# Emergence of broken-symmetry states at half-integer band fillings in twisted
bilayer graphene

Saisab Bhowmik<EMAIL_ADDRESS>Department of Instrumentation and Applied
Physics, Indian Institute of Science, Bangalore, 560012, India Bhaskar Ghawri
Department of Physics, Indian Institute of Science, Bangalore, 560012, India
Nicolas Leconte  Department of Physics, University of Seoul, Seoul 02504,
Korea Samudrala Appalakondaiah  Department of Physics, University of Seoul,
Seoul 02504, Korea Mrityunjay Pandey  Centre for Nano Science and Engineering,
Indian Institute of Science, Bangalore 560 012, India Phanibhusan S. Mahapatra
Department of Physics, Indian Institute of Science, Bangalore, 560012, India
Dongkyu Lee  Department of Physics, University of Seoul, Seoul 02504, Korea
Department of Smart Cities, University of Seoul, Seoul 02504, Korea K.
Watanabe  Research Center for Functional Materials, National Institute for
Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan T. Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan Jeil Jung
<EMAIL_ADDRESS>Department of Physics, University of Seoul, Seoul 02504,
Korea Department of Smart Cities, University of Seoul, Seoul 02504, Korea
Arindam Ghosh  Department of Physics, Indian Institute of Science, Bangalore,
560012, India Centre for Nano Science and Engineering, Indian Institute of
Science, Bangalore 560 012, India U. Chandni<EMAIL_ADDRESS>Department
of Instrumentation and Applied Physics, Indian Institute of Science,
Bangalore, 560012, India

###### pacs:

The dominance of Coulomb interactions over kinetic energy of electrons in
narrow, non-trivial moiré bands of magic-angle twisted bilayer graphene (TBG)
gives rise to a variety of correlated phases such as correlated insulators [1,
2, 3], superconductivity [4, 2, 5, 5, 6, 7], orbital ferromagnetism [2, 8, 9],
Chern insulators [10, 11, 12, 13] and nematicity [14]. Most of these phases
occur at or near an integer number of carriers per moiré unit cell.
Experimental demonstration of ordered states at fractional moiré band-fillings
at zero applied magnetic field $B$, is a challenging pursuit. In this letter,
we report the observation of states near half-integer band-fillings of
$\nu\approx 0.5$ and $\pm 3.5$ at $B\approx 0$ in TBG proximitized by tungsten
diselenide (WSe2) through magnetotransport and thermoelectricity measurements.
A series of Lifshitz transitions due to the changes in the topology of the
Fermi surface implies the evolution of van Hove singularities (VHSs) of the
diverging density of states (DOS) at a discrete set of partial fillings of
flat bands. Furthermore, at a band filling of $\nu\approx-0.5$, a symmetry-
broken Chern insulator emerges at high $B$, compatible with the band structure
calculations within a translational symmetry-broken supercell with twice the
area of the original TBG moiré cell. Our results are consistent with a
spin/charge density wave ground state in TBG in the zero $B$-field limit.

FIG. 1. Electrical characterization and low-field Hall measurements in device
D1, $\theta\approx 1.14^{\circ}$. a. Schematic of hBN encapsulated TBG/WSe2
heterostructure on SiO2/Si substrate. b. Four-probe longitudinal resistance
$R_{xx}$ as a function of carrier density $n$ measured in the temperature $T$
range $3-32$ K for $B=0$. The twist angle $\theta$ $\approx$ $1.14^{\circ}$
was calculated from the position of the resistance peak at a filling $\nu$
($n$/$n_{0}$) = 4 where $n_{0}$ is the density corresponding to one carrier
per moiré unit cell. c. Landau fan diagram at $T=5$ K for magnetic field $B$
up to 7.75 T where d$R_{xx}$/d$n$ is plotted as a function of ($\nu$, $B$).
The Landau levels (LL) diverging from the charge neutrality point (CNP) are
shown by solid black lines. The filling factors 4, 8, 12, and 16 are
determined from the slopes of these lines. In addition, a straight line with a
slope of 2 that nucleates from $\nu$ = 1 at B $\gtrsim$ 5 T is shown by the
solid green line. d. Density dependence of field-symmetrized Hall conductivity
$\sigma_{xy}$ at $B=1$ T and $T$ = 6 K. Zero-crossings appear at $\nu$ = 0,
0.5, 1, 3.5 and 4. $\sigma_{xy}$ suddenly drops to zero at $\nu$ = 2 and 3 and
rises again upon crossing $\nu$ = 2 and 3. The inset shows an additional peak
in $R_{xx}(n)$ around $\nu$ = 3.5, when $B=0$. This data in $R_{xx}$ was
recorded after a magnetic field sweep and the peak at $\nu$ = 3.5 persisted at
zero-field for the next few thermal cycles. e. Hall density $n_{H}$,
calculated from Hall resistance $R_{xy}$ exhibits similar behavior as
$\sigma_{xy}$.

