}{e}^{i\omega t})$. Explicitly, from Eq. (6) to (8),

$$\left\langle {J}_{y}^{2}({{{{\mathcal{E}}}}}_{x})\right\rangle = \, \int_{0}^{\tilde{T}}\frac{dt}{\tilde{T}}{\left[{J}_{y}^{0}+{J}_{y}^{\omega }\cos (\omega t)+{J}_{y}^{2\omega }\cos (2\omega t)\right]}^{2} \\ = \, {\left({J}_{y}^{0}\right)}^{2}+\frac{1}{2}{\left({J}_{y}^{\omega }\right)}^{2}+\frac{1}{2}{\left({J}_{y}^{2\omega }\right)}^{2}\\ =\, {\left[{\chi }_{yxx}^{(0)}{{{{\mathcal{E}}}}}_{x}^{2}\right]}^{2}+\frac{1}{2}{\left[{\chi }_{yx}^{(1)}{{{{\mathcal{E}}}}}_{x}\right]}^{2}+\frac{1}{2}{\left[{\chi }_{yxx}^{(2)}{{{{\mathcal{E}}}}}_{x}^{2}\right]}^{2}\\ \approx \, \frac{{{{{\mathcal{E}}}}}_{x}^{2}}{2}\left[{\sigma }_{H}^{2}+\frac{3{e}^{6}{\tau }^{2}{D}_{xz}^{2}}{4{\hslash }^{4}}{{{{\mathcal{E}}}}}_{x}^{2}\right],$$

(21)

where σ_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 A₀ 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 A₀, 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 A₀^77,78. Further discussion on the competition between the linear and nonlinear current contributions can be found in Methods Section VIE.

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 (shown in Fig. 3), as can be achieved in experiments on attosecond pulses of light^{9,13,14,79,80,81,82}. Without loss of generality, we employ right-handed circularly polarized light with φ = π/2 in the following discussions.

Fig. 3: Proposed experimental setup for Floquet quench.

In our nonlinear Hall material (taking the left figure for example), a lock-in amplifier is 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. ${I}_{x}^{\omega }$ with ultralow frequency ω, which is much lower than the optical driving frequency $\tilde{\omega }$. Left figure: In the duration time T₁, there is no illumination on the nonlinear Hall material. Right figure: In the duration time T₂, there is a high-frequency irradiated light (purple) with amplitude ${A}_{0}=\sqrt{2}{A}_{0c}$ and high frequency $\tilde{\omega }$ illuminating the nonlinear Hall material. The total periodic time is T = T₁ + T₂.

As shown in Fig. 3, we consider a periodic two-step quench with a total period of T₁ + T₂, such that each period consists of an odd (even) step governed by the Hamiltonian under light-off (light-on) polarized light described by ${{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}})$ [${{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}})$], for a duration of T₁ [T₂]. ${{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}})$ denotes the Hamiltonian without light, i.e., A₀ = 0, and ${{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}})$ denotes the Hamiltonian under right-handed circularly polarized light with φ = π/2 and ${A}_{0}=\sqrt{2}{A}_{0c}=\sqrt{2m/\left(\alpha +\eta \frac{{v}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right)}\approx 1.49071$ nm⁻¹, which from Fig. 1 is well within the normal insulating phase. The effective Floquet Hamiltonian is given by⁶¹

$${{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}({{{\bf{k}}}})\equiv \frac{i}{{T}_{1}+{T}_{2}}\ln [{e}^{-i{{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}}){T}_{2}}{e}^{-i{{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}}){T}_{1}}],$$

(22)

whose gap (pale yellow in Fig. 4) depends intimately on the values of both T₁ and T₂. For our choice of light amplitudes A₀ = 0 for ${{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}})$ and ${A}_{0}=\sqrt{2}{A}_{0c}$ for ${{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}})$, the gap closes at T₁ = T₂ (Fig. 4b1), which also induces a band inversion that separates the CI phase (with T₁ > T₂) from the NI phase (with T₁ < T₂). This is shown through the Berry curvature of the lower band ${\Omega }_{{{{\bf{k}}}},z}^{(-)}$ in Fig. 4a2–c2 and its corresponding linear Hall conductance in Fig. 4a3–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 T₁/T₂ (with T₂ fixed at 0.1ℏ/eV ≈ 6.58212 × 10⁻¹⁷s), rather than the laser amplitude A₀. Analogous results for left-handed circularly polarized optical driving (φ = − π/2) can be found in the Supplementary Note 3.

Fig. 4: Quench-induced topological band inversion and Berry curvature dipole (BCD) peaks.

a1–c1 Floquet energy bands (Eq. (22)) for the k_y = 0 slice. In (a1) where T₁ = 0.1T₂ and (c1) where T₂ = 0.1T₁, the ${{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}})$ with light amplitude ${A}_{0}=\sqrt{2}{A}_{0c}$ and ${{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}})$ with A₀ = 0 are respectively dominant. They respectively correspond to the normal insulator (NI) and Chern insulator (CI) phases and are both gapped (pale yellow). But in (b1) where T₁ = T₂, nontrivial contributions from ${{{{\mathcal{H}}}}}_{1}({{{\bf{k}}}})$ and ${{{{\mathcal{H}}}}}_{2}({{{\bf{k}}}})$ 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 T₁/T₂ 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 T₁ = T₂. The other parameters are identical to those in Fig. 1. Here, T₂ is fixed at 0.1ℏ/eV ≈ 6.58212 × 10⁻¹⁷ s.

With independent control over both T₁ and T₂ step durations, this Floquet quench approach offers even greater versatility in tuning the topological phase transition and approaching the BCD peak. Shown in Fig. 5 are the band gap Δ and BCD peak $\max | {D}_{xz}|$ in the parameter spaces of T₁ and T₂. Evidently, the phase boundary occurs along the T₁ = T₂ line, which also corresponds to the peaks in the BCD. By simultaneously adjusting T₁ and T₂, maximal nonlinear Hall response from the BCD term can be obtained.

