# Topological Floquet engineering using two frequencies in two dimensions

Yixiao Wang  Anne-Sophie Walter  Institute for Quantum Electronics, ETH
Zurich, 8093 Zurich, Switzerland  Gregor Jotzu  Max Planck Institute for the
Structure and Dynamics of Matter, 22761 Hamburg, Germany  Konrad Viebahn
<EMAIL_ADDRESS>Institute for Quantum Electronics, ETH Zurich, 8093
Zurich, Switzerland

###### Abstract

Using two-frequency driving in two dimensions opens up new possibilites for
Floquet engineering, which range from controlling specific symmetries to
tuning the properties of resonant gaps. In this work, we study two-band
lattice models subject to two-tone Floquet driving and analyse the resulting
effective Floquet bandstructures both numerically and analytically. On the one
hand, we extend the methodology of Sandholzer et al.
[10.1103/PhysRevResearch.4.013056] from one to two dimensions and find
competing topological phases in a simple Bravais lattice when the two resonant
drives at $1\omega$ and $2\omega$ interfere. On the other hand, we explore
driving-induced symmetry breaking in the hexagonal lattice, in which the
breaking of either inversion or time-reversal symmetry can be tuned
independently via the Floquet modulation. Possible applications of our work
include a simpler generation of topological bands for ultracold atoms, and the
realisation of non-linear Hall effects as well as Haldane’s parity anomaly in
inversion-symmetric parent lattices.

##  I Introduction

Topological phenomena emerge naturally for electrons under the effect of
strong magnetic fields, exemplified by the quantum Hall effect (QHE) Klitzing
_et al._ (1980). In these phenomena, the underlying physical mechanism is the
breaking of time-reversal symmetry due to the magnetic field. Several years
after the discovery of the QHE it was noticed that topological insulators can
also arise without an ambient magnetic field, such as the anomalous quantum
Hall state Haldane (1988) and the quantum spin Hall effect Kane and Mele
(2005). Today, anomalous topological phases can be realised by Floquet
engineering, that is, periodic driving of a quantum system. So far, Floquet
engineered topological phenomena have been largely limited to single-frequency
protocols, applied to optical lattices Holthaus (2016); Eckardt (2017); Cooper
_et al._ (2019); Weitenberg and Simonet (2021); Aidelsburger  _et al._ (2013);
Miyake  _et al._ (2013); Jotzu  _et al._ (2014); Tai  _et al._ (2017);
Fläschner  _et al._ (2018); Wintersperger  _et al._ (2020), real materials Oka
and Kitamura (2019); Rudner and Lindner (2020); Harper  _et al._ (2020);
McIver  _et al._ (2020), photonic and plasmonic waveguides Ozawa  _et al._
(2019); Rechtsman  _et al._ (2013); Fedorova  _et al._ (2020), and other
synthetic systems Fleury  _et al._ (2016); Nagulu  _et al._ (2022); Chen  _et
al._ (2021); Yang  _et al._ (2022). Recently, bichromatic and multi-frequency
Floquet engineering has emerged as a powerful strategy to enhance the driving
capabilities. These include time-reversal or spatial symmetry breaking
Schiavoni  _et al._ (2003); Struck  _et al._ (2012); Nag  _et al._ (2019);
Stammer  _et al._ (2020); Jiménez-Galán  _et al._ (2020); Heide  _et al._
(2021); Neufeld  _et al._ (2021); Yao  _et al._ (2022); Ikeda  _et al._
(2022); Trevisan  _et al._ (2022); Olin and Lee (2022) and interference
between different Floquet harmonics Zhuang  _et al._ (2013); Niu  _et al._
(2015); Grossert  _et al._ (2016); Görg  _et al._ (2019); Viebahn  _et al._
(2021); Sandholzer  _et al._ (2022); Castro  _et al._ (2022). The combination
of both two-path interference and time-reversal symmetry breaking has led to
the proposal Kang and Shin (2020) and experimental demonstration Minguzzi  _et
al._ (2022) of a topological pump in one-dimensional lattice. To our
knowledge, all experimental results in the multi-frequency regime have been
limited to one-dimensional driving patterns, motivating the need to find novel
strategies for two-dimensional driving.

In this work, we use the idea of two-tone Floquet engineering to generate
novel topological bandstructures and topological phase transitions from simple
parent lattices in two dimensions. On the one hand, we consider resonant
driving for the lowest two bands of a triangular lattice. Although this model
only features a single potential minimum per unit cell, we are able to drive
topological transitions by tuning the amplitude and relative phase between
$1\omega$ and $2\omega$ drives. The simplicity of this scheme could open up
new pathways for realising strongly correlated topological insulators, which
have remained out of reach in existing methods. On the other hand, we apply
two-frequency driving to an inversion-symmetric (graphene-like) hexagonal
lattice. This driving scheme enables the selective breaking of inversion
symmetry while maintaining time-reversal symmetry, giving access to a new
class of Floquet Hamiltoninans. Thus, we find that an external breaking of
inversion symmetry is not necessary to drive the celebrated parity anomaly in
the Haldane model Haldane (1988). The two-frequency approach generically
applies to Floquet-driven systems in lattices, ranging from condensed matter
to synthetic quantum matter.

The remainder of this paper is structured as follows. The relevant two-tone
driving waveforms, and possible experimental implementations, are introduced
in section II. Afterwards, we show how resonant $1\omega$–$2\omega$ driving in
a triangular lattice leads to topological phase transitions (section III).
Section IV discusses the effective breaking of time-reversal and inversion
symmetry under off-resonant driving in a hexagonal lattice.

Figure 1:  Single- and two-frequency Lissajous curves for two-dimensional
driving. The plots show the real-space modulation figures
$\textbf{r}_{m}(\tau)$ (Eq. 4). These shapes are topologically equivalent to
those of the time-dependent force $\textbf{F}(\tau)$, since the two are
related to one another by taking the second derivative (Eq. 2). (a-c) Circular
strong driving at $1\omega$ (a), circular weak driving $2\omega$ (b), or both
drives combined (c) can be used to resonantly address a higher-band
transition. The thin line in (c) represents the dominant $1\omega$ waveform,
the same as (a). (d-f) Superposition of an elliptical driving field at
$1\omega$ and a linear field at $2\omega$ with relative phases
$\varphi_{y1}=0$ (d), $0.2\pi$ (e), and $0.5\pi$ (f). (g-i) Superposition of
two circular drives with relative phase $\varphi_{x2}=0.9\pi$,
$\varphi_{y1}=0.5\pi$, $\varphi_{y2}=0.4\pi$, driving strength
$K_{x2}=K_{y2}=0.7$, and $K_{x1}=K_{y1}=0.5$ (g), $0.65$ (h), and $0.8$ (i).

##  II $1\omega$–$2\omega$ Floquet driving in two dimensions

Two-frequency driving can be applied, in princple, to any physical system
which has been employed for Floquet engineering so far Oka and Kitamura
(2019). As a generic starting point, we consider a particle confined to a
static potential $V_{\text{lat}}(r)$, and subject to an oscillating force

$\hat{H}_{\text{cm}}(\tau)=\frac{\hat{p}^{2}}{2m}+V_{\text{lat}}(r)-\textbf{F}(\tau)\cdot\hat{r}~{},$
(1)

with

$\textbf{F}(\tau)\equiv
e\textbf{E}(\tau)=-m\ddot{\textbf{r}}_{m}(\tau)=-\dot{\textbf{A}}(\tau)~{}.$
(2)

In solid state systems, the driving force $\mathbf{F}(\tau)$ can directly
correspond to the oscillating field $\textbf{E}(\tau)$ of a laser coupling to
free electrons Wang  _et al._ (2013); McIver  _et al._ (2020). Equivalently,
the application of periodic driving can be seen as ‘minimal coupling’ using
the vector potential as $q\rightarrow q-\textbf{A}(\tau)/\hbar$. Furthermore,
collective modes such as phonons Disa  _et al._ (2021) or magnons Vinas
Boström  _et al._ (2020) can also be involved. In waveguide-based synthetic
lattices, transverse movements of the waveguides provide an effective inertial
force Rechtsman  _et al._ (2013). For optical lattices, the driving force can
be provided by an oscillating optical or magnetic gradient Jotzu  _et al._
(2015). Alternatively, an inertial force can result from a periodic
displacement of the entire lattice potential, an approach known as ‘lattice
shaking’ Holthaus (2016); Eckardt (2017). In this case, the Hamiltonian in the
lab frame is

$\displaystyle\hat{H}_{\text{lab}}(\tau)$
$\displaystyle=\frac{\hat{p}^{2}}{2m}+V_{\text{lat}}\left[r-\textbf{r}_{m}(\tau)\right]~{},$
(3)

related to Eq. 2 by two unitary transformations Dalibard (2013). In summary,
the derivations in this work directly carry over from engineered quantum
platforms, such as ultracold atoms in optical lattices, to condensed matter
systems.

We define the two-dimensional driving pattern $\textbf{r}_{m}(\tau)$ as

$\displaystyle x_{m}(\tau)=$ $\displaystyle A_{x1}\cos(\omega\tau)$
$\displaystyle+A_{x2}\cos(2\omega\tau+\varphi_{x2})~{}$ (4) $\displaystyle
y_{m}(\tau)=$ $\displaystyle A_{y1}\cos(\omega\tau+\varphi_{y1})$
$\displaystyle+A_{y2}\cos(2\omega\tau+\varphi_{y2})~{}.$

These two-tone driving waveforms are visualised as Lissajous curves in Fig. 1
for the relevant choices of driving parameters (phases and amplitudes) used in
this work. The amplitude of the vector potential is given by the dimensionless
driving strength $K_{\alpha\beta}=m\omega_{\beta}A_{\alpha\beta}a/\hbar$,
where $a$ is the lattice spacing and $A_{\alpha\beta}$ is the real-space
amplitude. The index $\alpha\in\\{x,y\\}$ denotes the direction, whereas
$\beta\in\\{1,2\\}$ indicates the frequency.

