# Controllable Floquet edge modes in a multi-frequency driving system

HaRu K. Park<EMAIL_ADDRESS>Department of Physics, Korea Advanced
Institute of Science and Technology, Daejeon, 34141, Korea  Junmo Jeon
<EMAIL_ADDRESS>Department of Physics, Korea Advanced Institute of
Science and Technology, Daejeon, 34141, Korea  Gil Young Cho  Department of
Physics, Pohang University of Science and Technology (POSTECH), Pohang 37673,
Republic of Korea  Center for Artificial Low Dimensional Electronic Systems,
Institute for Basic Science (IBS), Pohang 37673, Republic of Korea  Asia
Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
SungBin Lee<EMAIL_ADDRESS>Department of Physics, Korea Advanced
Institute of Science and Technology, Daejeon, 34141, Korea

###### Abstract

A driven quantum system has been recently studied in the context of
nonequilibrium phase transitions and their responses. In particular, for a
periodically driven system, its dynamics are described in terms of the multi-
dimensional Floquet lattice with a lattice size depending on number of driving
frequencies and their rational or irrational ratio. So far, for a multi-
frequency driving system, the energy pumping between the sources of
frequencies has been widely discussed as a signature of topologically
nontrivial Floquet bands. However, the unique edge modes emerging in the
Floquet lattice has not been explored yet. Here, we discuss how the edge modes
in the Floquet lattice are controlled and result in the localization at
particular frequencies, when multiple frequencies are present and their
magnitudes are commensurate values. First, we construct the minimal model to
exemplify our argument, focusing on a two-level system with two driving
frequencies. For strong frequency limit, one can describe the system as a
quasi-one dimensional Floquet lattice where the effective hopping between the
neighboring sites depends on the relative magnitudes of potential for two
frequency modes. With multiple driving modes, there always exist the non-
trivial Floquet lattice boundaries via controlling the frequencies and this
gives rise to the states that are mostly localized at such Floquet lattice
boundaries, i.e. particular frequencies. We suggest the time-dependent Creutz
ladder model as a realization of our theoretical Hamiltonian and show the
emergence of controllable Floquet edge modes.

##  I Introduction

Topology becomes an essential concept in modern condensed matter physics Kane
and Mele (2005a, b); Bernevig  _et al._ (2006); Hasan and Kane (2010); Qi and
Zhang (2011); Xiao  _et al._ (2020); Helbig  _et al._ (2020). One such example
is the so-called Thouless pumping model, which is an adiabatic, time-dependent
model. The model is designed to pump an integer number of electric charge,
which is related to the topological winding numberThouless (1983); Switkes
_et al._ (1999); Kraus  _et al._ (2012); Lohse  _et al._ (2016); Nakajima  _et
al._ (2016). More recently, it has been noted that non-trivial topological
properties can also emerge in the time-periodic driving systemsKitagawa  _et
al._ (2010); Rudner  _et al._ (2013); Martin  _et al._ (2017); Rudner and
Lindner (2020). Such time-periodic models are called as the Floquet model,
which has been extensively investigated in the context of transports, laser
controlled atoms and electron-phonon coupled systemsShirley (1965); Sambe
(1973); Moskalets and Büttiker (2002); Grifoni and Hänggi (1998); Chu and
Telnov (2004); Murakami  _et al._ (2017); Karzig  _et al._ (2015); Dehghani
_et al._ (2014); Goldman and Dalibard (2014); Eckardt (2017). It has been also
noted that a $d$-spatial-dimensional time-dependent system under the
$D$-frequency drives can be classified by the topology of the static
Hamiltonian in $(d+D)$ spatial dimensions and the relevant examples are
studied.Long  _et al._ (2021a); Qi  _et al._ (2021); Nathan  _et al._ (2021)

In this paper, we will uncover another novel phenomena of the driven quantum
systems. Specifically, we will show that the multi-frequency driving system
can induce the localized “edge” mode in the frequency lattice, equivalently
the Floquet lattice (which will be defined below). These novel topological
modes are localized at a special frequency. The physical origin of such novel
mode can be understood as follows. When the ratio between the frequencies are
commensurate to each other, one can consider the multi-dimesional Floquet
lattice, which repeats along a certain direction as shown in Fig.1. This
naturally introduces an “edge” to the Floquet lattice, and the edge may trap
an interesting mode, which depends on the topology of the driven system. As a
proof of this claim, we will introduce an explicit, two-level model with two
commensurate driving frequency, which can explicitly demonstrate the desired
physics.

