# Moiré flat bands in alternating twisted MoTe2 multilayer

Miao Liang  School of Physics and Wuhan National High Magnetic Field Center,
Huazhong University of Science and Technology, Wuhan 430074, China  Shi-Ping
Ding  School of Physics and Wuhan National High Magnetic Field Center,
Huazhong University of Science and Technology, Wuhan 430074, China  Ming Wu
School of Physics and Wuhan National High Magnetic Field Center, Huazhong
University of Science and Technology, Wuhan 430074, China  Chen Zhao  School
of Physics and Wuhan National High Magnetic Field Center, Huazhong University
of Science and Technology, Wuhan 430074, China  Jin-Hua Gao
<EMAIL_ADDRESS>School of Physics and Wuhan National High Magnetic Field
Center, Huazhong University of Science and Technology, Wuhan 430074, China

###### Abstract

The long-awaited fractional quantum anomalous Hall (FQAH) effect recently has
been observed in the twisted $\mathrm{MoTe_{2}}$ homobilayers, causing a great
sensation. Here, we theoretically investigate the moiré band structures of a
closely related system, the alternating twisted multilayer $\mathrm{MoTe_{2}}$
(ATML-MoTe2), where the adjacent layers have opposite twist angles. We
illustrate that such ATML-MoTe2 is a very unique moiré system, exhibiting
multiple topological flat bands highly controllable by the layer number and
twist angle, which is not only an ideal platform to simulate Hubbard model,
but also may host FQAH states. Specifically, an N-layer ATML-MoTe2 ($N\geq 3$)
always possesses $N-2$ topological flat bands near Fermi energy $E_{f}$, which
has an odd-even dependent decomposition rule to understand the behaviors of
the moiré flat bands. We predict three intriguing examples: (1) The AT3L-MoTe2
($N=3$) has one isolated moiré flat band, which corresponds to a triangular
lattice Hubbard model, resembling the twisted TMD heterobilayers. (2) The
AT4L-MoTe2 ($N=4$) has two topological flat bands that are very similar to the
twisted $\mathrm{MoTe_{2}}$ homobilayers, implying the possible existence of
FQAH states. (3) When $N>4$, the giant density of states (DOS) induced by the
multiple moiré flat bands may induce exotic correlated states.

##  I Introduction

Recently, the fractional quantum anomalous Hall (FQAH) effect has been
successfully observed in twisted $\mathrm{MoTe_{2}}$ homobilayers, which has
sparked widespread research interest Cai  _et al._ (2023); Park  _et al._
(2023); Zeng  _et al._ (2023a); Kang  _et al._ (2024); Xu  _et al._ (2023).
The emergence of the FQAH states in twisted $\mathrm{MoTe_{2}}$ homobilayers
is attributed to its isolated topological moiré flat band Wu  _et al._ (2019);
Li  _et al._ (2021a); Reddy  _et al._ (2023); Wang  _et al._ (2024); Qiu  _et
al._ (2023); Goldman  _et al._ (2023); Xu  _et al._ (2024); Reddy and Fu
(2023); Yu  _et al._ (2024); Abouelkomsan  _et al._ (2024); Zeng  _et al._
(2023b). Meanwhile, it is further suggested that the uniformly distributed
Berry curvature is crucial for the topological flat bands to realize the FQAH
states Devakul  _et al._ (2021). All of these important theoretical and
experimental studies reveal the uniqueness of the $\mathrm{MoTe_{2}}$-based
moiré structures.

The alternating twisted multilayer moiré structure is a natural generalization
of the twisted homobilayers, where the twist angles of adjacent layers are
$\pm\theta$ with opposite rotation directions. The celebrated example is the
alternating twisted few-layer graphene (ATFLG)Khalaf  _et al._ (2019); Carr
_et al._ (2020); Lei  _et al._ (2021); Shin  _et al._ (2021); Wu  _et al._
(2021); Park  _et al._ (2021); Hao  _et al._ (2021); Cao  _et al._ (2021a);
Lopez-Bezanilla and Lado (2020); Xie  _et al._ (2022); Ma  _et al._ (2023);
Liang  _et al._ (2022); Ding  _et al._ (2023); Phong  _et al._ (2021);
Scammell  _et al._ (2022), which also host moiré flat bands very similar as
the twisted bilayer graphene (TBG)Cao  _et al._ (2018, 2018); Bistritzer and
MacDonald (2011); Moon and Koshino (2013); Cao  _et al._ (2021b).
Interestingly, it is proved that there is a decomposition rule, by which an
ATFLG can be decomposed into several effective TBGs with renormalized
parameters Khalaf  _et al._ (2019). Therefore, it can be expected that
correlated states in TBG will also exist in ATFLG. Note that superconductivity
in ATFLG has been observed in recent experiments Park  _et al._ (2021); Hao
_et al._ (2021); Park  _et al._ (2022); Zhang  _et al._ (2022); Burg  _et al._
(2022). Inspired by the success of ATFLG, a fascinating question arises: can
the alternating twisted multilayer $\mathrm{MoTe_{2}}$ (ATML-MoTe2) generate
richer moiré band structures, and thus induce more novel correlation and
topological effects, such as FQAH states ?

Figure 1: Schematic diagram of ATML-MoTe2. (a) Schematic of AT4L-MoTe2, with
the rotation directions of the 1st and 3rd layers (blue) opposite to those of
the 2nd and 4th layers (green). (b) Top view of an AA-stacked AT4L-MoTe2. The
inset shows a side view between adjacent layers. (c) Brillouin zones of the
odd-layer (blue), the even-layer (green), and the moiré Brillouin zone
(black).

In this work, we systematically study the moiré band structures of the ATML-
MoTe2 based on the continuum model method. We find that ATML-MoTe2 is a unique
moiré system hosting multiple topological flat bands, which are highly
adjustable by the layer number and twist angle. Specifically, an $N$-layer
ATML-MoTe2 ($N\geq 3$) always has $N-2$ topological flat bands at small twist
angle, and the intriguing features of these topological flat bands are
summarized as follows:

  1. 1.

_Multiple degenerate flat bands at small twist angle_. When $\theta\lesssim
1.4^{\circ}$, the $N-2$ flat bands of the $N$-layer ATML-MoTe2 are nearly
degenerate at $E_{f}$, with a large band gap separating these flat bands from
other low energy bands.

  2. 2.

_Isolated topological flat band at magic angle_. As $\theta$ increases, the
degeneracy of these flat bands is lifted and an isolated flat band appears
near the $E_{f}$. For even $N$, there is a magic angle near $2.02^{\circ}$, at
which an isolated topological flat band is achieved.

