# Fractional quantum anomalous Hall effects in rhombohedral multilayer
graphene in the moiréless limit and in Coulomb imprinted superlattice

Boran Zhou  Hui Yang  Ya-Hui Zhang  Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, Maryland 21218, USA

###### Abstract

The standard theoretical framework for fractional quantum anomalous Hall
effect (FQAH) assumes an isolated flat Chern band in the single particle
level. In this paper we challenges this paradigm for the FQAH recently
observed in the pentalayer rhombohedral stacked graphene aligned with hexagon
boron nitride (hBN). We show that the external moiré superlattice potential is
simply a perturbation in a model with continuous translation symmetry. Through
Hartree Fock calculation, we find that interaction opens a sizable remote band
gap, resulting an isolated narrow $C=1$ Chern band at filling $\nu=1$. From
exact diagonalization (ED) we identify FQAH phases at various fillings. But
they exist also in the calculations without any external moiré potential. We
suggest that the QAH insulator at $\nu=1$ should be viewed as an interaction
driven QAH-Wigner crystal, which is then pinned by a small moiré potential. In
the second part we propose a new setup with Coulomb generated moiré
superlattice. For example, we separate $n$-layer graphene and twisted bilayer
graphene (TBG) with a thin hBN and imprint the superlattice of the TBG to the
$n$-layer graphene. Now the superlattice potential is controlled by the
thickness $d$ of the hBN and the superlattice period is controlled by the
twist angle of the TBG. Overall in both setups the $C=1$ QAH-Wigner crystal is
robust with a crystal period around $10\mathrm{nm}$ in 4-layer, 5-layer,
6-layer and 7-layer graphene systems. Our work suggests a new direction to
explore the interplay of topology and FQAH with spontaneous Wigner crystal
formation in the vanishing moiré potential limit.

Introduction There have been lots of efforts in realizing fractional quantum
Hall (FQH) states[1, 2] from fractionally filling a narrow Chern band [3, 4,
5, 6, 7, 8, 9, 10, 11] on a lattice. Such a state is dubbed as fractional
Chern insulator (FCI). FCI has been experimentally realized at non-zero
magnetic field[12, 13]. Fractional quantum anomalous Hall effect (FQAH) phase,
a FCI at zero magnetic field, is more challenging. It was proposed that the
two dimensional moiré systems are wonderful platforms to host nearly flat
Chern band and thus FQAH states[14, 15, 16, 17, 18] following spontaneous
valley polarization[14, 19]. Indeed integer quantum anomalous Hall (QAH)
states[20, 21] were realized in twisted bilayer graphene(TBG) aligned with
hexagon boron nitride (hBN)[22, 23], in ABC stacked trilayer graphene moiré
with hBN alignment[24], and also in transition Metal Dichalcogenide (TMD)
bilayers[25, 26]. More recently, FQAH was finally observed in twisted MoTe2
homobilayer from optical and capacitance measurement[27, 28], further
supported directly from transport measurements[29, 30]. Theoretically the
existence of the FQAH (zero field FCI) in the twisted MoTe2 system is quite
natural due to the existence of isolated narrow Chern band[31, 32, 33]. Indeed
FQAH phases were predicted[34, 35] even before the experiment. Recent
theoretical works further confirm the existence of FQAH [36, 37, 38, 39] and
also composite Fermi liquid (CFL)[40, 41] at even denominator. Apparently
twisted MoTe2 system mimics the familiar lowest Landau level physics quite
well.

Possibility of FQAH was also discussed in graphene systems[14, 42, 43, 44, 45,
46, 47], but so far it has been reported only in pentalayer ($5$-layer)
rhombohedral stacked graphene aligned with hBN[48], which is actually a
surprise and unexpected theoretically. In the following we always refer
$n$-layer graphene to rhombohedral stacked multilayer graphene. It was
predicted by one of us[14, 49] that there is a $C\neq 0$ band in $n$-layer
graphene aligned with hBN for one sign of the displacement field $D$,
corresponding to the side where particles are pushed away from the hBN. See
also Ref. [50] for earlier discussions on possible narrow bands with non-zero
Berry curvature in the system. The superlattice potential from the hBN
alignment on top only applies on the top layer of the $n$-layer graphene.
Therefore, for the topological side of $D$, electrons fell a weaker
superlattice potential and the remote gap tends to be small. This effect is
more severe for larger $n_{\mathrm{layer}}$ because $D$ can polarize the layer
degree of freedom more easily. For the $5$-layer graphene, FQAH is found only
in the strong displacement region and we estimate that the moire superlattice
potential projected to the conduction band is only at order of
$0.05\mathrm{meV}$. Such a small superlattice potential is only a perturbation
term and we indeed notice that the band structures with and without moiré
potential are roughly the same. Hence it is a surprise that even QAH insulator
can be stabilized at $\nu=1$ filling per moiré unit cell.