The rest of this paper is organized as follows. First, we briefly review the
Floquet theory in a periodically driven system. Then we introduce our model, a
two-level system with two driving modes, which is transformed into two-
dimensional Floquet lattice. In the strong frequency limit, the Floquet
lattice can be mapped onto a quasi-one dimensional lattice having a non-
trivial edge mode. We also discuss the possible realization of our model in
the quasi-one dimensional Creutz lattice. Finally, we also support our
theoretical argument with various numerical results.

##  II Floquet Lattice & Floquet Theory

Here, we briefly review the basic physics of the Floquet theory,Floquet (1883)
describing a system under the multiple time-periodic driving. Our starting
point is the Hamiltonian $H$ which depends on the $D$-time periodic parameters
$(\theta_{1},\cdots,\theta_{D})=\vec{\theta}_{D}$, each depends on one time
parameter $t$. That is, with some constants
$(\Omega_{1},\cdots,\Omega_{D})=\vec{\Omega}$,

$H(\theta_{i})=H(\theta_{i}+2\pi),\quad\theta_{i}(t)=\Omega_{i}t.$ (1)

We attempt to solve the time-dependent Schrödinger equation,

$\displaystyle
i\partial_{t}|\psi(t)\rangle=H(\vec{\theta}(t))|\psi(t)\rangle.$ (2)

Performing the Fourier transformation, we obtain,

$\omega|\psi(\omega)\rangle=\sum_{\vec{m}}H_{\vec{m}}|\psi(\omega-\vec{m}\cdot\vec{\Omega})\rangle.$
(3)

Here, $|\psi(\omega)\rangle$ and $H_{\vec{m}}$ are the Fourier coefficients of
the wavefunction and the Hamiltonian respectively.

$|\psi(t)\rangle=\int d\omega e^{-i\omega t}|\psi(\omega)\rangle,\quad
H(\vec{\theta})=\sum_{\vec{m}}e^{-i\vec{m}\cdot\vec{\theta}}H_{\vec{m}}.$ (4)

Note that Eq.(3) only couples the frequency $\epsilon$ with other frequencies
$\omega=\epsilon+\vec{n}\cdot\vec{\Omega}$. Therefore, by indexing
$|\vec{n}\rangle\equiv|\psi(\epsilon+\vec{n}\cdot\vec{\Omega})\rangle,$ Eq.(3)
becomes,

$\epsilon|\vec{n}\rangle=\sum_{\vec{m}}(H_{\vec{n}-\vec{m}}-\vec{n}\cdot\vec{\Omega}\delta_{\vec{n},\vec{m}})|\vec{m}\rangle.$
(5)

Eq.(5) is mathematically equivalent with the $D$-dimensional tight-binding
model defined on a lattice at sites $\vec{n}$. We call this lattice as a
“Floquet lattice”. In this analogy, $|\vec{n}\rangle$ represents the state,
when a particle is exactly at the site $\vec{n}$ on the lattice. We will also
call $\epsilon$ as the quasi-energy. Physically, one can interpret the numbers
$n_{i}$ (the $i$-th component of the vector $\vec{n}$) as the number of
absorbed photons from the $i$-th drive. Therefore, the hopping terms,
$H_{\vec{n}}$ in Eq.(5), describe the process of absorbing and/or emitting
certain number of photons. Unlike an ordinary tight-binding model, Eq.(5) has
an additional on-site potential $-\vec{n}\cdot\vec{\Omega}$. This on-site
potential term corresponds to the electric field $\vec{\Omega}$ on the tight-
binding model, hence we call $\vec{\Omega}$ as the quasi-electric field.

##  III The Model

Let us consider a two-frequency-driven Hamiltonian on a two-level system in
the rest of this paper. The Hamiltonian is represented as,

$\displaystyle H(t)=B_{z}(k)\sigma_{z}$
$\displaystyle+\left[(\Delta-\delta(k))\cos\left(p\Omega t\right)\right.$
$\displaystyle\left.+(\Delta+\delta(k))\cos\left(q\Omega
t\right)\right]\sigma_{x}.$ (6)

Here, $k$ is another parameter (periodic in $2\pi$) which can be independently
tuned. This variable will be used to change the parameters $B_{z}$ and
$\delta$, which will be shown later to play the role of staggered potential
and alternating hopping in the Floquet lattice. $\sigma_{x}$ and $\sigma_{z}$
are the Pauli matrices, whose basis can be considered as a pseudo-spin states
$\\{|\uparrow\rangle,|\downarrow\rangle\\}$. $t$ represents the time and
$\Omega$ is the frequency of the system with coprime integers $p$ and $q$.

Eq.(III) describes a time-periodic system with the period $T=2\pi/\Omega$.
Hence, it can be Fourier transformed into the discrete frequency domain. After
performing the Fourier transformation on time, we can write out the following
tight-binding model on the Floquet lattice, which is equivalent to the
Schrödinger equation on the following Hamiltonian $H_{F}$.