  3. 3.

_Decomposition rules_. An even-odd dependent decomposition rule is proved,
which can well interpret the origin and behaviors of the $N-2$ topological
flat bands. Meanwhile, there exists a mirror symmetry decomposition for the
odd $N$ cases.

Based on our calculation, we can give three interesting predictions:

  1. 1.

The AT3L-MoTe2 ($N=3$) has one isolated moiré flat band near $E_{f}$, which
can be described as a triangular lattice Hubbard model, resembling the twisted
transition metal dichalcogenides (TMD) heterobilayersWu  _et al._ (2018a);
Tang  _et al._ (2020); Morales-Durán  _et al._ (2022).

  2. 2.

The AT4L-MoTe2 ($N=4$) has two topological moiré flat bands near $E_{f}$,
which exhibit similar properties to those of the twisted $\mathrm{MoTe_{2}}$
homobilayers, making it a promising moiré system for achieving the FQAH
effect.

  3. 3.

When $N>4$, the presence of multiple degenerate flat bands implies a giant DOS
at $E_{f}$, which is expected to promote the formation of various correlated
insulating states.

Our results indicate that the ATML-MoTe2 is an ideal platform for simulating
both the Hubbard model and FQAH states, promising to host a variety of exotic
topological and correlated quantum states like that in twisted TMD
homobilayers Cai  _et al._ (2023); Park  _et al._ (2023); Zeng  _et al._
(2023a); Xu  _et al._ (2023); Wu  _et al._ (2019); Li  _et al._ (2021a); Reddy
_et al._ (2023); Wang  _et al._ (2024); Qiu  _et al._ (2023); Goldman  _et
al._ (2023); Xu  _et al._ (2024); Reddy and Fu (2023); Yu  _et al._ (2024);
Abouelkomsan  _et al._ (2024); Zeng  _et al._ (2023b); Devakul  _et al._
(2021); Fang  _et al._ (2015); Wang  _et al._ (2017); Naik and Jain (2018);
Zhan  _et al._ (2020); Wang  _et al._ (2020); Zhang  _et al._ (2020); Ghiotto
_et al._ (2021); Li  _et al._ (2021b); Xu  _et al._ (2022). We expect that our
predictions can be immediately tested in future experiments.

The paper is organized as follows. In Sec. II, the model and methods of ATML-
MoTe2 used in the calculations are given. In Sec. III, we discuss the
calculation results. Sec. IV is a short summary.

##  II MODEL AND METHODS

###  II.1 The continuum model of ATML-MoTe2

In Fig. 1, we give a schematic of ATML-MoTe2. The continuum model of ATML-
MoTe2 can be given in a similar way to that of the twisted TMD homobilayers Wu
_et al._ (2019). For each $\rm{MoTe_{2}}$ monolayer, the valence electrons are
around the $\pm K$ valleys, which are mainly of the $d$-orbial character with
opposite spin due to spin-valley lockingXiao  _et al._ (2012); Wu  _et al._
(2019). Considering the layer-dependent potential and interlayer tunneling,
the Hamiltonian of $N$-layer ATML-MoTe2 of the $+K$ valley is

$\displaystyle\mathcal{H}=\left(\begin{array}[]{ccc}\mathcal{H}_{1}&T_{bm}&0\\\
T_{bm}^{+}&\mathcal{\widetilde{H}}_{m}&T_{tm}\\\
0&T_{tm}^{+}&\mathcal{H}_{N}\end{array}\right),$ (1)

where $\mathcal{H}_{1}$ and $\mathcal{H}_{N}$ describe the bottom (1st) and
top (Nth) TMD layer, respectively. $\widetilde{H}_{m}$ represents the middle
$N-2$ layers.

For the outermost two layers,

$\displaystyle\mathcal{H}_{1/N}=-\frac{\hbar^{2}\left(\mathbf{k}-\mathbf{\kappa}_{1/N}\right)^{2}}{2m^{*}}+\Delta\left(\mathbf{r},\theta_{1/N}\right),$
(2)

where $\theta_{1/N}$ is the twist angle of the 1st/Nth monolayer, and
$\mathbf{\kappa}_{1/N}$ is the corresponding momentum offset. As shown in Fig.
1, for the alternating twist case, we always assume the twist angle of the odd
(even) layer is $\theta/2$ ($-\theta/2$), and the momentum offset is
$\mathbf{\kappa}_{+}$ ($\mathbf{\kappa}_{-}$) for the odd (even) layer, where
$\mathbf{\kappa}_{\pm}=\left[4\pi/\left(3a_{M}\right)\right](-\sqrt{3}/2,\mp
1/2)$. $a_{M}=a_{0}/\theta$ is the period of the moiré potential, and $a_{0}$
is the lattice constant of the TMD monolayer.
$\Delta\left(\mathbf{r},\theta\right)$ describes the layer-dependent moiré
potential

$\begin{aligned}
\Delta(\mathbf{r},\theta)=2V\sum_{j=1,3,5}\cos\left(\mathbf{g}_{j}\cdot\mathbf{r}+\eta\psi\right)\end{aligned},$
(3)

where $\mathbf{g}_{j}$ is $(j-1)\pi/3$ counterclockwise rotation of the moiré
reciprocal lattice vector $\mathbf{g}_{1}=(4\pi/\sqrt{3}a_{M},0)$.
$\eta(\theta)=\mathrm{sign}(\theta)$ denotes the direction of the rotation,
e.g. $\eta(\theta/2)=1$. For $\rm{MoTe_{2}}$, we set
$a_{0}=3.472\mathring{\rm{A}}$, $V=8~{}\rm{meV}$, $\psi=-89.6^{\circ}$, and
the effective mass $m^{*}=0.62~{}m_{e}$ Wu  _et al._ (2019); Liu  _et al._
(2013).

For the middle $N-2$ layers, the Hamiltonian $\mathcal{\widetilde{H}}_{m}$ is,

$\displaystyle\mathcal{\widetilde{H}}_{m}=\left(\begin{array}[]{ccccc}\mathcal{H}_{2}&\Delta_{T}^{\dagger}(\mathbf{r})&0&\cdots&0\\\
\Delta_{T}(\mathbf{r})&\mathcal{H}_{3}&\Delta_{T}(\mathbf{r})&\cdots&0\\\
0&\Delta_{T}^{\dagger}(\mathbf{r})&\mathcal{H}_{4}&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\cdots&\mathcal{H}_{N-1}\end{array}\right)$ (4)

where $\mathcal{H}_{i}$ is the Hamiltonian of the i-th layer with $1<i<N$.

