# Uncover band topology via quantized drift in two-dimensional Bloch
oscillations

Bo Zhu  Institute of Mathematics and Physics, Central South University of
Forestry and Technology, Changsha 410004, China  Shi Hu  School of
Optoelectronic Engineering, Guangdong Polytechnic Normal University,
Guangzhou, 510665, China  Honghua Zhong<EMAIL_ADDRESS> Institute of
Mathematics and Physics, Central South University of Forestry and Technology,
Changsha 410004, China  Yongguan Ke<EMAIL_ADDRESS> Guangdong
Provincial Key Laboratory of Quantum Metrology and Sensing $\&$ School of
Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082,
China

(August 27, 2024)

###### Abstract

We propose to measure band topology via quantized drift of Bloch oscillations
in a two-dimensional Harper-Hofstadter lattice subjected to tilted fields in
both directions. When the difference between the two tilted fields is large,
Bloch oscillations uniformly sample all momenta, and hence the displacement in
each direction tends to be quantized at multiples of the overall period,
regardless of any momentum of initial state. The quantized displacement is
related to a reduced Chern number defined as a line integral of Berry
curvature in each direction, providing an almost perfect measurement of Chern
number. Our scheme can apply to detect Chern number and topological phase
transitions not only for the energy-separable band, but also for energy-
inseparable bands which cannot be achieved by conventional Thouless pumping or
integer quantum Hall effect.

##  I Introduction

Topological band theory Bansil  _et al._ (2016) provides a general framework
for exploring a variety of topological states and phenomena such as integer
quantum Hall effects Klitzing  _et al._ (1980); Thouless  _et al._ (1982),
topological insulators Kane and Mele (2005); Bernevig  _et al._ (2006); König
_et al._ (2007); Chang  _et al._ (2013); Xu  _et al._ (2014), and Thouless
pumping Thouless (1983); Ke  _et al._ (2016); Hu  _et al._ (2019); L.  _et
al._ (2020). A key notion is topological invariant (e.g., Chern number and
winding number) which characterizes the robust global property of Bloch states
of certain energy bands. Topological phase transition happens when a
topological invariant changes. Extending the topological states from solid-
state materials to ultracold atoms in optical lattices have attracted
tremendous interests in recent years Aidelsburger  _et al._ (2013); Miyake
_et al._ (2013); Jotzu  _et al._ (2014); Roy  _et al._ (2015). Ultracold
atomic system, as an excellent platform for quantum simulations, provides
unprecedented opportunities for studying topological states, in particular,
uncovering band topology.

Many schemes were proposed to explore band topology and some of them have been
realized in experiments of ultracold atoms. In momentum space, the bulk
topological invariants can be determined by linking numbers Wang  _et al._
(2017); Tarnowski  _et al._ (2019), band-inversion surfaces Sun  _et al._
(2018); Zhang  _et al._ (2018, 2019) in quench dynamics, and dynamic winding
number Zhu  _et al._ (2020) based on time-averaged spin textures. However,
these dynamical methods need to measure different observables involving all
momentum states at different time, which turn out to be redundancy and
inefficient. In real space, winding number can be measured by a mean chiral
displacement in quantum walks Zeuner  _et al._ (2015); Longhi (2018, 2019),
and Chern number can be extracted by a mean displacement in Thouless pumping
or transverse velocity in the integer quantum Hall effect Ke  _et al._ (2016);
Hu  _et al._ (2019). However, these methods in real space require that the
initial state uniformly occupies all the momentum states of a band, which in
general is a challenge. It becomes a dilemma about either suffering labored
measurements or overcoming difficulty in preparing an input state.

The above dilemma may be solved by utilizing Bloch oscillations under external
tilted fields, which have been proved to be a powerful method for exacting
various geometric features of Bloch bands, including Berry phase Atala  _et
al._ (2013); Duca  _et al._ (2015), Berry curvature Price and Cooper (2012);
Jotzu  _et al._ (2014); Wimmer  _et al._ (2017) and Chern number Aidelsburger
_et al._ (2015); Genkina  _et al._ (2019); Chalopin  _et al._ (2020); Ke  _et
al._ (2020). In particular, Berry curvature was proposed to be mapped from
semiclassical dynamics by using two-dimensional Bloch oscillations (i.e.,
applying external forces in two dimensions) Price and Cooper (2012). However,
to further extract Chern number, this protocol needs to measure the
displacement difference under positive and negative forces to cancel the
contribution of group velocity from energy dispersion, and then to average the
displacement difference under different initial momenta. Recently, we have
shown how to directly extract Chern number via a one-dimensional quantized
topological pumping assisted by Bloch oscillations that does not require
uniform band occupation Ke  _et al._ (2020). This is because Bloch
oscillations uniformly sample all the momenta. It is highly nontrivial to
directly and efficiently measure band topology via Bloch oscillations in two
dimensions.

In this paper, we reveal a quantized drift of Bloch oscillations as a _direct
measurement_ of Chern number in a two-dimensional Harper-Hofstadter-like
lattice subjected to external fields in both $x$ and $y$ directions. Such
scheme is readily accessible with typical experimental settings of ultracold
atomic gases in optical lattices Aidelsburger  _et al._ (2013, 2015). We give
analytical expressions for the mean displacement in both $x$ and $y$
directions via the adiabatic transport theorem. As the ratio between external
fields in $x$ and $y$ directions approaches to $0$ or $\infty$, we find that
the mean displacement in each direction is almost perfectly quantized and
independent of the initial momentum. The quantized displacement in each
direction is given by a time integral of Berry curvature dubbed as reduced
Chern number (RCN), which is directly related to the conventional Chern
number. According to the quantized drift in the overall period of Bloch
oscillations, one can successfully detect topological phase transitions and
Chern number for an energy-separable band. With the increase of ratio of tilts
in two directions, the accuracy of Chern number near the phase transition
point will be gradually improved. Importantly, we show a direct measurement of
the Chern number for an energy-inseparable band via RCN, which can never be
achieved by the conventional Hall-response scheme with tilt in only one
direction Aidelsburger  _et al._ (2015). Compared with Ref. Price and Cooper
(2012), our method has several advantages listed as follows. (i) Our protocol
automatically cancels the group velocity from energy dispersion, and directly
gives the Chern number via displacement in a single wavepacket dynamics, in
which there is no need to inverse the force to eliminate the contribution of
displacement from energy dispersion; (ii) We also reveal that the quantized
drift in Bloch oscillations is independent of the momentum of the initial
Gaussian wavepacket, reducing difficulty of the initial state preparation and
many wavepacket dynamics with different initial momenta; (iii) We give a
direct measurement of super-band topology which has never been discussed in
Ref. Price and Cooper (2012).

The rest of the paper is organized as follows. In Sec. II, we give a physical
description of the model. In Sec. III, we give the definition of RCN and
establish the relation between RCN and the quantized displacement in two-
dimensional Bloch oscillations. In Sec. IV, we show how to use Bloch
oscillations to extract Chern numbers for an energy-separable band in the
subsection IV.1 and for an energy-inseparable band in subsection IV.2. At
last, we give a conclusion and discussion in Sec. V.