$\displaystyle H_{F}=\sum_{\vec{n},\alpha,\beta}$
$\displaystyle\left[B_{z}(k)\sigma_{z}^{\alpha\beta}-\vec{n}\cdot\vec{\Omega}\delta^{\alpha\beta}\right]c_{\vec{n};\alpha}^{\dagger}c_{\vec{n};\beta}$
$\displaystyle+(\Delta-\delta(k))\sigma_{x}^{\alpha\beta}c_{\vec{n}+(1,0);\alpha}^{\dagger}c_{\vec{n};\beta}+h.c.$
$\displaystyle+(\Delta+\delta(k))\sigma_{x}^{\alpha\beta}c_{\vec{n}+(0,1);\alpha}^{\dagger}c_{\vec{n};\beta}+h.c.$
(7)

Here, $c_{\vec{n};\alpha}$ represents the annihilation operator for a pseudo-
spin $\alpha$ state with frequency $\epsilon+\vec{n}\cdot\vec{\Omega}$, where
$\epsilon$ is the quasi-energy. Here $\vec{\Omega}=\Omega(p,q)$.

Our primary goal here is to show that the Hamiltonian Eq.(III) can support the
topological “edge” modes, which are localized in the corresponding Floquet
lattice. We will demonstrate this in a few different ways.

(a)

(b)

Figure 1: LABEL:sub@fig:lattice-structure Floquet lattice from Eq.(III), with
$p=4$ and $q=5$. Each dots represents $|n,m\rangle$ state of the Floquet
system, which represents the frequency $(n,m)\cdot(p\Omega,q\Omega)$. Two
thick lines are the same line due to the translational symmetry
$(n,m)\mapsto(n-q,m+p)$. Dashed diagonal line is an equipotential line, and
the red and blue dots in the green box are the points near equipotential line,
representing up and down spins respectively. Due to the $\sigma_{z}$-term, the
on-site energy on red and blue dots has different sign, $\pm B_{z}(k)$. Red
horizontal lines and blue vertical lines represents the hopping terms, whose
strengths are $\Delta+\delta(k)$ and $\Delta-\delta(k)$, respectively.
LABEL:sub@fig:lattice-cylinder-structure By pasting two thick lines in
LABEL:sub@fig:lattice-structure, we get the cylindrical structure.

###  III.1 Instructive Limit

The non-trivial topology of Eq.(III) can be most cleanly manifested in the
strong driving limit, i.e., $\Delta\gg B_{z}(k),\delta(k)$, so the dominant
term is the hopping term with $\sigma_{x}$. This implies that the distribution
of the pseudo-spin is alternating as in Fig.1a. Furthermore, the strong quasi-
electric field $\vec{\Omega}$ localizes the ground state around an
equipotential line by the Stark localizationWannier (1962); Long  _et al._
(2021b). Thus, in this limit, one can choose the sites near the equipotential
line and interactions between those sites to construct an effective quasi-one
dimensional lattice. Although there are many equipotential lines in the
system, in strong frequency limit, the contribution near the zero frequency
mode is dominant. (See Appendix B for details.)

To exemplify, we first focus on our interest to $q=p+1$. In this case, the
sites (in the Floquet lattice) near the equipotential line interacts with the
sites connected by the alternating horizontal and vertical bonds, as shown in
Fig.1a. Since we can tune the strength of the vertical and horizontal hoppings
independently, the quasi-one dimensional lattice becomes an 1D lattice model
with a unit-cell which consists of the two sites. In this lattice, the pseudo-
spin directions of the two sites are opposite because of $\Delta\gg
B_{z}(k),\delta(k)$.

Having these in mind, the total Hamiltonian in Eq.(III) can be approximated
into the following form.

$\displaystyle H_{1d}(k)=$
$\displaystyle\sum_{n}\left[B_{z}(k)(c_{2n}^{\dagger}c_{2n}-c_{2n+1}^{\dagger}c_{2n+1})\right.$
$\displaystyle+(\Delta-\delta(k))c_{2n}^{\dagger}c_{2n+1}+h.c.$
$\displaystyle\left.+(\Delta+\delta(k))c_{2n+1}^{\dagger}c_{2n+2}+h.c.\right]$
(8)

Here, $n$ increases along the sites on the Floquet lattice near the
equipotential line, following this effective 1D lattice structure in Fig.1a.
Here, the parameter $B_{z}$ plays the role of staggered potential and $\delta$
tunes the alternating hopping strength.