We resolve this puzzle by performing a Hartree Fock calculation at filling
$\nu=1$ first. Assuming spin-valley polarization[19], we find a $C=1$ Chern
insualtor in a range of displacement field $D$ at twist angle
$\theta\in(0.66^{\circ},1.47^{\circ})$ with sizable band gap and narrow
bandwidth of the filled band. Then at fractionally filling of the Hartree Fock
(HF) renormalized band, we find FQAH insulators through exact diagonalization
(ED). There is also signature of composite Fermi liquid[51] at 1/2 filling. In
this calculation, the external moiré potential does not play any essential
role. The same result can be reproduced by a calculation with the moiré
potential turned off by hand. In this sense, we should view the $\nu=1$ Chern
insulator as a QAH Wigner crystal spontaneously breaking the approximate
continuous translation symmetry, which is then pinned by the small external
moiré potential. The FQAH is then realized at fractional filling of this
spontaneously formed crystal. Our work thus establishes the n-layer graphene
as a completely new platform for FQAH phase, distinct from twisted MoTe2. We
note that the picture of viewing moiré as a perturbation has already been
suggested in Ref.52 in the trilayer graphene system, though in that case there
is still controversy and the main focus is on the valence band[53]. When the
$n_{\mathrm{layer}}\geq 4$, the conduction band is more flat and the projected
moiré potential is clearly small, hence the physics is closer to the moiréless
limit with approximate continuous translation symmetry. This implies that it
may be possible even to realize the QAH Wigner crystal in the moiréless
n-layer graphene[54, 55, 56, 57]. Note that QAH Wigner crystal state has been
suggested in experiment of bilayer graphene[58], though there still lacks
unambiguous evidence of QAH effect there.

In the second part, we ask whether it is possible to generate a moiré
superlattice potential on the $n$-layer graphene without hBN alignment.
Especially it is desirable to tune the moiré superlattice potential
continuously and control the superlattice period separately. We propose a new
setup to achieve this by Coulomb induced superlattice. Specifically, we
consider a system with n-layer graphene separated from a twisted bilayer
graphene by a thin hBN. The $n$-layer graphene is misaligned with hBN, but it
feels the moiré superlattice potential from the hBN by inter-layer repulsion.
The moiré peirod $a_{M}$ is controlled by the twist angle of TBG, which does
not need to be close to the magic angle for our purpose. We will tune the
density of TBG to be $\nu_{\mathrm{TBG}}=4$ so it is in a band insulator. We
find Chern insulator with narrow $C=1$ band at $\nu=1$ in a range of the
parameter space $(a_{M},d,D)$ for $n_{\mathrm{layer}}=4,5,6,7$. Now narrow
$C=1$ band is possible in both signs of $D$ for $a_{M}\approx 10\mathrm{nm}$.
The same idea works if we replace the TBG with a monolayer graphene aligned
with the middle hBN. Given the large tunability of the system, we expect a
rich phase diagram to be found in future experimental studies of this new
class of moiré materials. Especially one can tune the moiré potential by the
distance $d$ to verify our conjecture that QAH and maybe even FQAH are
possible in the zero moiré potential limit.

Model We model the n-layer graphene aligned with hexagonal boron nitride (hBN)
as $H_{K}=H_{0}+H_{M}$, where $H_{0}$ is the Hamiltonian of $n$-layer graphene
and $H_{M}$ is the effective moiré potential from hBN alignment. It is by now
well established that the $\nu=1$ QAH insulator is spin-valley polarized from
the Coulomb exchange[19]. The real puzzle is how to obtain a Chern band within
one spin-valley flavor. We will thus assume spin-valley polarization and focus
on one valley and define
$\psi_{z}(\mathbf{k})=(f_{z;A}(\mathbf{k}),f_{z;B}(\mathbf{k}))^{T}$ for the
two sublattice at the layer $z$. Then the free Hamiltonian is written down as
a $2n_{\mathrm{layer}}$ band model:

$\displaystyle H_{0}=$
$\displaystyle\sum_{\mathbf{k}}\sum_{z=1}^{n_{\mathrm{layer}}}\psi^{\dagger}_{z}(\mathbf{k})\begin{pmatrix}V_{z,l}+u_{A,l}&-v_{1}(k_{x}-\mathrm{i}k_{y})e^{\mathrm{i}\theta_{3}}\\\
-v_{1}(k_{x}+\mathrm{i}k_{y})e^{-\mathrm{i}\theta_{3}}&V_{z,l}+u_{B,l}\end{pmatrix}\psi_{z}(\mathbf{k})+\sum_{\mathbf{k}}\sum_{z=1}^{n_{\mathrm{layer}}-2}\psi_{z}^{\dagger}({\bf
k})\begin{pmatrix}0&\frac{\gamma_{2}}{2}\\\ 0&0\end{pmatrix}\psi_{z+2}({\bf
k})+\mathrm{H.c}.$
$\displaystyle+\sum_{\mathbf{k}}\sum_{z=1}^{n_{\mathrm{layer}}-1}\psi_{z}^{\dagger}({\bf
k})\begin{pmatrix}-v_{4}(k_{x}-\mathrm{i}k_{y})e^{\mathrm{i}\theta_{3}}&-v_{3}(k_{x}+\mathrm{i}k_{y})e^{-\mathrm{i}\theta_{3}}\\\
\gamma_{1}&-v_{4}(k_{x}-\mathrm{i}k_{y})e^{\mathrm{i}\theta_{3}}\end{pmatrix}\psi_{z+1}({\bf
k})+\mathrm{H.c.}$