##  II Tilted Harper-Hofstadter lattice

We consider a single particle in a two-dimensional tilted superlattice in the
presence of a uniform flux per plaquette, see Fig. 1. The corresponding
Hamiltonian consists of three parts,

$\displaystyle H=H_{1}+H_{2}+H_{3},$ (1)

with

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle-\sum_{m,n}\tau_{x}c_{m+1,n}^{\dagger}c_{m,n}+\tau_{y}e^{i2\pi\beta
m}c_{m,n+1}^{\dagger}c_{m,n}+\mathrm{h.c.},$ $\displaystyle H_{2}$
$\displaystyle=$
$\displaystyle-\sum_{m,n}\frac{\delta}{2}[(-1)^{m}+(-1)^{n}]c_{m,n}^{\dagger}c_{m,n},$
$\displaystyle H_{3}$ $\displaystyle=$
$\displaystyle-\sum_{m,n}(F_{x}m+F_{y}n)c_{m,n}^{\dagger}c_{m,n}.$ (2)

Here, $H_{1}$ is the conventional Harper-Hofstadter Hamiltonian Hofstadter
(1976); Sorensen  _et al._ (2005); Motruk and Na (2020); Repellin  _et al._
(2020), and the Peierls phase accounts for the presence of a flux
$\phi=2\pi\beta$ per plaquette. $c_{m,n}^{\dagger}(c_{m,n})$ creates
(annihilates) a particle at site $(m,n)$, $\tau_{x}$ and $\tau_{y}$ are the
hopping strengths along $x$ and $y$ directions. $H_{2}$ describes a staggered
detuning $\delta$ between sublattices in two directions, which can be used to
induce topological phase transition Aidelsburger  _et al._ (2015). When
$\tau_{x}=\tau_{y}=\tau$, $|\delta|\geq 2\tau$ and $|\delta|<2\tau$ correspond
to topologically trivial and nontrival phases, respectively. $H_{3}$
represents the tilts of the square lattice, where $F_{x}$ and $F_{y}$ are the
tilted strengths along $x$ and $y$ directions, which may be realized by
applying a magnetic field gradient or subjecting the superlattice along the
gravity with an angle. Because the tilted strengths are very weak compared to
the coupling strengths, $H_{3}$ can be treated as a perturbation Chang and Niu
(1996), which means that $H_{3}$ does not destroy the topological properties
of the system.

Figure 1: Schematic diagram of the Hofstadter-like optical lattice with
additional tilts along $x$ and $y$ directions. $\tau_{x(y)}$ and $F_{x(y)}$
are the hopping strength and tilted strength along $x(y)$ direction,
$(A,B,C,D)$ denote the sublattices in a unit cell with onsite energies
$(-\delta,0,0,+\delta)$, and $\phi$ is the accumulated phase per plaquette due
to the gauge field.

The time evolution of state $|\psi(t)\rangle$ can be obtained by solving the
Schrödinger equation $i\hbar\frac{\partial}{\partial
t}|\psi(t)\rangle=H|\psi(t)\rangle$. Hereafter we set $\hbar=1$ for
simplicity. By making a unitary transformation
$|\tilde{\psi}(t)\rangle=\exp(iH_{3}t)|{\psi}(t)\rangle$, we can equivalently
deal with the problem in a rotational framework, and $|\tilde{\psi}(t)\rangle$
is governed by

$i\frac{\partial}{\partial
t}|\tilde{\psi}(t)\rangle=H_{rot}(t)|\tilde{\psi}(t)\rangle$ (3)

with the time-dependent Hamiltonian
$H_{rot}(t)=e^{iH_{3}t}(H_{1}+H_{2})e^{-iH_{3}t}$, which is explicitly given
by

$\displaystyle
H_{rot}(t)=-\frac{\delta}{2}\sum_{m,n}[(-1)^{m}+(-1)^{n}]c_{m,n}^{\dagger}c_{m,n}-$
$\displaystyle\sum_{m,n}\tau_{x}e^{iF_{x}t}c_{m+1,n}^{\dagger}c_{m,n}+\tau_{y}e^{iF_{y}t}e^{i2\pi\beta
m}c_{m,n+1}^{\dagger}c_{m,n}+\mathrm{h.c.}$

If $\beta=p/q$, where $p$ and $q$ are co-prime numbers, the Hamiltonian has
magnetic translational symmetry. In the whole paper, we consider $\beta=1/4$
without loss of generality. Under the periodic boundary condition, one can
obtain the Hamiltonian in quasimomentum space
$H_{rot}(t)=\sum_{k_{x},k_{y}}\Psi_{{k_{x},k_{y}}}^{\dagger}h(k_{x},k_{y},t)\Psi_{{k_{x},k_{y}}}$
with

$\displaystyle
h=2\left(\begin{array}[]{cccc}-\frac{\delta}{2}&-\tau_{y}\cos\kappa_{y}&-\tau_{x}\cos\kappa_{x}&0\\\
-\tau_{y}\cos\kappa_{y}&0&0&\tau_{x}\sin\kappa_{x}\\\
-\tau_{x}\cos\kappa_{x}&0&0&i\tau_{y}\sin\kappa_{y}\\\
0&\tau_{x}\sin\kappa_{x}&-i\tau_{y}\sin\kappa_{y}&\frac{\delta}{2}\end{array}\right).$
(9)

Here,
$\Psi_{\textit{{k}}}=\left(c_{\textit{{k}},A},c_{\textit{{k}},B},c_{\textit{{k}},C},c_{\textit{{k}},D}\right)^{T}$,
where $c_{\textit{{k}},\alpha}$ destroys a plane wave with quasimomentum
$\textit{{k}}\equiv(k_{x},k_{y})$ at $\alpha$ sublattice,
$\kappa_{x(y)}=k_{x(y)}-F_{x(y)}t$ and $k_{x(y)}\in[-\pi/2,\pi/2]$ (see
Appendix A for details). We can numerically obtain the eigenvalues
$E_{l}(k_{x},k_{y},t)$ and eigenstates $|\mu_{l}(k_{x},k_{y},t)\rangle$ by
diagonalizing the Hamiltonian,
$h(k_{x},k_{y},t)|\mu_{l}(k_{x},k_{y},t)\rangle=E_{l}(k_{x},k_{y},t)|\mu_{l}(k_{x},k_{y},t)\rangle$,
where $l$ is the band index.

Figure 2: Reduced Chern number of the first band as a function of
$(k_{x},k_{y})$. (a) $C_{1,x}(k_{x},k_{y})$ and (b) $C_{1,y}(k_{x},k_{y})$ for
tilts in both $x$ and $y$ directions ($F_{x}/F_{y}=\eta_{x}/\eta_{y}=10$,
$F_{x}=0.04,F_{y}=0.004$). (c) $C_{1,x}(k_{x},k_{y})$ and (d)
$C_{1,y}(k_{x},k_{y})$ for tilts in only $x$ direction
($F_{x}=0.0402,F_{y}=0$). The other parameters are chosen as
$\tau_{x}=\tau_{y}=0.2$, and $\delta=0$.