The topological property of this model in the Floquet lattice can be easily
understood by comparing it with the Rice-Mele modelRice and Mele (1982), which
is a time-dependent one-dimensional model with the staggered potential and
alternating hopping strength. In this point of view, the Hamiltonian is
topologically non-trivial, when the vector $(B_{z},\delta)(k)$ winds around
the origin of the parameter space. Thus, for example, if we set

$B_{z}(k)=\cos k,\quad\delta(k)=\sin k,$ (9)

then the topology of the Hamiltonian becomes nontrivial. More detailed
explanation about the Rice-Mele model is described in Appendix A.

Figure 2: The periodic structure of a quasi-one dimensional Floquet lattice,
with $p=4$ and $q=5$. The large black dots represent the points near
equipotential line, and horizontal and vertical red and blue lines represent
the interaction. Due to the periodicity, the quasi-one dimensional lattice on
large black dots goes periodically. Then naturally, the black dots emphasized
by red circles become a boundary between two different SSH Hamiltonians.

###  III.2 Boundary Localization

We now present the intuitive understanding of the edge mode in the Floquet
lattice. Our Floquet Hamiltonian in Eq.(III) is a periodic Hamiltonian under
translation $\vec{n}\rightarrow\vec{n}+(-q,p)$ and has a cylinderical
structure, as we can see in Fig.1b. This periodicity follows from the fact
that $\vec{n}\cdot\vec{\Omega}=0$ when $\vec{n}=(-q,p)$, that is, the system
is periodic under the direction perpendicular to the quasi-electric field
$\vec{\Omega}$. (Note that $\vec{\Omega}=\Omega(p,q)$.) Therefore, restricting
the whole system into the perpendicular direction of $\vec{\Omega}$ must also
give a periodic lattice system. However, because the alternating hopping on
the quasi-one dimensional lattice starts and ends with the same type of
hopping, there exists a boundary between two topologically distinct lattices.

For the limit we consider (See Fig.1a), the 1D lattice has an alternating
distribution of hopping term, with two different hoppings $\Delta\pm\delta$.
Fig.2 shows the periodic structure of the quasi-1D lattice. Here, the 1D
lattice ends and restarts both by the hopping strength $\Delta-\delta$. This
introduces a “solitonic configurion” in the hopping terms, and generates a
topological boundary on the periodic lattice. This topological boundary exists
due to the one-directional periodicity of the lattice, and thus only happens
when the ratio between frequencies are commensurate.

To concretely demonstrate the edge mode, we consider the easiest case
$B_{z}(k)=\cos k$ and $\delta(k)=\sin k$, and compare $k=\pi/2$ and $k=3\pi/2$
case. In both cases, we have $B_{z}(k)=0$. Hence the 1D lattice becomes a Su-
Schreiff-Heeger(SSH) lattice with hopping terms $\Delta\pm 1$, and the
boundary of the system can be considered as the joint between the trivial and
topological phases of the SSH chain. This creates the localized eigenstate
near the boundary. Because each site of the Floquet lattice represents the
frequency of the state, this corresponds to the state with a high occupation
on the frequency which represents the boundary. Since the small perturbation
does not affects much to the localization of the state, the localization
appears on every $k\in[0,2\pi]$ for $\delta(k)\neq 0$.

Until now we have only discussed $q=p+1$ case. In this case, the position of
the localized edge mode is not controllable, since there is only one boundary
point and the state is simply localized near it. To show the controllability
of the localized mode, we introduce $q=p+2$ case, which can be considered as
the case with multiple boundaries.

Figure 3: The periodic structure of a quasi-one dimensional Floquet lattice,
with $p=3$ and $q=5$. The large black dots represent the points near
equipotential line, and horizontal and vertical red and blue lines represent
the interaction. The red dashed circles represent the position of boundary on
quasi-one dimensional lattice.

Fig.3 shows an example of quasi-one dimensional lattice structure for $q=p+2$,
particularly when $p=3$ and $q=5$, i.e. $q=p+2$. In the vicinity of the
equipotential line, there are in total three different boundary points, marked
as dotted circles in Fig.3. Specifically, when the strength of the blue bond
is stronger than the red one, we expect the localization at $(0,0)$ point,
which gives the same result with $q=p+1$ case. On the other hand, when the
strength of the red bond is stronger then the blue one, the localization in
the middle of the quasi-one dimensional lattice would occur. Since the
strengths of red and blue bonds depend on the parameter $k$, we can control
the localization mode of the system by tuning $k$ value.

###  III.3 Physical Realization via Creutz Lattice

Before showing some numerical results which justify our heuristic
understandings above, we present a potential realization of our model in a
lattice model. So far, we have not assigned any particular physical meaning to
$k$, and hence, in principle, there could be many different ways to realize
the Hamiltonian in Eq.(III). Here, we will consider a time-dependent 1D
lattice, which is a variation of the so-called Creutz ladderCreutz (1999), as
a potential candidate to realize our model.

