# Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2

Aidan P. Reddy<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA  Faisal Alsallom
Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Yang Zhang  Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA  Min H. Kao
Department of Electrical Engineering and Computer Science, University of
Tennessee, Knoxville, Tennessee 37996, USA  Trithep Devakul  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Liang Fu<EMAIL_ADDRESS>Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

We demonstrate via exact diagonalization that AA-stacked TMD homobilayers host
fractional quantum anomalous Hall (FQAH) states with fractionally quantized
Hall conductance at fractional fillings $n=\frac{1}{3},\,\frac{2}{3}$ and zero
magnetic field. While both states are most robust at angles near
$\theta\approx 2^{\circ}$, the $n=\frac{1}{3}$ state gives way to a charge
density wave with increasing twist angle whereas the $n=\frac{2}{3}$ state
survives across a much broader range of twist angles. We show that the
competition between FQAH states and charge density wave or metallic phases is
primarily controlled by the wavefunctions and dispersion of the underlying
Chern band, respectively. Additionally, Ising ferromagnetism is found across a
broad range of fillings where the system is insulating or metallic alike. The
spin gap is enhanced at filling fractions where integer and fractional quantum
anomalous Hall states are formed.

The discovery of the integer and fractional quantum Hall effects (QHE) in two-
dimensional electron systems under a magnetic field ushered in the paradigm of
topological matter and electron fractionalization [1, 2] over forty years ago.
It was recognized shortly thereafter that, while broken time reversal symmetry
is a necessary condition for QHE, Landau levels are not: a Bloch band with a
nonzero Chern number suffices [3, 4]. The possibility of quantum Hall analogs
in which time reversal symmetry is broken _spontaneously_ at zero magnetic
field is a subject of fundamental importance and long-standing interest.
Advances in quantum materials have brought the search for such phases, known
collectively as quantum anomalous Hall (QAH) states, to the forefront of
condensed matter physics.

Following theoretical proposals, transport measurements have demonstrated the
integer QAH effect in a variety of material systems [5], including
magnetically doped topological insulator [6], intrinsic magnetic topological
insulator MnBi2Te4 [7], magic-angle twisted bilayer graphene [8], and
transition metal dichalcogenide heterobilayer MoTe2/WSe2 [9]. Beyond a proof
of principle, the experimental demonstration of QHE at zero magnetic field
opens a new path to microwave circulators [10], chiral topological
superconductivity, and Majorana fermion [11, 12].

Even more exciting is the fractional quantum anomalous Hall (FQAH) state, a
new phase of matter that exhibits fractionally quantized Hall conductance and
hosts fractional quasiparticles (anyons) at zero magnetic field. Physical
realization of FQAH states relies on the synergy between band topology, strong
correlation, and spontaneous time reversal symmetry breaking. These states can
host new types of fractionalization unseen before in Landau levels. Moreover,
proximity coupling between FQAH states and superconductors at zero magnetic
field may provide a promising route to topological quantum computing [13, 14].

Magic-angle twisted bilayer graphene is a theoretically interesting candidate
platform for the FQAH state [15, 16, 17, 18, 19]. Local compressibility
measurements demonstrate its fractional quantum Hall states at magnetic fields
above $5$ T [20]. However, at zero and small fields, the incompressible states
at fractional fillings are observed to be topologically trivial.

Recently, a new moiré system, AA-stacked twisted homobilayer of transition
metal dichalcogenide (TMD) semiconductors WSe2 or MoTe2, has also been
predicted to host FQAH [21, 22, 23]. Here, narrow moiré bands are formed at
small twist angles and acquire nontrivial topology from the layer pseudospin
structure of their Bloch wavefunctions [24]. In addition to band topology and
narrow bandwidth, strong atomic spin-orbit coupling in moiré TMDs results in
the locking of electrons’ spin to their valley degree of freedom. This makes
spontaneous Ising ferromagnetism possible at finite temperature, a key
requisite for the realization of FQAH states [23].

Very recently, signatures of integer and fractional QAH states in optical
measurements of twisted MoTe2 bilayers have been reported [25].
Photoluminescence spectra clearly show a reduction in intensity and a blue
shift in peak energy at integer and fractional fillings of the moiré unit cell
$n=-1$ and $-\frac{2}{3}$, indicating the emergence of correlated insulators.
Furthermore, magnetic circular dichroism measurements reveal robust
ferromagnetism over a wide range of hole fillings
$0.4\lessapprox|n|\lessapprox 1.2$ The coercive field determined from magnetic
hysteresis is distinctively enhanced at $n=-1$ and $-\frac{2}{3}$. Remarkably,
a linear shift in the carrier densities of the optically detected $n=-1$ and
$-\frac{2}{3}$ states with the applied magnetic field is found, with a slope
$\frac{\partial n}{\partial B}$ in units of $\frac{e}{hc}$ matching $C=-1$ and
$-\frac{2}{3}$ respectively, as expected from Streda formula
$\frac{\partial{n}}{\partial_{B}}|_{\mu}=\frac{\sigma_{H}}{ec}$ for states
with integer and fractionally quantized Hall conductance
$\sigma_{H}=C\frac{e^{2}}{h}$. Importantly, the linear dispersion persists
down to zero magnetic field. These observations taken altogether provide
clear, strong evidence for integer and fractional QAH in hole-doped twisted
bilayer MoTe2.

In an independent experiment around the same time, integer QAH states were
observed in twisted bilayer WSe2 at hole fillings $n=-1$ and $n=-3$ by
electronic compressibility measurements [26]. Here, the linear shift in the
density of the incompressible state reveals states with quantized Hall
conductance $C=+1$, which persist down to zero magnetic field. The topological
gap of the QAH state at $n=-1$ is found to be around $1$ meV.

The discovery of integer and fractional QAH states in twisted bilayer MoTe2
and WSe2 following theoretical prediction [21, 22, 23] demonstrates the
extraordinary richness at the intersection of band topology and electron
correlation. Many open questions remain to be answered. While prior
theoretical studies of FQAH in twisted TMD homobilayers have focused on the
filling factor $n=-\frac{1}{3}$ and at ultrasmall twist angles
$\theta\lessapprox 1.5^{\circ}$ [22, 23, 27], the newly observed FQAH state in
twisted MoTe2 occurs at the filling fraction $n=-\frac{2}{3}$ in a device with
a larger twist angle $\theta=3.7^{\circ}$. In addition, a weak feature
indicative of another FQAH state at $n=-\frac{3}{5}$ was observed.