Fig. 5: Tunable band gap and Berry curvature dipole (BCD) peaks from Floquet quench.

Our quenching protocol offers versatility in the tuning of the nonlinear Hall response through the two independent quench durations T₁ and T₂. a The band gap Δ (Eq. (22)) in the (T₂, T₁) parameter space. b The quench-enhanced BCD peak (Eq. (10)), which saliently peaks around the T₁ = T₂ line where the gap closes. Here NI denotes the normal insulator phase and CI denotes the Chern insulator phase. The other parameters are identical to those in Fig. 1.

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. 6a. Floquet driving is achieved through high-frequency laser pumping (purple), whose amplitude A₀ 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 ω, can be measured with a lock-in amplifier on its corresponding potential difference ${V}_{y}^{2\omega }$.

Fig. 6: 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. ${I}_{x}^{\omega }$ of ultralow frequency ω, which is much lower than the optical driving frequency $\tilde{\omega }$. High-frequency ($\tilde{\omega }$) irradiated light (purple) of amplitude A₀ ≈ A_0c produces the requisite Floquet-enhanced Berry curvature dipole (BCD). b The BCD (D_xz) (Eq. (10)) as a function of the Fermi energy E_F for various light amplitudes A₀ close to the critical value A_0c ≈ 1.0541 nm⁻¹ where a topological transition occurs, as labeled in (c). The BCD is dramatically enhanced nearest to A_0c, peaking at  ~ 3 nm that is comparable to experimentally measured values in Ref. ⁸. 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 A₀ nearest to A_0c. The other parameters are φ = π/2 (right-handed circularly polarized light), $\hslash \tilde{\omega }=1$ eV, t₀ = 0.05 eV ⋅ nm, v = 0.1 eV ⋅ nm, α = 0.1 eV ⋅ nm², m = 0.1 eV (Eq. (14)), k_BT = 0.003 eV, i.e., T ≈ 34.8 K, τ ≈ 4.124 × 10⁻¹⁴ s⁸, and ω = 17.77 Hz⁸.

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):

$${D}_{xz}=\frac{2{\hslash }^{2}(1+i\omega \tau ){J}_{y}^{2\omega }}{{e}^{3}\tau {\varepsilon }^{yxx}{{{{\mathcal{E}}}}}_{x}^{2}},$$

(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. 6a, W is the effective sample width, and the longitudinal conductance is of the order of σ ~ 10⁻⁵ S⁸. From the Drude formula $\sigma =\frac{{n}_{0}{e}^{2}\tau }{\tilde{m}}$, one can also estimate the relaxation time τ to be  ~ 10⁻¹⁴ s with the effective mass being $\tilde{m}=0.3{m}_{e}$, m_e the electron mass⁸, and n₀ the charge density. Tuning this charge density n₀ also modifies the Fermi energy E_F, as demonstrated by first-principle calculations in a related system⁸.

Experimentally, it suffices to use an ultralow a.c. frequency ω of the order of 10−10² Hz⁸, such that ωτ → 0 and we obtain

$${D}_{xz}=\frac{2{\hslash }^{2}{\sigma }^{3}{V}_{y}^{2\omega }W}{{e}^{3}\tau {({I}_{x}^{\omega })}^{2}}.$$

(24)

Based on these physical parameters, the BCD (D_xz) is plotted in Fig. 6b for various optical driving amplitudes A₀ near the critical value A_0c ≈ 1.0541 nm⁻¹. It exhibits peaks of up to 3 nm for A₀ = 1.0540 nm⁻¹ (blue curve in Fig. 6b), the same order as in the experimentally measured BCD in bilayer WTe₂⁸, even though our system is monolayer. The peak width increases with temperature and is about E_F ~ 10⁻² eV when calculated with the temperature of T ≈ 34.8 K taken from the experiment in⁸.

As evident, D_xz exhibits high sensitivity to the value of A₀ 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. 6c. 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 A₀ closest to A_0c (blue, pink).

Notice that there are two frequency scales that differ by several orders of magnitude in this work: the very slow driving frequency of the longitudinal electrical field (ω = 17.77 Hz⁸), and the rapid driving from the irradiation ($\hslash \tilde{\omega }=1$ eV). As shown in Fig. 6a, the driving frequency of the longitudinal electrical field (blue) E (i.e., ${E}_{x}^{\omega }$) in Eq. (4) is ω = 17.77 Hz⁸, which is very low. Therefore, the semiclassical current (Eq. (4)), which involves the Berry curvature, is valid. On the other hand, the high frequency $\hslash \tilde{\omega }=1$ eV is for our irradiated light (purple) as shown in Fig. 6a. Here, the rapid driving modifies the band structure and does not participate in the semiclassical theory, which is based on the effective Floquet model after “integrating out” the rapid driving. Furthermore, our irradiated light (purple) is along z direction as shown in Fig. 6a and Eq. (13) only provides the z component of the Berry curvature. Therefore, the irradiated light cannot contribute to the semiclassical current (Eq. (4)) with an additional term $\frac{e}{\hslash }{{{{\bf{E}}}}}_{{{{\rm{light}}}}}\times {{{{\mathbf{\Omega }}}}}_{{{{\bf{k}}}}}$, where E_light is the electrical field of our irradiated light and E_light × Ω_k = E_lightΩ_k,z(e_z × e_z) = 0. But the irradiated light can modify the Berry curvature through the electromagnetic gauge field (Eq. (15)) within the Floquet theory as shown in Eq. (13).

Conclusion

We find 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^{8,17,18,19,20}, MoTe₂^44,63, and other WTe₂-type materials⁷¹. 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 precisely assesses a topological transition and its accompanying BCD peak, but also enhances the sensitivity and reproducibility of nonlinear Hall measurements in the larger realm of nonlinear physics^83,84.