To realize the Hamiltonian in Eq.(III), we consider $k$ as the momentum
parameter. Furthermore, by considering two sublattices $A,B$ in real space, we
can choose the basis $\\{|\uparrow\rangle,|\downarrow\rangle\\}$ of
$\sigma_{x},\sigma_{z}$. We will assign the creation operator of the momentum
$k$ at $A,B$ sites in a way that
$|\uparrow\rangle_{k}=c_{k,A}^{\dagger}|0\rangle$ and
$|\downarrow\rangle_{k}=c_{k,B}^{\dagger}|0\rangle$. This allows us to write,

$\sigma_{x}\rightarrow
c_{k,A}^{\dagger}c_{k,B}+c_{k,B}^{\dagger}c_{k,A},~{}~{}~{}~{}\sigma_{z}\rightarrow
c_{k,A}^{\dagger}c_{k,A}-c_{k,B}^{\dagger}c_{k,B}.$ (10)

We can now write down the Hamiltonian with $B_{z}(k)=B_{z}\cos k$ and
$\delta(k)=\delta\sin k$ as following.

$\displaystyle H(k,t)=B_{z}\cos
k\left(c_{k,A}^{\dagger}c_{k,A}-c_{k,B}^{\dagger}c_{k,B}\right)$
$\displaystyle+\Delta(\cos(p\Omega t)+\cos(q\Omega
t))\left(c_{k,A}^{\dagger}c_{k,B}+c_{k,B}^{\dagger}c_{k,A}\right)$
$\displaystyle+\delta\sin k(-\cos(p\Omega t)+\cos(q\Omega
t))\left(c_{k,A}^{\dagger}c_{k,B}+c_{k,B}^{\dagger}c_{k,A}\right).$ (11)

Using the inverse Fourier transformation,
$c_{A/B,k}=\sum_{x}c_{A/B,x}e^{-ikx}$, one gets,

$H(t)\equiv\sum_{k}H(k,t)=H_{h}+H_{v}+H_{d}.$ (12)

Here, the horizontal interaction between dimers of the ladder, $H_{h}$, the
intra-interaction of the dimers, $H_{v}$, and the diagonal interaction between
dimers of the ladder, $H_{d}$, are represented as respectively,

$\displaystyle H_{h}$
$\displaystyle=\frac{B_{z}}{2}\sum_{x}\left[c_{x+1,A}^{\dagger}c_{x,A}+c_{x+1,B}^{\dagger}c_{x,B}+h.c.\right],$
$\displaystyle H_{v}$ $\displaystyle=\frac{\Delta}{2}(\cos(p\Omega
t)+\cos(q\Omega t))\sum_{x}\left[c_{x,A}^{\dagger}c_{x,B}+h.c.\right],$
$\displaystyle H_{d}$ $\displaystyle=\frac{1}{4}(\\!-\\!\cos(p\Omega
t)\\!+\\!\cos(q\Omega t))\sum_{x}\Big{[}ic_{x+1,A}^{\dagger}c_{x,B}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+ic_{x+1,B}^{\dagger}c_{x,A}+h.c.\Big{]}.$
(13)

This type of ladder is so called the Creutz ladder. After constructing time-
dependent Creutz ladder with coefficients given above, we may scan the system
on the momentum space $k$. This process experimentally measures the
localization of the state, which is the clue of the edge states in our model.

##  IV Numerical Demonstrations

In this section, we will present the numerical proof of our claim above. In
this numerical simulation, we solve the full Floquet problem and do not
restrict ourselves to the effective 1D Floquet Hamiltonian. We will find that
the above understanding based on the effective 1D lattice model in Eq.(III.1)
is indeed correct.

We first confirm the localization of the modes under the strong-enough quasi-
electric field $\vec{\Omega}$, or equivalently, on the strong frequency limit.
Numerically, we show the localization near equipotential line when the energy
level of the frequency $\Omega$ is comparable with the hopping term $\Delta$.
This is reasonable because if the quasi-electric field is weaker than the
hopping term, then the hopping term leads the state spread over all the
Floquet lattice. Detailed numerical studies are explained in Appendix B.