$\displaystyle\mathcal{H}_{i}=-\frac{\hbar^{2}\left(\mathbf{k}-\mathbf{\kappa}_{i}\right)^{2}}{2m^{*}}+2\Delta\left(\mathbf{r},\theta_{i}\right)$
(5)

where $i=2,3,...,N-1$ is the layer index. $\mathbf{\kappa}_{i}$ and
$\theta_{i}$ are the momentum offset and twist angle of the ith layer as
defined before. Note that the $i$th layer here feels the moiré potential from
both the $(i-1)$th and $(i+1)$th layers, which thus is distinct from the top
and bottom layers in Eq. (2). $\Delta_{\mathrm{T}}$ is the interlayer
tunneling

$\displaystyle\Delta_{T}(\mathbf{r})=w\left(1+e^{-i\cdot\mathbf{g}_{2}\cdot\mathbf{r}}+e^{-i\cdot\mathbf{g}_{3}\cdot\mathbf{r}}\right).$
(6)

where $w=-8.5$ meV is the interlayer tunneling strength Wu  _et al._ (2019).

The interlayer tunneling terms in Eq. (1) are

$\displaystyle T_{bm}=\left(\Delta_{T}(\mathbf{r}),0,\ldots,0\right)$ (7)
$\displaystyle T_{tm}=\left(0,\ldots,0,\Delta_{T}(\mathbf{r})\right)$

###  II.2 Decomposition rules

Firstly, for an $N$-layer ATML-MoTe2, it should be noted that its electronic
states near $E_{f}$, i.e., the moiré flat bands, mainly originate from the
middle $N-2$ layers $\widetilde{H}_{m}$, with little influence from the
outermost two layers. It is because, as shown in Eq. (2) and (5), the moiré
potential applied on the outermost two layers is $\Delta$, while that of the
middle layers is $2\Delta$. The difference of moiré potential is significant,
causing the electronic states of the outermost two layers to be far from the
$E_{f}$. Therefore, it is a good approximation to use the electronic structure
of $\widetilde{H}_{m}$ to understand the moiré band structures of ATML-MoTe2
near $E_{f}$, where the influence of the outermost two layers can be
disregarded. The numerical results show that such an approximation works quite
well when $\theta$ is small. Hereafter, we always use $N$ to refer to the
total number of layers in an ATML-MoTe2.

Very interestingly, $\widetilde{H}_{m}$ here has an even-odd dependent
decomposition rule, due to its alternating twist configuration. The
decomposition rule here is very similar to that of the ATFLG. First, the
middle $N-2$ layers in $\widetilde{H}_{m}$ can be divided into two sets, i.e.
$\\{H_{i}|i\in even\\}$ and $\\{H_{i}|i\in odd\\}$, where $H_{i}$ are
identical in each set due to the alternating twist structure. As is proved in
Ref. Ledwith  _et al._ , 2021, such kind of Hamiltonian matrix can be
decomposed using singular value decomposition scheme. This decomposition
depends on whether the layer number of $\widetilde{H}_{m}$, i.e. $(N-2)$, is
even or odd:

$\mathcal{\widetilde{H}}^{dec}_{m}=\begin{cases}\oplus^{n}_{\nu=1}h(\lambda_{\nu}),&\text{if
N-2 is even},\\\
\oplus^{n}_{\nu=1}h(\lambda_{\nu})\oplus\mathcal{H}_{2},&\text{if N-2 is
odd},\end{cases}$ (8)

where $\lambda_{\nu}=2\cos\frac{\pi\nu}{N-1}$ with $\nu=1,...,n$ and
$n=[(N-2)/2]$. Here, $h(\lambda_{\nu})$ is an effective twisted TMD
homobialyer with coupling parameter $\lambda_{\nu}$

$\displaystyle
h(\lambda_{\nu})=\left(\begin{array}[]{cc}\mathcal{H}_{2}&\lambda_{\nu}\Delta_{T}(\mathbf{r})^{\dagger}\\\
\lambda_{\nu}\Delta_{T}(\mathbf{r})&\mathcal{H}_{3}\end{array}\right).$ (9)

where $H_{2}$ actually represents a twisted heterobilayers, i.e. an effective
TMD monolayer under a moiré potential as given in Eq. (5). Note that, in Eq.
(8) and (9), $H_{2}$ ($H_{3}$) is used to denote an effective TMD monolayer
with a twist angle $-\theta/2$ ($\theta/2$). The decomposition above implies
that, near $E_{f}$, the moiré bands of $N$-layer ATML-MoTe2 can be
approximately decoupled into those of several twisted TMD homobilayers (and a
TMD monolayer with a moiré potential) if $N$ is even (odd).

Furthermore, each $h(\lambda_{\nu})$, as a twisted TMD homobilayers, will
provide two moiré flat bands near $E_{f}$, and the $H_{2}$ in Eq. (8) will
offer one isolated moiré flat band at $E_{f}$, similar as the twisted TMD
heterobilayers. So, the decomposition rules in Eq. (8) actually imply that,
with a small $\theta$, an N-layer ATML-MoTe2 should always exhibit $N-2$ moiré
flat bands near $E_{f}$.

In addition to the decomposition rule with Eq. (8), an $N$-layer ATML-MoTe2
with odd $N$ exhibits mirror symmetry relative to the middle monolayer,
resulting in a mirror symmetry decomposition as wellDing  _et al._ (2023);
Zhang  _et al._ (2023). It should be noted that the mirror symmetry
decomposition here is exact, but the decomposition in Eq. (8) is only valid
for the moiré bands very near $E_{f}$.

##  III RESULTS AND DISCUSSION

In this section, we show our numerical results about the ATML-MoTe2, where the
cases of even and odd $N$ are discussed separately.

###  III.1 Odd-layer ATML-MoTe2

####  III.1.1 AT3L-MoTe2

We first consider the case of AT3L-MoTe2 ($N=3$). As we will see, an
AT3L-MoTe2 always has one moiré flat band near $E_{f}$, which is approximately
equivalent to a triangle lattice Hubbard model resembling the twisted TMD
heterobilayers when $\theta$ is small.