Particularly, we find that our work and ref. ⁸⁵ focus on different physics that is distinguished by the magnitude of the frequency of the pumping light. The previous work⁸⁵ focuses on the nonlinear optical susceptibilities of semiconductors. Notice that the optical susceptibilities require that the frequency of the driving light be in the same order of magnitudes as the bandwidth. It is called a resonant light when its frequency is about the same as the bandwidth. Therefore, the electrons can transition between the conduction band and the valence band, driven by the resonant light. However, in our work, we focus on the off-resonant light, whose frequency is much larger than the bandwidth. In this case, the electrons cannot transition between the conduction band and the valence band, driven by the off-resonant light. Therefore, an effective Floquet Hamiltonian based on a high-frequency expansion method provided in our work is applicable for this off-resonant light situation.

Methods

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^{2,3,4,5,8,42,44,48}.

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 = σ_xyE_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 ε^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

$${J}_{i}={\sigma }_{ij}{E}_{j}+{\chi }_{ijk}{E}_{j}{E}_{k}+{\xi }_{ijkl}{E}_{j}{E}_{k}{E}_{l}+\cdots \,,$$

(25)

where {i, j, k, l} ∈ {x, y, z}, 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 χ_ijk for the nonlinear Hall effect.

Equations of motion

Under an electric field E, the equations of motion of semi-classical electronic wavepackets are given by^56,57,86,87

$$\dot{{{{\bf{r}}}}}=\frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}-\dot{{{{\bf{k}}}}}\times {{{{\mathbf{\Omega }}}}}_{{{{\bf{k}}}}},$$

(26)

$$\dot{{{{\bf{k}}}}}=-\frac{e}{\hslash }{{{\bf{E}}}},$$

(27)

where both the position r and wave vector k simultaneously characterize the center-of-mass and momentum of a wavepacket, $\dot{{{{\bf{r}}}}}$ and $\dot{{{{\bf{k}}}}}$ are their time derivatives, E is the external longitudinal electric field, ϵ_k is the energy band dispersion, and Ω_k is the Berry curvature defined in k space^56,57.

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

$${\tilde{{{{\bf{v}}}}}}_{{{{\bf{k}}}}}=\dot{{{{\bf{r}}}}}={{{{\bf{v}}}}}_{{{{\bf{k}}}}}+\frac{e}{\hslash }{{{\bf{E}}}}\times {{{{\mathbf{\Omega }}}}}_{{{{\bf{k}}}}},$$

(28)

where we have defined the wavepacket velocity as⁵⁶

$${{{{\bf{v}}}}}_{{{{\bf{k}}}}}=\frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}.$$

(29)

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

$${f}_{0}({\epsilon }_{{{{\bf{k}}}}})=\frac{1}{{e}^{\beta ({\epsilon }_{{{{\bf{k}}}}}-\mu )}+1},$$

(30)

where β = 1/(k_BT) with Boltzmann constant k_B and temperature T, and μ is the chemical potential.

When collisions exist, we consider the non-equilibrium distribution function f(r, k, t)

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

(31)

Further, we expand the non-equilibrium distribution as

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

(32)

and obtain

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

(33)

Thus, the Boltzmann equation acts as

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

(34)

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

Under an external electric field E, the drift term of the Boltzmann equation (Eq. (34)) becomes

$$\frac{\partial f}{\partial t}+\dot{{{{\bf{r}}}}}\cdot \frac{\partial f}{\partial {{{\bf{r}}}}}+\dot{{{{\bf{k}}}}}\cdot \frac{\partial f}{\partial {{{\bf{k}}}}}=\frac{\partial f}{\partial t}-\frac{e}{\hslash }{{{\bf{E}}}}\cdot \frac{\partial f}{\partial {{{\bf{k}}}}},$$

(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

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

(36)

Then, the Boltzmann equation (34) becomes

$$f-{f}_{0}=-\tau \frac{\partial f}{\partial t}+\frac{e\tau }{\hslash }{{{\bf{E}}}}\cdot \frac{\partial f}{\partial {{{\bf{k}}}}}=-\tau \frac{\partial f}{\partial t}+e\tau {{{\bf{E}}}}\cdot \frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}\frac{\partial f}{\partial {\epsilon }_{{{{\bf{k}}}}}}-\tau \frac{\partial f}{\partial t}+e\tau {{{\bf{E}}}}\cdot {{{{\bf{v}}}}}_{{{{\bf{k}}}}}\frac{\partial f}{\partial {\epsilon }_{{{{\bf{k}}}}}},$$

(37)

where ${{{{\bf{v}}}}}_{{{{\bf{k}}}}}=\frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}$.

Furthermore, the Boltzmann equation (37) becomes

$$f-{f}_{0}=-\tau \frac{\partial f}{\partial t}+e\tau {{{{\bf{v}}}}}_{{{{\bf{k}}}}}\frac{\partial f}{\partial {\epsilon }_{{{{\bf{k}}}}}}\cdot {{{\bf{E}}}}=-\tau \frac{\partial f}{\partial t}+\frac{e\tau }{\hslash }{{{\bf{E}}}}\cdot \frac{\partial f}{\partial {{{\bf{k}}}}}=-\tau \frac{\partial f}{\partial t}+\frac{e\tau }{\hslash }{E}_{a}\frac{\partial f}{\partial {k}_{a}}.$$

(38)

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

$$f = \, \frac{{f}_{0}}{1+\tau {\partial }_{t}-\frac{e\tau }{\hslash }{E}_{a}{\partial }_{{k}_{a}}}\approx \sum\limits_{n=0}^{\infty }{\left(-\tau {\partial }_{t}+\frac{e\tau }{\hslash }{E}_{a}{\partial }_{{k}_{a}}\right)}^{n}{f}_{0}={f}_{0}+{f}_{1}+{f}_{2} \\ + \cdots \approx {f}_{0}+{f}_{1}+{f}_{2},$$