where the parameters $v_{i}=\frac{\sqrt{3}}{2}t_{i}$, with [59, 60]
$t_{1}=-2600\mathrm{meV}$, $t_{3}=293\mathrm{meV}$, $t_{4}=144\mathrm{meV}$,
$\gamma_{1}=358\mathrm{meV}$, $\gamma_{2}=-8.3\mathrm{meV}$ correspond to the
hoppings. $V_{z,l}$ is the layer-dependent potential from the displacement
field with $V_{z,l}=\frac{D(l-1-n_{\mathrm{layer}}/2)}{n_{\mathrm{layer}}-1}$,
where $D$ is the displacement field, and $l=1,2,\cdots$ is the $l$-th layer.
$u_{A,1}=u_{B,n_{\mathrm{layer}}}=0,u_{B,1}=u_{A,n_{\mathrm{layer}}}=12.2\mathrm{meV},u_{A,l}=u_{B,l}=-16.4\mathrm{meV}$
for other terms. The phase $\theta_{3}$ is introduce to rotate the graphene.
We have $\theta_{3}=\arctan\frac{\theta}{\delta}$ with $\theta$ as the twist
angle between graphene and hBN.
$\delta=\frac{a_{\mathrm{hBN}}-a_{\mathrm{G}}}{a_{\mathrm{hBN}}}=0.017$,
$a_{\mathrm{G}}$(0.246nm) and $a_{\mathrm{hBN}}$(0.25025nm) are the lattice
constant of graphene and hBN respectively.

Figure 1: Numerical results of the 5-layer graphene/hBN system at
$\theta=0.77^{\circ}(a_{M}=11.36\mathrm{nm})$, $\epsilon=6$,
$D=-160\mathrm{meV}$. (a) Conduction band structure before and after
performing HF approximation, the HF calculation is performed by using
$30\times 30$ points in the moiré Brillouin zone (MBZ). The solid line and the
dashed line corresponds to $V_{M}=1,0$ respectively. They are basically
identical to each other. (b) ED calculation at $\nu=2/3$ filling with a system
size of $5\times 6$. We project to the lowest band in the Hartree fock
calculation using hole picture at hole filling $\nu_{h}=1-\nu=1/3$. We use the
dispersion of the band from the Hartree Fock calculation. $k_{1}$ and $k_{2}$
is defined as $\mathbf{k}=k_{1}\mathbf{G_{1}}/N_{1}+k_{2}\mathbf{G_{2}}/N_{2}$
with the system size of $N_{1}\times N_{2}$. We also show the spectral flow
under flux insertion along $k_{2}$ direction. The flux insertion calculation
is performed with system size of $4\times 6$. Clearly there is a permutation
of the degenerate states under a flux of $\phi=2\pi$. (c) ED calculation at
$\nu=3/5$ filling with system size $5\times 5$. In the lower part we show a
zoom in of the five nearly degenerate states.

The moiré potential term is
$H_{M}=\sum_{\mathbf{k}}\sum_{j=1}^{6}\psi^{\dagger}_{1}(\mathbf{k})H_{M}(\mathbf{G_{j}})\psi_{1}(\mathbf{k+G_{j}})$
which only acts on the first layer aligned with the hBN. Here the moiré
reciprocal lattice vectors are
$\mathbf{G_{j}}=\frac{4\pi}{\sqrt{3}a_{M}}(\cos(\frac{j\pi}{3}-\frac{5\pi}{6}),\sin(\frac{j\pi}{3}-\frac{5\pi}{6}))^{T},j=1,2\cdots,6$.
The moiré potential parameters are listed [61, 60, 59] in the supplementary.
In the following we will also introduce a parameter $V_{M}$ to scale the moiré
potential term by $H_{M}\rightarrow V_{M}H_{M}$. We mainly focus on
$V_{M}=1,0$.

In addition to the kinetic energy, we also have the Coulomb interaction:

$H_{V}=\frac{1}{2A}\sum_{l,l^{\prime}}\sum_{\mathbf{q}}V_{ll^{\prime}}(q):\rho_{l}(q)\rho_{l^{\prime}}(-q):$
(2)

where $A$ is the area of the sample, $\rho_{l}(q)$ is the density operator in
the layer $l$. $V_{ll^{\prime}}(\mathbf{q})=\frac{e^{2}\tanh
q\lambda}{2\epsilon_{0}\epsilon q}e^{-q|l-l^{\prime}|d_{\mathrm{layer}}}$,
$\lambda=30\mathrm{nm}$ is the screening length and
$d_{\mathrm{layer}}=0.34\mathrm{nm}$ is the distance between nearest two
layers.

Numerical evidence of QAH and FQAH We focus on the conduction band on the
large and negative $D$, so electrons in the conduction band are mainly staying
in the bottom, far away from the aligned hBN on the top. In Fig. 1(a) we can
clearly see that there is no remote band gap for the 5-layer graphene at
$\theta=0.77^{\circ}$, $D=-160$ meV. Actually the band structure is basically
the same as the band of the moiréless 5-layer graphene folded into the moiré
Brillouin zone (MBZ). Then with a Hartree Fock calculation (see the
supplementary) keeping the lowest $N_{b}=3$ number of conduction bands at the
filling $\nu=1$ per moiré unit cell, we find a sizable band gap opening and a
filled nearly flat $C=1$ band (see Fig. 1(a)). This means that the $\nu=1$
filling is an interaction driven QAH insulator. Then we perform exact
diagonalization (ED) projected to the Hartree Fock renormalized lowest band.
Note that we need to use the hole picture with Hartree-Fock renormalized
dispersion relative to the fully filled band insulator. The electron picture
is different due to the normal ordering of the interaction term. In Fig.
1(b)(c) we can see low lying 3-fold and 5-fold states, consistent with FCI at
electron filling $\nu=2/3$ and $\nu=3/5$ respectively. At $\nu=\frac{1}{2}$,
we also find signature of composite Fermi liquid (CFL) (see the
supplementary).