##  III Reduced Chern number

Consider that the initial state is prepared as a Bloch state with
quasimomentum ($k_{x},k_{y}$) in the $l$th band. If the tilted strengths are
so weak then the particle adiabatically follows the trajectory
$[\kappa_{x}(t),\kappa_{y}(t)]$ in the $l$th band. According to the theorem of
adiabatic transport Xiao  _et al._ (2010); Ho and Gong (2012), the group
velocity of the particle along the $x(y)$ direction comes from the energy
dispersion and the Berry curvature,

$v_{l,x(y)}(k_{x},k_{y},t)=\frac{\partial E_{l}(k_{x},k_{y},t)}{\partial
k_{x(y)}}+\mathcal{F}_{l,x(y)}(k_{x},k_{y},t),$ (11)

with the Berry curvature given by

$\mathcal{F}_{l,x(y)}=-2\operatorname{Im}\sum_{l^{\prime}\neq
l}\frac{\left\langle
u_{l}\left|\partial_{k_{x(y)}}h\right|u_{l^{\prime}}\right\rangle\left\langle
u_{l^{\prime}}\left|\partial_{t}h\right|u_{l}\right\rangle}{\left(E_{l}-E_{l^{\prime}}\right)^{2}}.$
(12)

Because the energy is a (quasi-)periodic function of time, if the band
topology is completely trivial, the first term of group velocity will
periodically oscillate and hence induce the conventional Bloch oscillations in
both directions. In the following subsection, we will show how the nontrival
Berry curvature induces quantized drift in the Bloch oscillations.

###  III.1 Rational case

We first consider the case that the tilts in two directions are nonzero and
commensurate $F_{x}/F_{y}=\eta_{x}/\eta_{y}$, where $\eta_{x}$ and $\eta_{y}$
are coprime integers. If $F_{x}=F_{y}$, the period of Bloch oscillations in
$x$ direction $T_{x}=2\pi/F_{x}$ is the same as that in $y$ direction
$T_{y}=2\pi/F_{y}$. However, if $F_{x}\neq F_{y}$, $T_{x}$ and $T_{y}$ may be
not the period in their own directions. The coexistence of the tilts in two
directions leads to an overall period $T_{o}=\eta_{x}T_{x}=\eta_{y}T_{y}$.

For a Bloch state with quasimomentum $(k_{x},k_{y})$, the mean displacement in
$x$ and $y$ directions $(\Delta X,\Delta Y)$ at time $t$ can be given by the
semi-classical expression

$\displaystyle\Delta X(k_{x},k_{y},t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}v_{l,x}(k_{x},k_{y},t^{\prime})dt^{\prime},$
$\displaystyle\Delta Y(k_{x},k_{y},t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}v_{l,y}(k_{x},k_{y},t^{\prime})dt^{\prime}.$ (13)

In the commensurate case, because the energy bands are periodic functions of
time, the integral of dispersion velocity is exactly zero in the overall
period. Due to the nontrivial Berry curvature, the anomalous group velocity
thus plays a determinant role in the mean displacement. We define the time
integral of Berry curvature over the overall period as

$C_{l,x(y)}(k_{x},k_{y})=\frac{1}{q}\int_{0}^{T_{o}}\mathcal{F}_{l,x(y)}(k_{x},k_{y},t)dt,$
(14)

When $\eta_{x}/\eta_{y}$ approaches $0$ or $\infty$, $C_{l,x}(k_{x},k_{y})$
and $C_{l,y}(k_{x},k_{y})$ respectively tend to be integer numbers
$C_{l,x}^{0}$ and $C_{l,y}^{0}$, independent of the initial momentum value of
a Bloch state (see Appendix C for details). Interestingly, the integer numbers
$C_{l,x}^{0}$ and $C_{l,y}^{0}$ are related to the conventional Chern number
of the $l$th band (see Appendix B for details),

$\displaystyle C_{l}=C_{l,x}^{0}/\eta_{y}=-C_{l,y}^{0}/\eta_{x}.$ (15)

For this reason, we name the line integral of Berry curvature $C_{l,x(y)}$ as
reduced Chern number. Then the displacement in an overall period tends to be
quantized values

$\displaystyle\Delta X(k_{x},k_{y},T_{o})$ $\displaystyle=$ $\displaystyle
qC_{l,x}(k_{x},k_{y}),$ $\displaystyle\Delta Y(k_{x},k_{y},T_{o})$
$\displaystyle=$ $\displaystyle qC_{l,y}(k_{x},k_{y}).$ (16)

Combining Eqs. (15) and (III.1), we find that the mean displacement in $x$ and
$y$ directions will satisfy

$\mathop{\lim}\limits_{{\eta_{x}(\eta_{y})}\to\infty}(\Delta
X(T_{o})\eta_{x}+\Delta Y(T_{o})\eta_{y})\rightarrow 0.$ (17)

To verify the above analytical results, we respectively show $C_{1,x}$ and
$C_{1,y}$ as functions of $(k_{x},k_{y})$ in Figs. 2(a) and 2(b), with the
parameters $F_{x}/F_{y}=\eta_{x}/\eta_{y}=10$, $F_{x}=0.04,F_{y}=0.004$ ,
$\tau_{x}=\tau_{y}=0.2$ and $\delta=0$. When the tilts in both $x$ and $y$
directions are present and their ratio is small or large enough, both
$C_{1,x}$ and $C_{1,y}$ are extremely flat for all momenta and close to
quantized values, $C_{1,x}^{0}=1$ and $C_{1,y}^{0}=-10$, respectively.
Considering that the Chern number $C_{1}=1$, it is easy to find that the
formula (15) is satisfied. The fact that $C_{1,x}/\eta_{y}$ and
$C_{1,y}/\eta_{x}$ are almost independent of the initial momentum means that
one can use a single momentum state to detect Chern number. This is strikingly
different from the case that a tilt is present in only $x$ direction, see
Figs. 2(c) and 2(d), where $F_{x}=0.0402,F_{y}=0$ and the overall period
becomes $T_{o}=T_{x}$. $F_{x}$ and $F_{y}$ have the same norm
$\sqrt{F_{x}^{2}+F_{y}^{2}}$ as that in Fig. 2(a) and 2(b). $C_{1,x}$ is
exactly zero, while $C_{1,y}$ maintains the same along $k_{x}$ direction for a
fixed $k_{y}$ but periodically varies along the $k_{y}$ direction. This is
because the Berry curvature $\mathcal{F}_{1,x}=0$ at any time and
$\mathcal{F}_{1,y}$ is a periodic function of $k_{y}$.

Obviously, in the case of tilt in a single direction, we cannot use an
arbitrary single momentum state to detect Chern number. However, the average
of $C_{1,x}(k_{x},k_{y})$ over $k_{y}$ is still quantized, i.e.,
$\frac{1}{\pi}\int_{-\pi/2}^{\pi/2}C_{1,y}(k_{x},k_{y})dk_{y}=1$, which is
consistent with the first band Chern number $C_{1}=1$. This means that if the
initial state is an equal superposition of momentum states involving all
$k_{y}$, the Chern number of the band can also be detected by mean
displacement for the system with the tilt in a single direction. It is worth
noting that the experimental scheme of detecting Chern numbers of Hofstadter
bands for the tilt in a single direction has been realized Aidelsburger  _et
al._ (2015), where it requires that the topological bands are very flat. The
essential reason is that the flat band makes it easier to uniformly fill the
band.