(39)

where f_n refers to the contribution to f that is of the n-order in the field E (and also of eτ/ℏ). Here, we noted that ∂_tf₀ = 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 ω, we have the linear contribution f₁⁴

$${f}_{1} = \, \sum\limits_{n=0}^{\infty }{(-\tau {\partial }_{t})}^{n}\frac{e\tau }{\hslash }{{{\rm{Re}}}}({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}){\partial }_{{k}_{a}}{f}_{0} \\ =\, \frac{e\tau }{\hslash }\sum\limits_{n=0}^{\infty }{(-\tau {\partial }_{t})}^{n}{{{\rm{Re}}}}({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}){\partial }_{{k}_{a}}{f}_{0}\\ =\, \frac{e\tau }{\hslash }\sum\limits_{n=0}^{\infty }{{{\rm{Re}}}}[{{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}{(-i\omega \tau )}^{n}]{\partial }_{{k}_{a}}{f}_{0}=\frac{e\tau }{\hslash }{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}}{1+i\omega \tau }\right){\partial }_{{k}_{a}}{f}_{0},$$

(40)

where we noted that

$$\sum\limits_{n=1}^{\infty }{(-\tau {\partial }_{t})}^{n-1}\frac{e\tau }{\hslash }{E}_{a}^{\omega }{\partial }_{{k}_{a}}=\frac{e\tau }{\hslash }\sum\limits_{{n}^{{\prime} }=0}^{\infty }{(-\tau {\partial }_{t})}^{{n}^{{\prime} }}{E}_{a}^{\omega }{\partial }_{{k}_{a}}\propto {E}_{a}^{\omega }$$

(41)

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

$${f}_{2} = \, \sum\limits_{n=0}^{\infty }{(-\tau {\partial }_{t})}^{n}\frac{e\tau }{2\hslash }{{{\rm{Re}}}}({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}+{{{{\mathcal{E}}}}}_{a}^{* }{e}^{-i\omega t}){\partial }_{{k}_{a}}\left[\sum\limits_{m=0}^{\infty }{(-\tau {\partial }_{t})}^{m}\frac{e\tau }{2\hslash }{{{\rm{Re}}}}({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}+{{{{\mathcal{E}}}}}_{a}^{* }{e}^{-i\omega t}){\partial }_{{k}_{a}}\right]{f}_{0}\\ =\, \frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{2}}\sum\limits_{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\}\\ =\, \frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{2}}\sum\limits_{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}\\ =\, \frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{2}}\sum\limits_{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}\\ =\, \frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{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}\\ =\, \frac{{e}^{2}{\tau }^{2}}{2{\hslash }^{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 τ is taken to be spatially constant, i.e., ${\partial }_{{k}_{a}}\tau =0$. Note that f₂ contains two distinct types of nonlinear contributions, one oscillating at the doubled frequency 2ω and the other static.

Electric current density

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

$${{{\bf{J}}}}({{{\bf{E}}}})=\frac{-e}{V} {\sum}_{{{{\bf{k}}}}}\dot{{{{\bf{r}}}}}{f}_{{{{\bf{k}}}}}\,\approx \,\frac{-e}{V} {\sum}_{{{{\bf{k}}}}}\dot{{{{\bf{r}}}}}({f}_{0}+{f}_{1}+{f}_{2}),$$

(43)

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

$${{{\bf{J}}}}({{{\bf{E}}}})\approx \frac{-e}{V} {\sum}_{{{{\bf{k}}}}}\dot{{{{\bf{r}}}}}({f}_{0}+{f}_{1}+{f}_{2})=\frac{-e}{V}{\sum}_{{{{\bf{k}}}}}\left({{{{\bf{v}}}}}_{{{{\bf{k}}}}}+\frac{e}{\hslash }{{{\bf{E}}}}\times {{{{\mathbf{\Omega }}}}}_{{{{\bf{k}}}}}\right)({f}_{0}+{f}_{1}+{f}_{2}).$$

(44)

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

$${J}_{b} \approx \, \frac{-e}{V}\sum\limits_{{{{\bf{k}}}}}\left({v}_{{k}_{b}}+\frac{e}{\hslash }{\varepsilon }^{bac}{E}_{a}^{\omega }{\Omega }_{{{{\bf{k}}}},c}\right)({f}_{0}+{f}_{1}+{f}_{2})\\ = \, \frac{-e}{V}\sum\limits_{{{{\bf{k}}}}}\left[{v}_{{k}_{b}}+\frac{e}{2\hslash }{\varepsilon }^{bac}{{{\rm{Re}}}}\left({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}+{{{{\mathcal{E}}}}}_{a}^{* }{e}^{-i\omega t}\right){\Omega }_{{{{\bf{k}}}},c}\right]\\ \times \left\{{f}_{0}+\frac{e\tau }{2\hslash }{{{\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} \right. \\ \left. + \frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{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.\\ \left. +\frac{{e}^{2}{\tau }^{2}}{4{\hslash }^{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\}\\ =\, \frac{-e}{V}\sum\limits_{{{{\bf{k}}}}}\left[{v}_{{k}_{b}}+\frac{e}{\hslash }{\varepsilon }^{bac}{{{\rm{Re}}}}\left({{{{\mathcal{E}}}}}_{a}{e}^{i\omega t}\right){\Omega }_{{{{\bf{k}}}},c}\right]\\ \times \left\{{f}_{0}+\frac{e\tau }{\hslash }{{{\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{\hslash }^{2}}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1+i\omega \tau }\right){\partial }_{{k}_{a}}{\partial }_{{k}_{a}}{f}_{0} \right.\\ \left. + \frac{{e}^{2}{\tau }^{2}}{2{\hslash }^{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\}\\ \equiv \, {{{\rm{Re}}}}({J}_{b}^{0}+{J}_{b}^{\omega }{e}^{i\omega t}+{J}_{b}^{2\omega }{e}^{i2\omega t}),$$

(45)