Resonant circular $1\omega$–$2\omega$ driving. The combination of a strong
circular $1\omega$ drive (Fig. 1a, $\varphi_{y1}=\pi/2$) with a weak circular
$2\omega$ drive (Fig. 1b, $\varphi_{x2}=0$, $\varphi_{y2}=\pi/2$) leads to a
modulated circular pattern (Fig. 1c). Due to two-path interference when
addressing the $s$–$p$ resonance of the triangular lattice, this driving
pattern can be used to drive topological transitions in two dimensions,
similar to the topological pump of ref. Minguzzi  _et al._ (2022).

Off-resonant $1\omega$–$2\omega$ driving to access spatial and temporal
symmetries. Alternatively, an elliptical driving field at $1\omega$ and a
linearly polarised laser at $2\omega$ ($A_{x2}=0,\varphi_{y2}=0$) leads to a
competition between the breaking of inversion symmetry and time-reversal
symmetry. For instance, the choice $\varphi_{y1}=0$ results in a ‘boomerang’
pattern (Fig. 1d), which breaks inversion symmetry but not time-reversal
symmetry. Conversely, the case of $\varphi_{y1}=\pi/2$ (Fig. 1f) gives a
rounded ‘kite’ shape in which the breaking of time-reversal symmetry dominates
over the breaking of inversion symmetry. Tuning the phase $\varphi_{y1}$ thus
allows to interpolate between the two symmetry breaking situations (Fig. 1e),
thereby accessing novel topological regimes.

Instead of combining linear and elliptical drives, the competition between
inversion-breaking and time-reversal-breaking can be achieved by superimposing
two circular drives of opposite helicity (Fig. 1g-i). The way in which
inversion symmetry is broken in this case depends on the relative orientation
of the driving pattern to the lattice geometry. This can be tuned by
increasing $\varphi_{x2}$ and $\varphi_{y2}$ by the same amount (i.e. by
delaying the two frequency drives with respect to each other), which rotates
the driving pattern around the origin (Fig. 1g, see also refs. Jiménez-Galán
_et al._ (2020); Trevisan  _et al._ (2022); Ikeda  _et al._ (2022)). Driving a
topological transition via changing only a relative phase can be
experimentally advantageous in condensed matter systems where changing
polarisation or amplitude can lead to spurious effects.

Experimental implementation. In this work we focus on a triangular and a
hexagonal lattice. A corresponding optical lattice potential, and the
resulting tight-binding parameters, can be found in Appendix A. In waveguide-
based systems, the lattice geometry can generally be chosen freely Rechtsman
_et al._ (2013). The time-dependence of the driving term is encoded in an
additional spatial coordinate, which can accommodate multi-frequency driving.
Finally, in solid state systems, the laser field corresponding to the driving
term at $2\omega$ can be generated via second-harmonic generation, ensuring
that it is phase-stable with respect to the $1\omega$ term. The relative phase
between the two can then be tuned using a dispersive material of varying
thickness.

Figure 2:  Resonant band coupling in the triangular lattice (a), with relevant
parameters $t_{1}=-0.05$, $t_{2}=-0.07$, $t_{3}=-0.07$, $\varepsilon=-0.66$ in
units of $E_{rec}$ and dimensionless inter-band coupling $\eta_{sp}=0.19$.
Resonant one-photon or two-photon processes lead to the coupling of lower and
higher bands, corresponding to the green and black arrows in (b),
respectively. Floquet driving lifts the degeneracy of the static band
structure in the rotating frame (c) and opens a band gap (d). The ring-shape
minima of the band structure leads to a ring-shape Berry curvature (e). (f)
shows a trivial spatially localized Berry curvature under two-frequency
driving, which features a strong Berry curvature dipole.

Tight-binding Hamiltonians for two-dimensional Floquet engineering. By
expanding the original Hamiltonian (Eq. 1) in a Wannier basis and transforming
it to quasimomentum space, we get the following expression

$\displaystyle\sum_{n}\biggl{\\{}\varepsilon_{n}\hat{a}_{n}^{\dagger}\hat{a}_{n}-\sum_{i=1}^{Z}\left[t_{n,i}\left(e^{i(\theta_{i}(\tau)+\textbf{q}\cdot\textbf{b}_{i})}\hat{a}_{n}^{\dagger}\hat{a}_{n}+h.c.\right)\right]$
(5)
$\displaystyle-\textbf{F}(\tau)\cdot\hat{r}\sum_{n^{\prime}}\eta_{nn^{\prime}}\hat{a}_{n}^{\dagger}\hat{a}_{n^{\prime}}\biggl{\\}}~{}.$

Here, $\varepsilon_{n}$, $t_{n,i}$, $\eta_{nn^{\prime}}$ are the band centre
energies, nearest-neighbour tunnelling matrix elements, and interband coupling
elements, respectively, for bands $n$, $n^{\prime}$ (Appendix B).
$\textbf{b}_{i}=j\textbf{a}_{1}+k\textbf{a}_{2}$ are the nearest-neighbour
tunneling vectors and $i$ indexes the in-equivalent neighbours up to the
coordination number $Z$; $\textbf{a}_{1}$, $\textbf{a}_{2}$ correspond to the
primitive translation vectors in two dimensions. The time-periodic Peierls
phases are
$\theta_{i}(\tau)=\frac{m}{\hbar}\textbf{b}_{i}\cdot\dot{\textbf{r}}_{m}(\tau)$.
In second quantisation, $\hat{a}_{n}$ denotes the annihilation operator in
band $n$. The first line of Eq. 5 describes static and time-dependent
contributions to intra-band processes whereas the second line is the inter-
band coupling that arises from periodically forcing the system. The coupling
elements $\eta_{nn^{\prime}}$ can be understood as dipole matrix elements of
Bloch states Sandholzer  _et al._ (2022). Consequently, the first line is
diagonal in quasimomentum and band index $n$, whereas the second line
contributes off-diagonal elements to the Hamiltonian.

We use the Hamiltonian in Eq. 5 as the starting point for the Floquet
analysis. If the periodic drive resonantly addresses the gap between $s$ and
$p$ bands, we additionally employ a unitary transformation to ‘rotate away’
the energy difference (sec. III). In order to evaluate the effective Floquet
Hamiltonian, we employ two complementary methods. On the one hand, we
calculate the effective Hamiltonian analytically in inverse powers of the
driving frequency $\omega$ using the well-known high-frequency expansion (HFE)
Rahav  _et al._ (2003); Goldman and Dalibard (2014); Eckardt and Anisimovas
(2015); Bukov  _et al._ (2015); Mikami  _et al._ (2016). Since we work in the
weak driving regime the HFE remains valid. On the other hand, we can
numerically obtain the effective Hamiltonian via the Trotter decomposition
Trotter (1959). We use this numerical method to validate the results of the
analytic calculation and to justify the truncation of the high-frequency
expansion a posteriori. To characterise the band topologies, the Berry
curvature is evaluated via the eigenstates in the discretised two-dimensional
Brillouin zone. The resulting Chern number is the surface integral of the
Berry curvature over the closed Brillouin torus Fukui  _et al._ (2005); Asbóth
_et al._ (2016).

##  III Resonantly driven triangular lattice

The first situation we consider is a simple triangular lattice in which the
lowest two bands ($s$ and $p$) are resonantly coupled with two frequencies,
inspired by ref. Kang and Shin (2020) and building on previous works using
single-frequency driving Baur  _et al._ (2014); Zheng and Zhai (2014); Zhang
and Zhou (2014).

In the non-shaken case the static Hamiltonian for a triangular tight-binding
model is

$\hat{H}=\left[\varepsilon+\sum_{i=1}^{3}t_{i}\cos(\textbf{q}\cdot\textbf{b}_{i})\right]\sigma_{z}~{}.$
(6)

The nearest-neighbour tunneling vectors are $\textbf{b}_{1}=(-1,1)b$,
$\textbf{b}_{2}=(1,0)b$, $\textbf{b}_{3}=(0,1)b$ and
$\textbf{q}=(q_{x},q_{y})$ in the rotated coordinate
$\hat{e}_{x}^{\prime}=\frac{\hat{e}_{x}+\hat{e}_{y}}{\sqrt{2}}$,
$\hat{e}_{y}^{\prime}=\frac{-\hat{e}_{x}+\hat{e}_{y}}{\sqrt{2}}$ (Fig. 2a).
The simplest way to couple $s$ and $p$ bands is to apply single-frequency
driving, parametrised by Eq. 4 with $A_{x2}=A_{y2}=0$ and denoting
$\varphi_{y1}$ as $\varphi$.