In contrast to previous schemes with tilt in a single direction, according to
the definition of RCN, our proposed scheme with tilts in two directions has
greater advantages for experimental detection of Chern number. Firstly, this
scheme does not require a flat topological band; Secondly, this scheme also
does not require the initial states being equal superposition of all momentum
states, Therefore, RCN may provide an experimentally-friendly way for
detecting the topological phase transition. To this end, we first show the
topological phase diagram in the parameter plane $(\tau_{y},\delta)$ by using
conventional Chern number, see Fig. 3(a). The Chern numbers of the first band
are $C_{1}=1$ and $0$ in the blue and white regions, where the green dashed-
dot line corresponds to the case of hopping constant $\tau_{x}=\tau_{y}$ and
the phase transition point is $\delta/\tau_{x}=2$. Along the green dashed-dot
line, we give the RCN $C_{1,x}^{0}/\eta_{y}$ as a function of $\delta$ for
different ratios of tilts $\eta_{x}/\eta_{y}$ in Fig. 3(b). The RCNs are
consistent with the first band Chern number $C_{1}$ away from the phase
transition point. As the ratio of tilts increases, the change of RCN around
$\delta/\tau_{x}=2$ becomes sharper, marking the position of the transition
point. Therefore, the RCN can apply to detect the topological phase transition
by choosing $\eta_{x}/\eta_{y}\rightarrow\infty$, or equivalently
$\eta_{x}/\eta_{y}\rightarrow 0$.

Figure 3: (a) Topological phase diagram. (b) RCN as a function of $\delta$ for
different commensurate ratios $\eta_{x}/\eta_{y}$ (fixed
$\sqrt{F_{x}^{2}+F_{y}^{2}}=0.0402$ and $k_{x}=k_{y}=0$), the other parameters
are chosen as $\tau_{x}=\tau_{y}=0.2$, corresponding to the green dashed line
in (a).

###  III.2 Irrational case

When the tilts in two directions are incommensurate, both the energy spectrum
and Berry curvature are quasi-periodic function of time, and hence there is no
such an overall period as the rational case. However, we can use a sequence of
rational ratios $\\{\eta_{x}^{n}/\eta_{y}^{n}\\}$ to approach the irrational
ratio $F_{x}/F_{y}$, and treat the irrational case in a similar way as the
rational case. When $\eta_{y}^{n}$ tends to infinite integer, we can reach the
RCN and the relation in Eq. (III.1) still holds. Taking
$F_{x}/F_{y}=(\sqrt{5}+1)/2$ as example, we can use a continued fraction
representation for the golden ratio, which is given by
$\\{\eta_{x}^{n}/\eta_{y}^{n}\\}=(1/1,2/1,3/2,5/3,8/5,...,F_{j+1}/F_{j},...)$
with the Fibonacci sequence $F_{j}$. We show how the derivation between RCN
and the conventional Chern number changes with the increase of $\eta_{y}^{n}$
in Fig. 4. As expected, the derivation quickly decays as $\eta_{y}^{n}$
increases, indicating that the irrational case can be practically treated as
the rational case with large $\eta_{y}^{n}$.

Figure 4: Derivation between $C_{1,x}/\eta_{y}^{n}$ and $C_{1}$ as
$\eta_{y}^{n}$ increases for the irrational $F_{x}/F_{y}=(\sqrt{5}+1)/2$. The
other parameters are chosen as $F_{y}=0.004,\tau_{x}=\tau_{y}=0.2$, $\delta=0$
and $k_{x}=k_{y}=0$.

##  IV Two-dimensional quantized-drift Bloch oscillations

In the previous section, we have shown how the Chern number of a given band is
related to the quantized drift in the Bloch oscillations based on
semiclassical analysis. To verify such relation, in this section we study the
quantum dynamics of a Gaussian wavepacket centered in $(m_{0},n_{0})$ with
arbitrary mean quasimomentum $(k_{x_{0}},k_{y_{0}})$ in the $l$th band. The
initial wavefunction at the site $(m,n)$ hence is given by

$\displaystyle\psi_{m,n}(0)=\zeta
e^{-\frac{(m-m_{0})^{2}+(n-n_{0})^{2}}{4D^{2}}}\widetilde{\mu}_{m,n}^{l}(k_{x_{0}},k_{y_{0}})e^{i(k_{x_{0}}m+k_{y_{0}}n)},$

where $\zeta$ is a normalization factor, $D$ is the initial wavepacket width,
$\widetilde{\mu}_{m,n}^{l}(k_{x_{0}},k_{y_{0}})$ is the amplitude of the real-
space representation of the Bloch state
$|\mu_{l}(k_{x_{0}},k_{y_{0}},0)\rangle$. Such type of wavepacket can be
prepared by applying an additional harmonic trap Lu  _et al._ (2016). In the
following calculations, the time evolution of the wavepacket is given by
$|\psi(t)\rangle=\exp(-iHt)|\psi(0)\rangle$ according to the Hamiltonian Eq.
(1). One may alternatively calculate the wavepacket dynamics in the rotating
framework with time-dependent Hamiltonian, which gives the same density
distribution but increases the calculation resources due to the update of
Hamiltonian at each time step. We then examine the density distribution
profile $|\psi_{m,n}(t)|^{2}$ of the time-evolving wavepacket $\psi_{m,n}(t)$,
and the two components of mean displacement in $x$ and $y$ directions

$\displaystyle\Delta X(t)$ $\displaystyle=$ $\displaystyle X(t)-X(0),$
$\displaystyle\Delta Y(t)$ $\displaystyle=$ $\displaystyle Y(t)-Y(0),$ (18)

where $X(t)=\sum_{m,n}m\left|\psi_{m,n}(t)\right|^{2}$ and
$Y(t)=\sum_{m,n}n\left|\psi_{m,n}(t)\right|^{2}$. The dynamical results are
shown in subsection IV.1 for the energy-separable band and in subsection IV.2
for the energy-inseparable band.

###  IV.1 Direct measurement of separable-band topology

Figure 5: Wavepacket dynamics in topologically nontrivial phase. (a) Density
distribution of initial states $(t=0)$ and final states $(t=T_{o})$ in real
space. (b) and (c) correspond to the density evolution project to $x$ and $y$
directions, respectively. (d) Drift $\Delta X$ and $\Delta Y$ versus time $t$.
The green and red solid lines are obtained from quantum dynamics, and the
black dashed line is obtained via Eq. (III.1). The other parameters are chosen
as $\tau_{x}=\tau_{y}=0.2$, $D=6$, $\eta_{x}/\eta_{y}=10$,
$F_{x}=0.04,F_{y}=0.004$, $\delta=0$ and $k_{x_{0}}=k_{y_{0}}=0$.  Figure 6:
Wavepacket dynamics in topologically trivial phase. (a) Density distribution
of initial states $(t=0)$ and final states $(t=T_{o})$ in real space. (b) and
(c) correspond to the density evolution projected to $x$ and $y$ directions,
respectively. (d) Drift $\Delta X$ and $\Delta Y$ versus time $t$. The green
and red solid lines are obtained from quantum dynamics, and the black dashed
line is obtained via Eq. (III.1). The other parameters are chosen as
$\tau_{x}=\tau_{y}=0.2$, $D=6$, $\eta_{x}/\eta_{y}=10$,
$F_{x}=0.04,F_{y}=0.004$, $\delta=4\tau_{x}$ and $k_{x_{0}}=k_{y_{0}}=0$.