Phase diagram Next we study the phase diagram in the $(\theta,D)$ space for
the 5-layer graphene/hBN system. In Fig. 2(a) we show a color map of
$\frac{|C|\Delta}{W}$ with $C,\Delta,W$ as the Chern number, the band gap and
the bandwidth from the Hartree Fock calculation at the filling $\nu=1$. The
phase diagram is dominated by $|C|=0,1$. We find that we always have $C=0$ for
twist angle $\theta<0.65^{\circ}$. The ideal region with large
$\frac{|C|\Delta}{W}$ is stabilized with $a_{M}\in(8,12)\mathrm{nm}$,
corresponding to $\theta\in(0.66^{\circ},1.47^{\circ})$.

Figure 2: The HF calculation is performed by using $12\times 18$ points in
MBZ. (a) $\epsilon=6$, dependence of $\frac{|C|\Delta}{W}$ on $D$ and the
moire lattice constant $a_{M}$. $|C|$ is the Chern number, $\Delta$ is the
band gap at filling $n=1$ and $W$ is the band width of the lowest band after
Hartree Fock. When $a_{M}>12\mathrm{nm}$, the C=1 QAH crystal disappears. (b)
At $\theta=0.9^{\circ}(a_{M}=10.63\mathrm{nm}),\epsilon=6$, dependence of the
Hartree Fock renormalized band gap and the bandwidth on $D$. (c)At
$\theta=0.9^{\circ}(a_{M}=10.63\mathrm{nm}),\epsilon=6$, dependence of the
many body gap of the FQAH insulator at $\nu=2/3$ filling on $D$. The many body
gap is defined as the energy difference between the fourth and the third
states in the ED calculation. The ED calculation is performed by using
$4\times 6$ points in MBZ. We have introduced a factor $V_{M}$ to the moiré
potential term $H_{M}$. $V_{M}=0$ means that the moiré potential is turned
off. (d) Similar to (c), but with different value of $\epsilon$ and $V_{M}=0$.

In Fig. 2(b) we further check the existence of a narrow $C=1$ band with
sizable band gap at $\epsilon=6$ and $\theta=0.9^{\circ}$, with and without
the moiré potential term. In Fig. 2(c) we show the corresponding many body gap
of the FQAH at $\nu=2/3$. One can see a dome of FQAH phase in a wide range of
the displacement field $D$. We find $\theta=0.9^{\circ}$ is more ideal than
$\theta=0.77^{\circ}$. Interestingly, we find that $V_{M}=0$ and $V_{M}=1$
show basically the same result, implying that the moiré potential term is not
important in our calculation. In Fig. 2(d) we show the many body gap at
different dielectric constant $\epsilon=10,15,20$. One can see the stable
region of FQAH is diminished at larger $\epsilon$, but there is still a gap at
order $0.2$ meV at $\epsilon=20$, which is actually closer to the experimental
value.

The role of the moiré potential Our result in Fig. 2 clearly shows that the
external moiré potential term is not essential for QAH and FQAH phases in this
system. We project the moiré potential into the conduction band, the matrix
element is around $0.03\mathrm{meV}$ and $0.05\mathrm{meV}$ at $K,K^{\prime}$
points respectively, which is obviously just a tiny perturbation. Therefore,
we conclude that the narrow Chern band shown in Fig. 1(a) is completely from
the Coulomb interaction. Our $\nu=1$ QAH insulator survives to the
$V_{M}\rightarrow 0$ limit and should be viewed as a Wigner crystal
spontaneously breaking the continuous translation symmetry. The FQAH phases
are then realized at fractional filling of this crystal. The external moiré
potential may still be necessary to pin the crystal and stabilize the FQAH
phases, but it plays no roles in producing the narrow Chern band. It is
interesting to see whether a QAH Wigner crystal can be realized in a really
moiréless system in future experimental investigations.

Figure 3: Illustration of the $n$-layer graphene/hBN/TBG setup. The Coulomb
interaction between the TBG and the $n$-layer graphene will effectively
generate a moiré potential in the $n$-layer graphene. We fix the density of
TBG in the fully filled filling $\nu_{\mathrm{TBG}}=4$. The TBG can also be
replaced with a monolayer graphene (MG) aligned with the middle hBN. Figure 4:
Dependence of $\frac{|C|\Delta}{W}$ on $D$ and $a_{M}$ of the $n$-layer
graphene/hBN/TBG setup for (a) 4-layer, (b) 5-layer, (c) 6-layer, (d) 7-layer.
$|C|$, $\Delta$ and $W$ follow the same definition as in Fig. 2(a). Here we
use $\epsilon=6$, and the distance between the TBG and $n$-layer graphene is
$d=1\mathrm{nm}$. Figure 5: (a) The band gap, the bandwidth and Chern number
for the 5-layer graphene/hBN/TBG system at filling $\nu=1$ as a function of
the distance $d$ between the TBG and 5-layer graphene in the setup of Fig. 3.
Here we use $D=180\mathrm{meV}$. (b) Many body gap for $D=180\mathrm{meV}$, at
$\nu=\frac{2}{3}$ with system size $4\times 6$. We use twisted angle
$\theta=1.5^{\circ}$ ($a_{M}\approx 9.4\mathrm{nm}$), and dielectric constant
$\epsilon=6$. One can see that FQAH is possible now also at the $D>0$ side and
survives to the $d\rightarrow+\infty$ limit.