In this work, we study ferromagnetism, FQAH, and competing states in AA
stacked TMD homobilayers. We begin with a detailed discussion of the system’s
single particle physics, which evolves dramatically with twist angle. We
present original ab initio calculations for $t$MoTe2 band structure. Next, we
demonstrate robust Ising ferromagnetism across a broad range of carrier
densities in the lowest moiré band, independent of whether the system is
metallic or insulating at a given carrier density. For a range of twist angles
and realistic Coulomb interaction, we demonstrate FQAH states at filling
factors $n=\frac{1}{3},\frac{2}{3}$. These states are fully spin/valley
polarized and exhibit fractionally quantized Hall conductance at zero magnetic
field. We find for both fillings that the FQAH gap is largest near
$\theta\approx 2^{\circ}$. As twist angle increases, the $n=\frac{1}{3}$ state
gives way to a charge density wave (CDW) while the $n=\frac{2}{3}$ state
survives to significantly higher twist angles, ultimately giving way to a
metal. We find that the angle at which the FQAH-CDW transition for
$n=\frac{1}{3}$ occurs is only weakly dependent on interaction strength,
whereas that at which the FQAH-metal transition occurs for $n=\frac{2}{3}$ is
strongly interaction-strength-dependent. This suggests that the former is
controlled primarily by a change in the wavefunctions of the lowest-band Bloch
states whereas the latter is controlled primarily by a change in their
dispersion. within the single-band-projected model, we find that the
suppression of charge density wave at $n=\frac{2}{3}$ compared to
$n=\frac{1}{3}$ is due to interaction-induced renormalization of band
dispersion.

Figure 1: (a) Schematic of AA-stacked homobilayer moiré superlattice and (b)
Wannier diagrams showing quantum anomalous Hall states in $t$MoTe2 and
$t$WSe2, which have opposite Chern numbers in a given valley.

##  I Topological moiré bands

_Continuum model –_ The valence band edges of a TMD monolayer are located at
the $K$ and $K^{\prime}$ points of the Brillouin zone and have a large
effective mass in the range of $0.5-1$ $m_{e}$. Due to strong atomic spin-
orbit coupling, holes at $K$ and $K^{\prime}$ have opposite spin so that spin
and valley degrees of freedom lock into a single two-component “spin” degree
of freedom. When two identical $K$-valley TMD layers are stacked with
$0^{\circ}$ alignment (AA stacking), holes at a given valley have the same
spin in both layers and therefore direct spin-conserving, intra-valley, inter-
layer tunneling is present.

Rotation between the two layers modifies the dispersion of low-energy holes by
introducing an intra-layer superlattice potential and inter-layer tunneling
that vary spatially with the superlattice periodicity. As shown by Wu et al
[24], the continuum model Hamiltonian for the spin-$\uparrow$ component takes
the form of a $2\times 2$ matrix in layer space:

$\displaystyle\mathcal{H}_{\uparrow}=\begin{pmatrix}\frac{\hbar^{2}(-i\nabla-\kappa_{+})^{2}}{2m}+V_{1}(\bm{r})&t(\bm{r})\\\
t^{{\dagger}}(\bm{r})&\frac{\hbar^{2}(-i\nabla-\kappa_{-})^{2}}{2m}+V_{2}(\bm{r})\end{pmatrix}$
(1)

with $\mathcal{H}_{\downarrow}$ its time reversal conjugate. Note that this
single-particle Hamiltonian is defined with charge neutrality as its vacuum.
Accordingly, we have chosen a natural convention for $\mathcal{H}$ to write
the continuum model Hamiltonian in terms of the hole operators directly:

$\displaystyle H_{0}=\sum_{\sigma=\uparrow,\downarrow}\int
d{\bm{r}}\;\psi^{\dagger}_{\sigma}\mathcal{H}_{\sigma}\psi_{\sigma},$ (2)

where $\psi^{\dagger}_{\sigma}$ creates a hole in the valence band. As such,
the single-particle energy spectrum of $H_{0}$ is bounded from below. We also
define $n$ to be the number of holes per unit cell relative to charge
neutrality so that $n$ is positive, opposite to the definition commonly used
in experiments.

Here, the kinetic energy of holes in a given layer follows a quadratic energy-
momentum dispersion centered about its $K$ point. The $K$ points of the two
layers are displaced due to the interlayer twist and fold into the corners of
the moiré Brillouin zone, $\kappa_{+}$ and $\kappa_{-}$. We choose our moiré
reciprocal lattice vectors to be
$\bm{g}_{i}=\frac{4\pi}{\sqrt{3}a_{M}}(\cos\frac{\pi(i-1)}{3},\sin\frac{\pi(i-1)}{3})$
and $\kappa_{-}=\frac{\bm{g}_{1}+\bm{g}_{6}}{3}$,
$\kappa_{+}=\frac{\bm{g}_{1}+\bm{g}_{2}}{3}$. Here
$a_{M}=\frac{a_{0}}{2\sin(\theta/2)}\approx\frac{a_{0}}{\theta}$ where $a_{0}$
is the atomic lattice constant.

The parameters of the moiré potential $V_{l}(\bm{r}),\,t(\bm{r})$ can be
fitted to first-principles density functional theory calculations given
symmetry constraints that we now discuss. The most general Fourier expansion
of the intra-layer potential to the lowest harmonic is
$V_{l}(\bm{r})=-\sum_{i=1}^{6}V_{\bm{g}_{i}l}e^{i\phi_{\bm{g}_{i}l}}e^{i\bm{g}_{i}\cdot\bm{r}}$
where $V_{\bm{g}_{i}}$ is real and the reality of $V_{l}(\bm{r})$ requires
$\phi_{\bm{g}_{i}l}=-\phi_{-\bm{g}_{i}l}$. It follows from $C_{3z}$ symmetry
that

$\displaystyle
V_{l}(\bm{r})=-2V\sum_{i=1,3,5}\cos(\bm{g}_{i}\cdot\bm{r}+\phi_{l}).$ (3)

Here, the origin of $\bm{r}$ is defined to be at the center of an MM stacking
region. Additionally, symmetry under a twofold rotation that interchanges the
two layers requires $V_{l}(\bm{r})=V_{\bar{l}}(-\bm{r})$ and, in turn,
$\phi_{2}=-\phi_{1}\equiv\phi$. The same symmetry consideration also applies
to the inter-layer tunneling term, which must take the general form

$\displaystyle
t(\bm{r})=w(1+e^{i\bm{g}_{2}\cdot\bm{r}}+e^{i\bm{g}_{3}\cdot{\bm{r}}}).$ (4)

This model Hamiltonian has spin $U(1)$ symmetry $([S_{z},\mathcal{H}]=0)$, but
_not_ spin $SU(2)$ symmetry, a property that we will see enables robust Ising
type ferromagnetism.