When a particle in a seperable band is driven by tilted potentials, the
adiabatic condition can be easily satisfied if the tilted potentials are weak
enough. The displacement can be predicted by the semiclassical formula based
on the theorem of adiabatic transport. We first consider the quantum dynamics
of a Gaussian wavepacket in the first band which is topologically nontrivial.
The parameters are set as $\tau_{x}=\tau_{y}=0.2$, $D=6$ and $\delta=0$, which
correspond to the first band with Chern number $C_{1}=1$.
$F_{x}=0.04,F_{y}=0.004$ are chosen according to the method proposed above, so
that the large ratio $\eta_{x}/\eta_{y}=10$ supports well quantization of
displacement and meanwhile the small $F_{x}$ and $F_{y}$ can ensure well
adiabatic dynamics. Fig. 5 shows the density distributions of initial state
$(t=0)$ and final state $(t=T_{o})$ in real space, the time-evolution of
wavefunction and the drift of the wavepacket center as a function of time
along $x$ and $y$ directions.

First, the wavepacket always maintains well spatial localization in an overall
period, see Fig. 5(a). By tracing its dynamical evolution along $x$ and $y$
directions, the wavepacket is shifted rightward in $x$ direction and
lefttoward in $y$ direction, see Figs. 5(b) and 5(c). For clarity, the drifts
$\Delta X$ and $\Delta Y$ versus time $t$ are given in Fig. 5(d). We can see
that the results obtained by the semi-classical expressions of Eq. (III.1)
agree well with the ones directly obtained by wavepacket dynamical
calculation. Especially, at $t=T_{o}$, we can obtain $\Delta X(T_{o})=4$ and
$\Delta Y(T_{o})=-40$, and the RCNs
$C_{1,x}^{0}/\eta_{y}=-C_{1,y}^{0}/\eta_{x}=1$ by combining Eqs. (15) and
(III.1), which are consistent with the first band Chern number $C_{1}=1$. By
contrast, for the topologically trivial phase, the wavepackets of the initial
state at $(t=0)$ and the final state at $(t=T_{o})$ in real space also
maintain spatial localization, and the density distributions of the initial
and final states are overlapped in space. It means that the final state comes
back to the position of the initial state, see Fig. 6(a). The Bloch
oscillations of the wavepacket in $x$ and $y$ directions have different
periods and amplitudes, as shown in Figs. 6(b) and 6(c). Indeed, at $t=T_{o}$,
one can obtain $\Delta X(T_{o})=0$ and $\Delta Y(T_{o})=0$, and the RCNs
$C_{1,x}^{0}/\eta_{y}=-C_{1,y}^{0}/\eta_{x}=0$ are consistent with the first
band Chern number $C_{1}=0$, see Fig. 6(d). To verify the mean quasimomentum
of an initial Gaussian wavepacket can be arbitrarily chosen, we also give the
drift $\Delta X$ and $\Delta Y$ versus time $t$ for different
($k_{x_{0}},k_{y_{0}}$) in Appendix D, the results are consistent with the
theoretical prediction. Therefore, we can directly measure band topology via
the quantized drift of Bloch oscillations.

###  IV.2 Direct measurement of super-band topology

In this subsection, we will show the possibility of measuring band topology
for a composite super-band which consists of two bands touching at several
degenerate points. Since the initial state is mostly localized at momentum
$(k_{x_{0}},k_{y_{0}})$ of the $l$th band, we can better understand the
dynamics in the momentum space. According to Eq. (9), in the four-band systems
the momentum $(\kappa_{x},\kappa_{y})=(k_{x_{0}}-F_{x}t,k_{y_{0}}-F_{y}t)$ is
linearly driven by a constant force
$(\dot{\kappa}_{x},\dot{\kappa}_{y})=-(F_{x},F_{y})$ Lim  _et al._ (2015). If
the constant force is large enough, Landau-Zener transitions will always
happen especially near anticrossing points of energy bands. We characterize
the Landau-Zener transition by the transition probability from $l$th to
$l^{\prime}$th bands,

$P_{l,l^{\prime}}(t)=|\langle\psi(t)|\mu_{l^{\prime}}(k_{x},k_{y},t)\rangle|^{2},$
(19)

where $l,l^{\prime}=1,2,3,4$ and $|\mu_{l^{\prime}}(k_{x},k_{y},t)\rangle$ is
the $l^{\prime}$th eigenstate of $h(k_{x},k_{y},t)$. To suppress Landau-Zener
transition, the system needs to take a shorter time to travel across the anti-
crossing point than the Zener tunneling time
$T_{lz}=\sqrt{\xi}\max(1,\sqrt{\xi})/\Delta$, where
$\xi=\Delta^{2}/(4\sqrt{F_{x}^{2}+F_{y}^{2}})$ is the adiabaticity parameter
and $\Delta$ is the minimal energy gap Shevchenko  _et al._ (2010); He and Li
(2020). Therefore, for an isolated band, due to a finite energy gap
$\Delta\neq 0$, one can always find proper weak force ($F_{x},F_{y}$) that
makes the system evolve adiabatically. For the composite super-band, it seems
impossible to satisfy the adiabatic condition for individual bands because
there exist gap closing points. Indeed, it is true for the conventional scheme
with a tilted field in only one direction such as $x$-direction, in which the
initial state needs to uniformly occupy the band involving all $k_{y}$.
Consequently, Landau-Zener transition is unavoidable when sweeping the gap
closing points under the tilted force in $x$-direction. However, based on our
scheme with tilts in both $x$ and $y$ directions, there is no requirement of
uniform band occupation. The trajectory of $(\kappa_{x},\kappa_{y})$ may avoid
the degenerated points in two-dimensional Brillouin zone by selecting the
appropriate initial momentum $(k_{x_{0}},k_{y_{0}})$, and hence it is possible
to measure the Chern number for an energy-inseparable super-band via the RCN.

Figure 7: (a) Energy spectrum for the Harper-Hofstadter Hamiltonian. (b)
Eigenvalues change with the trajectory $(\kappa_{x}(t),\kappa_{y}(t))$. (c)
Transition probability for Landau-Zener tunneling as a function of evolution
time $t$. (d) Drift $\Delta X$ and $\Delta Y$ versus time $t$. The other
parameters are chosen as $\tau_{x}=\tau_{y}=0.2$, $\eta_{x}/\eta_{y}=4$,
$F_{x}=0.002,F_{y}=0.0005$, $\delta=0$ and $k_{x_{0}}=k_{y_{0}}=\pi/4$.