Coulomb induced moire In the second part, we propose a different route to
generate superlattice potential in the $n$-layer graphene. As is illustrated
in Fig. 3, we have $n$-layer graphene separated with a twisted bilayer
graphene (TBG) by a thin hBN with thickness $d$. TBG has moiré lattice
constant $a_{M}=a_{\mathrm{G}}/(2\sin\frac{\theta}{2})$ controlled by its
twist angle $\theta$. The Coulomb interaction between the TBG layer and the
$n$-layer graphene will effectively induce a moiré potential in the $n$-layer
graphene layers. We will fix the density of the TBG to be
$\nu_{\mathrm{TBG}}=4$ so it is in a band insulator. Then the TBG has a charge
profile, which generates superlattice potential felt by each layer of the
$n$-layer graphene: $V_{l}({\bf
G}_{i})=\frac{2}{\sqrt{3}}\frac{e^{2}}{2\epsilon\epsilon_{0}a_{M}}e^{-G_{M}{(d+ld_{\mathrm{layer}})}}\frac{1}{G_{M}a_{M}}\langle\rho({\bf
G}_{M})\rangle$, where $\langle\rho({\bf G}_{M})\rangle$ is the Fourier
transformation of the density profile in the TBG and
$d_{\mathrm{layer}}=0.34\mathrm{nm}$ is the distance between the adjacent
graphene layers. $l=0,1,...$ is the layer index count from the one closest to
the TBG. The Hamiltonian also depends on the rotation angle of the n-layer
graphene relative to the TBG through the $\theta_{3}$ term in Eq. Fractional
quantum anomalous Hall effects in rhombohedral multilayer graphene in the
moiréless limit and in Coulomb imprinted superlattice. For now we set
$\theta_{3}=0$ for simplicity.

Narrow $C=1$ Chern band in the n-layer graphene/hBN/TBG system We perform the
Hartree Fock calculation and draw a phase diagram in Fig. 4. The result is
quite similar to Fig. 2(a) of the 5-layer graphene/hBN system. We find that a
narrow $C=1$ Chern band is possible for 4-layer, 5-layer, 6-layer and 7-layer
graphene system with $a_{M}$ in a range smaller than $11\mathrm{nm}$. Now the
$D>0$ side also hosts a narrow Chern band. We can also replace TBG with a
monolayer graphene (MG) aligned with the middle hBN. At $d=1\mathrm{nm}$, for
$\epsilon=6$, the moiré potentials $V_{0}(G_{M})$ in the n-layer
graphene/hBN/TBG and n-layer graphene/hBN/MG are, $17.9$meV and $0.84$meV,
respectively. In our Hartree Fock calculation we can get the QAH Wigner
crystal in both cases as the moiré potential does not make any difference. But
it is known that the Hartree Fock calculation may overestimate the strength of
an insulating state, we leave to future experiment to see whether the weak
moiré potential from the n-layer graphene/hBN/MG system is enough to stabilize
the QAH crystal and FQAH phases. In this new setup we can control the moiré
potential by tuning the distance $d$. In Fig. 5, we find that QAH-Wigner
crystal phase, together with the FQAH phase, survive to the
$d\rightarrow+\infty$ limit also for $D>0$, consistent with our conclusion
that the external moiré potential is not essential for the Chern band physics.

Summary In conclusion, we studied the recently observed FQAH in the pentalayer
rhombohedral stacked graphene aligned with hBN and in related systems. In
contrast to the standard framework for other moiré systems, we find that the
external moiré potential is negligible and the QAH and FQAH phases should be
understood as from a spontaneously formed topological Wigner crystal due to
interaction in a model with continuous translation symmetry. In future it is
interesting to study the interplay between the almost gapless phonon modes of
the Wigner crystal with the small moiré potential and disorder. As the QAH
Wigner crystal is stabilized with a period around $10\mathrm{nm}$, it may be
possible to stabilize it at $\nu=1/2$ filling of moiré unit cell with a moiré
lattice constant of around $5\mathrm{nm}$. Besides, we propose a new setup
with moiré in the n-layer graphene generated from the Coulomb repulsion of
twist bilayer graphene separated by a thin hBN. We find similar narrow $C=1$
Chern band in $n_{\mathrm{layer}}=4,5,6,7$. The ability to separately tune the
superlattice potential and the moiré period should reveal a rich phase diagram
in future experimental studies. It is also interesting to dope the TBG layer,
which screens the Coulomb in the n-layer graphene and may drive a transition
out of the quantum Hall phases[62].

Note added: We note two other papers[63, 64] on the same topic also appeared
roughly at the same time. Our results agree with each other when they overlap.