Starting from Eq. 5 the time-dependent Hamiltonian can be written as

$\displaystyle\hat{H}(\tau)=$
$\displaystyle\left\\{\varepsilon+\sum_{i=1}^{3}t_{i}\cos\left[\textbf{q}\cdot\textbf{b}_{i}+\theta_{i}(\tau)\right]\right\\}\sigma_{z}+$
(7) $\displaystyle\left[\eta_{sp}\hbar\omega
K_{x}\cos(\omega\tau)+\eta_{sp}\hbar\omega
K_{y}\cos(\omega\tau+\varphi)\right]\sigma_{x},$

where $\varepsilon=(\varepsilon_{s}-\varepsilon_{p})/2=-\Delta\varepsilon/2$
with $\Delta\varepsilon$ being the gap size. The other parameters are
$t_{i}=t_{s,i}-t_{p,i}$ with $i={1,2,3}$, the Peierls phases are
$\theta_{i}(\tau)=-\frac{1}{\hbar}\int_{0}^{\tau}\textbf{F}(\tau^{\prime})\cdot\textbf{b}_{i}\,d\tau^{\prime}=-[K_{x}\sin(\omega\tau)\hat{x}+K_{y}\sin(\omega\tau+\varphi)\hat{y}]\cdot\textbf{b}_{i}/b$,
and the Pauli matrices are $\sigma_{j}$ with $j=x,y,z$. The dimensionless
driving strengths are denoted by $K_{x}=m\omega A_{x1}a/\hbar$, $K_{y}=m\omega
A_{y1}a/\hbar$, respectively. The tight-binding parameters, including the
tunneling amplitudes, inter-band coupling and band center energies, are
obtained from a triangular lattice potential by evaluating the Wannier
functions (Appendix B, Fig. 2a).

Figure 3:  Phase diagrams of the triangular lattice under resonant driving.
Single-frequency driving on resonance ($\hbar\omega=25.6t_{1}=1.28E_{rec}$,
$K_{y}=0.5$, $\varepsilon+\hbar\omega/2\simeq 0$) leads to trivial bands (a),
whereas driving on half-resonance ($\hbar\omega=12.8t_{1}=0.64E_{rec}$,
$K_{y}=0.5$, $\varepsilon+\hbar\omega\simeq 0$) induces topological bands (b).
The trivial regions in (b) are caused by bands being shifted out of resonance
due to the $ac$-Stark effect. (c) shows the competition between the two
resonant processes induced by the two-frequency driving with
$K_{x1}=0.4,K_{y1}=0.5,K_{x2}=0.02,K_{y2}=0.02$. (d) resulting band gap
evaluated for a cut through (c) along the red line.

Single-frequency circular driving resonant with the gap. If the frequency
$\omega$ is near-resonant with the gap, corresponding to the dominant one-
photon process, we apply the unitary
$\mathcal{U}_{R}(\tau)=\exp(-i\omega\tau\sigma_{z}/2)$ to eliminate the energy
gap between the two bands (Fig. 2d). This gives

$\displaystyle\hat{H}(\tau)=$
$\displaystyle\left\\{\varepsilon+\hbar\omega/2+\sum_{i=1}^{3}t_{i}\cos\left[\textbf{q}\cdot\textbf{b}_{i}+\theta_{i}(\tau)\right]\right\\}\sigma_{z}$
(8) $\displaystyle+\left[\eta_{sp}\hbar\omega
K_{x}\cos(\omega\tau)+\eta_{sp}\hbar\omega
K_{y}\cos(\omega\tau+\varphi)\right]$
$\displaystyle\times\left[\sigma_{x}\cos(\omega\tau)-\sigma_{y}\sin(\omega\tau)\right]/2.$

The terms proportional to the identity matrix can be omitted without affecting
the topology of the bands.

By analysing Eq. 8 in the high-frequency expansion (Appendix C), we find a
ring-shaped gap opening (Fig. 2e, see also refs. Wintersperger  _et al._
(2020); Bracamontes  _et al._ (2022)). However, the off-diagonal terms are
dominanted by constants. Physically, this corresponds to on-site couplings
between the $s$ and $p$ bands, which yield topologically trivial bands (Fig.
3a).

Single-frequency elliptical driving resonant with half the gap. Driving the
system at half the gap energy ($\hbar\omega=12.8t_{1}=0.64E_{rec}$) leads to
neighbouring-site interband couplings which can give rise to topological
bands. Starting from the waveforms in Eq. 4 with $K_{x2}=K_{y2}=0$ and
applying the unitary $\mathcal{U}_{R}(\tau)=\exp(-i\omega\tau\sigma_{z})$
gives

$\displaystyle\hat{H}(\tau)$
$\displaystyle=\left\\{\varepsilon+\hbar\omega+\sum_{i=1}^{3}t_{i}\cos\left[\textbf{q}\cdot\textbf{b}_{i}+\theta_{i}(\tau)\right]\right\\}\sigma_{z}$
(9) $\displaystyle+[\eta_{sp}\hbar\omega
K_{x}\cos(\omega\tau)+\eta_{sp}\hbar\omega K_{y}\cos(\omega\tau+\varphi)]$
$\displaystyle\times[\sigma_{x}\cos(2\omega\tau)-\sigma_{y}\sin(2\omega\tau)]/2,$

where $\varepsilon+\hbar\omega\approx 0$. In this case, the only non-zero
Fourier components of the drive are $\pm 1\omega$ and $\pm 3\omega$. Contrary
to the direct one-photon resonance, the inter-band couplings happen via two-
photon processes (Fig. 2b). We now find quasimomentum-dependent $\sigma_{x}$
and $\sigma_{y}$ terms in the effective Hamiltonian (Appendix C), leading to
topologically non-trivial regions in the phase diagram (Fig. 3b, see also ref.
Zhang and Zhou (2014)). Lines at $K_{x}=0$ and $\varphi=0$ have zero extent in
the phase diagram of Fig. 3b, preventing the observation of a transition from
$C\neq 0$ to $C=0$ in a realistic experiment. While large driving amplitudes
(beyond $K_{x}=0.5$) render the model topologically trivial, this effect is
not due to an interference between different processes, but rather due to the
$ac$-Stark effect that shifts the bands out of resonance Holthaus (2016).
Combining the above two resonant schemes at $1\omega$ and $2\omega$ introduces
genuine topological transitions as function of driving phase, as outlined in
the following.

Two-frequency resonant elliptical driving. The competition between resonant
$1\omega$ and $2\omega$ driving results in an interplay between topological
phases with $C=0$ and $C=\pm 1$. We now consider the full two-tone waveforms
of Eq. 4 for fixed $[K_{x1},K_{x2},K_{y1},K_{y2}]$, and tuneable
$\varphi_{y1}$ and $\varphi_{y2}$ ($\varphi_{x2}=0$). This Floquet scheme
allows to cross topological transitions purely by changing the phase between
$1\omega$ and $2\omega$ drives, as is evident from the distinct regions of
$C=0$ and $C\neq 0$ in the phase diagram (Fig. 3c). The values of the induced
band gaps are on the order of $0.1$ to $0.2$ tunnelling energies, which can be
enlarged by choosing stronger driving amplitudes.

Figure 4:  Off-resonant driving in the hexagonal lattice (a) with tunnelings
$t_{1}=t_{2}=t_{3}=0.06$ in units of $E_{rec}$. The static band structure
features two degenerate Dirac points (b). Floquet driving lifts the degeneracy
and opens a band gap (c) by effectively breaking symmetries.  Figure 5:  Phase
diagrams of the hexagonal lattice under off-resonant driving
($\hbar\omega=20.8t_{1}=1.25E_{rec}$). Single-frequency elliptical driving
($K_{y1}=0.9$, $K_{x2}=K_{y2}=0$) can break time-reversal symmetry and result
in a non-zero Chern number (a). The breaking of the inversion and time-
reversal symmetries can be independently controlled by adding a linearly
polarized field ($K_{y1}=K_{y2}=0.9$, $K_{x2}=0$ and
$\varphi_{x2}=\varphi_{y2}=0$) (b), as well as using two circular fields with
opposite polarization ($K_{x1}=K_{y1}$, $K_{x2}=K_{y2}=0.7$,
$\varphi_{y1}=\pi/2$ and $\varphi_{y2}=\varphi_{x2}-\pi/2$) (c). The
transition paths I - II - III marked in (b) and (c) correspond to the
Lissajous curves in Fig. 1d-f and Fig. 1g-i, respectively.

Realising topological band structures in a Bravais lattice, such as the simple
triangular lattice considered here, has important implications. For instance,
it simplifies the generation of topological bands in the context of ultracold
atoms. Up to now, two-dimensional topological models in optical lattices have
been realised either in bipartite lattices Jotzu  _et al._ (2014); Fläschner
_et al._ (2018); Zhang and Zhou (2014), square lattices with moving
superlattices Aidelsburger  _et al._ (2013); Miyake  _et al._ (2013); Tai  _et
al._ (2017), or by employing spin-orbit coupling Sun  _et al._ (2018). These
implementations rely on the relative phase stability between lattice or Raman
laser beams, leading to a significant technological overhead. An insufficient
phase stability may have been an obstacle for realising strongly correlated
phases in two-dimensional topological lattices. The triangular lattice, on the
contrary, can be built by simply superimposing three standing waves in the
plane with no active stabilisation. Therefore, the $1\omega$–$2\omega$ Floquet
scheme could provide an avenue towards realising correlated topological states
of matter. In addition to its conceptual simplicity, the two-tone driving
leads to the appearance of Berry curvature along a ring-shaped gap, which can
be asymmetrically distributed (Fig. 2f), particularly in the topologically
trivial regions of Fig. 3c. This supports a non-zero Berry curvature dipole,
enabled by periodic driving, although the underlying lattice is trivial and
inversion-symmetric. This finding is relevant beyond the field of ultracold
atoms and it could enable the observation of nonlinear Hall effects which so
far have relied on materials with broken inversion symmetry Du  _et al._
(2021).