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

$${{{\rm{Re}}}}\left({J}_{b}^{0}\right) =\, \frac{{e}^{2}\tau }{4{\hslash }^{2}}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left[{\varepsilon }^{bac}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }+\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1-i\omega \tau }\right){\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}} \right. \\ + \left. \tau {v}_{{k}_{b}}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }+\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1-i\omega \tau }\right){\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right]{f}_{0}\\ =\, \frac{{e}^{2}\tau }{4{\hslash }^{2}}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left[{\varepsilon }^{bac}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }+\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1-i\omega \tau }\right){\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}\right. \\ + \left.\tau {v}_{{k}_{b}}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }+\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1-i\omega \tau }\right){\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right]{f}_{0}\\ =\, \frac{{e}^{2}\tau }{4{\hslash }^{2}}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }+\frac{{{{{\mathcal{E}}}}}_{a}^{* }{{{{\mathcal{E}}}}}_{a}}{1-i\omega \tau }\right)\left({\varepsilon }^{bac}{\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}+\tau {v}_{{k}_{b}}{\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right){f}_{0}\\ =\, \frac{{e}^{2}\tau }{2{\hslash }^{2}}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}{{{\rm{Re}}}}\left(\frac{{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* }}{1+i\omega \tau }\right)\left({\varepsilon }^{bac}{\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}+\tau {v}_{{k}_{b}}{\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right){f}_{0},$$

(46)

$${{{\rm{Re}}}}\left({J}_{b}^{\omega }\right)=\, \frac{e}{\hslash }\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left[{\varepsilon }^{bac}{{{{\mathcal{E}}}}}_{a}{\Omega }_{{{{\bf{k}}}},c}+{{{\rm{Re}}}}\left(\frac{\tau {v}_{{k}_{b}}{{{{\mathcal{E}}}}}_{a}}{1+i\omega \tau }\right){\partial }_{{k}_{a}}\right]{f}_{0}\\ =\, \frac{e{{{{\mathcal{E}}}}}_{a}}{\hslash }\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left[{\varepsilon }^{bac}{\Omega }_{{{{\bf{k}}}},c}+{{{\rm{Re}}}}\left(\frac{\tau {v}_{{k}_{b}}}{1+i\omega \tau }\right){\partial }_{{k}_{a}}\right]{f}_{0},$$

(47)

$${{{\rm{Re}}}}\left({J}_{b}^{2\omega }\right) =\, \frac{{e}^{2}\tau }{2{\hslash }^{2}}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left\{{{{\rm{Re}}}}\left(\frac{{\varepsilon }^{bac}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}}{1+i\omega \tau }\right){\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}+{{{\rm{Re}}}}\left[\frac{\tau {v}_{{k}_{b}}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}}{(1+i\omega \tau )(1+i2\omega \tau )}\right]{\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right\}{f}_{0}\\ =\, {{{\rm{Re}}}}\left[\frac{{e}^{2}\tau }{2{\hslash }^{2}(1+i\omega \tau )}\right]\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left[{{{\rm{Re}}}}({\varepsilon }^{bac}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}){\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}+{{{\rm{Re}}}}\left(\frac{\tau {v}_{{k}_{b}}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}}{1+i2\omega \tau }\right){\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right]{f}_{0}\\ =\, {{{\rm{Re}}}}\left[\frac{{e}^{2}\tau {{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}}{2{\hslash }^{2}(1+i\omega \tau )}\frac{(-e)}{V}\sum\limits_{{{{\bf{k}}}}}\left({\varepsilon }^{bac}{\Omega }_{{{{\bf{k}}}},c}{\partial }_{{k}_{a}}+\frac{\tau {v}_{{k}_{b}}}{1+i2\omega \tau }{\partial }_{{k}_{a}}{\partial }_{{k}_{a}}\right){f}_{0}\right].$$

(48)

By defining

$${J}_{b}^{0}={\chi }_{baa}^{(0)}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a}^{* },\quad {J}_{b}^{\omega }={\chi }_{ba}^{(1)}{{{{\mathcal{E}}}}}_{a},\quad {J}_{b}^{2\omega }={\chi }_{baa}^{(2)}{{{{\mathcal{E}}}}}_{a}{{{{\mathcal{E}}}}}_{a},$$

(49)

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

$${\chi }_{baa}^{(0)}=\frac{{e}^{3}\tau }{2{\hslash }^{2}(1+i\omega \tau )}\left({\varepsilon }^{bac}{D}_{ac}+\tau {U}_{baa}\right)\approx \frac{{e}^{3}\tau {\varepsilon }^{bac}{D}_{ac}}{2{\hslash }^{2}(1+i\omega \tau )},$$

(50)

$${\chi }_{ba}^{(1)}=-\frac{{e}^{2}}{\hslash }\left[{\varepsilon }^{bac}{A}_{c}+\frac{\tau }{(1+i\omega \tau )}{C}_{ba}\right]\approx -\frac{{e}^{2}}{\hslash }{\varepsilon }^{bac}{A}_{c}=-{\varepsilon }^{bac}{\sigma }_{c},$$

(51)

$${\chi }_{baa}^{(2)}=\frac{{e}^{3}\tau }{2{\hslash }^{2}(1+i\omega \tau )}\left[{\varepsilon }^{bac}{D}_{ac}+\frac{\tau }{(1+i2\omega \tau )}{U}_{baa}\right]\approx \frac{{e}^{3}\tau {\varepsilon }^{bac}{D}_{ac}}{2{\hslash }^{2}(1+i\omega \tau )},$$

(52)

where ${{{{\bf{v}}}}}_{{{{\bf{k}}}}}=\frac{1}{\hslash }{\nabla }_{{{{\bf{k}}}}}{\epsilon }_{{{{\bf{k}}}}}$, ${v}_{a}=\frac{1}{\hslash }{\partial }_{{k}_{a}}{\epsilon }_{{{{\bf{k}}}}}$, and

$${A}_{c}={\sum}_{n}\int\frac{{d}^{D}{{{\bf{k}}}}}{{(2\pi )}^{D}}{\Omega }_{{{{\bf{k}}}},c}^{(n)}{f}_{0},$$