To be clear, we consider such a system with super-band by choosing the
parameters as $\tau_{x}=\tau_{y}=0.2$ and $\delta=0$, whose energy bands are
shown in Fig. 7(a). We find that the second and third bands form a composite
super-band with energy degeneracies at some momenta. Such super-band is well
separated from the first and fourth energy bands. The Chern numbers of
individual bands in the super-band are not well-defined due to the difficulty
in separating the two bands at the degenerated points. A common Chern number
has been used to describe the overall topology of the composite super-band
Nakahara (2018),

$C_{s}=\frac{1}{{2\pi
i}}\int_{BZ}dk_{y}{dk_{x}\textrm{Tr}[\mathcal{F}({k_{x},k_{y}})]}.$ (20)

Here,
$[\mathcal{F}({k_{x},k_{y}})]=\partial_{k_{x}}A_{k_{y}}-\partial_{k_{y}}A_{k_{x}}+i[A_{k_{x}},A_{k_{y}}]$
is the non-Abelian Berry curvature with Berry connection elements
$[A_{\mu}]_{m,n}=\langle\psi_{m}|\nabla_{\mu}|\psi_{n}\rangle$, where $m,n$
are the band indices in the super-band. When the super-band only includes a
single band, because the commutator vanishes, the non-Abelian generalization
of Chern number is reduced to the conventional one [Eq. (41)]. Even when the
super-band contains more gapless bands with finite degenerated points, if we
carefully exclude the Berry curvature of the degenerated points, the Chern
number receives contribution only from the Abelian part of Berry curvatures
Mochol-Grzelak  _et al._ (2018). Alternatively, the Chern number of the super-
band is taken as if the sum of effective “Chern numbers” (excluding the Berry
curvature of degenerated points) of the second and third bands. Eq. (41) gives
the same value of $C=-1$ for the second and third bands excluding the
degenerated points. The Chern numbers $C_{1},C_{s},C_{4}$ corresponding to the
first band, super-band and the fourth band take the values $\\{1,-2,1\\}$,
respectively. We should further emphasize that the Chern number of the super-
band $C_{s}$ can not be directly measured by conventional Thouless pumping Ke
_et al._ (2016) or using the tilt in only one direction Aidelsburger  _et al._
(2015), where all momenta are equally involved. This is because these methods
will lead to non-adiabatic transitions near the degenerated points, and then
hinder the measurement of topological invariants.

However, the RCN only depends on the trajectory of
$(\kappa_{x}(t),\kappa_{y}(t))$ in two-dimensional Brillouin zone. By choosing
appropriate initial momentum $(k_{x_{0}},k_{y_{0}})$ and tilted fields, one
can ensure that energy gaps of the four energy bands remain open along the
trajectory $(\kappa_{x}(t),\kappa_{y}(t))$, where the super-band can be
decomposed to two separable energy bands (i.e. the second and third bands), as
shown in Fig. 7(b). Without lossing of generality, we show how to measure RCN
of the second band by preparing initial state in the second band,
$|\psi(t_{0})\rangle=|\mu_{2}(k_{x_{0}},k_{y_{0}},0)\rangle$, with initial
momentum $(k_{x_{0}},k_{y_{0}})=(\pi/4,\pi/4)$. To ensure that the system is
adiabatically evolving, we choose weak tilted fields as $F_{x}=0.002$ and
$F_{y}=0.0005$. Fig. 7(c) shows the transition probability from the second
band to the $l$th band, $P_{2,l}$, along the trajectory
$(\kappa_{x}(t),\kappa_{y}(t))$. It is clear that the transition probability
$P_{2,2}$ is always equal to one, which means that the system is adiabatically
following the eigenstates in the second band. Then, the drifts $\Delta X$ and
$\Delta Y$ versus time $t$ are given in Fig. 7(d). At $t=T_{o}$, we can obtain
$\Delta X(T_{o})=-4,\Delta Y(T_{o})=16$, and the RCN
$C_{2,x}^{0}/\eta_{y}=-C_{2,y}^{0}/\eta_{x}=-1$ by using Eq. (III.1).
Similarly, by choosing the initial state
$|\psi(t_{0})\rangle=|\mu_{3}(k_{x_{0}},k_{y_{0}},0)\rangle$, we also can
obtain the RCN $C_{3,x}^{0}/\eta_{y}=-C_{3,y}^{0}/\eta_{x}=-1$, as a direct
measurement of the effect “Chern number” excluding the degenerated points.
Then, the Chern number of the super-band ($C_{s}=-2$) is consistent with the
summation of reduced Chern numbers of the second and third bands,
$(C_{2,x}^{0}+C_{3,x}^{0})/\eta_{y}=-2$. Therefore, we provide a direct method
to measure the Chern number of the super-band via the quantized drift in two-
dimensional Bloch oscillations.

##  V Conclusion and discussion

In summary, we reveal an almost perfectly quantized drift of two-dimensional
Bloch oscillations in a topological Chern insulator. The quantized drift is
related to a reduced Chern number defined by line integral of Berry curvature
in each direction, as an effective measurement of the conventional Chern
number. Our scheme can apply to extracting Chern number of both energy-
separable and energy-inseparable bands. Compared with schemes of Thouless
pumping or integer quantum Hall effect, our scheme does not require equal
superposition of states involving all momenta, but any particular momentum
state in a band can be chosen as the initial state. Meanwhile, the
introduction of tilts also well suppresses the diffusion of Gaussian
wavepackets and improves the measurement accuracy of the Chern number near the
phase transition point.

In this work, we show how to use Bloch oscillations to extract Chern number of
a super-band even when there exist several degenerate points. This is because
the Bloch oscillations in the momentum space avoid the degenerate points and
the non-Abelian part of the Berry curvature takes no effect. If the super-band
contains completely degenerated bands, there is no way to avoid the degeneracy
and non-Abelian Berry curvature must play an important role in the wavepacket
dynamics. It is unclear how to apply our method to measure topological
invariants such as second Chern number of a completely degenerated super-band
Sugawa  _et al._ (2018); Mochol-Grzelak  _et al._ (2018). Inversely, it is
also very interesting to explore how to tail the Bloch oscillations with such
nontrivial topological super-band.

Recently, we became aware of two new works studying the center-of-mass drift
in a fractional Chern insulator subjected to a tilted field in only one
direction Motruk and Na (2020); Repellin  _et al._ (2020). However, the drifts
are not quite well quantized, which may come from the fluctuation of Berry
curvature in different center-of-mass momenta. If the tilted fields are
introduced along both directions, the center-of-mass momentum may be sampled
in a more uniform way and the fluctuation may be wiped away. We believe our
method can be extended to interacting topological systems Rachel (2018) and
could potentially provide more accuracy measurement of fractional topological
states.

###### Acknowledgements.

The authors are thankful to Chaohong Lee for enlightening suggestions and
helpful discussions. This work is supported by the National Natural Science
Foundation of China under Grant No.11805283, the Hunan Provincial Natural
Science Foundation under Grant No.2019JJ30044, the Scientific Research Fund of
Hunan Provincial Education Department under Grant No.19A510 and the Talent
project of Central South University of Forestry and Technology under Grant
No.2017YJ035. Y.K. is supported by the National Natural Science Foundation of
China (Grant No.11904419).