The AT3L-MoTe2 has a mirror symmetry relative to the middle monolayer. So, as
mentioned in the last section, we can do a mirror symmetry decomposition for
its moiré bands. To do the mirror symmetry decomposition, we first have to
build a parity-resolved basis
$\\{|\phi_{1,3},e\rangle,|\phi_{2},e\rangle,|\phi_{1,3},o\rangle\\}$ according
to the mirror symmetry, where

$\displaystyle|\phi_{1,3},e\rangle=\frac{1}{\sqrt{2}}(|\phi_{1}\rangle+|\phi_{3}\rangle),$
(10) $\displaystyle|\phi_{2},e\rangle=|\phi_{2}\rangle,$
$\displaystyle|\phi_{1,3},o\rangle=\frac{1}{\sqrt{2}}(|\phi_{1}\rangle-|\phi_{3}\rangle).$

Here, $e/o$ denotes the even/odd parity relative to the middle monolayer.
$|\phi_{1,2,3}\rangle$ are the original basis functions of the 1st, 2nd, 3rd
monolayers, used in the Hamiltonian (1). This parity-resolved basis gives rise
to a transformation matrix,

$\displaystyle U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}1&0&1\\\
0&\sqrt{2}&0\\\ 1&0&-1\end{array}\right).$ (11)

Under this transformation, we get a decoupled Hamiltonian
$\mathcal{H}_{mirr}^{\prime}=U^{-1}\mathcal{H}U$, where

$\displaystyle\mathcal{H}_{mirr}^{\prime}=\left(\begin{array}[]{ccc}\mathcal{H}_{1}&\sqrt{2}\Delta_{T}(\mathbf{r})&0\\\
\sqrt{2}\Delta_{T}^{\dagger}(\mathbf{r})&\mathcal{H}_{2}&0\\\
0&0&\mathcal{H}_{3}\end{array}\right).$ (12)

Figure 2: (a)-(d) The moiré band structures of AT3L-MoTe2 for (a)
$\theta=1^{\circ}$, (b) $1.43^{\circ}$, (c) $2.13^{\circ}$, and (d)
$3^{\circ}$. Blue (red) lines represent the bands with even (odd) parity. (e)
The bandwidth W for the 1st band, and the energy gap $\epsilon_{12}$
($\epsilon_{23}$) between the 1st (2nd) and 2nd (3rd) bands as a function of
$\theta$. By using different colors to identify the various regions of
$\theta$, the top two moiré bands have distinct Chern numbers $[C_{1},C_{2}]$
within these regions. (f) shows the 1st flat band wave function distribution
in each layer of the AT3L-MoTe2, which correspond to the $\gamma$ point for
$\theta=2.13^{\circ}$. (g) We present the wave function distribution of the
middle layer. (h) The blue solid line is the 1st band of AT3L-MoTe2 at
$\theta=2.13^{\circ}$, which is the enlarged view of the 1st band in (c). The
orange dashed line is a tight-binding-model fit to this band that includes
hopping up to the 3rd nearest-neighbor. (i) The yellow solid line is a
characteristic interaction energy scale for $\epsilon=10$, and the green solid
line is the nearest-neighbor hopping parameter $t_{1}$ as a function of moiré
period $a_{M}$.

We see that the Hamiltonian of AT3L-MoTe2 is now decoupled into two subsystems
with opposite parity. Formally, the part with even parity is very like twisted
TMD homobilayers with the interlayer tunneling scaled by $\sqrt{2}$, while the
part with odd parity is equivalent to a TMD monolayer with an applied moiré
potential, i.e. $H_{3}$. An important fact is that the moiré potentials in
$H_{1,3}$ are largely different from that in $H_{2}$, so that the electron
states from $H_{1,3}$ are pushed far away from the $E_{f}$, especially when
$\theta$ is small. Therefore, at a small twist angle, the moiré bands of
AT3L-MoTe2 near $E_{f}$ mainly originate the middle monolayer $H_{2}$ with
even parity. If we further note that $H_{2}$ here is nearly identical to the
Hamiltonian of a twisted TMD heterobilayers (e.g. $\mathrm{WSe_{2}/WS_{2}}$)
in form, which hosts one isolated moiré flat band at $E_{f}$, described by a
triangle lattice Hubbard modelWu  _et al._ (2018a); Tang  _et al._ (2020);
Morales-Durán  _et al._ (2022). It can be expected that, when $\theta$ is
small, an AT3L-MoTe2 will have a single isolated moiré flat band (even parity)
near $E_{f}$, similar to the one in $\mathrm{WSe_{2}/WS_{2}}$ system, which
can be described by a triangle lattice Hubbard model.

However, we emphasize that this picture of the triangle lattice Hubbard model
is only valid for small twist angle regions. It is because, as $\theta$
approaches $3^{\circ}$, the moiré band from $H_{1,3}$ in Eq. (12) will
approach that of $H_{2}$, and the hybridization between them cannot be
ignored. In other words, when $\theta$ becomes large enough, the part with
even parity in Eq. (12) will behave more like a twisted TMD homobilayers,
instead of a twisted TMD heterobilayers, and the 1st moiré band near $E_{f}$
will get a nonzero Chern number then.

The numerical results of the moiré bands for AT3L-MoTe2 with various twist
angles are given in Fig. 2, where blue (red) lines represent the bands with
even (odd) parity. Fig. 2 (a) shows the moiré band structure at
$\theta=1^{\circ}$. As we expected, at such small $\theta$, we do get an
isolated moiré flat band near $E_{f}$, which is separated from other moiré
bands by a large gap of about $20~{}\rm meV$. Meanwhile, the odd parity bands
(red lines) are far away from the $E_{f}$, so that the three topmost moiré
bands are all with even parity (blue lines). According to the analysis above,
we in fact can consider that the three topmost moiré bands mainly originate
from an equivalent twisted TMD heterobilayers like $\mathrm{WSe_{2}/WS_{2}}$,
where the first one is a moiré $s$-band. The numerical results indicate that
these three moiré bands are topologically trivial, i.e. Chern number is zero,
when $\theta$ is small.

The bandwidth of the 1st moiré band $W$ increases monotonically as $\theta$
increases, see Fig. 2 (e). We see that when $\theta<1.5^{\circ}$, the 1st
moiré band is almost completely flat as shown in Fig. 2 (b). In Fig. 2 (e), we
also plot the gap between the 1st moiré band and other low energy bands, i.e.
$\epsilon_{12}$, as a function of $\theta$. And the gap between the 2nd and
3rd moiré bands $\epsilon_{23}$ is also shown in Fig. 2 (e) as well. As
$\theta$ becomes larger, $\epsilon_{12}$ nearly monotonically decreases, and
it becomes zero when $\theta$ approaches $3^{\circ}$. Thus, in the regime of
$\theta<3^{\circ}$, the 1st moiré band remains isolated and its Chern number
stays at zero, see Fig. 2 (c).