Figure 2:  $a)$ The interlayer distance of the twisted MoTe2 structure
obtained from DFT is shown, demonstrating a large variation between the MM and
XM/MX regions. $b)$ The continuum band structure (blue lines) is plotted in
comparison with large scale DFT calculations (black dots) at twist angle
$\theta=4.4^{\circ}$, showing excellent agreement. Note that additional bands
in DFT calculations come from $\Gamma$ valley states.

_First principles calculations –_ We now compare the moiré band structure of
the continuum model with first-principles calculations on twisted bilayer
MoTe2. We perform large-scale density functional theory (DFT) calculations
with the SCAN density functional [28] with dDsC dispersion correction method,
which captures the intermediate-range vdW interaction through its semilocal
exchange term. We find that lattice relaxation has a dramatic effect on moiré
bands. Our DFT calculations at $\theta=4.4^{\circ}$ with 1014 atoms per unit
cell show that the layer distance $d$ varies significantly in different
regions of the moiré superlattice, as shown in Fig 2(a). $d=7.0$Å is smallest
in MX and XM stacking regions, where the metal atom on top layer is aligned
with chalcogen atom on the bottom layer and vice versa, while $d=7.8$Å is
largest in MM region where metal atoms of both layers are aligned. With the
fully relaxed structure, the low-energy moiré valence bands of twisted bilayer
MoTe2 are found to come from the $\pm K$ valley (shown in Fig.1b).

In Fig 2c, we compare the band structures of twisted bilayer MoTe2 at
$\theta=4.4^{\circ}$ computed by large-scale DFT and by the continuum model.
Remarkably, the low-energy part of DFT band structure is well fitted with the
continuum model band structure with parameters stated in Table 1. Importantly,
our direct large-scale DFT calculation reveals a significantly narrower moiré
bandwidth than reported in the previous model study [24]. Correspondingly, the
intralayer potential $V$ and interlayer tunneling strength $w$ is
significantly larger than previously thought.

Materials |  $\phi$ (deg) |  $V$ (meV) |  $w$ (meV) |  $m$ ($m_{e}$) |  $a_{0}$ (Å)
---|---|---|---|---|---
$t$MoTe2 | -91 | 11.2 | -13.3 | 0.62 | 3.52
$t$WSe2 | 128 | 9 | -18 | 0.43 | 3.32
Table 1: Continuum model parameters extracted from density functional theory
calculations. Parameters for $t$MoTe2 are from this work and those for $t$WSe2
are from [21]. Figure 3: (a) Continuum model hole bands of $t$MoTe2 in valley
$K$ at several twist angles. Chern numbers of the first and second lowest
bands are labeled in blue and black respectively. (b) Corresponding particle
number density associated with the lowest band,
A$n(\bm{r})=A\sum_{\bm{k}}|\psi_{1\bm{k}\uparrow}(\bm{r})|^{2}$ where $A$ is
the moiré unit cell area. The lowest band, onto which we project the continuum
model Hilbert space in the exact diagonalization calculation, is highlighted
in blue.

_Twist angle dependence –_ As we showed recently [21], the moiré band
structure of twisted TMD homobilayers is highly tunable by the twist angle
$\theta$, which controls the moiré period $a_{M}$ and thereby the ratio of
kinetic to moiré potential energy. As the twist angle decreases, the moiré
band structure evolves from nearly-free-electron-like to the tight-binding
regime. In Fig. 3, we show continuum model band structures and the
corresponding charge densities at several twist angles. At
$\theta=1.2^{\circ}$, the lowest two moiré bands have narrow widths $\sim 1$
meV and are well described by the Kane-Mele type tight binding model on the
honeycomb lattice as we will elaborate later. The charge density of the lowest
band exhibits sharp maxima at MX and XM stacking sites, which are local
extrema of the intra-layer moiré potential and form a honeycomb lattice. We
note that the layer polarization of these charge density peaks is strong and
opposite between the two sublattices.

In the large angle limit where kinetic energy dominates, the charge density is
more uniform. It exhibits shallower peaks on a triangular lattice of MM
stacking regions where inter-layer tunneling is of maximum amplitude. The
marked difference in the moiré band structures at low and high twist angles is
evidenced by the lowest moiré band minimum changing from $\gamma$ to
$\kappa_{+}/\kappa_{-}$. As we showed recently [21], the crossover between the
two regimes dictates the existence of a “magic angle” at which the lowest
moiré band becomes extremely flat.

In Fig. 4, we also show the evolution of the width of the first band $W_{1}$
as well as the difference between the average energies of the first two bands
$\Delta_{12}\equiv\sum_{\bm{k}}(\varepsilon_{2\bm{k}\uparrow}-\varepsilon_{1\bm{k}\uparrow})/\sum_{\bm{k}}1$
as a function of twist-angle for both WSe2 and MoTe2. For angles $\gtrapprox
3^{\circ}$, the lowest moiré bands acquire significant dispersion $>10$ meV.
The bands of MoTe2 are narrower than those of WSe2. As we will later
elaborate, as long as this bandwidth is small compared to the system’s
characteristic interaction energy, it plays an insignificant role in
determining the many-body ground state. $\Delta_{12}$ in both cases
monotonically increases with twist angle.

Figure 4: (a) Bandwidth of the lowest moiré band $(W_{1})$ and difference in
the average energy of two lowest moiré bands ($\Delta_{12}$) as a function of
inter-layer twist angle $\theta$ for MoTe2 and WSe2. Also shown is a
characteristic interaction energy scale $\frac{e^{2}}{\epsilon a_{M}}$ for
$\epsilon=10$. (b) Chern number of the lowest moiré band as a function of the
dimensionless ratio of the inter- and intra-layer moiré potential strengths
$w/V$ as well as the phase parameter $\phi$ demonstrating that WSe2 and MoTe2
have opposite Chern numbers in a given valley.