##  Appendix A The Hamiltonian in momentum space

According to the Hamiltonian (II), the eigenvalue equation can be written as,

$\displaystyle-E|m,n\rangle$ $\displaystyle=$
$\displaystyle\tau_{x}e^{iF_{x}t}|m+1,n\rangle+\tau_{y}e^{iF_{y}t}e^{i2\pi\beta
m}|m,n+1\rangle$ (21) $\displaystyle+$
$\displaystyle\tau_{x}e^{-iF_{x}t}|m-1,n\rangle+\tau_{y}e^{-iF_{y}t}e^{-i2\pi\beta
m}|m,n-1\rangle$ $\displaystyle+$
$\displaystyle\frac{\delta}{2}\left[(-1)^{m}+(-1)^{n}\right]|m,n\rangle.$

Here we consider $\beta=1/4$ which describes hopping on the square lattice in
the presence of a magnetic flux $2\pi\beta=\pi/2$ per plaquette. To solve this
equation, we make the following ansatz for the wave function:

$\displaystyle|m,n\rangle=e^{ik_{x}m}e^{ik_{y}n}\left\\{\begin{array}[]{ll}\psi_{A},&\text{
for }m,n\text{ odd };\\\ \psi_{B}e^{im\pi/2}&\text{ for }m\text{ even
},n\text{ odd };\\\ \psi_{C}&\text{ for }m\text{ odd, }n\text{ even };\\\
\psi_{D}e^{im\pi/2}&\text{ for }m,n\text{ even }.\end{array}\right.$ (23)

Here, $k_{x},k_{y}$ are defined within the first magnetic Brillouin zone.
Inserting this ansatz into the Schrödinger equation we obtain the following
$4\times 4$ eigenvalue equation

$\displaystyle h(k_{x},k_{y},t)\left(\begin{array}[]{l}\psi_{A}\\\ \psi_{B}\\\
\psi_{C}\\\
\psi_{D}\end{array}\right)=E(k_{x},k_{y},t)\left(\begin{array}[]{c}\psi_{A}\\\
\psi_{B}\\\ \psi_{C}\\\ \psi_{D}\end{array}\right),$ (32)

with

$\displaystyle
h=2\left(\begin{array}[]{cccc}-\frac{\delta}{2}&-\tau_{y}\cos\kappa_{y}&-\tau_{x}\cos\kappa_{x}&0\\\
-\tau_{y}\cos\kappa_{y}&0&0&\tau_{x}\sin\kappa_{x}\\\
-\tau_{x}\cos\kappa_{x}&0&0&i\tau_{y}\sin\kappa_{y}\\\
0&\tau_{x}\sin\kappa_{x}&-i\tau_{y}\sin\kappa_{y}&\frac{\delta}{2}\end{array}\right),$
(37)

where $\kappa_{x}=k_{x}-F_{x}t$, $\kappa_{y}=k_{y}-F_{y}t$,
$k_{x}\in[-\pi/2,\pi/2]$ and $k_{y}\in[-\pi/2,\pi/2]$.

##  Appendix B Relation between reduced Chern number and Chern number

When the tilt is absent, the model (1) is reduced to the Harper-Hofstadter-
like topological model, $\widetilde{H}=H_{1}+H_{2}$, whose Hamiltonian in
momentum space is given by $\widetilde{h}(k_{x},k_{y})=h(k_{x},k_{y},0)$.
Compared to the case with tilts, the corresponding bands and eigenstates can
be obtained by replacing $\kappa_{x}$ by $k_{x}$ and $\kappa_{y}$ by $k_{y}$.
The conventional Chern number can be defined by the integral of Berry
curvature in the two-dimensional Brillouin zone as

$\displaystyle
C_{l}=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}dk_{x}\int_{-\pi/2}^{\pi/2}dk_{y}\mathcal{F}_{l}(k_{x},k_{y}),$
(39)

where

$\mathcal{F}_{l}(k_{x},k_{y})=-2\operatorname{Im}\sum_{l^{\prime}\neq
l}\frac{\left\langle
u_{l}\left|\partial_{k_{x}}\widetilde{h}\right|u_{l^{\prime}}\right\rangle\left\langle
u_{l^{\prime}}\left|\partial_{k_{y}}\widetilde{h}\right|u_{l}\right\rangle}{\left(E_{l}-E_{l^{\prime}}\right)^{2}}.$
(40)

The Chern number can be further written as

$\displaystyle C_{l}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}dk_{x}\int_{-\pi/2}^{\pi/2}d\kappa_{y}\mathcal{F}_{l}(k_{x},\kappa_{y})$
(41)
$\displaystyle=\frac{1}{4\pi}\int_{-\pi/2}^{\pi/2}dk_{x}\int_{0}^{T_{y}}\mathcal{F}_{l,x}(k_{x},k_{y},t)dt$
$\displaystyle=\frac{1}{\eta_{y}\pi}\int_{-\pi/2}^{\pi/2}dk_{x}C_{l,x}(k_{x},k_{y}).$

When $\eta_{x}/\eta_{y}$ approaches $0$ or $\infty$, $C_{l,x}(k_{x},k_{y})$
tends to be an integer $C_{l,x}^{0}$, almost independent of both $k_{x}$ and
$k_{y}$ (see Appendix C for details). Hence we can get rid of the average over
$k_{x}$ in the above equation and yield $C_{l}=C_{l,x}^{0}/\eta_{y}$. In the
similar way, the Chern number can be also written as

$\displaystyle
C_{l}=-\frac{1}{\eta_{x}\pi}\int_{-\pi/2}^{\pi/2}dk_{y}C_{l,y}(k_{x},k_{y})=-C_{l,y}^{0}/\eta_{x}.$
(42)

As a consequence, the Chern number is related to $C_{l,x}^{0}$ and
$C_{l,y}^{0}$ via

$\displaystyle C_{l}=C_{l,x}^{0}/\eta_{y}=-C_{l,y}^{0}/\eta_{x}.$ (43)

Indeed, $C_{l,x}$ and $C_{l,y}$ defined by a one dimensional integral can be
effectively regarded as a reduced expression for the Chern number $C_{l}$,
which we call Reduced Chern number(RCN) Ke  _et al._ (2020).

##  Appendix C Convergence of reduced Chern number when
$F_{x}/F_{y}=\eta_{x}/\eta_{y}$ approaches $0$ or $\infty$

When $F_{x}/F_{y}=\eta_{x}/\eta_{y}$ approaches $0$ or $\infty$,
$C_{l,x}(k_{x},k_{y})$ and $C_{l,y}(k_{x},k_{y})$ respectively tend to be
integer numbers $C_{l,x}^{0}$ and $C_{l,y}^{0}$, independent of the initial
momentum value of a Bloch state. It is mainly because the Bloch oscillations
sample the Berry curvature in an uniform way, and the sampling becomes more
uniform when $\eta_{x}/\eta_{y}$ tends to be $0$ or $\infty$. Without loss of
generality, we will explain why $C_{l,x}(k_{x},k_{y})$ are convergent to a
constant RCN in the limit of $\eta_{x}/\eta_{y}\rightarrow\infty$.