Acknowledgement YHZ thanks Zhihuan Dong and T. Senthil for many inspiring
discussions and previous collaborations on moiré systems. We especially thank
Zhihuan Dong for helps on improving the efficiency of the Hartree Fock
calculation. We also thank Long Ju and Sankar Das Sarma for discussions and
useful comments. The numerical simulation was carried out at the Advanced
Research Computing at Hopkins (ARCH) core facility (rockfish.jhu.edu), which
is supported by the National Science Foundation (NSF) grant number OAC
1920103. This work was supported by the National Science Foundation under
Grant No. DMR-2237031.

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##  Appendix A The moiré potential form

The moiré potential parameters used in our calculation are listed in the
following:

$\displaystyle H_{z}(\mathbf{G_{1}})$
$\displaystyle=H_{z}(\mathbf{G_{3}})=H_{z}(\mathbf{G_{3}})=C_{z}e^{\mathrm{i}\phi_{z}}$
$\displaystyle=H_{z}^{*}(\mathbf{G_{2}})=H_{z}^{*}(\mathbf{G_{4}})=H_{z}^{*}(\mathbf{G_{6}}),$
(3) $\displaystyle H_{0}(\mathbf{G_{1}})$
$\displaystyle=H_{0}(\mathbf{G_{3}})=H_{0}(\mathbf{G_{5}})=C_{0}e^{\mathrm{i}\phi_{0}}$
$\displaystyle=H_{0}^{*}(\mathbf{G_{2}})=H_{0}^{*}(\mathbf{G_{4}})=H_{0}^{*}(\mathbf{G_{6}}),$
(4) $\displaystyle H_{AB}(\mathbf{G_{1}})$
$\displaystyle=H_{AB}^{*}(\mathbf{G_{4}})=C_{AB}e^{\mathrm{i}(2\pi/3-\phi_{AB})},$
(5) $\displaystyle H_{AB}(\mathbf{G_{3}})$
$\displaystyle=H_{AB}^{*}(\mathbf{G_{2}})=C_{AB}e^{-\mathrm{i}\phi_{AB}},$ (6)
$\displaystyle H_{AB}(\mathbf{G_{5}})$
$\displaystyle=H_{AB}^{*}(\mathbf{G_{6}})=C_{AB}e^{\mathrm{i}(-2\pi/3-\phi_{AB})},$
(7)

where $C_{0}=-10.13\mathrm{meV}$, $\phi_{0}=86.53^{\circ}$,
$C_{z}=-9.01\mathrm{meV}$, $\phi_{z}=8.43^{\circ}$,
$C_{AB}=11.34\mathrm{meV}$, $\phi_{AB}=19.60^{\circ}$. Here
$H_{AA}(\mathbf{G_{j}})=H_{0}(\mathbf{G_{j}})+H_{z}(\mathbf{G_{j}})$ and
$H_{BB}(\mathbf{G_{j}})=H_{0}(\mathbf{G_{j}})-H_{z}(\mathbf{G_{j}})$ for each
$\mathbf{G_{j}}$.

##  Appendix B Hartree Fock calculation

We perform Hartree Fock approximation by keeping the first $N_{b}$ conduction
bands in the MBZ. The interaction term is written as:

$H_{V}=\frac{1}{2A}\sum_{\mathbf{q}}\sum_{l,l^{\prime}}V_{l,l^{\prime}}(\mathbf{q}):\rho_{l}(\mathbf{q})\rho_{l^{\prime}}(\mathbf{-q}):,$
(8)

where
$\rho_{l}(\mathbf{q})=\sum_{\mathbf{k},m,m^{\prime}}c^{\dagger}_{m}(\mathbf{k+q})\Lambda^{l}_{m,m^{\prime}}(\mathbf{k},\mathbf{q})c_{m^{\prime}}(\mathbf{k})$
and $\Lambda^{l}_{m,m^{\prime}}(\mathbf{k},\mathbf{q})=\langle
u_{m}({\mathbf{k+q}})|P_{l}|u_{m^{\prime}}(\mathbf{k})\rangle$, $P_{l}$ is the
projection operator to layer $l$, $m$ and $m^{\prime}$ are band indices.
$V_{l,l^{\prime}}(\mathbf{q})=\frac{e^{2}e^{-q|l-l^{\prime}|d_{\mathrm{layer}}}\tanh{(q\lambda)}}{2\epsilon\epsilon_{0}|q|}$,
where $d_{\mathrm{layer}}$ is the distance between nearest two layers,
$\lambda$ is the screening length. In our calculation, we choose that
$d_{\mathrm{layer}}=0.34\mathrm{nm}$ and $\lambda=30\mathrm{nm}$. The
interaction can be decoupled into Hartree and Fock terms respectively, leading
to a mean field theory:

$\begin{split}H_{V}=&\sum_{\mathbf{k_{1}},\mathbf{k_{2}}}\sum_{\mathbf{q}}\sum_{m,m^{\prime},n,n^{\prime}}V_{m,m^{\prime},n,n^{\prime}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q})c^{\dagger}_{m}(\mathbf{k_{1}+q})c^{\dagger}_{n}(\mathbf{k_{2}-q})c_{n^{\prime}}(\mathbf{k_{2}})c_{m^{\prime}}(\mathbf{k_{1}})\\\
=&\sum_{\mathbf{k_{1}},\mathbf{k_{2}}}\sum_{m,m^{\prime},n,n^{\prime}}\left(V_{m,m^{\prime},n,n^{\prime}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{0})-V_{n,m^{\prime},m,n^{\prime}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{2}-k_{1}})\right)\langle
c^{\dagger}_{m}(\mathbf{k_{1}})c_{m^{\prime}}(\mathbf{k_{1}})\rangle
c_{n}^{\dagger}(\mathbf{k_{2}})c_{n^{\prime}}(\mathbf{k_{2}}).\end{split}$ (9)