_Band topology –_ As first pointed out in the seminal work of Wu et al [24],
moiré bands of a given spin component in the continuum model for twisted
bilayer TMD have finite Chern numbers that satisfy
$C_{\uparrow}=-C_{\downarrow}\equiv C$ due to time reversal symmetry. As we
will show later, the existence of topological bands in combination with small
bandwidth at small twist angles is crucial for ferromagnetism and (integer and
fractional) QAH states in this system.

Remarkably, the topological character of moiré bands depends on the twist
angle and shows two distinct regimes (see Fig. 3). Previous theoretical
studies have mostly focused on the ultra-small twist angle regime
($\theta<1.5^{\circ}$), where low-energy states are localized on a honeycomb
lattice. Correspondingly, the lowest two moiré bands are isolated from higher
bands and well described by a honeycomb lattice tight-binding model with Kane-
Mele spin-orbit coupling [29]. The first and second band of spin $\uparrow$
states in valley $K$ have Chern numbers $C_{\uparrow}=(-1,+1)$ respectively
for MoTe2 [24].

We now show that topological bands at larger twist angles have a different
origin. This regime can be understood from a nearly-free-electron analysis. We
treat the spatially varying intralayer moiré potential $V(\bm{r})$ and
interlayer tunneling $t(\bm{r})$ as perturbations to the free particle gas.
The leading effect of these perturbations is to induce superlattice gaps where
free particle states with momenta $\bm{k}$ and $\bm{k}+\bm{g}$ are degenerate
and coupled by $V(\bm{r})$, $t(\bm{r})$, leading to the formation of moiré
bands. We calculate the superlattice gap and Bloch wavefunction of moiré bands
at high symmetry points with a degenerate perturbation theory approach, first
introduced in Ref.[30], enabling us to determine the Chern number of moiré
bands in a given valley in terms of the superlattice parameters $V,w,\phi$.

Fig. 4 shows the Chern number $C$ thus obtained as a function of continuum
model parameters $w/V$ and $\phi$ (without loss of generality $V$ is chosen to
be positive). We further confirm by numerical calculation that the energy gap
between the first and second moiré band remains finite over the entire range
of twist angles from large to small, despite that the moiré band structure
changes dramatically. Trivial bands ($C=0$) are found when the minima of the
moiré potential $V(\bm{r})$ are located at MM sites (which form a triangular
lattice), whereas topological bands ($|C|=1$) are formed when (1) the intra-
layer potential minima are located at MX/XM sites (which form a honeycomb
lattice) and (2) the inter-layer tunneling $w$ is not too large compared to
the intra-layer potential $V$.

For a given valley/spin, the Chern numbers of the lowest moiré band at $\phi$
and $-\phi$ are opposite. Our large-scale DFT calculations find
$\phi=-91^{\circ}$ for twisted bilayer MoTe2 as presented above, and
$\phi=128^{\circ}$ for twisted bilayer WSe2 as shown in Ref.[21]. Therefore,
our theory predicts that the lowest moiré bands in twisted MoTe2 and WSe2
homobilayers have opposite Chern numbers in a given valley.

Our conclusion about band topology is further confirmed by examining the Bloch
wavefunction of moiré bands in large-scale DFT calculation. The Chern number
$\mod 3$ can be computed from the symmetry eigenvalues of spin-$\frac{1}{2}$
Bloch states under $C_{3z}$ rotation [31]. $C\mod
3=\frac{3}{2\pi}\arg(-\lambda_{\kappa_{+}}\lambda_{\kappa_{-}}\lambda_{\gamma})$,
where $\mathcal{R}_{\theta}$ is a rotation operator about the $z$-axis, and
$\lambda_{\bm{k}}=\bra{\psi_{\bm{k}}}\mathcal{R}_{2\pi/3}\ket{\psi_{\bm{k}}}$
at three high symmetry points ${\bm{k}}=\kappa_{+},\kappa_{-},\gamma$ are
$C_{3z}$ symmetry eigenvalues. The symmetry eigenvalues $\lambda$ for twisted
bilayer MoTe2 and WSe2 are determined from DFT calculations (see Table 2). The
Chern number is then determined from the symmetry eigenvalues to be $C=-1$ and
$1$ respectively.

Materials | Band, Valley | $\kappa_{+}$ | $\kappa_{-}$ | $\gamma$
---|---|---|---|---
$t$MoTe2 | 1, $K$ | $e^{i\pi/3}$ | $e^{i\pi/3}$ | $e^{-i\pi/3}$
| 1, $K^{\prime}$ | $e^{-i\pi/3}$ | $e^{-i\pi/3}$ | $e^{i\pi/3}$
$t$WSe2 | 1, $K$ | $e^{i\pi/3}$ | $e^{i\pi/3}$ | $e^{i\pi}$
| 1, $K^{\prime}$ | $e^{-i\pi/3}$ | $e^{-i\pi/3}$ | $e^{i\pi}$
Table 2: $C_{3z}$ eigenvalues of the topmost moiré bands from each valley,
computed from large-scale DFT wavefunctions at high symmetry momentum points.

Importantly, this difference in Chern number has observable consequences.
According to the Streda formula, the Chern number determines the slope of the
linear shift in carrier density with an applied magnetic field:
$\frac{\partial n}{\partial B}=\frac{e}{hc}C$. We note that the measured sign
of the $n$-$B$ slope in $t$MoTe2 or $t$WSe2 depends not only on the valley
Chern number $C$, but also on the sign of the valley $g$-factor which
determines the sign of the valley polarization under an applied $B$ field.
Assuming that both $t$MoTe2 and $t$WSe2 have valley g-factors of the same
sign, our theory predicts that QAH states in the two systems will exhibit
$n$-$B$ slopes of _opposite_ sign, see Fig. 1.

Finally, we note that, as twist angle increases, the second band’s Chern
number changes sign from $-C$ to $C$ ($C$ is the Chern number of the first
band) due to inversion with the third band at $\gamma$ (see Fig. 3) [24, 21].
This applies to both $t$MoTe2 and $t$WSe2. Our DFT calculation shows that both
the first and second bands of $t$MoTe2 have the same Chern number in the twist
angle range studied experimentally [25]. As a result, we expect a double
quantum spin Hall state at the filling of $n=4$ holes per unit cell [21],
which can be experimentally detected through edge-state transport and current-
induced edge spin polarization.