Due to the similarity with the $\mathrm{WSe_{2}/WS_{2}}$ system, we expect
that the 1st moiré band here should be approximately equivalent to a triangle
lattice Hubbard model when $\theta$ is small. To verify this prediction, we
first calculate the wave function of the 1st moiré flat band ($\gamma$ point,
$\theta=2.13^{\circ}$), as shown in Fig. 2 (f) and (g). The numerical results
show that the wave function distribution of the 1st moiré band is mainly
localized in the middle monolayer, and the wave function distribution of
clearly forms a triangle lattice, as we expect. In this situation, the
influence of the outermost two layers, i.e. $H_{1,3}$ in Eq.(12), is small, so
that the triangle lattice Hubbard model is a rather good approximation. This
is also demonstrated clearly by the fitting of the 1st moiré band with a tight
binding model of triangle lattice, as given in Fig. 2 (h). The characteristic
Coulomb energy $\frac{e^{2}}{\epsilon a_{M}}$, which is proportional to the
on-site Hubbard $U_{0}$, is plotted in Fig. 2 (i). And, the nearest-neighbor
(NN) hopping parameter $t_{1}$ of the triangle lattice model got by fitting
the 1st moiré band as a function of moiré period $a_{M}$ is given in Fig. 2
(i) as well. Note that the exotic correlated states in twisted TMD
heterobilayers have been intensively studied recentlyWu  _et al._ (2018a);
Seyler  _et al._ (2019); Jin  _et al._ (2019); Tran  _et al._ (2019); Wu  _et
al._ (2018b, 2017); Xu  _et al._ (2020); Slagle and Fu (2020); Tang  _et al._
(2020); Huang  _et al._ (2021); Li  _et al._ (2021c); Regan  _et al._ (2020);
Kennes  _et al._ (2021); Li  _et al._ (2021d); Morales-Durán  _et al._ (2022);
Wang  _et al._ (2022); Xie  _et al._ (2023).

With a larger $\theta$, the 2nd and 3rd moiré bands can have a nonzero Chern
number once touching with other moiré bands occurs. Note that the Chern number
hereafter is for one valley, and the other valley has the opposite Chern
number due to the time reversal symmetry. In Fig. 2 (b), the Chern number of
the 3rd moiré band becomes $C_{3}=-1$ when $\theta=1.43^{\circ}$. And with
$\theta=2.13^{\circ}$, the 2nd moiré band has a nonzero Chern number
$C_{2}=1$, see Fig. 2 (c). Interestingly, as $\theta$ increases and crosses
$3^{\circ}$, the gap $\epsilon_{12}$ will first close and reopen, so that the
1st moiré band gets a nonzero Chern number $C_{1}=-1$ once $\theta>3^{\circ}$,
see Fig. 2 (d). As analyzed above, in this case, the triangle lattice Hubbard
model becomes invalid for the 1st moiré band. Based on our numerical
calculations above, we then can identify various regions of $\theta$ in Fig. 2
(e), denoted by different colors, in which the moiré bands have distinct Chern
numbers $[C_{1},C_{2}]$ (the 3rd moiré band is hard to be accessed in the
experiment).

Finally, we would like to mention that the moiré band structures of AT3L-MoTe2
have been also discussed in two other recent papers, but the numerical results
are analyzed from a completely different perspective Li  _et al._ (2024);
AlBuhairan and Vogl (2023). The decomposition of moiré bands, the analysis of
the origination of the 1st moiré band, and the prediction about the equivalent
Hubbard model are the focus of this work. Meanwhile, we also notice the
twisted TMD homotrilayer systems have been successfully fabricated in recent
experiments Zheng  _et al._ (2023); Liu  _et al._ (2023); Leng  _et al._
(2021); Liao  _et al._ (2020); Zhong  _et al._ (2024).

Figure 3: The moiré band structures of AT5L-MoTe2 for (a) $\theta=1^{\circ}$,
(b) $1.43^{\circ}$, (c) $2.13^{\circ}$, and (d) $3.48^{\circ}$. Blue (red)
lines represent the bands with even (odd) parity. In (e), we plot $W$,
$\epsilon_{12}$ and $\epsilon_{24}$ as functions of $\theta$. $W$ is the
bandwidth of the 1st band. $\epsilon_{12}$ ($\epsilon_{24}$) is the gap
between the 1st (2nd) and 2nd (4th) bands. Here, the 1st, 2nd and 4th bands
refer to the three topmost even parity bands (blue lines), while the 3rd band
is the topmost odd parity band (red lines). The regions with different Chern
numbers $[C_{1},C_{2},C_{3}]$ are denoted by different colors.

####  III.1.2 AT5L-MoTe2

The case of AT5L-MoTe2 ($N=5$) is more complex. According to the decomposition
rule in Eq. (8), we can expect that, when $\theta$ is not too large,
AT5L-MoTe2 should have three moiré flat bands near $E_{f}$ isolated from other
low energy bands, which are mainly from the middle three monolayers.

Note that all the ATML-MoTe2 with odd N have mirror symmetry relative to the
middle monolayer. So, we can also do the mirror symmetry decomposition for the
AT5L-MoTe2. Similarly, we first needed to build a parity-resolved basis
$\\{|\phi_{1,5},e\rangle,|\phi_{2,4},e\rangle,|\phi_{3},e\rangle,|\phi_{2,4},o\rangle|\phi_{1,5},o\rangle\\}$,
where

$\displaystyle|\phi_{1,5},e\rangle=\frac{1}{\sqrt{2}}(|\phi_{1}\rangle+|\phi_{5}\rangle)$
(13) $\displaystyle|\phi_{3},e\rangle=|\phi_{3}\rangle$
$\displaystyle|\phi_{1,5},o\rangle=\frac{1}{\sqrt{2}}(|\phi_{1}\rangle-|\phi_{5}\rangle)$

Based on this basis, we get the following transformation matrix,

$\displaystyle U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccccc}1&0&0&0&1\\\
0&1&0&1&0\\\ 0&0&\sqrt{2}&0&0\\\ 0&1&0&-1&0\\\ 1&0&0&0&-1\end{array}\right).$
(14)

After the transformation, a decoupled Hamiltonian is achieved

$\displaystyle\mathcal{H}_{mirr}^{\prime}$ (15) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}\mathcal{H}_{1}&\Delta_{T}(\mathbf{r})&0&0&0\\\
\Delta_{T}^{\dagger}(\mathbf{r})&\mathcal{H}_{2}&\sqrt{2}\Delta_{T}^{\dagger}(\mathbf{r})&0&0\\\
0&\sqrt{2}\Delta_{T}(\mathbf{r})&\mathcal{H}_{3}&0&0\\\
0&0&0&\mathcal{H}_{4}&\Delta_{T}^{\dagger}(\mathbf{r})\\\
0&0&0&\Delta_{T}(\mathbf{r})&\mathcal{H}_{5}\end{array}\right).$

In this decoupled Hamiltonian, the part with even parity is equivalent to an
alternating twisted TMD trilayer with asymmetric interlayer tunneling, where
the one between $H_{2}$ and $H_{3}$ is scaled by $\sqrt{2}$. Meanwhile, the
odd parity part is an equivalent twisted TMD bilayer.