(53)

$${\sigma }_{c}=\frac{{e}^{2}2\pi }{h}{A}_{c}=\frac{{e}^{2}2\pi }{h}{\sum}_{n}\int\frac{{d}^{D}{{{\bf{k}}}}}{{(2\pi )}^{D}}{\Omega }_{{{{\bf{k}}}},c}^{(n)}{f}_{0},$$

(54)

$${C}_{ba}=\frac{2\pi }{h}{\sum}_{n}\int\frac{{d}^{D}{{{\bf{k}}}}}{{(2\pi )}^{D}}\left({\partial }_{{k}_{b}}{\epsilon }_{{{{\bf{k}}}}}^{(n)}\right)({\partial }_{{k}_{a}}{f}_{0}),$$

(55)

$${D}_{ac}=-{\sum}_{n}\int\frac{{d}^{D}{{{\bf{k}}}}}{{(2\pi )}^{D}}\left({\partial }_{{k}_{a}}{\epsilon }_{{{{\bf{k}}}}}^{(n)}\right){\Omega }_{{{{\bf{k}}}},c}^{(n)}\frac{\partial {f}_{0}}{\partial {\epsilon }_{{{{\bf{k}}}}}^{(n)}},$$

(56)

$${U}_{baa}=-\frac{2\pi }{h}{\sum}_{n}\int\frac{{d}^{D}{{{\bf{k}}}}}{{(2\pi )}^{D}}\left({\partial }_{{k}_{b}}{\epsilon }_{{{{\bf{k}}}}}^{(n)}\right)({\partial }_{{k}_{a}}{\partial }_{{k}_{a}}{f}_{0}).$$

(57)

Here, we can use

$$\frac{\partial {f}_{0}}{\partial {\epsilon }_{{{{\bf{k}}}}}^{(n)}}=\frac{-{e}^{\left({\epsilon }_{{{{\bf{k}}}}}^{(n)}-{E}_{F}\right)/({k}_{B}T)}}{{\left[1+{e}^{\left({\epsilon }_{{{{\bf{k}}}}}^{(n)}-{E}_{F}\right)/({k}_{B}T)}\right]}^{2}{k}_{B}T}=\frac{-1}{2{k}_{B}T\left\{1+\cosh \left[\left({\epsilon }_{{{{\bf{k}}}}}^{(n)}-{E}_{F}\right)/({k}_{B}T)\right]\right\}}.$$

(58)

In the above, we have approximated the expressions to the leading order in a small dimensionless parameter $\frac{e\tau }{\hslash }{{{{\mathcal{E}}}}}_{a}a$ (with the lattice constant a) that is proportional to τ. 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.

Expression for the Floquet Hamiltonian 17

In this subsection, we derive the concrete analytical expressions for the time Fourier components ${{{{\mathcal{H}}}}}_{0}$, ${{{{\mathcal{H}}}}}_{-n}$, and ${{{{\mathcal{H}}}}}_{n}$ 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 ${A}_{0}=e{E}_{0}/(\hslash \tilde{\omega })$, we have the photon-dressed effective Hamiltonian as

$${{{\mathcal{H}}}}({{{\bf{k}}}},t) =\, {{{\mathcal{H}}}}\left({{{\bf{k}}}}-\frac{e}{\hslash }{{{\bf{A}}}}(t)\right)\\ =\, {t}_{0}[{k}_{x}-{A}_{0}\sin (\tilde{\omega }t)]{\sigma }_{0}+v[{k}_{y}-{A}_{0}\sin (\tilde{\omega }t+\varphi )]{\sigma }_{x}+\eta v[{k}_{x}-{A}_{0}\sin (\tilde{\omega }t)]{\sigma }_{y}\\ +\left\{m-\alpha \left\{{[{k}_{x}-{A}_{0}\sin (\tilde{\omega }t)]}^{2}+{[{k}_{y}-{A}_{0}\sin (\tilde{\omega }t+\varphi )]}^{2}\right\}\right\}{\sigma }_{z}\\ =\, {t}_{0}{k}_{x}{\sigma }_{0}+v{k}_{y}{\sigma }_{x}+\eta v{k}_{x}{\sigma }_{y}+m{\sigma }_{z}-{A}_{0}\frac{{t}_{0}}{2i}\left({e}^{i\tilde{\omega }t}-{e}^{-i\tilde{\omega }t}\right){\sigma }_{0}\\ -{A}_{0}\frac{v}{2i}\left[{e}^{i(\tilde{\omega }t+\varphi )}-{e}^{-i(\tilde{\omega }t+\varphi )}\right]{\sigma }_{x}-{A}_{0}\eta \frac{v}{2i}\left({e}^{i\tilde{\omega }t}-{e}^{-i\tilde{\omega }t}\right){\sigma }_{y}\\ -\alpha \left\{{\left[{k}_{x}-\frac{{A}_{0}}{2i}\left({e}^{i\tilde{\omega }t}-{e}^{-i\tilde{\omega }t}\right)\right]}^{2}+{\left[{k}_{y}-\frac{{A}_{0}}{2i}\left({e}^{i(\tilde{\omega }t+\varphi )}-{e}^{-i(\tilde{\omega }t+\varphi )}\right)\right]}^{2}\right\}{\sigma }_{z}.$$

(59)

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

$${{{{\mathcal{H}}}}}_{0}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t)dt={t}_{0}{k}_{x}{\sigma }_{0}+v{k}_{y}{\sigma }_{x}+\eta v{k}_{x}{\sigma }_{y}+\left(m-\alpha {A}_{0}^{2}-\alpha {k}^{2}\right){\sigma }_{z},$$