##  II Ising ferromagnetism and spin gap

Figure 5: Finite sized clusters used in the exact diagonalization calculations
in real space (a) and momentum space (b). The 27-unit-cell cluster can be
viewed as a $9$-unit-cell cluster with a tripled unit cell.

We now turn to the many-body problem of a finite density of doped holes in
twisted TMD homobilayers that interact with each other through Coulomb
repulsion. The many-body continuum model Hamiltonian is given by

$\displaystyle\begin{split}H&=H_{0}+V\\\
V&=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\int
d{\bm{r}}d{\bm{r}}^{\prime}\psi^{\dagger}_{\sigma}(\bm{r})\psi^{\dagger}_{\sigma^{\prime}}(\bm{r}^{\prime})V(\bm{r}-\bm{r}^{\prime})\psi_{\sigma^{\prime}}(\bm{r}^{\prime})\psi_{\sigma}(\bm{r}).\end{split}$
(5)

Here we use a long-range Coulomb interaction $V(\bm{r})=\frac{e^{2}}{\epsilon
r}$ in contrast to previous studies of FQAH in TMD moiré superlattices using a
dual-gate screened Coulomb interaction [22, 23, 27]. We perform an exact
diagonalization calculation within the Fock space of the lowest Bloch band.
Upon band projection, the many-body continuum model Hamiltonian is most
conveniently written in momentum space as

$\displaystyle\begin{split}\tilde{H}&=\sum_{\bm{k},\sigma}\varepsilon_{\bm{k}\sigma}c^{{\dagger}}_{\bm{k}\sigma}c_{\bm{k}\sigma}\\\
&+\frac{1}{2}\sum_{\bm{k}^{\prime}\bm{p}^{\prime}\bm{k}\bm{p},\sigma\sigma^{\prime}}V_{\bm{k}^{\prime}\bm{p}^{\prime}\bm{k}\bm{p};\sigma\sigma^{\prime}}c^{{\dagger}}_{\bm{k}^{\prime}\sigma}c^{{\dagger}}_{\bm{p}^{\prime}\sigma^{\prime}}c_{\bm{p}\sigma^{\prime}}c_{\bm{k}\sigma}\end{split}$
(6)

where $c^{{\dagger}}_{\bm{k}\sigma}$ creates a Bloch state in the lowest band
at crystal momentum $\bm{k}$ and spin/valley $\sigma$ with corresponding
single-particle energy $\varepsilon_{\bm{k}\sigma}$.
$V_{\bm{k}^{\prime}\bm{p}^{\prime}\bm{k}\bm{p};\sigma\sigma^{\prime}}\equiv\bra{\bm{k}^{\prime}\sigma;\bm{p}^{\prime}\sigma^{\prime}}\hat{V}\ket{\bm{k}\sigma;\bm{p}\sigma^{\prime}}$
are the corresponding matrix elements of the Coulomb interaction. Our choice
to use the unscreened Coulomb interaction was motivated by simplicity and
generality. We detail our methods in the supplement.

Projection to the lowest band neglects band mixing and therefore is
quantitatively accurate only when the ratio of the characteristic Coulomb
energy $\frac{e^{2}}{\epsilon a_{M}}$ to the moiré band gap is small. However,
band projection is known to be qualitatively correct in the study of
fractional quantum Hall states in lowest Landau level, even when this
dimensionless parameter is not small [32, 33]. A follow up study of twisted
TMD bilayers addressing band mixing is being prepared and will be presented
elsewhere.

In performing the calculation, we take advantage of the model’s charge-$U(1)$,
spin-$U(1)$, and translation symmetries to diagonalize within common
eigenspaces of $N_{h}$, $S_{z}$, and crystal momentum. We use three clusters
of different sizes and geometries illustrated in Fig. 5 with periodic boundary
conditions.

Figure 6: $\Delta_{S}\equiv
E_{\text{min}}(S_{z})-E_{\text{min}}(S_{z\text{max}})$ across all possible
$S_{z}$ values within $15$ meV cutoff as a function of the filling factor
$n\equiv N_{h}/N_{uc}$ on the 15-unit-cell cluster at fixed
$\epsilon^{-1}=0.1$ and several $\theta$. min($\Delta_{S}$) is the minimum
value of $\Delta_{S}$ for $S_{z}<S_{z\text{max}}$).

We begin with an analysis of magnetism in $t$MoTe2 across a broad range of
filling factors $n\leq 1$. In Fig. 6, we plot $\Delta_{S}\equiv
E_{\text{min}}(S_{z})-E_{\text{min}}(S_{z\text{max}})$ for $S_{z}\geq 0$ as a
function of the filling factor $n=N_{h}/N_{uc}$ on the 15-unit-cell torus with
a fixed value of $\epsilon^{-1}=0.1$ and several twist angles. Here
$E_{\text{min}}(S_{z})$ is the minimum energy within a given $S_{z}$ sector,
$N_{h}$ is the number of holes, and $N_{uc}$ is the number of moiré unit
cells.

At $\theta=2^{\circ}$, the lowest energy state is fully spin-polarized at all
filling factors $0.27\leq n\leq 1$ (the lower bound precision is limited by
system size), showing robust spin/valley ferromagnetism in $t$MoTe2. The spin
gap (defined as the minimum of $\Delta_{S}$ with $S_{z}\neq S_{z\text{max}}$),
which controls the Curie temperature and coercive field, is maximum at $n=1$
and generally decreases with decreasing $n$. Notably, we find the spin gap at
$n=1$ exceeds 10 meV. A similar conclusion was reached for twisted TMD
bilayers at smaller twist angles [21, 23]. Here, we find that Ising
ferromagnetism and large spin gap ($>10$ meV) at $n=1$ persist to much larger
twist angles, as shown $\theta=2.5^{\circ}$ and $3.5^{\circ}$. On the other
hand, ferromagnetism at low filling $n<0.4$ is less robust and disappears at
$\theta=3.5^{\circ}$ for $\epsilon^{-1}=0.1$.

At $\theta=2^{\circ}$, the spin gap clearly exhibits local maxima at filling
factors $n=\frac{1}{3}$ and $\frac{2}{3}$, where FQAH states are formed as we
will soon see. Notably, the spin gap at $n=\frac{2}{3}$ is much larger than at
$n=\frac{1}{3}$. For $\epsilon^{-1}=0.1$, at $\theta=2.5^{\circ}$, the FQAH
state only appears at $n=\frac{2}{3}$ where the spin gap is still weakly
enhanced, but not at $n=\frac{1}{3}$. At $\theta=3.5^{\circ}$, the ground
state at $n=\frac{2}{3}$ is a fully polarized Fermi liquid whose spin gap does
not show any prominent feature, while the state at $n=\frac{1}{3}$ is non-
magnetic.