Similar to that in AT3L-MoTe2, the moiré potentials in the outermost two
layers ($H_{1,5}$) are significantly different from those in the middle three
monolayers, which makes the electronic states from $H_{1}$ and $H_{5}$ far
from the $E_{f}$, especially at small twist angle. So, in the small twist
angle region, the even parity part will behave like the twisted TMD
homobilayers, which offers a pair of moiré flat bands near $E_{f}$. Similarly,
the odd parity part will host one isolated moiré flat band at $E_{f}$, which
originates from the $H_{4}$. Therefore, the mirror symmetry decomposition here
indicates that AT5L-MoTe2 exhibits three moiré flat bands at $E_{f}$ for a
small twist angle, which is in good agreement with the deduction of the
decomposition rule in Eq. (8). Actually, Eq. (8) also indicates that
AT5L-MoTe2 can be decoupled into one twisted TMD homobilayers
$h(\lambda_{\nu=1})$ and one twisted heterobilayers described as $H_{4}$.

The numerical results of the moiré bands of AT5L-MoTe2 are given in Fig. 3. In
Fig. 3 (a-d), we plot the moiré bands of AT5L-MoTe2 for different twist
angles, where the blue (red) lines are for the even (odd) parity. As expected,
when $\theta$ is small, AT5L-MoTe2 does have three moiré flat bands at
$E_{f}$, isolated from other low energy bands by an obvious gap, see Fig. 3
(a-c). Here, two of the three moiré flat bands with even parity (blue lines)
behave like that of a twisted MoTe2 homobilayers, while the left one (red
lines, odd parity) corresponds to the moiré flat band of a twisted TMD
heterobilayers.

At a small twist angle $\theta=1^{\circ}$, the three moiré flat bands are
nearly degenerate at $E_{f}$, see Fig. 3 (a). Such threefold degeneracy of the
moiré flat bands here will give rise to a giant DOS at $E_{f}$, which may
induce exotic correlated insulating states in experiments. Increasing
$\theta$, the bandwidth of the moiré flat bands increases and a gap between
the two even parity flat bands (blue lines) becomes observable, as shown in
Fig. 3 (b) and (c). In the region $3.2^{\circ}>\theta>1.8^{\circ}$ [Fig. 3 (b)
and (c)], the Chern number of the two even parity flat bands becomes nonzero,
i.e. $[C_{1},C_{2}]=[-1,1]$ like that in twisted MoTe2 homobilayers, while
that of the odd parity flat band (red lines) remains zero, i.e. $C_{3}=0$.
When $\theta$ is large enough, e.g. $\theta=3.48^{\circ}$ in Fig. 3 (d), the
three moiré flat bands are no longer isolated from other low energy bands, and
the Chern numbers are changed accordingly, i.e.
$[C_{1},C_{2},C_{3}]=[-1,-2,-1]$.

In Fig. 3 (e), we plot the $W$, $\epsilon_{12}$ and $\epsilon_{24}$ as
functions of $\theta$. Here, $W$ is the bandwidth of the 1st moiré flat band,
$\epsilon_{12}$ is the gap between the two even parity moiré flat bands, and
$\epsilon_{24}$ represents the gap between the 2nd moiré flat band and the 4th
moiré band (blue, even parity). We also denote the regions of $\theta$ with
different Chern numbers by different colors, as shown in Fig. 3 (e). We see
that, though $W$ increases monotonically with $\theta$, the AT5L-MoTe2 does
not have a clear magic angle. In principle, in the region
$1.8^{\circ}<\theta<2.2^{\circ}$, we can get an isolated moiré flat band at
$E_{f}$, separated from other moiré bands with a small but detectable gap.

###  III.2 Even-layer ATML-MoTe2

In this section, we then discuss the moiré band structures of the ATML-MoTe2
with even $N$.

Figure 4: (a)-(f) The moiré band structures of AT4L-MoTe2 for twist angle
$\theta=1^{\circ},1.43^{\circ},1.89^{\circ},2.04^{\circ},2.65^{\circ},3^{\circ}$.
(g) The bandwidth W of the 1st band(solid blue line), the gap $\epsilon_{12}$
between the 1st and 2nd bands (dashed red line), and the gap $\epsilon_{23}$
between the 2nd and 3rd bands (dashed green line) as a function of twist
angle. Different colored regions represent different Chern numbers of the top
two bands. (h)-(k) show the distribution of Berry curvature $\mathcal{F}$ for
(h) $\theta=1.89^{\circ}$, (i) $\theta=2.04^{\circ}$, (j)
$\theta=2.65^{\circ}$, and (k) $\theta=3.48^{\circ}$.

####  III.2.1 AT4L-MoTe2

We first consider the case of AT4L-MoTe2. The numerical results are shown in
Fig. 4, where the moiré band structures of AT4L-MoTe2 at different twist
angles are given, as illustrated in Fig. 4 (a)-(f).

For AT4L-MoTe2, the most prominent feature is that, at a small twist angle, it
always exhibits two moiré flat bands at $E_{f}$, which are separated from the
remaining moiré bands by a large gap. This is quite like the twisted
$\mathrm{MoTe_{2}}$ homobilayers, as indicated by the decomposition rules in
Eq. (8). Note that the ATML-MoTe2 with even $N$ does not have mirror symmetry
anymore. The two moiré flat bands exhibit distinct behaviors across different
regions of $\theta$, and three regions of $\theta$ are identified according to
their distinct topological features:

When $\theta\lesssim 1.4^{\circ}$, the two moiré flat bands are nearly
degenerate at the $E_{f}$, which are separated from other bands by a large
gap. The Chern numbers of the first two flat bands are $[C_{1},C_{2}]=[0,0]$
in this region, as indicated in Fig. 4 (a) and (g). The case of
$\theta=1^{\circ}$ is plotted in Fig. 4 (a). As shown in Fig.4(g), the minimum
value of the bandwidth $W$ of the first flat band occurs around
$\theta=1^{\circ}$. Meanwhile, $\epsilon_{23}$ has its maximum value about
$21.48~{}\rm{meV}$ around $\theta\approx 1.1^{\circ}$.