(60)

$${{{{\mathcal{H}}}}}_{-1}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{-i\tilde{\omega }t}dt=i\frac{{A}_{0}{t}_{0}}{2}{\sigma }_{0}+i\frac{{A}_{0}v}{2}{e}^{i\varphi }{\sigma }_{x}+\eta i\frac{{A}_{0}v}{2}{\sigma }_{y}-i{A}_{0}\alpha \left({k}_{x}+{k}_{y}{e}^{i\varphi }\right){\sigma }_{z},$$

(61)

$${{{{\mathcal{H}}}}}_{1} = \, \frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{i\tilde{\omega }t}dt=-i\frac{{A}_{0}{t}_{0}}{2}{\sigma }_{0}-i\frac{{A}_{0}v}{2}{e}^{-i\varphi }{\sigma }_{x}-\eta i\frac{{A}_{0}v}{2}{\sigma }_{y} \\ +i{A}_{0}\alpha \left({k}_{x}+{k}_{y}{e}^{-i\varphi }\right){\sigma }_{z},$$

(62)

$${{{{\mathcal{H}}}}}_{-2}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{-i2\tilde{\omega }t}dt=\frac{1}{4}{A}_{0}^{2}\alpha (1+{e}^{2i\varphi }){\sigma }_{z},$$

(63)

$${{{{\mathcal{H}}}}}_{2}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{i2\tilde{\omega }t}dt=\frac{1}{4}{A}_{0}^{2}\alpha (1+{e}^{-2i\varphi }){\sigma }_{z},$$

(64)

$${{{{\mathcal{H}}}}}_{-n < -2}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{-in\tilde{\omega }t}dt=0,$$

(65)

$${{{{\mathcal{H}}}}}_{n\ > \ 2}=\frac{1}{T}\int_{0}^{T}{{{\mathcal{H}}}}({{{\bf{k}}}},t){e}^{in\tilde{\omega }t}dt=0.$$

(66)

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

$$\frac{[{{{{\mathcal{H}}}}}_{-1},{{{{\mathcal{H}}}}}_{1}]}{\hslash \tilde{\omega }} =\, \frac{{A}_{0}^{2}}{\hslash \tilde{\omega }}\left\{\eta \frac{{v}^{2}}{4}{e}^{i\varphi }[{\sigma }_{x},{\sigma }_{y}]+\eta \frac{{v}^{2}}{4}{e}^{-i\varphi }[{\sigma }_{y},{\sigma }_{x}]\right.\\ -\frac{v\alpha }{2}{e}^{i\varphi }\left({k}_{x}+{k}_{y}{e}^{-i\varphi }\right)[{\sigma }_{x},{\sigma }_{z}]-\frac{v\alpha }{2}{e}^{-i\varphi }\left({k}_{x}+{k}_{y}{e}^{i\varphi }\right)[{\sigma }_{z},{\sigma }_{x}]\\ \left.-\eta \frac{v\alpha }{2}\left({k}_{x}+{k}_{y}{e}^{-i\varphi }\right)[{\sigma }_{y},{\sigma }_{z}]-\eta \frac{v\alpha }{2}\left({k}_{x}+{k}_{y}{e}^{i\varphi }\right)[{\sigma }_{z},{\sigma }_{y}]\right\}\\ =\, \frac{{A}_{0}^{2}}{\hslash \tilde{\omega }}\left\{\eta \frac{{v}^{2}}{4}{e}^{i\varphi }[{\sigma }_{x},{\sigma }_{y}]+\eta \frac{{v}^{2}}{4}{e}^{-i\varphi }[{\sigma }_{y},{\sigma }_{x}]\right.\\ -\frac{v\alpha }{2}\left({k}_{x}{e}^{i\varphi }+{k}_{y}\right)[{\sigma }_{x},{\sigma }_{z}]-\frac{v\alpha }{2}\left({k}_{x}{e}^{-i\varphi }+{k}_{y}\right)[{\sigma }_{z},{\sigma }_{x}]\\ \left.-\eta \frac{v\alpha }{2}\left({k}_{x}+{k}_{y}{e}^{-i\varphi }\right)[{\sigma }_{y},{\sigma }_{z}]-\eta \frac{v\alpha }{2}\left({k}_{x}+{k}_{y}{e}^{i\varphi }\right)[{\sigma }_{z},{\sigma }_{y}]\right\}\\ =\, \frac{{A}_{0}^{2}}{\hslash \tilde{\omega }}\left\{i\eta \frac{{v}^{2}}{2}({e}^{i\varphi }-{e}^{-i\varphi }){\sigma }_{z}+iv\alpha ({e}^{i\varphi }-{e}^{-i\varphi }){k}_{x}{\sigma }_{y} \right. \\ + \left. i\eta v\alpha ({e}^{i\varphi }-{e}^{-i\varphi }){k}_{y}{\sigma }_{x}\right\}\\ = -\frac{{A}_{0}^{2}(\sin \varphi )}{\hslash \tilde{\omega }}\left(\eta {v}^{2}{\sigma }_{z}+2v\alpha {k}_{x}{\sigma }_{y}+2\eta v\alpha {k}_{y}{\sigma }_{x}\right),$$

(67)

$$\frac{[{{{{\mathcal{H}}}}}_{-2},{{{{\mathcal{H}}}}}_{2}]}{2\hslash \tilde{\omega }}=\frac{{A}_{0}^{2}\alpha }{8\hslash \tilde{\omega }}\left\{(1+{e}^{2i\varphi })(1+{e}^{-2i\varphi })[{\sigma }_{z},{\sigma }_{z}]\right\}=0,$$

(68)

$$\frac{[{{{{\mathcal{H}}}}}_{-n < -2},{{{{\mathcal{H}}}}}_{n\ > \ 2}]}{n\hslash \tilde{\omega }}=0,$$

(69)

where we have used [σ_a, σ_b] = 2iε_abcσ_c.