##  IV Off-resonantly driven hexagonal lattice

The second application of the two-frequency driving method considers the
hexagonal lattice in brick-wall configuration, shown in Fig. 4a. This
potential can be realised in the same setup as the triangular lattice of sec.
III, but the derivation also applies to $120^{\circ}$ honeycomb lattices
Fläschner  _et al._ (2018) as well as real graphene McIver  _et al._ (2020).

We start from the tight-binding Hamiltonian

$\displaystyle\hat{H}_{\bm{q}}(\tau)=$
$\displaystyle\left\\{\sum_{i=1}^{3}t_{i}\cos\left[\textbf{q}\cdot\textbf{b}_{i}+\theta_{i}(\tau)\right]\right\\}\sigma_{x}+$
(10)
$\displaystyle\left\\{\sum_{i=1}^{3}t_{i}\sin\left[\textbf{q}\cdot\textbf{b}_{i}+\theta_{i}(\tau)\right]\right\\}\sigma_{y},$

The nearest-neighbour tunneling vectors are $\textbf{b}_{1}=(0,1)b$,
$\textbf{b}_{2}=(-1,1)b$ and $\textbf{b}_{3}=(-1,0)b$ in the rotated
coordinate system $\\{\hat{e}_{x}^{\prime},\hat{e}_{y}^{\prime}\\}$ (Fig. 4a).
The waveforms of Eq. 4 result in the following Peierls phases:

$\displaystyle\theta_{1}(\tau)=-K_{y1}\sin(\omega\tau+\varphi_{y1})-K_{y2}\sin(2\omega\tau+\varphi_{y2})=-\theta_{3}(\tau),$
(11)
$\displaystyle\theta_{2}(\tau)=K_{x1}\sin(\omega\tau)+K_{x2}\sin(2\omega\tau+\varphi_{x2}),$

with the driving strengths being $K_{x1}=m\omega A_{x1}a/\hbar$,
$K_{x2}=2m\omega A_{x2}a/\hbar$, $K_{y1}=m\omega A_{y1}a/\hbar$, and
$K_{y2}=2m\omega A_{y2}a/\hbar$, as before. Since we do not consider inter-
banding couplings here, there is no need to apply unitaries and the high-
frequency expansion can be directly conducted.

Single-frequency elliptical driving. The choice $K_{x2}=K_{y2}=0$, $K_{x1}>0$
corresponds to elliptical driving, giving rise to topological or trivial
bands. For any $K_{y1}>0$ and $\varphi_{y1}\mod\pi\neq 0$ we recover the well-
known topological gap opening at both Dirac points due to time-reversal
symmetry breaking (Fig. 4b-c) Oka and Aoki (2009). The limits of $K_{y1}=0$ or
$\varphi_{y1}\mod\pi=0$ correspond to linearly polarised driving, leaving
time-reversal symmetry intact and both Dirac points remain closed. However,
the trivial region with $C=0$ is only a ‘line’ of zero extent in the phase
diagram (Fig. 5a, as before in Fig. 3b). Usually, a manifestly broken
inversion symmetry of the underlying lattice is required to cross a
topological phase transition and induce Haldane’s ‘parity anomaly’, thereby
extending the trivial ‘lines’ in the phase diagram to finite regions Haldane
(1988); Jotzu  _et al._ (2014). In the following, we demonstrate that
inversion symmetry can be broken by employing the two-tone Floquet drive only.
Therefore, a topological phase transition between trivial and non-trivial
regions can be achieved purely via driving.

Two-frequency driving. Starting from the above case of single-frequency
elliptical driving, we add linearly polarized ‘light’ by setting $K_{x2}=0$
while $K_{y2}\neq 0$. The driving field further breaks inversion symmetry and
the competition between the breaking of inversion and time-reversal symmetry
can be observed in Fig. 5b-c. The topological phase transition can simply be
induced by tuning the phase along the marked path, corresponding to the
Lissajous curves in Fig. 1d-f.

Alternatively, constraining $K_{x1}=K_{y1}$, $K_{x2}=K_{y2}$,
$\varphi_{y1}=\pi/2$, and $\varphi_{y2}=\varphi_{x2}-\pi/2$ in Eq. 4, the
$1\omega-2\omega$ driving scheme corresponds to two counter-rotating,
circularly polarized fields. Clearly, when one circular driving field is much
larger than the other, the resulting Chern number will correspond to the
helicity of the dominant field. However, when the amplitude of one circular
field approaches the other one, the Lissajous curve becomes a ‘cloverleaf’
pattern and the breaking of time-reversal symmetry is suppressed (Fig. 1g-i).
This competition allows to interpolate between topologically trivial and non-
trivial settings (with either positive or negative mass terms) purely by
changing driving phases. The resulting gap sizes are similar to the known
results in the Haldane model Jotzu  _et al._ (2014). Interestingly, we found
that a pure Dirac Hamiltonian subject to two-frequency driving does not show a
corresponding gap opening in the case of broken inversion symmetry. While
time-reversal symmetry breaking via circular driving can be captured within
the approximate Dirac Hamiltonian McIver  _et al._ (2020), the inversion
symmetry breaking relies on the presence of a lattice potential.
Mathematically, this can be seen from the fact that the driven Dirac
Hamiltonian remains linear and its Fourier components at $\omega$ and
$2\omega$ will not mix in the high-frequency expansion. Conceptually, a
specific driving pattern favours a gap opening specific to one sublattice of
the honeycomb, which does not apply to the Dirac Hamiltonian.

We also studied the behaviour of inversion symmetry breaking beyond the
validity of the high-frequency expansion, using numerically calculated band
structures. Here, the gap opening scales roughly as $1/\omega^{4}$ which
becomes dominant in the low-frequency regime. Therefore, the
$1\omega$–$2\omega$ scheme could be particularly interesting for phase-
controlled topological transitions in laser-driven graphene, which could be
readily implemented in setups such as used in ref. McIver  _et al._ (2020).

##  V Conclusion

We have shown that two-frequency Floquet driving in two dimensions gives rise
to rich topological phase diagrams, both for resonant and off-resonant
modulation. We identify the relative phase between the $1\omega$ and $2\omega$
drives as an important control parameter over the symmetry properties and the
resulting topologies in the effective Floquet Hamiltonians. In many
experiments, phase-only control is advantageous over other parameters, such as
frequencies or amplitudes. For periodically driven real materials, such as
graphene McIver  _et al._ (2020), incoherent excitations and electron
relaxation can strongly affect the measurements, potentially obscuring any
topological response. Incoherent effects often depend on the polarisation of
the incident light, especially close to the contacts of a sample. Several of
the driving schemes proposed in this work, particularly the case of counter-
rotating, circular drives, depend only on the relative phase between the light
fields. Therefore, we believe that our approach will lead to novel
applications in manipulation and detection of topological phenomena. By
combining the schemes considered here with spin-dependent Floquet engineering
Jotzu  _et al._ (2015), it may become possible to access the entire ‘periodic
table’ of topological insulators Kitaev (2009).

## Acknowledgments

We acknowledge funding by the Swiss National Science Foundation (Grants No.
182650 and NCCR-QSIT) and European Research Council advanced grant TransQ
(Grant No. 742579).