Increasing $\theta$ to the region $1.4^{\circ}<\theta<3.3^{\circ}$, the
degeneracy of the two moiré flat bands is lifted, and the gap $\epsilon_{12}$
becomes obvious. Meanwhile, the Chern numbers of the two moiré bands become
nonzero, i.e. $[-1,1]$. We thus can get an isolated topological flat band near
$E_{f}$. In this region, we plot $\theta=1.43^{\circ}$ in Fig. 4(b), where a
sudden change of the Chern number occurs near $\theta\approx 1.4^{\circ}$. The
gap between the two moiré flat bands is observable when $\theta=1.89^{\circ}$,
see Fig. 4(c). Most interestingly, we find a magic angle around $\theta\approx
2.02^{\circ}$, where the bandwidth of the 1st moiré flat band becomes as small
as $0.1$ meV, as shown in Fig. 4(d) and (g). Further increasing $\theta$ to
$2.65^{\circ}$, the two moiré bands become more dispersive, see Fig. 4(e).
When $\theta$ approaches about $3.1^{\circ}$, the gap $\epsilon_{23}$ becomes
zero, see Fig. 4 (g).

In the region of $\theta>3.3^{\circ}$, we find a change of Chern number from
$[-1,1]$ to $[-1,-1]$, see Fig. 4(f) and (g). In this region, the bandwidth of
the first two moiré bands will increase monotonically as $\theta$ increases.
Consequently, the two bands gradually begin to overlap in energy, i.e.
$\epsilon_{12}=0$, though they are still isolated from each other.

Figure 5: (a)-(f) The moiré band structures of AT6L-MoTe2 at different twist
angles, corresponding to
$\theta=1^{\circ},1.43^{\circ},1.89^{\circ},2.02^{\circ},2.44^{\circ},3^{\circ}$,
respectively. In (a) and (b), the four topmost moiré flat bands can be
approximately divided into two equivalent twisted TMD homobilayers (blue and
red). (g) The bandwidth W of the 1st band, the gap $\epsilon_{12}$ between the
1st and 2nd bands, and the gap $\epsilon_{45}$ between the 4th and 5th bands
as a function of $\theta$. (h)-(k) show the distribution of $\mathcal{F}$ for
(h) $\theta=1.89^{\circ}$, (i) $2.02^{\circ}$, (j) $2.44^{\circ}$, and (k)
$3^{\circ}$.

As well known, in order to realize FQAH state, the presence of a topological
flat band is essential Tang  _et al._ (2011); Neupert  _et al._ (2011); Sheng
_et al._ ; Sun  _et al._ (2011); Regnault and Bernevig (2011); Song  _et al._
(2015); Zhang  _et al._ (2019); Crépel and Fu (2023); Morales-Durán  _et al._
(2023). And recent works indicates that the uniformity of Berry curvature may
also play an important role Devakul  _et al._ (2021). Thus, we then discuss
Berry curvature $\mathcal{F}(\mathbf{k})$ of the top flat band at different
$\theta$. The $\mathcal{F}(\mathbf{k})$ in Fig. 4(h)-(k) corresponds to the
band structures depicted in Fig. 4(c)-(f), respectively. With a small twist
angle, e.g. $\theta=1.89^{\circ}$ in Fig. 4(h), $\mathcal{F}$ is sharply
peaked at the $\kappa_{\pm}$ points. As $\theta$ increases, $\mathcal{F}$ will
first spread along the ”bond” directions between the adjacent $\kappa_{\pm}$
points, as shown in Fig. 4(i) with $\theta=2.04^{\circ}$. Increasing $\theta$
further, $\mathcal{F}$ will shift from $\kappa_{\pm}$ to $\gamma$. Around
$\theta=2.65^{\circ}$, $\mathcal{F}$ becomes almost uniform in the whole moiré
Brillouin zone, see Fig. 4(j). This uniform distribution of $\mathcal{F}$
reminds us of Landau levels, which is also similar to the twisted
$\mathrm{MoTe_{2}}$ homobilayers Devakul  _et al._ (2021). In Fig. 4(k), we
plot $\mathcal{F}$ at $\theta=3.48^{\circ}$, where $\mathcal{F}$ is peaked
around $\gamma$ point.

Based on the results above, we see that AT4L-MoTe2 shares similarities with
the twisted $\mathrm{MoTe_{2}}$ homobilayers, while also exhibiting
difference. The similarities are they both have a pair of topological flat
bands, which behave similarly as $\theta$ increases. The differences are: (1)
The AT4L-MoTe2 has a larger magic angle $\theta\approx 2.04^{\circ}$, while
the magic angle of the twisted $\mathrm{MoTe_{2}}$ homobilayers is about
$1.4^{\circ}$; (2) The Chern number of the two flat bands of the AT4L-MoTe2
are $[0,0]$ when $\theta<1.4^{\circ}$, but the Chern numbers of the twisted
$\mathrm{MoTe_{2}}$ homobilayers are always nonzero.

Most importantly, our results imply that AT4L-MoTe2 should also host FQAH
state around the magic angle $\theta\approx 2.04^{\circ}$ like the twisted
$\mathrm{MoTe_{2}}$ homobilayers, because that their moiré flat bands have
nearly the same topological features, e.g. the flat dispersion and uniformity
of the Berry curvature near the magic angle. Meanwhile, owing to the
similarity between AT4L-MoTe2 and twisted TMD homobilayers, this system could
also serve as an ideal platform for realizing a Hubbard model Wu  _et al._
(2019); Devakul  _et al._ (2021); Pan  _et al._ (2020a, b); Zhang  _et al._
(2021); Xu  _et al._ (2022).

####  III.2.2 AT6L-MoTe2

We then consider the case of AT6L-MoTe2, where $N=6$. According to the
decomposition rules in Eq. (8), its moiré bands near $E_{f}$ can approximately
be viewed as originating from two independent twisted TMD homobilayers, i.e.
$h(\lambda_{1})$ and $h(\lambda_{2})$, with different scaling factors
$\lambda_{1,2}$. Here, each equivalent twisted TMD homobilayers will offer two
moiré flat bands at $E_{f}$, thus, in this case, we should obtain four
isolated moiré flat bands in the small twist angle region.

The numerical results of AT6L-MoTe2 are given in Fig. 5. In Fig. 5 (a)-(f), we
plot the moiré bands with different $\theta$. Clearly, as anticipated, the
AT6L-MoTe2 does exhibit four moiré flat bands near $E_{f}$ at a small twist
angle, which are separated from other bands by a large gap and nearly
degenerate when $\theta<1.4^{\circ}$. As analyzed above, when $\theta$ is
small [Fig. 5 (a), (b)], the four moiré flat bands can be further
approximately divided into two groups (blue and red), with two bands in each
group, where each group here belongs to an equivalent twisted TMD
homobilayers, i.e. $h(\lambda_{1,2})$. For a large $\theta$, this
approximation does not work well, since the coupling between the two groups of
moiré flat bands will obviously alter the band shape and the Chern number.