The interaction vertex
$V_{m,m^{\prime},n,n^{\prime}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q})$ can
be calculated as:

$V_{m,m^{\prime},n,n^{\prime}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q})=\sum_{l,l^{\prime}}V_{l,l^{\prime}}(\mathbf{q})\Lambda^{l}_{m,m^{\prime}}(\mathbf{k_{1}},\mathbf{q})\Lambda^{l}_{n,n^{\prime}}(\mathbf{k_{2}},\mathbf{-q}).$
(10)

In the above $m,m^{\prime},n,n^{\prime}=1,2,...,N_{b}$ are the band indexes.
In the calculation we solve $\rho_{mm^{\prime}}(\mathbf{k})=\langle
c^{\dagger}_{m}(\mathbf{k})c_{m^{\prime}}(\mathbf{k})\rangle$ self
consistently from randomized initial ansatz. We try $40$ number of randomized
initial ansatz and choose the one with lowest energy.

##  Appendix C ED calculation

We perform all the ED calculation in the hole picture. We use $h(k)$ as the
hole annihilation operator. The total Hamiltonian is written as:

$H=-\sum_{\mathbf{k}}\epsilon_{\mathrm{HF}}(\mathbf{k})h^{\dagger}(\mathbf{k})h(\mathbf{k})+\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q}}V(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q})h^{\dagger}(\mathbf{k_{1}+q})h^{\dagger}(\mathbf{k_{2}-q})h(\mathbf{k_{2}})h(\mathbf{k_{1}}),$
(11)

where $\epsilon_{\mathrm{HF}}(\mathbf{k})$ is the first conduction band
dispersion after HF calculation. The interaction vertex can be calculated as:

$V(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q})=\sum_{l,l^{\prime}}V_{l,l^{\prime}}(\mathbf{q})\left(\Lambda^{l}_{\mathrm{HF}}(\mathbf{k_{1}},\mathbf{q})\Lambda^{l^{\prime}}_{\mathrm{HF}}(\mathbf{k_{2}},\mathbf{-q})\right)^{*},$
(12)

where $\Lambda^{l}_{\mathrm{HF}}(\mathbf{k_{1}},\mathbf{q})=\langle
u_{\mathrm{HF}}(\mathbf{k_{1}+q})|P_{l}|u_{\mathrm{HF}}(\mathbf{k_{1}})\rangle$
is the form factor of the bloch wavefunction on layer $l$ after performing HF
approximation.

##  Appendix D Composite Fermi liquid at $\nu=1/2$ filling

With parameters $\theta=0.9^{\circ}$($a_{M}=10.63\mathrm{nm}$), $\epsilon=6$,
$D=-160\mathrm{meV}$, we perform an ED calculation at $\nu=1/2$ filling in
system size of $4\times 5$ and $4\times 6$. The result is shown in Fig. 6. In
$4\times 5$ size case, we perform a HF calculation at a $20\times 20$ system
and extracts a $4\times 5$ submesh to perform the ED calculation. In $4\times
6$ case, we perform a HF calculation at a $24\times 24$ system and extracts a
$4\times 6$ submesh to perform the ED calculation. As shown in Fig. 6, we do
not see any many body gap in the ED spectrum, implying a gapless state.
Meanwhile the momentum distribution $n(\mathbf{k})$ is relatively flat.
Therefore the phase is consistent with the composite Fermi liquid (CFL) state.

Figure 6: (a) ED calculation at $\nu=1/2$ filling with system size of $4\times
5$ and the momentum distribution $n(\mathbf{k}=\langle
c^{\dagger}(\mathbf{k})c(\mathbf{k})\rangle$. (b) Similar with (a), but the
system size is $4\times 6$ instead.

##  Appendix E More results

We use $N_{b}=3$ to perform the Hartree Fock calculation in the main text, in
which the bare band structure is obtained by using free electron approximation
in $5\times 5$ moiré Brillouin zones. We also do HF calculation in $N_{b}=4,5$
cases, the result is shown in Fig. 7. It is verifed that the many body gap
have already converged at $N_{b}=3$. At $\epsilon=6$ and $N_{b}=4$, dependence
of $\frac{|C|\Delta}{W}$ on $D$ and $a_{M}$ is shown in Fig. 8(d), which is
nearly the same as Fig.2(a) in the main text.

We also add more details about Fig.2(a) in the main text in Fig. 8(a)-(c).

Figure 7: ED calculation of the $5$-layer graphene/hBN system at
$\theta=0.77^{\circ}$($a_{M}=11.36\mathrm{nm}$), $\epsilon=6$,
$D=-160\mathrm{meV}$. The system size is fixed to be $4\times 6$. Figure 8:
The same parameters as in Fig.2(a) in the main text, but in (d) we use
$N_{b}=4$ instead. In (a)-(c), we still use $N_{b}=3$. (a) Absolute value of
lowest band’s Chern number after HF: $|C|$. The blank area corresponds to the
metal phase. (b) The band gap after HF: $\Delta$. (c) The band width of the
lowest band after HF: $W$. (d) $\frac{|C|\Delta}{W}$.