Consistent with our numerical findings, magnetic circular dichroism
measurements on twisted bilayer MoTe2 [25] observed robust Ising
ferromagnetism over a broad range of hole fillings between $n\sim 0.4$ and
$1$, with a maximum Curie temperature of $15$ K at $n=1$. Moreover, the
coercive field is enhanced at $n=\frac{2}{3}$. This agrees with the spin gap
shown in Fig. 6 and is associated with the formation of a FQAH state as we
demonstrate below.

Our calculation shows that Ising ferromagnetism in $t$MoTe2 appears not only
at $n=1,\frac{2}{3}$ and $\frac{1}{3}$, but throughout a broad range of
filling factors below $n=1$ where the system is insulating or metallic alike.
As a consequence of Ising ferromagnetism and Berry flux in moiré bands, we
predict an anomalous Hall effect over a broad of fillings at and below $n=1$
(as found in $t$WSe2 [23]). In particular, quantized anomalous Hall effect is
expected at $n=1$ and certain fractional filling factors that support FQAH
insulators.

From now on, we systematically study the many-body spectrum in the fully spin
polarized sector at $n=\frac{2}{3}$ and $\frac{1}{3}$, for various twist
angles and interaction strengths $\epsilon^{-1}$. We note that, at
$n=\frac{1}{3}$ and large twist angles, the ground state may not be fully
spin-polarized at zero field (see Fig. 6). We leave further investigation of
spin physics at $n=\frac{1}{3}$ to future study.

Figure 7: (a) Many-body spectra of $t$MoTe2 within the fully polarized sector
$(S_{z}=S_{\text{max}})$ on the 30-unit-cell cluster at $n=\frac{1}{3}$,
$\frac{2}{3}$. We use $\theta=2.0^{\circ}$, $\epsilon=10$. At the top we show
the ground state manifold’s spectral flow under flux insertion demonstrating
its FQAH nature. The 4 lowest states within each crystal momentum sector are
shown. (b) Same as (a) except at a larger twist angle $\theta=2.5^{\circ}$ and
on the 27-unit-cell cluster. The lowest 6 states within each momentum sector
are shown. The spectrum at $n=\frac{2}{3}$ indicates FQAH whereas at
$n=\frac{1}{3}$ indicates a CDW.

##  III FQAH and competing phases

In Fig. 7 (a), we show the many-body spectra obtained for $t$MoTe2 on the
30-unit-cell cluster at $\theta=2^{\circ}$ and $n=\frac{1}{3}$, $\frac{2}{3}$
as a function of crystal momentum. We assign each crystal momentum
$\bm{k}=k_{1}\bm{T}_{1}+k_{2}\bm{T}_{2}$ an integer index $k=k_{1}+N_{1}k_{2}$
where $N_{i}$ is the number of crystal momenta along axis $i$. Here
$\bm{T}_{i}=\frac{2\pi\epsilon_{ij}\bm{L}_{j}\times\hat{z}}{A}$ is a basis
vector of crystal momentum, $\bm{L}_{i}$ defines the periodic boundary
condition in real space, and $A=|\bm{L}_{1}\times\bm{L}_{2}|$ is the system
area. At both fillings, we find 3 nearly degenerate ground states separated by
a sizable energy gap $\sim 2$ meV from excited states. The approximate ground
state degeneracy matches the expected topological degeneracy of a fractional
quantum Hall state on a torus. We note that imperfect ground state degeneracy
is expected in a finite system. We have tested several cluster sizes with all
other parameters fixed and find that the gap remains $\sim 2$ meV, indicating
its presence in the thermodynamic limit. The many-body crystal momenta of the
$n=\frac{1}{3}$, $\frac{2}{3}$ states – having linear indices (5,15,25) and
(0,10,20) respectively – are in precise agreement with the generalized Pauli
principle discussed in Ref. [34].

In addition to the threefold ground state degeneracy, a necessary property of
an $n=\frac{p}{q}$ fractional quantum Hall state is that its ground states on
a torus permute upon insertion of one magnetic flux quantum such that each
state returns to itself only after insertion of $q$ flux quanta. Flux
insertion induces a shift in one component of the kinetic momentum
$\bm{\pi}=\bm{p}+\hbar\Phi\bm{T}_{i}$ where $\Phi\equiv\frac{\phi}{\phi_{0}}$,
$\phi$ is the inserted flux, $\phi_{0}=\frac{hc}{e}$ is the flux quantum. In
Fig. 7(a), we show that both $n=\frac{1}{3}$ and $\frac{2}{3}$ exhibit this
spectral flow, providing definitive evidence of their FQAH nature.

A change in the the twist angle $\theta$ induces a change in (1) the Bloch
wavefunctions of the lowest band, (2) the band dispersion and width, and (3)
the system’s characteristic interaction energy scale $\frac{e^{2}}{\epsilon
a_{M}}$. The band dispersion governs the kinetic energy $H_{0}$. At large
twist angles where the lowest moiré band is highly dispersive, the ground
state at fractional fillings is expected to be a Fermi liquid. The Bloch
wavefunctions determine the form of the band-projected interaction $V$ through
the Coulomb matrix elements. Therefore, a given filling factor, the system can
exhibit distinct many-body ground states as a function of twist angle even
when the band dispersion is neglected altogether. Thus, the influence of twist
angle is multifold and needs systematic study.

An obvious candidate ground state in the presence of strong, long-range
Coulomb repulsion is a charge density wave (CDW). Such states are
experimentally observed in TMD moiré hetero-bilayers where they are known as
generalized Wigner crystals [35, 36, 37, 38]. To address the possible
competition between FQAH and CDW with exact diagonalization, it is essential
to choose a cluster that accommodates a tripled unit cell or, equivalently,
samples $\gamma,\kappa_{+}$, and $\kappa_{-}$. The 27-unit-cell cluster
depicted in Fig. 5 satisfies this criterion. In Fig. 7(b) we show spectra
obtained at a larger twist angle $\theta=2.5^{\circ}$ using this cluster. At
$n=\frac{2}{3}$, we find three nearly degenerate ground states at $\gamma$,
indicative of FQAH. On the other hand, at $n=\frac{1}{3}$, we find three
nearly degenerate states with one at each of $\gamma,\kappa_{+}$ and
$\kappa_{-}$ (the center and corners of the moiré Brillouin zone,
respectively). These are the momenta appropriate to a charge density wave with
a tripled unit cell because they fold back to $\gamma$ in the symmetry-broken
Brillouin zone.