(a)

(b)

Figure 4: Distribution of the state on the equipotential line of the Floquet
lattice at LABEL:sub@fig:localization-k0 $k=0$ and LABEL:sub@fig:localization-
kpi $k=\pi/2$. Here, the calculation is performed with
$\Delta=10,\Omega=10,\delta(k)=\sin k,B_{z}(k)=\cos k$, and $p=33,q=34.$

Now let us explore the physics of the topological edge mode. We plot the
distribution of the states on the quasi-one dimensional Floquet lattice, at
$k=0$ (Fig.4a) and $k=\pi/2$ (Fig.4b) respectively. For $k=0$, i.e.,
$\delta=0$, it exhibits the extended state in addition to the localized mode
at zero frequency. Whereas, for $k=\pi/2$ i.e., $\delta\neq 0$, the localized
state at zero frequency only survives, which can be easily understood from the
SSH model as discussed above. Fig.4 shows the distribution of the states
projected on the alternating pseudo-spin state, for example,
$|\uparrow\downarrow\uparrow\downarrow\cdots\rangle$. Suppose that we
projected the state on another alternating pseudo-spin configuration,
$|\downarrow\uparrow\downarrow\uparrow\cdots\rangle$. Because this process
changes up state and down state, we may consider the total Hamiltonian
transforms as $\sigma_{z}\rightarrow-\sigma_{z}$. By absorbing the sign change
of $\sigma_{z}$ into $B_{z}(k)=B_{z}\cos(k)$ parameter, we may consider this
transformation as following.

$\displaystyle B_{z}\cos(k)$ $\displaystyle\rightarrow-
B_{z}\cos(k)=B_{z}\cos(\pi-k),$ $\displaystyle\delta\sin(k)$
$\displaystyle\rightarrow\delta\sin(k)=\delta\sin(\pi-k)$ (14)

Therefore, flipping the sign is equivalent to transforming $k$ into $\pi-k$.
This shows that $k=0$ case in Fig.4a and $k=\pi$ case changes in the different
alternating pseudo-spin projection case. Notice that this also implies the
opposite sign of winding numbers of $(B_{z},\delta)$ and $(-B_{z},\delta)$,
showing the boundary is created by the joint between two lattices with
different topological properties.

To see the controllable edge mode, we discuss the case with $q=p+2$ with $p,q$
both odd, which also have a quasi-one dimensional lattice near the
equipotential line with different structure.

(a)

(b)

(c)

(d)

Figure 5: Distribution of the state on the equipotential line of the Floquet
lattice at LABEL:sub@fig:localization-k0-oddodd $k=\pi/2$,
LABEL:sub@fig:localization-kpi-oddodd $k=3\pi/2$,
LABEL:sub@fig:localization-k0eps-oddodd $k=\pi/2+\eta$, and
LABEL:sub@fig:localization-kpieps-oddodd $k=3\pi/2+\eta$. Here,
$\Delta=10,\Omega=10,\delta(k)=\sin(k),B_{z}(k)=\cos(k),\eta=0.001,$ and
$p=33,q=35$.

Figs.5a and 5b represent the exact probability distributions on the quasi-one
dimensional lattice, for $k=\pi/2$ and $k=3\pi/2$ respectively. Theoretically,
we have expected the localized mode at the edge of quasi-one dimensional
lattice when $k=\pi/2$, and at the center when $k=3\pi/2$. However, numerical
result does not show the localization at the center for both $k=\pi/2$ and
$3\pi/2$. Instead, if we measure on $k=3\pi/2+\eta$ with small $\eta$ value,
then we get the localization behavior at the center as desired as shown in
Fig.5d. The absence of the localized mode at the center for $k=3\pi/2$ is
originated from the on-site energy induced by the quasi-electric field.
Ignoring the on-site energy induced by the quasi-electric field, the states on
quasi-one dimensional lattice are three-fold degenerate excluding the spin
degrees of freedom: two of them are localized at the center of the quasi-one
dimensional lattice, and one of them are localized at the edge. Because the
edge of the quasi-one dimensional lattice is closer to the equipotential line,
the state localized at the edge is more stable compared to the other ones when
$k=3\pi/2$. Thus, we observe the localization dominant at the edge, as shown
in Fig.5b.

However, as the value $k$ deviates from $3\pi/2$, the degeneracy ignoring
quasi-electric field breaks down, and for non-zero quasi energy, one can
observe the states localized at the center. Thus, when $k$ sufficiently
deviates from $3\pi/2$, the localized state at the center is shown for non-
zero quasi energy despite the presence of the quasi-electric field (See
Fig.5d.). Notice that because only a single boundary exist at the edge for
$k=\pi/2$ case, the localization at the center does not appear although we
slightly change the $k$-value from $\pi/2$, as we can see in 5c.

In summary, we have shown that the localization modes at particular
frequencies can be controlled via multi-frequency ratio and their magnitudes,
and this can be explained by the transformation of our model into the Floquet
SSH model on a quasi-one dimensional lattice.

##  V Conclusions

In this research, we designed a one dimensional Creutz ladder model with two-
driving modes. Under strong frequency, the Floquet version of this model can
be reduced into quasi-one dimensional model with nontrivial topological
properties. Due to the construction, this quasi-one dimensional model is
mathematically equivalent to the time-driven SSH model with boundary.
Therefore the localization on the Floquet lattice becomes the evidence of the
topological property, generated by the junction of two SSH models with
different topological properties, which is experimentally measurable.