## References

  * Klitzing  _et al._ (1980) K. v. Klitzing, G. Dorda, and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
  * Haldane (1988) F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”,” Physical Review Letters 61, 2015–2018 (1988).
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,” Phys. Rev. Lett. 95, 226801 (2005).
  * Holthaus (2016) Martin Holthaus, “Floquet engineering with quasienergy bands of periodically driven optical lattices,” Journal of Physics B: Atomic, Molecular and Optical Physics 49, 013001 (2016).
  * Eckardt (2017) André Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 11004 (2017).
  * Cooper  _et al._ (2019) N. R. Cooper, J. Dalibard, and I. B. Spielman, “Topological bands for ultracold atoms,” Reviews of Modern Physics 91, 015005 (2019).
  * Weitenberg and Simonet (2021) Christof Weitenberg and Juliette Simonet, “Tailoring quantum gases by Floquet engineering,” Nat. Phys. (2021), 10.1038/s41567-021-01316-x.
  * Aidelsburger  _et al._ (2013) M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 111, 185301 (2013).
  * Miyake  _et al._ (2013) Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, and Wolfgang Ketterle, “Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices,” Phys. Rev. Lett. 111, 185302 (2013).
  * Jotzu  _et al._ (2014) Gregor Jotzu, Michael Messer, Rémi Desbuquois, Martin Lebrat, Thomas Uehlinger, Daniel Greif, and Tilman Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
  * Tai  _et al._ (2017) M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Tim Menke, Dan Borgnia, Philipp M. Preiss, Fabian Grusdt, Adam M. Kaufman, and Markus Greiner, “Microscopy of the interacting Harper–Hofstadter model in the two-body limit,” Nature 546, 519–523 (2017).
  * Fläschner  _et al._ (2018) N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, “Observation of dynamical vortices after quenches in a system with topology,” Nature Physics 14, 265–268 (2018).
  * Wintersperger  _et al._ (2020) Karen Wintersperger, Christoph Braun, F. Nur Ünal, André Eckardt, Marco Di Liberto, Nathan Goldman, Immanuel Bloch, and Monika Aidelsburger, “Realization of an anomalous Floquet topological system with ultracold atoms,” Nat. Phys. 16, 1058–1063 (2020).
  * Oka and Kitamura (2019) Takashi Oka and Sota Kitamura, “Floquet Engineering of Quantum Materials,” Annu. Rev. Condens. Matter Phys. 10, 387–408 (2019).
  * Rudner and Lindner (2020) Mark S. Rudner and Netanel H. Lindner, “Band structure engineering and non-equilibrium dynamics in Floquet topological insulators,” Nature Reviews Physics 2, 229–244 (2020).
  * Harper  _et al._ (2020) Fenner Harper, Rahul Roy, Mark S. Rudner, and S.L. Sondhi, “Topology and Broken Symmetry in Floquet Systems,” Annual Review of Condensed Matter Physics 11 (2020), 10.1146/annurev-conmatphys-031218-013721.
  * McIver  _et al._ (2020) J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri, “Light-induced anomalous Hall effect in graphene,” Nature Physics 16, 38–41 (2020).
  * Ozawa  _et al._ (2019) Tomoki Ozawa, Hannah M. Price, Alberto Amo, Nathan Goldman, Mohammad Hafezi, Ling Lu, Mikael C. Rechtsman, David Schuster, Jonathan Simon, Oded Zilberberg, and Iacopo Carusotto, “Topological photonics,” Reviews of Modern Physics 91, 015006 (2019).
  * Rechtsman  _et al._ (2013) Mikael C. Rechtsman, Julia M. Zeuner, Yonatan Plotnik, Yaakov Lumer, Daniel Podolsky, Felix Dreisow, Stefan Nolte, Mordechai Segev, and Alexander Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
  * Fedorova  _et al._ (2020) Zlata Fedorova, Haixin Qiu, Stefan Linden, and Johann Kroha, “Observation of topological transport quantization by dissipation in fast Thouless pumps,” Nat Commun 11, 3758 (2020).
  * Fleury  _et al._ (2016) Romain Fleury, Alexander B Khanikaev, and Andrea Alù, “Floquet topological insulators for sound,” Nat Commun 7, 11744 (2016).
  * Nagulu  _et al._ (2022) Aravind Nagulu, Xiang Ni, Ahmed Kord, Mykhailo Tymchenko, Sasank Garikapati, Andrea Alù, and Harish Krishnaswamy, “Chip-scale Floquet topological insulators for 5G wireless systems,” Nat Electron 5, 300–309 (2022).
  * Chen  _et al._ (2021) Bing Chen, Shuo Li, Xianfei Hou, Feifei Ge, Feifei Zhou, Peng Qian, Feng Mei, Suotang Jia, Nanyang Xu, and Heng Shen, “Digital quantum simulation of Floquet topological phases with a solid-state quantum simulator,” Photon. Res. 9, 81 (2021).
  * Yang  _et al._ (2022) Kai Yang, Shaoyi Xu, Longwen Zhou, Zhiyuan Zhao, Tianyu Xie, Zhe Ding, Wenchao Ma, Jiangbin Gong, Fazhan Shi, and Jiangfeng Du, “Observation of Floquet topological phases with large Chern numbers,” Phys. Rev. B 106, 184106 (2022).
  * Schiavoni  _et al._ (2003) M. Schiavoni, L. Sanchez-Palencia, F. Renzoni, and G. Grynberg, “Phase Control of Directed Diffusion in a Symmetric Optical Lattice,” Phys. Rev. Lett. 90, 094101 (2003).
  * Struck  _et al._ (2012) J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, “Tunable Gauge Potential for Neutral and Spinless Particles in Driven Optical Lattices,” Physical Review Letters 108, 225304 (2012).
  * Nag  _et al._ (2019) Tanay Nag, Robert-Jan Slager, Takuya Higuchi, and Takashi Oka, “Dynamical synchronization transition in interacting electron systems,” Phys. Rev. B 100, 134301 (2019).
  * Stammer  _et al._ (2020) Philipp Stammer, Serguei Patchkovskii, and Felipe Morales, “Evidence of ac-Stark-shifted resonances in intense two-color circularly polarized laser fields,” Phys. Rev. A 101, 033405 (2020).
  * Jiménez-Galán  _et al._ (2020) Á. Jiménez-Galán, R. E. F. Silva, O. Smirnova, and M. Ivanov, “Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation,” Nat. Photonics 14, 728–732 (2020).
  * Heide  _et al._ (2021) Christian Heide, Tobias Boolakee, Timo Eckstein, and Peter Hommelhoff, “Optical current generation in graphene: CEP control vs. $\omega$ \+ 2$\omega$ control,” Nanophotonics 10, 3701–3707 (2021).
  * Neufeld  _et al._ (2021) Ofer Neufeld, Nicolas Tancogne-Dejean, Umberto De Giovannini, Hannes Hübener, and Angel Rubio, “Light-Driven Extremely Nonlinear Bulk Photogalvanic Currents,” Phys. Rev. Lett. 127, 126601 (2021).
  * Yao  _et al._ (2022) Kai-Xuan Yao, Zhendong Zhang, and Cheng Chin, “Domain-wall dynamics in Bose–Einstein condensates with synthetic gauge fields,” Nature 602, 68–72 (2022).
  * Ikeda  _et al._ (2022) Yuya Ikeda, Sota Kitamura, and Takahiro Morimoto, “Floquet engineering of electric polarization with two-frequency drive,” Prog. Theor. Exp. Phys. , 04A101 (2022).
  * Trevisan  _et al._ (2022) Thaís V. Trevisan, Pablo Villar Arribi, Olle Heinonen, Robert-Jan Slager, and Peter P. Orth, “Bicircular Light Floquet Engineering of Magnetic Symmetry and Topology and Its Application to the Dirac Semimetal Cd 3 As 2,” Phys. Rev. Lett. 128, 066602 (2022).
  * Olin and Lee (2022) Sam Olin and Wei-Cheng Lee, “Topological phase transition in commensurate multi-frequency Floquet Su-Schrieffer-Heeger model,”  (2022).
  * Zhuang  _et al._ (2013) Chao Zhuang, Christopher R. Paul, Xiaoxian Liu, Samansa Maneshi, Luciano S. Cruz, and Aephraim M. Steinberg, “Coherent Control of Population Transfer between Vibrational States in an Optical Lattice via Two-Path Quantum Interference,” Physical Review Letters 111, 233002 (2013).
  * Niu  _et al._ (2015) Linxiao Niu, Dong Hu, Shengjie Jin, Xiangyu Dong, Xuzong Chen, and Xiaoji Zhou, “Excitation of atoms in an optical lattice driven by polychromatic amplitude modulation,” Opt. Express 23, 10064 (2015).
  * Grossert  _et al._ (2016) Christopher Grossert, Martin Leder, Sergey Denisov, Peter Hänggi, and Martin Weitz, “Experimental control of transport resonances in a coherent quantum rocking ratchet,” Nat Commun 7, 10440 (2016).
  * Görg  _et al._ (2019) Frederik Görg, Kilian Sandholzer, Joaquín Minguzzi, Rémi Desbuquois, Michael Messer, and Tilman Esslinger, “Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter,” Nature Physics (2019), 10.1038/s41567-019-0615-4.
  * Viebahn  _et al._ (2021) Konrad Viebahn, Joaquín Minguzzi, Kilian Sandholzer, Anne-Sophie Walter, Manish Sajnani, Frederik Görg, and Tilman Esslinger, “Suppressing Dissipation in a Floquet-Hubbard System,” Physical Review X 11, 011057 (2021).
  * Sandholzer  _et al._ (2022) Kilian Sandholzer, Anne-Sophie Walter, Joaquín Minguzzi, Zijie Zhu, Konrad Viebahn, and Tilman Esslinger, “Floquet engineering of individual band gaps in an optical lattice using a two-tone drive,” Phys. Rev. Research 4, 013056 (2022).
  * Castro  _et al._ (2022) Alberto Castro, Umberto de Giovannini, Shusuke A. Sato, Hannes Hübener, and Angel Rubio, “Floquet engineering the band structure of materials with optimal control theory,”  (2022).
  * Kang and Shin (2020) Jin Hyoun Kang and Yong-il Shin, “Topological Floquet engineering of a one-dimensional optical lattice via resonant shaking with two harmonic frequencies,” Physical Review A 102, 063315 (2020).
  * Minguzzi  _et al._ (2022) Joaquín Minguzzi, Zijie Zhu, Kilian Sandholzer, Anne-Sophie Walter, Konrad Viebahn, and Tilman Esslinger, “Topological Pumping in a Floquet-Bloch Band,” Phys. Rev. Lett. 129, 053201 (2022).
  * Wang  _et al._ (2013) Y H Wang, H Steinberg, P Jarillo-Herrero, and N Gedik, “Observation of Floquet-Bloch States on the Surface of a Topological Insulator,” Science 342, 453 (2013).
  * Disa  _et al._ (2021) Ankit S. Disa, Tobia F. Nova, and Andrea Cavalleri, “Engineering crystal structures with light,” Nat. Phys. 17, 1087–1092 (2021).
  * Vinas Boström  _et al._ (2020) Emil Vinas Boström, Martin Claassen, James McIver, Gregor Jotzu, Angel Rubio, and Michael Sentef, “Light-induced topological magnons in two-dimensional van der Waals magnets,” SciPost Phys. 9, 061 (2020).
  * Jotzu  _et al._ (2015) Gregor Jotzu, Michael Messer, Frederik Görg, Daniel Greif, Rémi Desbuquois, and Tilman Esslinger, “Creating State-Dependent Lattices for Ultracold Fermions by Magnetic Gradient Modulation,” Phys. Rev. Lett. 115, 073002 (2015).
  * Dalibard (2013) Jean Dalibard, “Réseaux dépendant du temps,” Collège de France, Cours 4 (2013).
  * Rahav  _et al._ (2003) Saar Rahav, Ido Gilary, and Shmuel Fishman, “Effective Hamiltonians for periodically driven systems,” Phys. Rev. A 68, 013820 (2003).
  * Goldman and Dalibard (2014) N. Goldman and J. Dalibard, “Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields,” Physical Review X 4, 31027 (2014).
  * Eckardt and Anisimovas (2015) André Eckardt and Egidijus Anisimovas, “High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective,” New Journal of Physics 17, 093039 (2015).
  * Bukov  _et al._ (2015) Marin Bukov, Luca D’Alessio, and Anatoli Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering,” Advances in Physics 64, 139–226 (2015).
  * Mikami  _et al._ (2016) Takahiro Mikami, Sota Kitamura, Kenji Yasuda, Naoto Tsuji, Takashi Oka, and Hideo Aoki, “Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators,” Phys. Rev. B 93, 144307 (2016).
  * Trotter (1959) H. F. Trotter, “On the Product of Semi-Groups of Operators,” Proceedings of the American Mathematical Society 10, 545–551 (1959), publisher: American Mathematical Society.
  * Fukui  _et al._ (2005) Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki, “Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances,” Journal of the Physical Society of Japan 74, 1674–1677 (2005).
  * Asbóth  _et al._ (2016) János K. Asbóth, László Oroszlány, and András Pályi,  _A short course on topological insulators: band structure and edge states in one and two dimensions_ , Lecture notes in physics, Volume 919 (Springer, Cham, 2016).
  * Baur  _et al._ (2014) Stefan K. Baur, Monika H. Schleier-Smith, and Nigel R. Cooper, “Dynamic optical superlattices with topological bands,” Phys. Rev. A 89, 051605(R) (2014).
  * Zheng and Zhai (2014) Wei Zheng and Hui Zhai, “Floquet topological states in shaking optical lattices,” Phys. Rev. A 89, 061603(R) (2014).
  * Zhang and Zhou (2014) Shao-Liang Zhang and Qi Zhou, “Shaping topological properties of the band structures in a shaken optical lattice,” Phys. Rev. A 90, 051601(R) (2014).
  * Bracamontes  _et al._ (2022) C. A. Bracamontes, J. Maslek, and J. V. Porto, “Realization of a Floquet-Engineered Moat Band for Ultracold Atoms,” Phys. Rev. Lett. 128, 213401 (2022).
  * Sun  _et al._ (2018) Wei Sun, Bao-Zong Wang, Xiao-Tian Xu, Chang-Rui Yi, Long Zhang, Zhan Wu, Youjin Deng, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan, “Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases,” Phys. Rev. Lett. 121, 150401 (2018).
  * Du  _et al._ (2021) Z. Z. Du, Hai-Zhou Lu, and X. C. Xie, “Nonlinear Hall effects,” Nat Rev Phys 3, 744–752 (2021).
  * Oka and Aoki (2009) Takashi Oka and Hideo Aoki, “Photovoltaic Hall effect in graphene,” Phys. Rev. B 79, 081406 (2009).
  * Kitaev (2009) Alexei Kitaev, “Periodic table for topological insulators and superconductors,” in  _AIP Conference Proceedings_ (AIP, Chernogolokova (Russia), 2009) pp. 22–30.
  * Uehlinger  _et al._ (2013) Thomas Uehlinger, Gregor Jotzu, Michael Messer, Daniel Greif, Walter Hofstetter, Ulf Bissbort, and Tilman Esslinger, “Artificial Graphene with Tunable Interactions,” Physical Review Letters 111, 185307 (2013).
  * Marzari and Vanderbilt (1997) Nicola Marzari and David Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847–12865 (1997).
  * Bissbort (2013) Ulf Bissbort,  _Dynamical effects and disorder in ultracold bosonic matter_ , Ph.D. thesis, Johann Wolfgang Goethe-Universität (2013).