We start from the property of Berry curvatures, that is,
$\mathcal{F}_{l}(k_{x},k_{y},t)$ is a periodic function of quasi-momentum and
time. Thus, the integral of Berry curvature over an overall period of time is
independent of the initial time,

$\displaystyle{C_{l,x}}({k_{x}},{k_{y}})$ $\displaystyle=$
$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x}}({k_{x}},{k_{y}},t)dt}$ $\displaystyle=$
$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x(y)}}({k_{x}},{k_{y}},t-\Delta{k_{x}}/{F_{x}})dt}.$

We know that the quasi-momentum $({\kappa_{x}},{\kappa_{y}})$ change with time
as ${\kappa_{x(y)}}={k_{x(y)}}-{F_{x(y)}}t$. When we shift the time to
$t-\Delta{k_{x}}/{F_{x}}$, it is equivalent to shift $k_{x}$ to
${k_{x}}+\Delta{k_{x}}$ and $k_{y}$ to ${k_{y}}+\Delta{k_{y}}$ while maintain
the time as $t$. Eq. (C) can be further written as

$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x(y)}}({k_{x}},{k_{y}},t-\Delta{k_{x}}/{F_{x}})dt}$ $\displaystyle=$
$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x(y)}}({k_{x}}+\Delta{k_{x}},{k_{y}}+{F_{y}}/{F_{x}}\Delta{k_{x}},t)dt}.$

It means that

$\displaystyle{C_{l,x}}({k_{x}},{k_{y}})={C_{l,x}}({k_{x}}+\Delta{k_{x}},{k_{y}}+{F_{y}}/{F_{x}}\Delta{k_{x}}).$
(46)

When $F_{x}/F_{y}=\eta_{x}/\eta_{y}\rightarrow\infty$, for arbitrary $\Delta
k_{x}$, we can make an approximation,

$\displaystyle{C_{l,x}}({k_{x}},{k_{y}})$ $\displaystyle=$
$\displaystyle{C_{l,x}}({k_{x}}+\Delta{k_{x}},{k_{y}}+{F_{y}}/{F_{x}}\Delta{k_{x}})$
(47) $\displaystyle\approx$
$\displaystyle{C_{l,x}}({k_{x}}+\Delta{k_{x}},{k_{y}}).$

Making $\Delta k_{x}={F_{x}}/{F_{y}}\Delta{k_{y}}$, one can have

$\displaystyle{C_{l,x}}({k_{x}},{k_{y}})\approx{C_{l,x}}({k_{x}}+{F_{x}}/{F_{y}}\Delta{k_{y}},{k_{y}}).$
(48)

Similarly, Eq. (C) also can be written as

$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x(y)}}({k_{x}},{k_{y}},t-\Delta{k_{y}}/{F_{y}})dt}$ $\displaystyle=$
$\displaystyle\frac{1}{q}\int_{0}^{{T_{o}}}{{{\cal
F}_{l,x(y)}}({k_{x}}+{F_{x}}/{F_{y}}\Delta{k_{y}},{k_{y}}+\Delta{k_{y}},t)dt}.$

It means that

$\displaystyle{C_{l,x}}({k_{x}},{k_{y}})=C({k_{x}}+{F_{x}}/{F_{y}}\Delta{k_{y}},{k_{y}}+\Delta{k_{y}}).$
(50)

Combining Eqs. (48) and (50), One can immediately obtain

${C_{l,x}}({k_{x}^{\prime}},{k_{y}})\approx{C_{l,x}}({k_{x}^{\prime}},{k_{y}}+\Delta
k_{y}).$ (51)

for any arbitrary $\Delta k_{y}$. From Eqs. (47) and (50), it is found that
$C_{l,x}(k_{x},k_{y})$ are almost the same and denoted as $C_{l,x}^{0}$,
independent of both $k_{x}$ and $k_{y}$ in the limit of
$\eta_{x}/\eta_{y}\rightarrow\infty$. The Chern number is related to the RCN
as

${C_{l}}=\frac{1}{{{\eta_{y}}\pi}}\int_{-\pi/2}^{\pi/2}d{k_{x}}{C_{l,x}}({k_{x}},{k_{y}})=\frac{{C_{l,x}^{0}}}{{{\eta_{y}}}}.$
(52)

By swapping $k_{x}$ and $k_{y}$, we can also prove the relation Eq. (43) in
the limit of $\eta_{x}/\eta_{y}$ approaches $0$ or $\infty$.

As an example, we give the distribution of Berry curvature
$\mathcal{F}_{1}\left(k_{x},k_{y}\right)$ in the two-dimensional Brillouin
zone; see Fig. 8(a). Fig. 8(b)-(d) correspond to sampling trajectories of
${\mathcal{F}_{1,x}}/{F_{y}}$ in the momentum space for $\eta_{x}/\eta_{y}=6$,
$10$ and $40$, respectively. We clearly see that the sampling density
increases with the increase of $\eta_{x}/\eta_{y}$. When the sampling density
is large enough, the two-dimensional integral about the conventional Chern
number can be replaced by the linear integral of sampling trajectory. Because
the initial momentum value only determines the initial position of the
sampling trajectory, without affecting sampling density, we have
$C_{1,x}\left(k_{x},k_{y}\right)/\eta_{y}=C_{l,x}^{0}/\eta_{y}$. We have to
emphasize that the almost quantized RCN is quite general and it does not
require the uniform distribution of Berry curvature.

Figure 8:  (a) The distribution of Berry curvature
$\mathcal{F}_{1}\left(k_{x},k_{y}\right)$ in two-dimensional Brillouin zone.
(b)-(d) Sampling trajectories of ${\mathcal{F}_{1,x}}/{F_{y}}$ respectively
correspond to $F_{x}/F_{y}=\eta_{x}/\eta_{y}=6$, $10$ and $40$ with fixed norm
$\sqrt{F_{x}^{2}+F_{y}^{2}}=0.0402$. The other parameters are chosen as
$\tau_{x}=\tau_{y}=0.2,\delta=0$.  Figure 9: Drift $\Delta X$ and $\Delta Y$
versus time $t$ for different ($k_{x_{0}},k_{y_{0}}$). (a) Topologically
nontrivial phase ($C_{1}=1$, $\delta=0$). (b)Topologically trivial phase
($C_{1}=0$, $\delta=4\tau_{x}$). The other parameters are chosen as
$\tau_{x}=\tau_{y}=0.2$, $\eta_{x}/\eta_{y}=10$, $F_{x}=0.04$ and
$F_{y}=0.004$.

##  Appendix D The Influence of initial quasimomentum on quantized drift

To verify that the mean quasimomentum of an initial Gaussian wavepacket can be
arbitrarily chosen, we give the drift $\Delta X$ and $\Delta Y$ versus time
$t$ for different initial quasimomentum ($k_{x_{0}},k_{y_{0}}$) in Fig. 9.
Despite the drift trajectories of $\Delta X$ and $\Delta Y$ have a significant
difference for different mean quasimomentum, but these trace lines will
converge at one point when $t=T_{o}$. Fig. 9(a) corresponds to the
topologically nontrivial phase, no matter how we choose the initial mean
quasimomentum, we always obtain $\Delta X(T_{o})=4,\Delta Y(T_{o})=-40$, and
the RCNs $C_{1,x}^{0}/\eta_{y}=-C_{1,y}^{0}/\eta_{x}=1$ by combining the Eqs.
(15) and (III.1), which are consistent with the first band Chern number
$C_{1}=1$. Similarly, one can also obtain $\Delta X(T_{o})=0,\Delta
Y(T_{o})=0$ in Fig. 9(b), and the RCNs
$C_{1,x}^{0}/\eta_{y}=-C_{1,y}^{0}/\eta_{x}=0$ are consistent with the first
band Chern number $C_{1}=0$. These results prove again that our method is
rather insensitive and robust to the choice of initial quasimomentum.

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