Therefore, the Floquet Hamiltonian can be written as

$${{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}) = \, {t}_{0}{k}_{x}{\sigma }_{0}+v{k}_{y}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{x} \\ +\eta v{k}_{x}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{y}+\left(m-\alpha {A}_{0}^{2}-\eta \frac{{v}^{2}{A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}-\alpha {k}^{2}\right){\sigma }_{z}.$$

(70)

Inversion symmetry

We show that the Floquet Hamiltonian (70) (or Eq. (17) in the main text) is inversion-symmetric only when t₀ = 0. As such, a nonzero t₀ is necessary for the BCD.

The Floquet Hamiltonian (70) under inversion transformation becomes

$${{{\mathcal{I}}}}{{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}){{{{\mathcal{I}}}}}^{-1} = \, {t}_{0}{k}_{x}{\sigma }_{0}-v{k}_{y}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{x}-\eta v{k}_{x} \\ \times\left(1-\frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{y} +\left(m-\alpha {A}_{0}^{2}-\eta \frac{{v}^{2}{A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}-\alpha {k}^{2}\right) {\sigma }_{z} \\ = \, {{{{\mathcal{H}}}}}^{(F)}(-{{{\bf{k}}}})+2{t}_{0}{k}_{x}{\sigma }_{0},$$

(71)

where ${{{\mathcal{I}}}}={\sigma }_{z}$⁸⁸ is the inversion operator, and we have used

$$({\sigma }_{z})({\sigma }_{0}){({\sigma }_{z})}^{-1}={\sigma }_{0},$$

(72)

$$({\sigma }_{z})({\sigma }_{x}){({\sigma }_{z})}^{-1}=-{\sigma }_{x},$$

(73)

$$({\sigma }_{z})({\sigma }_{y}){({\sigma }_{z})}^{-1}=-{\sigma }_{y},$$

(74)

$$({\sigma }_{z})({\sigma }_{z}){({\sigma }_{z})}^{-1}={\sigma }_{z}.$$

(75)

As a result of Eq. (71), if t₀ = 0, we have ${{{\mathcal{I}}}}{{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}){{{{\mathcal{I}}}}}^{-1}={{{{\mathcal{H}}}}}^{(F)}(-{{{\bf{k}}}})$ which satisfies inversion symmetry. However, for t₀ ≠ 0, Eq. (71), the extra 2t₀k_xσ₀ term appears due to broken inversion symmetry.

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 t₀ and A₀.

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

$${{{\mathcal{T}}}}{{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}){{{{\mathcal{T}}}}}^{-1} = \, {t}_{0}{k}_{x}{\sigma }_{0}-v{k}_{y}\left(1-\eta \frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{x} \\ -\eta v{k}_{x}\left(1-\frac{2\alpha {A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}\right){\sigma }_{y} \\ -\left(m-\alpha {A}_{0}^{2}-\eta \frac{{v}^{2}{A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}-\alpha {k}^{2}\right){\sigma }_{z} \\ = \, {{{{\mathcal{H}}}}}^{(F)}(-{{{\bf{k}}}})+2{t}_{0}{k}_{x}{\sigma }_{0}-2\left(m-\alpha {A}_{0}^{2}-\eta \frac{{v}^{2}{A}_{0}^{2}\sin \varphi }{\hslash \tilde{\omega }}-\alpha {k}^{2}\right){\sigma }_{z},$$

(76)

where ${{{\mathcal{T}}}}=i{\sigma }_{y}{{{\mathcal{K}}}}$⁸⁸ is the time-reversal operator with the complex conjugate operator ${{{\mathcal{K}}}}$ such that ${{{\mathcal{K}}}}{{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}){{{{\mathcal{K}}}}}^{-1}={{{{\mathcal{H}}}}}^{{(F)}^{* }}({{{\bf{k}}}})$, and

$$(i{\sigma }_{y})({\sigma }_{0}){(i{\sigma }_{y})}^{-1}={\sigma }_{0},$$

(77)

$$(i{\sigma }_{y})({\sigma }_{x}){(i{\sigma }_{y})}^{-1}=-{\sigma }_{x},$$

(78)

$$(i{\sigma }_{y})(K{\sigma }_{y}{K}^{-1}){(i{\sigma }_{y})}^{-1}=-{\sigma }_{y},$$

(79)

$$(i{\sigma }_{y})({\sigma }_{z}){(i{\sigma }_{y})}^{-1}=-{\sigma }_{z}.$$

(80)

Due to the presence of the σ₀ and σ_z terms, Eq. (76) shows that the time-reversal symmetry is broken, i.e., ${{{\mathcal{T}}}}{{{{\mathcal{H}}}}}^{(F)}({{{\bf{k}}}}){{{{\mathcal{T}}}}}^{-1}\ne {{{{\mathcal{H}}}}}^{(F)}(-{{{\bf{k}}}})$ whether t₀ and A₀ 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 t₀ vanishes.

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. 7, 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.

Fig. 7: 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, φ = π/2 (right-handed circularly polarized light), $\hslash \tilde{\omega }=1$ eV, t₀ = 0.05 eV ⋅ nm, v = 0.1 eV ⋅ nm, α = 0.1 eV ⋅ nm², m = 0.1 eV, η = − 1, k_BT = 0.003 eV, i.e., T ≈ 34.8136 K, τ ≈ 4.12434 × 10⁻¹⁴ s⁸, and ω = 17.777 Hz⁸.

When A₀ is sufficiently small such that the system is in the Chern insulator phase, i.e., A₀ = 0.5 nm⁻¹, there is an obvious nonzero flat $\sqrt{\langle {J}_{y}^{2}\rangle }$ region as shown in Fig. 7a, 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 A₀ tends to the critical value A_0c ≈ 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 Fig. 7b and 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 A₀ is exactly such a very tunable parameter. When A₀ becomes even larger such that the system is in the topologically trivial phase with vanishing Chern number, i.e., A₀ = 1.5 nm⁻¹, there is a flat region at zero as shown in Fig. 7d, where both the linear and nonlinear Hall effects essentially vanish.