To reveal the influence of twist angle on many-body ground states at
$n=\frac{1}{3}$ and $\frac{2}{3}$, in Fig. 8 we plot the energy gap
$E_{\text{gap}}=E_{4}-E_{3}$ as a function of $\theta$, where $E_{i}$ is the
$i^{th}$ lowest energy with a fixed $N_{h}$ and maximum spin $S_{z}$. Two
values of the dielectric constant, $\epsilon^{-1}=0.1,\,0.2$ are used. When
the system is a correlated insulator with threefold ground state degeneracy
such as FQAH or CDW, $E_{\text{gap}}$ is indicative of its robustness.

For $\epsilon^{-1}=0.1$, we see that both the $n=\frac{1}{3},\,\frac{2}{3}$
states exhibit maxima in $E_{\text{gap}}$ near $\theta=1.8^{\circ}$. Beyond
$\theta\approx 1.8^{\circ}$, $E_{\text{gap}}$ decreases at both fractions, but
more rapidly so at $n=\frac{1}{3}$ where it reaches zero near $\theta\approx
2.3^{\circ}$ and then increases again. The many-body spectra on both sides of
this gap-closing transition (not shown) have three nearly degenerate ground
states. However, the ground states at $\theta<2.3^{\circ}$ have the crystal
momenta of the FQAH state as shown in Fig. 7(a), whereas those at
$\theta>2.3^{\circ}$ have the crystal momenta of the CDW state as shown in
Fig. 7(b). Thus, we conclude that at the fractional filling $n=\frac{1}{3}$, a
quantum phase transition between FQAH and CDW occurs around $\theta\approx
2.3^{\circ}$.

The situation is markedly different at $n=\frac{2}{3}$. In this case,
$E_{\text{gap}}$ remains finite until $\theta\approx 3.0^{\circ}$, beyond
which point it is small. The many-body spectrum shows a continuum of states at
low energy, indicating a metallic phase in the thermodynamic limit. These
results clearly show that the FQAH state at $n=\frac{2}{3}$, previously
overlooked in theoretical studies of twisted homobilayer TMD [22, 23],
persists to a substantially higher twist angle than that at $n=\frac{1}{3}$.

Figure 8: $E_{\text{gap}}\equiv E_{4}-E_{3}$ where $E_{i}$ is the fully spin-
polarized state of $i^{th}$ lowest energy as a function of twist angle
$\theta$ for $n=\frac{1}{3},\,\frac{2}{3}$ and two values of
$\epsilon^{-1}=0.1,\,0.2$.

When $\epsilon^{-1}=0.2$, the dependence of the $n=\frac{1}{3}$ state on
$\theta$ is largely similar to when $\epsilon^{-1}=0.1$, save for an expected
increase of $E_{\text{gap}}$ due to the increased Coulomb interaction. On the
other hand, at $n=\frac{2}{3}$, the increased interaction pushes the FQAH-
metal transition to $\theta\approx 3.8^{\circ}$, thereby significantly
expanding the twist angle range of the FQAH state.

These numerical results provide valuable insight into the competition between
FQAH, CDW, and metallic phases. At small twist angles, the bands are narrow
enough (see Fig. 4) that for both $\epsilon^{-1}=0.1$ and $0.2$ the system is
in its flat band limit $\frac{e^{2}}{\epsilon a_{M}}/W\gg 1$. The many-body
ground state is thus determined primarily by the projected interaction term
which is in turn determined by the Bloch wavefunctions. For $\theta\lessapprox
2.3^{\circ}$, FQAH state is preferred by at both fillings. On the other hand,
at large twist angles, the bandwidth becomes sizable and is crucial in the
competition between FQAH and metallic phase at $n=\frac{2}{3}$.

To disentangle the effect of bandwidth from that of Bloch wavefunction, we
study the FQAH-metal transition at $n=\frac{2}{3}$ for a fixed
$\theta=3.5^{\circ}$, tuned by the interaction strength $\epsilon^{-1}$.
Changing $\epsilon^{-1}$ does not affect the band wavefunction, but tunes the
ratio of bandwidth and interaction energy. Fig. 9 shows the many-body spectra
at $\epsilon^{-1}=0.4,0.2,0.05$. While $\epsilon^{-1}=0.4$ is likely larger
than experimental values, it provides useful insight into the strong coupling
limit in a similar spirit to the band-flattening approach [39].

Starting from the strong interaction limit $\epsilon^{-1}=0.4$, it is clear
that $n=\frac{2}{3}$ exhibits FQAH with $3$ well isolated, nearly degenerate
states at $\gamma$ as expected from the generalized Pauli principle rules
applied to this cluster geometry. As the interaction decreases, the energy gap
at momenta $\kappa_{+}$ and $\kappa_{-}$ softens ($\epsilon^{-1}=0.2$), before
the metallic state with a continuum of low-lying states appears
($\epsilon^{-1}=0.05$). The nature of this FQAH-metal transition at
$n=\frac{2}{3}$ is an interesting and important question that calls for
further study.

Figure 9: Low-lying spectra at $\theta=3.5^{\circ}$ for several values of the
inverse dielectric constant $\epsilon^{-1}$ at $n=\frac{2}{3}$. For small
$\epsilon^{-1}=0.1$ (weak interaction) the system is not in a FQAH state
whereas for $\epsilon^{-1}\gtrapprox 0.2$, it is. All energy levels in the
fully spin-polarized sector and the given window are shown.

##  IV $n=\frac{1}{3}$ versus $\frac{2}{3}$

We have shown by exact diagonalization study that AA-stacked TMD moiré
homobilayers exhibit robust Ising spin/valley ferromagnetism across a wide
range of carrier densities within the lowest moiré band. Since the valley-
polarized moiré bands carry finite Chern number, anomalous Hall effect is
expected throughout. At particular fractional filling factors
$n=\frac{p}{q}=\frac{1}{3}$, $\frac{2}{3}$ we predict fractional quantum
anomalous Hall states with corresponding quantized Hall conductances
$\sigma_{H}=\frac{p}{q}\frac{e^{2}}{h}$. Using continuum model parameters
obtained from our first-principles band structure calculations, our study
finds that the topological gap of FQAH states in twisted bilayer MoTe2 is
largest near $\theta\approx 2^{\circ}$.