Below, we only focus on the two topmost moiré flat bands out of the four. It
is because, on one hand, the two topmost moiré flat bands are the most easily
accessible bands in experiments, and on the other hand, these two bands can be
isolated via adjusting $\theta$, allowing their Chern numbers to be well-
defined. Thus, as illustrated in Fig. 5 (g), we identify three distinct
regions for $\theta$, according to the Chern numbers of the two topmost moiré
flat bands:

When $\theta\lesssim 1.4^{\circ}$, the Chern numbers of the top two flat bands
in this region are $[C_{1},C_{2}]=[0,0]$. In this case, the four moiré flat
bands are nearly degenerate near $E_{f}$, as depicted in Fig. 5 (a), which are
separated from other bands by a large gap. However, the first two moiré flat
bands are actually isolated, albeit with very small gaps. Therefore, the
calculations can yield a well-defined Chern number for the two topmost moiré
bands here. In comparison with cases where $N<6$, the four nearly degenerate
moiré flat bands in AT6L-MoTe2 will lead to a significantly larger DOS near
$E_{f}$, potentially resulting in exotic correlated insulating states in this
region.

When $1.4\lesssim\theta\lesssim 2.1^{\circ}$, the top two flat bands exhibit
non-zero Chern numbers $[C_{1},C_{2}]=[-1,1]$ in this range. First, as shown
in Fig. 5 (b)-(d), the degeneracy of the moiré flat bands is lifted as
$\theta$ increases. With a larger $\theta$, the gap between the two topmost
moiré bands, namely $\epsilon_{12}$, becomes more apparent in this region, as
illustrated in Fig. 5 (g). Consequently, the 1st moiré flat band ($C_{1}=-1$)
is isolated, as seen in Fig. 5 (c) and (d). Meanwhile, the bandwidth of the
1st moiré band, $W$, remains small. In Fig. 5 (g), we plot $W$ and
$\epsilon_{12}$ as functions of $\theta$. This indicates that $\theta\approx
2.02^{\circ}$ is a magic angle for the 1st moiré flat band, see Fig. 5 (d),
which is quite like the case of AT4L-MoTe2. And the gap between the 4nd and
5rd moiré bands $\epsilon_{45}$ is also shown in Fig. 5 (g) as well. As
$\theta$ becomes larger, $\epsilon_{45}$ nearly monotonically decreases, and
it becomes zero when $\theta$ approaches $3^{\circ}$.

When $\theta>2.1^{\circ}$, the Chern numbers of the top two moiré flat bands
becomes $[C_{1},C_{2}]=[-1,-1]$, as seen in Fig. 5 (g). The cases with
$\theta=2.44^{\circ}$ and $\theta=3^{\circ}$ are plotted in Fig. 5 (e) and
(f), respectively. In these cases, the top two moiré bands become more
dispersive but remain isolated.

The numerical results above imply that the top moiré flat band behaves very
like those in twisted MoTe2 homobilayers and AT4L-MoTe2, suggesting it is also
an ideal platform for realizing FQAH states. We thus further study the
distribution of the Berry curvature $\mathcal{F}$ of the top moiré flat band
at different $\theta$, as illustrated in Fig. 5 (h)-(k). At
$\theta=1.89^{\circ}$, the peak of $\mathcal{F}$ distribution is located at
the $\kappa_{\pm}$ points, see Fig. 5 (h). When $\theta$ increases to the
magic angle $2.02^{\circ}$, the $\mathcal{F}$ distribution will gradually
shift along the edges of the moiré BZ, as illustrated in Fig. 5 (i). When
$\theta=2.44^{\circ}$, the distribution of $\mathcal{F}$ becomes uniform, see
Fig. 5 (j). At a large $\theta$, e.g. $\theta=3^{\circ}$ in Fig. 5 (k),
$\mathcal{F}$ mainly distributes around the $\gamma$ points. The distribution
of $F$ here is also very similar to that in twisted MoTe2 bilayer and
AT4L-MoTe2, suggesting that the moiré flat bands in these systems will share
similar topological features, and thus may host FQAH states.

##  IV Summary

In conclusion, we theoretically investigated the moiré band structures of
ATML-MoTe2, where the MoTe2 monolayers are stacked in an alternatingly twist
way.

We first propose an even-odd dependent decomposition rule for the ATML-MoTe2,
which works quite well when $\theta$ is small. The decomposition rules first
indicate that an $N$-layer ATML-MoTe2 exhibits $N-2$ moiré flat bands near
$E_{f}$, which relies on the important fact that the outermost two MoTe2
monolayers actually have little influence on the electronic structures near
$E_{f}$, since their moiré potentials are significantly different from the
middle ones. This decomposition rule also indicates that when $N$ is even
(odd), ATML-MoTe2 can be approximately decoupled into several twisted TMD
homobilayers (several twisted TMD homobilayers together with one twisted TMD
heterobilayers). Meanwhile, we also reveal that the ATML-MoTe2 also has a
mirror symmetry decomposition when $N$ is odd.

Then, we numerically calculate the moiré band structures of the ATML-MoTe2
with $N=3,4,5,6$. All the numerical results are in good agreement with the
proposed decomposition rules. Most importantly, our numerical calculations
give three intriguing predictions:

  1. 1.

The AT3L-MoTe2 ($N=3$) has one isolated moiré flat band, which corresponds to
a triangular lattice Hubbard model, resembling the twisted TMD heterobilayers.

  2. 2.

The AT4L-MoTe2 ($N=4$) has two topological flat bands that are very similar to
the twisted $\mathrm{MoTe_{2}}$ homobilayers, implying the possible existence
of FQAH states.

  3. 3.

When $N>4$, the giant DOS induced by the multiple degenerate moiré flat bands
may induce exotic correlated states.

Our works reveal that the ATML-MoTe2 is a promising moiré platform, which not
only can be used to simulate the triangle lattice Hubbard model, but also may
host the exotic FQAH states, analogous to those observed in twisted MoTe2
bilayer. We expect that our predictions can be immediately tested in future
experiments.

###### Acknowledgements.

We appreciate helpful discussions with Professor Fengcheng Wu. This work was
supported by the National Natural Science Foundation of China (Grants No.
12141401), the National Key Research and Development Program of China (Grants
No. 2022YFA1403501).

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