##  Appendix A Optical lattice potential

Lattices of various geometric configurations can be readily realised in the
setup of ref. Uehlinger  _et al._ (2013), that is,

$\displaystyle V_{\text{lat}}(x,y)=$ (12) $\displaystyle-
V_{\overline{x}}\cos(k_{L}x+\theta/2)^{2}-V_{x}\cos(k_{L}x)^{2}-V_{y}\cos(k_{L}y)^{2}$
$\displaystyle-2\sqrt{V_{x}V_{y}}\cos(k_{L}x)\cos(k_{L}y)\cos(\phi),$

where $k_{L}=2\pi/\lambda$. The spacing between neighbouring minima of a one-
dimensional standing wave is $a=\lambda/2$, the recoil energy is
$E_{rec}=(\hbar k_{L})^{2}/(2m)$, the wavelength $\lambda$ is
$1064\text{\,}\mathrm{n}\mathrm{m}$, and the atoms are potassium-40. The
following choice of parameters: $V_{\overline{x}}=0.6$, $V_{x}=0.3$,
$V_{y}=2.8$ all in units of $E_{rec}$, $\theta=\pi$, and $\phi=0$ yields a
triangular lattice. The hexagonal lattice can be realized by setting
$V_{\overline{x}}=12$, $V_{x}=0.8$ and $V_{y}=4.65$. Compared to the tight-
binding lattices, we have the length of the unit cells
$b=\sqrt{2}a=\lambda/\sqrt{2}$. The tight-binding parameters can be evaluated
from the realistic lattice potential (Appendix B), yielding $t_{1}=-0.05$,
$t_{2}=-0.07$, $t_{3}=-0.07$, $\varepsilon=-0.66$ in units of $E_{rec}$
(triangular lattice, $\eta_{sp}=0.19$) and $t_{1}=t_{2}=t_{3}=0.06$ in units
of $E_{rec}$ (hexagonal lattice).

##  Appendix B Derivation of the Hamiltonian

We start from Hamiltonian in co-moving frame,

$\hat{H}_{cm}(\tau)=\frac{\hat{p}^{2}}{2m}+V_{lat}(\hat{r})-\textbf{F}(\tau)\cdot\hat{r}~{},$
(13)

where $\hat{p}$ and $\hat{r}$ are momentum and position operators,
respectively, and
$\textbf{F}=-m\ddot{\textbf{r}}_{m}(\tau)=-\dot{\textbf{A}}(\tau)$. The
Hamiltonian has two parts. The first two terms include the static lattice
Hamiltonian and the last term represents a time-dependent dispersion.

Next, we expand $\hat{H}_{cm}(\tau)$ in Wannier basis Marzari and Vanderbilt
(1997),

$w_{n,\textbf{R}}(\textbf{r})\equiv
w_{n}(\textbf{r}-\textbf{R})=\frac{V}{(2\pi)^{2}}\int_{BZ}e^{-i\textbf{k}\cdot\textbf{R}}\phi_{n,\textbf{k}}(\textbf{r})\,d\textbf{k}~{},$
(14)

where $n$ is band index,
$\phi_{n,\textbf{k}}(\textbf{r})=e^{i\textbf{k}\textbf{r}}u_{n,\textbf{k}}(\textbf{r})$
is Bloch state, $\textbf{R}=n_{1}\textbf{a}_{1}+n_{2}\textbf{a}_{2}$ is
arbitrary lattice vector, and $\textbf{a}_{1}$, $\textbf{a}_{2}$ are primitive
translation vectors in two dimensions.

In second quantization, we can rewrite the field operator in Wannier basis as

$\hat{\psi}(\textbf{r})=\sum_{n,\textbf{R}}w_{n,\textbf{R}}^{*}(\textbf{r})\hat{a}_{n,\textbf{R}}~{},$
(15)

where $\hat{a}_{n,\textbf{R}}$ is the annihilation operator.

The first part of the co-moving Hamiltonian can be expanded as

$\hat{H}_{0}(\tau)=\sum_{n,m,\textbf{R},\textbf{R'}}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{m,\textbf{R'}}\int
w_{n,\textbf{R}}(\textbf{r})\left[\frac{\hat{p}^{2}}{2m}+V_{lat}(\hat{r})\right]w_{m,\textbf{R'}}^{*}(\textbf{r})\,d\textbf{r}~{}.$
(16)

Using the fact that the static Hamiltonian cannot mix different bands and the
Wannier functions are exponentially localized, we only consider the onsite and
nearest-neighbour couplings in individual bands,

$\displaystyle\hat{H}_{0}(\tau)=\sum_{n,\textbf{R}}\left\\{\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}}\varepsilon_{n}+\sum_{\textbf{b}_{i}}\left[\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}+\textbf{b}_{i}}t_{i}+h.c.\right]\right\\}~{},$
(17) $\varepsilon_{n}=\int
w_{n,\textbf{R}}(\textbf{r})\left[\frac{\hat{p}^{2}}{2m}+V_{lat}(\hat{r})\right]w_{n,\textbf{R}}^{*}(\textbf{r})\,d\textbf{r}~{},$
(18) $t_{i}=\int
w_{n,\textbf{R}}(\textbf{r})\left[\frac{\hat{p}^{2}}{2m}+V_{lat}(\hat{r})\right]w_{n,\textbf{R}+\textbf{b}_{i}}^{*}(\textbf{r})\,d\textbf{r}~{}.$
(19)

$\textbf{b}_{i}=j\textbf{a}_{1}+k\textbf{a}_{2}$ are the nearest-neighbour
tunneling vectors. In the configuration of triangular and hexagonal lattices,
$i\in\\{1,2,3\\}$. $\varepsilon_{n}$ is the band center energy and $t_{i}$ is
the nearest-neighbour tunneling amplitude.

Now we consider the driving term which can couple different bands on-site but
has negligible inter-site effects, i.e.