##  Appendix F The moiré potential from TBG or MG

We consider a different way to generate the moiré potential in the $n$-layer
graphene through Coulomb interaction. We separate it with twisted bilayer
graphene (TBG) or monolayer graphene (aligned with the middle hBN) through a
thin hBN with thickness $d$. The coulomb interaction between TBG (or monolayer
graphene) and the graphene layers is:

$\displaystyle H_{V}=\frac{1}{A}\sum_{\mathbf{q},l}V_{l}({\bf
q})\rho_{t,l}({\bf q})\rho_{b}(-{\bf q}),$ (13)

where $\rho_{b}({\bf q})=\sum_{\bf k}f_{b}^{\dagger}({\bf k}+{\bf
q})f_{b}({\bf k})$ and $\rho_{t,l}(-{\bf q})=\sum_{\bf
k}f_{t,l}^{\dagger}({\bf k}-{\bf q})f_{t,l}({\bf k})$ correspond to the
density in the twisted bilayer graphene, and the density in each layer in the
multilayer graphene respectively.
$A=\frac{4\pi^{2}N_{\mathrm{cell}}}{|\bf{G}_{1}\times{\bf G}_{2}|}$ is the
sample area. $V_{l}({\bf q})=\frac{e^{2}}{2\epsilon\epsilon_{0}|{\bf
q}|}e^{-|{\bf q}|(d+ld_{\mathrm{layer}})}$ is the screened Coulomb potential
in the materials, and $\epsilon$ is the dielectric constant,
$d_{\mathrm{layer}}$ is the interlayer distance of the graphene multilayers.
In the discretized momentum space, the interaction $H_{V}$ becomes:

$\displaystyle H_{V}$
$\displaystyle=\frac{2}{\sqrt{3}}\frac{1}{N_{\mathrm{cell}}}\sum_{{\mathbf{q},l}}\frac{e^{2}}{2\epsilon\epsilon_{0}a_{M}}\frac{1}{|{\bf
q}|a_{M}}e^{-|{\bf q}|(d+ld_{\mathrm{layer}})}\rho_{b}({\bf
q})\rho_{t,l}(-{\bf q})$
$\displaystyle=\frac{2}{\sqrt{3}}\frac{1}{N_{\mathrm{cell}}}\sum_{{\mathbf{q},l}}\frac{e^{2}}{2\epsilon\epsilon_{0}a_{M}}\frac{1}{|{\bf
q}|a_{M}}e^{-|{\bf q}|(d+ld_{\mathrm{layer}})}\rho_{b}({\bf q})\sum_{{\bf
k}_{2}}f_{t,l}^{\dagger}({\bf k}_{2}-{\bf q})f_{t,l}({\bf k}_{2}).$ (14)

The density in the twisted bilayer graphene can be approximated as:

$\displaystyle\rho_{b}({\bf q})=\sum_{\bf k}c_{b}^{\dagger}({\bf k}+{\bf
q})\Lambda_{b}({\bf k},{\bf q})c_{b}({\bf k}),$ (15)

where $\Lambda_{b}({\bf k},{\bf q})=\langle
u_{b}(\mathbf{k+q})|u_{b}(\mathbf{k})\rangle$ is the form factor,
$c_{b}(\mathbf{k})$ is the annihilation operator of the electrons in the
conduction band. The potential generated to the $n$-layer graphene is:

$\displaystyle V_{l}({\bf
G}_{M})=\frac{2}{\sqrt{3}}\frac{e^{2}}{2\epsilon\epsilon_{0}a_{M}}e^{-G_{M}{(d+ld_{\mathrm{layer}})}}\frac{1}{G_{M}a_{M}}\langle\rho_{b}({\bf
G}_{M})\rangle,$ (16)

where $d$ is also the distance between the TBG (or MG) and the closest layer
of multilayer graphene. Finally, we find the potential in the layer
$l=0,1,...,n_{\mathrm{layer}}-1$ is $V_{l}({\bf G}_{M})=V_{0}({\bf
G}_{M})e^{-G_{M}ld_{\mathrm{layer}}}$, where $V_{0}({\bf G}_{M})$ is the
strength of the potetial in the first graphene layer and it depends on the
twisted angle and the distance between the hBN and the graphene layer.

We have shown the phase diagram for the case with TBG. We can also use
monolayer graphene aligned with the middle hBN. But this time the density
profile $\rho_{b}(\mathbf{G}_{M})$ is weaker by a factor of 10. At
$d=1\mathrm{nm}$ and $\epsilon=6$, the moiré potential $V_{0}(G_{M})$
imprinted on the $n$-layer graphene is only around $0.8$ meV. However, in our
Hartree-Fock calculation, we still find robust region of narrow $C=1$ Chern
band at $\nu=1$, as shown in Fig.9. This is expected because the moiré
potential is negligible and not essential anyway in our Hartree Fock
calculation.

Figure 9: $\frac{|C|\Delta}{W}$ of 5-layer graphene/hBN/MG system. $a_{M}$ is
controlled by the twist angle of the MG and the middle hBN. We use
$\epsilon=6$, $d=5\mathrm{nm}$.