In multi-frequency system with their commensurate frequency ratio, our work
suggested a new method to build the boundary on Floquet lattice. This
localization can be measured experimentally, when the frequency scale is
comparable to the amplitude of driving modes, by investigating the frequency
profile of the eigenstate. Because we have shown the topological property
which occurs due to the interplay between spatial dimension and frequency
modes, the extension of our system can be done not only by adding driving
modes but also increasing the spatial dimension. This makes the possibility of
the topological system on spatial multi-dimension with multi-frequency drives,
suggesting new kinds of topological Floquet systems.

###### Acknowledgements.

Acknowledgments.— This work is supported by the National Research Foundation
Grants (NRF- 2020R1A4A3079707, NRF grand 2021R1A2C1093060). GYC acknowledges
the support of the National Research Foundation of Korea (NRF) funded by the
Korean Government No. 2020R1C1C1006048 and the support by IBS-R014-D1. This
work is also supported by the Air Force Office of Scientific Research under
Award No. FA2386-20-1-4029. GYC acknowledges financial support from Samsung
Science and Technology Foundation under Project Number SSTF-BA2002-05.

##  Appendix A Rice-Mele model

The Rice-Mele model is a time-periodic model with adiabatic charge pumping,
with non-trivial topological property. As a one-dimensional lattice with a
unit cell containing two sublattices, its hamiltonian can be written as
following.

$\displaystyle H(t)=$ $\displaystyle\sum_{n}\left[\delta
h(t)(c_{n,1}^{\dagger}c_{n,1}-c_{n,2}^{\dagger}c_{n,2})\right.$
$\displaystyle+\frac{v+\delta v(t)}{2}c_{n,1}^{\dagger}c_{n,2}+h.c.$
$\displaystyle\left.+\frac{v-\delta
v(t)}{2}c_{n,2}^{\dagger}c_{n+1,1}+h.c.\right].$ (15)

(a)

(b)

Figure 6: LABEL:sub@fig:rm-structure Visualized structure of the Rice-Mele
model. Green rectangle shows the unit cell which contains two sublattices with
staggered potential $\pm\delta h$, marked by blue and red dots. Blue and red
lines connecting two sites represent the alternating hopping $v\pm\delta v$.
LABEL:sub@fig:rm-trace Trace of the parameters $(\delta h,\delta v)$ when
$\delta h(t)=\cos(2\pi t/T)$ and $\delta v(t)=\sin(2\pi t/T)$. In a period,
the trace of parameters $(\delta h,\delta v)$ winds origin once along the
anti-clockwise direction.

Here, we modulate a staggered potential $\delta h(t)$ and an alternating
hopping $\delta v(t)$ as the time $t$ changes, while the constant hopping $v$
never changes. The creation and annihilation operators on $n$-th unit cell,
with sublattice $1$ and $2$, has been written as
$c_{n,1/2}^{\dagger},c_{n,1/2}$, respectively. Visualized structure of the
Rice-Mele model is shown in Fig.6a.

In the case when $|\delta h|,|\delta v|<|v|$ and at half-filling, as the path
of the parameter $(\delta h,\delta v)(t)$ winds around the origin of the
parameter space, this model pumps charges with time period. For example, when
$\delta h(t)$ and $\delta v(t)$ are given as,

$\delta h(t)=\cos\left(\frac{2\pi t}{T}\right),\quad\delta
v(t)=\sin\left(\frac{2\pi t}{T}\right),$ (16)

the charge pumping occurs in a period $T$, since the parameter set $(\delta
h,\delta v)$ winds around the origin as in Fig.6b.

The number of charge pumped by the Rice-Mele model coincides with the winding
number of the parameter space, or equivalently, the Chern number of the
Hamiltonian,

$C=\frac{1}{2\pi}\sum_{n}\int_{0}^{T}dt\int dkF_{t,k}^{n}.$ (17)

Here, the summation $\sum_{n}$ runs around the eigenstates below the energy
gap, and $F_{t,k}^{n}$ represents the Berry curvature of $n$-th eigenstate, at
time $t$ and momentum $k$. Because the Chern number is a topological property
of the system, this shows the reason why the amount of charge is quantized.

##  Appendix B Localization near equipotential line

The relation between strength of frequency and the localization on
equipotential line is a key point of the research. In this paragraph we show
that the localization indeed occurs when the frequency is high enough.