$\displaystyle\hat{H}_{1}(\tau)$
$\displaystyle=-\sum_{n,m,\textbf{R},\textbf{R'}}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{m,\textbf{R'}}\int
w_{n,\textbf{R}}(\textbf{r})\left[\textbf{F}(\tau)\cdot\hat{r}\right]w_{m,\textbf{R'}}^{*}(\textbf{r})\,d\textbf{r}$
(20)
$\displaystyle=-\sum_{n,\textbf{R}}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}}\int
w_{n,\textbf{R}}(\textbf{r})\left[\textbf{F}(\tau)\cdot\hat{r}\right]w_{n,\textbf{R}}^{*}(\textbf{r})\,d\textbf{r}-\sum_{n,m\neq
n,\textbf{R}}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{m,\textbf{R}}\int
w_{n,\textbf{R}}(\textbf{r})\left[\textbf{F}(\tau)\cdot\hat{r}\right]w_{m,\textbf{R}}^{*}(\textbf{r})\,d\textbf{r}$
$\displaystyle=-\sum_{n,\textbf{R}}\left[\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}}\textbf{F}(\tau)\cdot\textbf{R}+\sum_{m\neq
n}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{m,\textbf{R}}a\textbf{F}(\tau)\cdot\eta_{nm}\hat{r^{\prime}}\right]~{}.$

Here we define the inter-band coupling as

$\eta_{nm}=\frac{1}{b}\left\lvert\int
w_{n,\textbf{R}}(\textbf{r})\hat{r}w_{m,\textbf{R}}^{*}(\textbf{r})\,d\textbf{r}\right\rvert.$
(21)

By numerically evaluating $\eta_{sp}$ according to the method in Bissbort
(2013); Uehlinger  _et al._ (2013), we justify that the dipole-like coupling
term $\int
w_{m,\textbf{R}}(\textbf{r})\hat{r}w_{n,\textbf{R}}^{*}(\textbf{r})\,d\textbf{r}$
between s and p band is along the direction
$\hat{r^{\prime}}=\hat{r}=\hat{e}_{x}+\hat{e}_{y}$. In addition, we evaluate
tunnelings and band center energies through calculating the Wannier functions
of atoms in a realistic lattice potential.

The on-site driving term
$\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}}\textbf{F}(\tau)\cdot\textbf{R}$
breaks translation symmetry, so we need to rotate it away by applying the
unitary

$\hat{U}(\tau)=\exp\left(-i\sum_{\textbf{R}n}\chi_{\textbf{R}(\tau)}\hat{a}_{n,\textbf{R}}^{\dagger}\hat{a}_{n,\textbf{R}}\right),$
(22)
$\hat{H}^{\prime}(\tau)=\hat{U}^{\dagger}(\tau)\hat{H}(\tau)\hat{U}(\tau)-i\hbar\hat{U}^{\dagger}(\tau)\frac{\partial}{\partial\tau}\hat{U}(\tau),$
(23)

where
$\chi_{\textbf{R}}(\tau)=\frac{m}{\hbar}\textbf{R}\cdot\dot{\textbf{r}}_{m}(\tau)$.
Then, transforming the Hamiltonian into quasi-momentum space and only
considering the lowest two bands, we get the final Hamiltonian,

$\displaystyle\hat{H}_{\textbf{q}}(\tau)=\sum_{n\in\\{s,p\\}}\biggl{\\{}\varepsilon_{n}\hat{a}_{n}^{\dagger}\hat{a}_{n}+\sum_{i=1}^{Z}\left[t_{n,i}\left(e^{i(\theta_{i}(\tau)+\textbf{q}\cdot\textbf{b}_{i})}\hat{a}_{n}^{\dagger}\hat{a}_{n}+h.c.\right)\right]-\textbf{F}(\tau)\cdot\hat{r}\sum_{m\in\\{s,p\\}\neq
n}\eta_{nm}\hat{a}_{n}^{\dagger}\hat{a}_{m}\biggl{\\}},$ (24)

with the time-periodic Peierls phase
$\theta_{i}(\tau)=\chi_{\textbf{R}+\textbf{b}_{i}}(\tau)-\chi_{\textbf{R}}(\tau)=\frac{m}{\hbar}\textbf{b}_{i}\cdot\dot{\textbf{r}}_{m}(\tau)$.

##  Appendix C Analytical Hamiltonian by the high-frequency expansion

In the following, we write down the effective Floquet Hamiltonian,
analytically derived using the high-frequency expansion (HFE) in tight-binding
approximation. The idea of the HFE is to decompose the original time-dependent
Hamiltonian into Fourier components and obtain the effective Hamiltonians,
then neglect the small high frequency terms. The effective Hamiltonian is
decomposed into terms of different orders,

$\hat{H}_{\mathrm{eff}}=\sum_{n=0}^{\infty}\hat{H}_{\mathrm{eff}}^{(n)},$ (25)

which is in inverse power of $\omega$, i.e.
$\hat{H}_{\mathrm{eff}}^{(n)}\sim\omega^{-n}$. If the frequency is much higher
than other energy scales in the system, the high-order effective Hamiltonian
can be truncated. Here we only use the first two orders Eckardt and Anisimovas
(2015),

$\hat{H}_{\mathrm{eff}}^{(0)}=\frac{1}{T}\int_{0}^{T}\hat{H}(\tau)\,d\tau\equiv\hat{H}_{0}~{},$
(26)
$\hat{H}_{\mathrm{eff}}^{(1)}=\frac{1}{\hbar\omega}\sum_{l=1}^{\infty}\frac{1}{l}\left[\hat{H}_{l},\hat{H}_{-l}\right]=\sum_{l=1}^{\infty}\hat{H}_{\mathrm{eff},l}^{(1)}~{}.$
(27)

Here, $\hat{H}_{l}$ are the Fourier components of the time-dependent
Hamiltonian,
$\hat{H}(\tau)=\sum_{l=-\infty}^{\infty}\hat{H}_{l}e^{il\omega\tau}$.

$\hat{H}_{\mathrm{eff}}^{(0)}$ | $\sigma_{x}\eta_{sp}\hbar\omega[K_{x}+K_{y}\cos(\varphi)]/2$
---|---
$\sigma_{y}\eta_{sp}\hbar\omega K_{y}\sin(\varphi)/2$
$\sigma_{z}\\{\varepsilon+\hbar\omega/2+t_{1}\cos[(q_{x}-q_{y})b]J_{0}(2K_{x})+t_{2}\cos(q_{x}b)[J_{0}(K_{x})J_{0}(K_{y})-2J_{1}(K_{x})J_{1}(K_{y})\cos(\varphi)+$
$2J_{2}(K_{x})J_{2}(K_{y})\cos(2\varphi)]+t_{3}\cos(q_{y}b)[J_{0}(K_{x})J_{0}(K_{y})+2J_{1}(K_{x})J_{1}(K_{y})\cos(\varphi)+2J_{2}(K_{x})J_{2}(K_{y})\cos(2\varphi)]\\}$
$\hat{H}_{\mathrm{eff},2}^{(1)}$ |  $\sigma_{x}\eta_{sp}\\{-t_{1}\cos[(q_{x}-q_{y})b]J_{2}(2K_{x})[K_{x}+K_{y}\cos(\varphi)]-t_{2}\cos(q_{x}b)\\{J_{1}(K_{x})J_{1}(K_{y})[K_{y}+K_{x}\cos(\varphi)]+$ $J_{0}(K_{y})J_{2}(K_{x})[K_{x}+K_{y}\cos(\varphi)]+J_{0}(K_{x})J_{2}(K_{y})[K_{y}\cos(\varphi)+K_{x}\cos(2\varphi)]\\}+t_{3}\cos(q_{y}b)\\{J_{1}(K_{x})J_{1}(K_{y})$ $[K_{y}+K_{x}\cos(\varphi)]-J_{0}(K_{y})J_{2}(K_{x})[K_{x}+K_{y}\cos(\varphi)]-J_{0}(K_{x})J_{2}(K_{y})[K_{y}\cos(\varphi)+K_{x}\cos(2\varphi)]\\}\\}/2$
$\sigma_{y}\eta_{sp}\sin(\varphi)\\{t_{1}\cos[(q_{x}-q_{y})b]J_{2}(2K_{x})K_{y}-t_{2}\cos(q_{x}b)\\{J_{1}(K_{x})J_{1}(K_{y})K_{x}-J_{0}(K_{y})J_{2}(K_{x})K_{y}+J_{0}(K_{x})J_{2}(K_{y})$
$[K_{y}+2K_{x}\cos(\varphi)]\\}+t_{3}\cos(q_{y}b)\\{J_{1}(K_{x})J_{1}(K_{y})K_{x}+J_{0}(K_{y})J_{2}(K_{x})K_{y}-J_{0}(K_{x})J_{2}(K_{y})[K_{y}+2K_{x}\cos(\varphi)]\\}\\}/2$
$\sigma_{z}\eta^{2}\hbar\omega[K_{x}^{2}+K_{y}^{2}+2K_{x}K_{y}\cos(\varphi)]/8$
Table 1: Effective Hamiltonian for the triangular lattice under single
frequency driving resonant with the gap

Table 1 is the effective Hamiltonian for the triangular lattice when the
driving field is resonant with the gap energy. The constant terms are much
larger than the quasimomentum dependent terms in off-diagonal, which means the
on-site inter-band couplings are dominant. We safely neglect the Bessel
functions higher than the second order as the driving strength $K\leq 1$
remains in weak driving regime.

Table 2 shows the case of single-frequency driving resonant with the half the
gap. Now the off-diagonal terms are dominantly quasimomentum dependent. The
Hamiltonian for the triangular lattice in two-frequency driving scheme are not
listed here.