At larger twist angles, the $n=\frac{1}{3}$ state gives way to a CDW near
$\theta\approx 2.3^{\circ}$ whereas the $n=\frac{2}{3}$ state persists to a
larger angle, eventually becoming metallic. As interaction strength increases,
the FQAH regime at $n=\frac{2}{3}$ extends to higher angles, suggesting that
the FQAH-metal transition is primarily bandwidth controlled. On the other
hand, the critical angle of the FQAH-CDW transition at $n=\frac{1}{3}$ is
weakly dependent on interaction strength, indicating that it is instead
controlled by a change in the Bloch wavefunctions.

The difference between $n=\frac{1}{3}$ and $\frac{2}{3}$ filling states is
noteworthy and interesting. Recall that the ground states of a Landau level at
filling factors $n$ and $1-n$ are simply related by a particle-hole
transformation that leaves the projected Hamiltonian invariant. This is not
the case in our system. In particular, at large twist angles, the
$\frac{1}{3}$\- and $\frac{2}{3}$-filling ground states are distinct phases of
matter: CDW and Fermi liquid respectively.

To understand the contrast between $n=\frac{1}{3}$ and $\frac{2}{3}$, we note
that the band-projected Hamiltonian within the fully spin-polarized sector
$S_{z}=S_{z\text{max}}$ is not symmetric under particle-hole transformation
$c_{\uparrow}^{\dagger}(\bm{r})\rightarrow d_{\uparrow}(\bm{r})$. In
particular, unlike a single particle added to an otherwise empty moiré band, a
single particle removed from an otherwise full moiré band has an interaction-
induced dispersion present even when the bare bandwidth $W$ vanishes [40, 41,
42].

We now show that the interaction-induced asymmetry between particle and hole
dispersion provides a natural explanation of the difference between
$n=\frac{1}{3}$ and $\frac{2}{3}$ filling states found in our exact
diagonalization study. Notably, we find that at large twist angles, a single
hole at $n=1$ is more dispersive than a single particle at $n=0$. Therefore,
in the presence of Coulomb interaction, the system at low filling $\delta\ll
1$ is more susceptible to Wigner crystallization into a CDW state than at the
filling $1-\delta$. This explains our finding of CDW at $n=\frac{1}{3}$ and
Fermi liquid at $\frac{2}{3}$.

##  V Discussion

We have also obtained strong numerical evidence for FQAH states at filling
factors $n=\frac{2}{5},\frac{3}{5}$. For $\theta=2^{\circ}$,
$\epsilon^{-1}=0.1$ and at both fillings, calculations on the 30-unit cell-
cluster show fivefold nearly degenerate ground states separated from the
continuum by an energy gap $>1$ meV.

As noted above, previous exact diagonalization studies of FQAH states in AA-
stacked TMD homo-bilayers have focused on the ultra-low twist-angle regime
$\theta<1.5^{\circ}$ [22, 23, 27]. This is where the lowest moiré band
satisfies various conditions purported in the literature to support FQAH
states, including nearly vanishing bandwidth and quantum geometric
properties—Berry curvature uniformity and “trace condition” [43, 44, 45].

In comparison with previous studies, our work explores larger twist angles and
additional filling fraction $n=\frac{2}{3}$ where the FQAH state has recently
been observed. In addition, we elucidate the origin of band topology at large
twist angle through degenerate perturbation theory analysis. Our results
clearly show that FQAH states in twisted bilayer MoTe2 extend to significantly
larger twist angles where 1) the lowest moiré band has significant dispersion
and 2) the first and second moiré bands within a given valley have the same
Chern number contrary to the Kane-Mele regime at small twist angle.
Nonetheless, the Coulomb energy scale $e^{2}/(\epsilon a_{M})$ suffices to
support CDW and FQAH phases. Remarkably, at larger twist angles, FQAH state is
found at $\frac{2}{3}$ filling, whereas CDW is found at $\frac{1}{3}$. In
addition, we find that FQAH states are accompanied by spin gap enhancements,
in agreement with the experimental observations of Ref. [25]. Our findings
point to the surprising richness and robustness of FQAH physics beyond flat
band and ideal quantum geometry.

Consistent with recent experimental observations [25, 26], our band-projected
exact diagonalization study also shows a robust integer quantum anomalous Hall
effect at $n=1$ protected by a large spin gap. Here, we also note the
possibility of topologically trivial, layer-polarized states at $n=1$ as well
as fractional fillings, especially at small twist angles where the band gap is
small. To faithfully describe such states requires the inclusion of at least
two lowest bands [46], which goes beyond our single-band calculation. As
mentioned above, an investigation of band mixing effects is currently underway
and results will be presented elsewhere.

A straightforward extension of our work is to understand the influence of
displacement field on the QAH states. Generally speaking, stronger
displacement field should drive the system into topologically trivial, layer-
polarized states [24]. Very recently, the ability to tune a topological phase
transition from the integer QAH state to a topologically trivial state at
$n=1$ has been demonstrated experimentally in twisted bilayer WSe2 [26].

Finally, we highlight the prospect of QAH beyond the lowest moiré band.
Indeed, integer QAH at $n=3$ has already been observed in $t$WSe2 [26], and
the possibility of $n\geq 1$ fractional states is enticing.

##  VI Acknowledgement

We thank Xiaodong Xu for collaboration on a closely related experiment on
$t$MoTe2 [25], Ben Foutty and Ben Feldman for collaboration on a closely
related experiment on $t$WSe2 [26], Valentin Crépel for previous collaboration
on related theoretical works [21, 23], as well as Di Luo and Patrick Ledwith
for helpful discussions. This work was supported by the Air Force Office of
Scientific Research (AFOSR) under award FA9550-22-1-0432 and the David and
Lucile Packard Foundation. Y.Z. acknowledges support from the start-up fund at
the University of Tennessee. F. A acknowledges support from the KAUST Gifted
Students Program and the Undergraduate Research Opportunities Program at MIT.

Note added: we recently became aware of independent work on similar topics
[47]. Also, a related experimental development on integer and fractional
quantum anomalous Hall states in $t$MoTe2 has been reported [48].

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