For the parameter of localization, we choose the variation of the distance
from equipotential line $\Delta x^{2}$. Specifically, we choose an
equipotential line and define $x$ as an operator measuring the distance from
the equipotential line to each cite. By calculating variance $\Delta
x^{2}=\langle x^{2}\rangle-\langle x\rangle^{2}$ of the state, we get the
dispersion of the state around the equipotential line. Notice that the
variance $\Delta x^{2}$ does not depend on the position of the equipotential
line we take.

Figure 7: The log value of the variance of distance from the equipotential
line $\Delta x^{2}=\langle x^{2}\rangle-\langle x\rangle^{2}$ on each $k$. As
the frequency increases from $\Omega=0.01$ (black dots) to $\Omega=0.1$ (green
dots) and $\Omega=1$ (red dots), the variance of distance decreases. The data
has been plotted on $\Delta=3,\delta(k)=\sin k,B_{z}(k)=0.05\cos k$, and
$p=16,q=17$.

As we can see in Fig.7, the variation $\Delta x$ becomes smaller as the
frequency becomes higher. This indicates that the state has a contribution
from more sites near equipotential line. This corresponds to the fact that the
localization on the equipotential line occurs due to the quasi-electric field,
whose strength depends on the frequency. As frequency increases, the strength
of quasi-electric field on the Floquet lattice also increases, which makes the
Stark effect even stronger.

## References

  * Kane and Mele (2005a) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005a).
  * Kane and Mele (2005b) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005b).
  * Bernevig  _et al._ (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, science 314, 1757 (2006).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Reviews of modern physics 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057 (2011).
  * Xiao  _et al._ (2020) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nature Physics 16, 761 (2020).
  * Helbig  _et al._ (2020) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. Molenkamp, C. Lee, A. Szameit, M. Greiter, and R. Thomale, Nature Physics 16, 747 (2020).
  * Thouless (1983) D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
  * Switkes  _et al._ (1999) M. Switkes, C. Marcus, K. Campman, and A. Gossard, Science 283, 1905 (1999).
  * Kraus  _et al._ (2012) Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
  * Lohse  _et al._ (2016) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, Nature Physics 12, 350 (2016).
  * Nakajima  _et al._ (2016) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Nature Physics 12, 296 (2016).
  * Kitagawa  _et al._ (2010) T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
  * Rudner  _et al._ (2013) M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Phys. Rev. X 3, 031005 (2013).
  * Martin  _et al._ (2017) I. Martin, G. Refael, and B. Halperin, Phys. Rev. X 7, 041008 (2017).
  * Rudner and Lindner (2020) M. S. Rudner and N. H. Lindner, Nature reviews physics 2, 229 (2020).
  * Shirley (1965) J. H. Shirley, Phys. Rev. 138, B979 (1965).
  * Sambe (1973) H. Sambe, Phys. Rev. A 7, 2203 (1973).
  * Moskalets and Büttiker (2002) M. Moskalets and M. Büttiker, Phys. Rev. B 66, 205320 (2002).
  * Grifoni and Hänggi (1998) M. Grifoni and P. Hänggi, Physics Reports 304, 229 (1998).
  * Chu and Telnov (2004) S.-I. Chu and D. A. Telnov, Physics reports 390, 1 (2004).
  * Murakami  _et al._ (2017) Y. Murakami, N. Tsuji, M. Eckstein, and P. Werner, Physical Review B 96, 045125 (2017).
  * Karzig  _et al._ (2015) T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, Phys. Rev. X 5, 031001 (2015).
  * Dehghani  _et al._ (2014) H. Dehghani, T. Oka, and A. Mitra, Phys. Rev. B 90, 195429 (2014).
  * Goldman and Dalibard (2014) N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014).
  * Eckardt (2017) A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).
  * Long  _et al._ (2021a) D. M. Long, P. J. D. Crowley, and A. Chandran, Phys. Rev. Lett. 126, 106805 (2021a).
  * Qi  _et al._ (2021) Z. Qi, G. Refael, and Y. Peng, Phys. Rev. B 104, 224301 (2021).
  * Nathan  _et al._ (2021) F. Nathan, R. Ge, S. Gazit, M. Rudner, and M. Kolodrubetz, Phys. Rev. Lett. 127, 166804 (2021).
  * Floquet (1883) G. Floquet, Annales scientifiques de l’École Normale Supérieure 12, 47 (1883).
  * Wannier (1962) G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962).
  * Long  _et al._ (2021b) D. M. Long, P. J. Crowley, and A. Chandran, arXiv preprint arXiv:2108.04834 (2021b).
  * Rice and Mele (1982) M. J. Rice and E. J. Mele, Phys. Rev. Lett. 49, 1455 (1982).
  * Creutz (1999) M. Creutz, Phys. Rev. Lett. 83, 2636 (1999).