$\hat{H}_{\mathrm{eff}}^{(0)}$ |  $\sigma_{z}\\{\varepsilon+\hbar\omega+t_{1}\cos[(q_{x}-q_{y})b]J_{0}(2K_{x})+t_{2}\cos(q_{x}b)[J_{0}(K_{x})J_{0}(K_{y})-2J_{1}(K_{x})J_{1}(K_{y})\cos(\varphi)+$ $2J_{2}(K_{x})J_{2}(K_{y})\cos(2\varphi)]+t_{3}\cos(q_{y}b)[J_{0}(K_{x})J_{0}(K_{y})+2J_{1}(K_{x})J_{1}(K_{y})\cos(\varphi)+2J_{2}(K_{x})J_{2}(K_{y})\cos(2\varphi)]\\}$
---|---
$\hat{H}_{\mathrm{eff},1}^{(1)}$ |  $\sigma_{x}\eta_{sp}\\{-t_{1}\sin[(q_{x}-q_{y})b]J_{1}(2K_{x})K_{y}\sin(\varphi)-t_{2}\sin(q_{x}b)\sin(\varphi)\\{K_{y}J_{0}(K_{y})J_{1}(K_{x})+K_{x}J_{1}(K_{y})J_{2}(K_{x})-$ $J_{1}(K_{x})J_{2}(K_{y})[K_{y}+2K_{x}\cos(\varphi)+2K_{y}\cos(2\varphi)]+J_{0}(K_{x})J_{1}(K_{y})[K_{x}+2K_{y}\cos(\varphi)]\\}$ $-t_{3}\sin(q_{y}b)\sin(2\varphi)\\{K_{y}J_{0}(K_{x})J_{1}(K_{y})+J_{1}(K_{x})J_{2}(K_{y})[K_{x}+2K_{y}\cos(\varphi)]\\}\\}$
$\sigma_{y}\eta_{sp}\\{t_{1}\sin[(q_{x}-q_{y})b]J_{1}(2K_{x})[K_{x}+K_{y}\cos(\varphi)]+t_{2}\sin(q_{x}b)\\{J_{0}(K_{y})J_{1}(K_{x})[K_{x}+K_{y}\cos(\varphi)]-$
$J_{1}(K_{y})J_{2}(K_{x})[K_{y}+K_{x}\cos(\varphi)]+J_{1}(K_{y})J_{0}(K_{x})[K_{x}\cos(\varphi)+K_{y}\cos(2\varphi)]-J_{1}(K_{x})J_{2}(K_{y})$
$[K_{x}\cos(2\varphi)+K_{y}\cos(3\varphi)]\\}+t_{3}\sin(q_{y}b)\sin(2\varphi)\\{-J_{0}(K_{y})J_{1}(K_{x})[K_{x}+K_{y}\cos(\varphi)]-J_{1}(K_{y})J_{2}(K_{x})$
$[K_{y}+K_{x}\cos(\varphi)]+J_{1}(K_{y})J_{0}(K_{x})[K_{x}\cos(\varphi)+K_{y}\cos(2\varphi)]+J_{1}(K_{x})J_{2}(K_{y})[K_{x}\cos(2\varphi)+K_{y}\cos(3\varphi)]\\}\\}$
$\sigma_{z}\eta_{sp}^{2}\hbar\omega[K_{x}^{2}+K_{y}^{2}+2K_{x}K_{y}\cos(\varphi)]/4$
$\hat{H}_{\mathrm{eff},3}^{(1)}$ |  $\sigma_{x}\eta_{sp}\\{-t_{2}\sin(q_{x}b)\\{K_{x}J_{1}(K_{y})J_{2}(K_{x})+J_{1}(K_{x})J_{2}(K_{y})[K_{y}+2K_{x}\cos(\varphi)]\\}$ $+t_{3}\sin(q_{y}b)\\{-K_{x}J_{1}(K_{y})J_{2}(K_{x})+J_{1}(K_{x})J_{2}(K_{y})[K_{y}+2K_{x}\cos(\varphi)]\\}\\}/3$
$\sigma_{y}\eta_{sp}\\{t_{2}\sin(q_{x}b)\\{J_{1}(K_{y})J_{2}(K_{x})[K_{y}+K_{x}\cos(\varphi)]+J_{1}(K_{x})J_{2}(K_{y})[K_{y}\cos(\varphi)+K_{x}\cos(2\varphi)]\\}$
$+t_{3}\sin(q_{y}b)\\{J_{1}(K_{y})J_{2}(K_{x})[K_{y}+K_{x}\cos(\varphi)]-J_{1}(K_{x})J_{2}(K_{y})[K_{y}\cos(\varphi)+K_{x}\cos(2\varphi)]\\}\\}/3$
$\sigma_{z}\eta_{sp}^{2}\hbar\omega[K_{x}^{2}+K_{y}^{2}+2K_{x}K_{y}\cos(\varphi)]/12$
Table 2: Effective Hamiltonian for the triangular lattice under single frequency driving resonant with the half the gap $\hat{H}_{\mathrm{eff}}^{(0)}$ | $\sigma_{x}\\{[t_{1}\cos(q_{y}b)+t_{3}\cos(q_{x}b)]J_{0}(K_{y1})+t_{2}\cos[(q_{x}-q_{y})b]J_{0}(K_{x1})\\}$
---|---
$\sigma_{y}\\{[t_{1}\sin(q_{y}b)-t_{3}\sin(q_{x}b)]J_{0}(K_{y1})-t_{2}\sin[(q_{x}-q_{y})b]J_{0}(K_{x1})\\}$
$\hat{H}_{\mathrm{eff}}^{(1)}$ |  $-\sigma_{z}\sum_{n=1}^{\infty}\frac{4}{n\hbar\omega}t_{2}J_{n}(K_{x1})J_{n}(K_{y1})\left[(-1)^{n+1}t_{1}\sin(q_{x}b)+t_{3}\sin(q_{y}b)\right]\sin(n\varphi_{y1})$
Table 3: Effective Hamiltonian for the hexagonal lattice under single
frequency off-resonant driving

For the hexagonal lattice, Table 3 gives the general Hamiltonian under single-
frequency off-resonant driving, while Table 4 is the Hamiltonian under two-
frequency off-resonant driving. Note that the second order Bessel functions
are omitted in $\hat{H}_{\mathrm{eff}}^{(1)}$ in Table 4 but kept in the
numerical calculations.

$\hat{H}_{\mathrm{eff}}^{(0)}$ |  $\sigma_{x}\\{[t_{1}\cos(q_{y}b)+t_{3}\cos(q_{x}b)]J_{0}(K_{y1})J_{0}(K_{y2})+t_{2}\cos[(q_{x}-q_{y})b]J_{0}(K_{x1})J_{0}(K_{x2})-$ $2[t_{1}\sin(q_{x}b)+t_{3}\sin(q_{y}b)]J_{1}(K_{y2})J_{2}(K_{y1})\sin(2\varphi_{y1}-\varphi_{y2})+2t_{2}\sin[(q_{x}-q_{y})b]J_{1}(K_{x2})J_{2}(K_{x1})\sin(\varphi_{x2})\\}$
---|---
$\sigma_{y}\\{[t_{1}\sin(q_{y}b)-t_{3}\sin(q_{x}b)]J_{0}(K_{y1})J_{0}(K_{y2})-t_{2}\sin[(q_{x}-q_{y})b]J_{0}(K_{x1})J_{0}(K_{x2})+$
$2[t_{1}\cos(q_{x}b)-t_{3}\cos(q_{y}b)]J_{1}(K_{y2})J_{2}(K_{y1})\sin(2\varphi_{y1}-\varphi_{y2})+2t_{2}\cos[(q_{x}-q_{y})b]J_{1}(K_{x2})J_{2}(K_{x1})\sin(\varphi_{x2})\\}$
$\hat{H}_{\mathrm{eff}}^{(1)}$ |  $\sigma_{z}\frac{2}{3\hbar\omega}\\{6J_{0}(K_{y2})J_{1}(K_{y1})[t_{2}J_{1}(K_{x1})J_{1}(K_{x2})(t_{1}\cos(q_{x}b)-t_{3}\cos(q_{y}b))\cos(\varphi_{x2}-\varphi_{y1})+(t_{1}^{2}-t_{3}^{2})$ $J_{1}(K_{y1})J_{1}(K_{y2})\cos(2\varphi_{y1}-\varphi_{y2})]-6t_{2}J_{0}(K_{x2})J_{1}(K_{x1})[t_{2}J_{1}(K_{x1})J_{1}(K_{x2})\cos(\varphi_{x2})+J_{1}(K_{y1})J_{1}(K_{y2})$ $(t_{1}\cos(q_{x}b)+t3\cos(q_{y}b))\cos(\varphi_{y1}-\varphi_{y2})-J_{1}(K_{y1})J_{0}(K_{y2})(t_{1}\sin(q_{x}b)+t3\sin(q_{y}b))\sin(\varphi_{y1})]$ $-t_{2}J_{1}(K_{x2})J_{1}(K_{y2})[3J_{0}(K_{x1})J_{0}(K_{y1})(t_{1}\sin(q_{x}b)+t_{3}\sin(q_{y}b))\sin(\varphi_{x2}-\varphi_{y2})-2J_{1}(K_{x1})J_{1}(K_{y1})$ $(t_{1}\sin(q_{x}b)-t_{3}\sin(q_{y}b))(\sin(\varphi_{x2}-\varphi_{y1}-\varphi_{y2})+3\sin(\varphi_{x2}+\varphi_{y1}-\varphi_{y2}))]\\}$
Table 4: Effective Hamiltonian for the hexagonal lattice under two frequency
off-resonant driving