FIG. 2. Symmetry-broken states at $\nu\approx-0.5$ and $\pm 3.5$ in device D2,
$\theta\approx 1.16^{\circ}$. a. $n_{H}$ vs. $\nu$ at $B=2$ T and $T=5$ K
shows sign-changes at $\nu=0,\pm 3.5$ and reset at $\nu=\pm 2$. b. $R_{xx}$ as
a function of $\nu$ for different $B$ up to 9 T and the color bar shown here
is in logarithmic scale. The minima in $R_{xx}$ emanating from $\nu=0$ have
slopes of $\pm 2,\pm 4$, while those from $\nu=1$ and $\pm 2$ have slopes $3$
and $\pm 2$, respectively. An additional state develops at $B\gtrsim 6.5$ T
whose position in density at $B=9$ T is marked by the red arrow on the top
axis. c. $R_{xy}$ vs ($\nu,B$) at $T=5$ K shows multiple states at different
$\nu$. The red and blue arrows in the bottom axis mark $\nu=+3.5$ and
$\nu=-3.5$ respectively, where $R_{xy}$ changes sign. In the bottom panel,
these states are labelled by ($C,\nu$) = ($\pm 4,0),(\pm 2,0)$ (solid blue),
($1,3),(\pm 2,2),(3,1)$ (solid green) and ($-4,-0.5$) (solid red). d-e.
$n_{H}$ exhibits sign changes at $\nu=\mp 3.5$ for high magnetic fields (5-7
T). f. $R_{xy}$ at $\nu\approx-0.5$ for different $B$, along the red line in
the lower panel in (c) that approaches the quantized limit $R_{xy}=h/4e^{2}$
as the field is increased.

FIG. 3. Density-dependence of zero-field thermoelectricity. a. Second harmonic
component of thermo-voltage $V_{2\omega}$ as a function of $n$ for heating
currents $I_{h}=300-400$ nA, with a step size of 20 nA, measured at $T=5$ K. A
schematic of the heating geometry and measurement of $V_{2\omega}$ can be
found in the inset. For all $I_{h}$, the values of $V_{2\omega}/I_{h}^{2}$ are
constant, which indicates the linear response regime in the range of $I_{h}$
used for heating. Sign-reversals at $\nu$ = 1, 2, 3 are marked by vertical
dotted lines (grey). An additional zero-crossing appears at $\nu$ = 3.5 (red
dotted line). ($1/R_{xx})dR_{xx}/dn$ is shown in the bottom panel. Both
$V_{2\omega}$ and $R_{xx}$ are measured independently in the same thermal
cycle. c. Temperature dependence of $V_{2\omega}$ in the range $5-40$ K. We
observe a series of sign changes as $n$ is swept across the different
correlated phases. Two vertical regions with opposite signs in $V_{2\omega}$
on the low and high density sides of $\nu$ = 3.5 are outlined by a rectangular
window for clarity. The zero-crossing at $\nu$ = 3.5 disappears around $T$
$\gtrsim$ 18 K.

In two-dimensional electron systems, a quantizing magnetic field $B$ is an
essential ingredient for the experimental realization of both integer and
fractional quantum Hall states. The first proposal for a zero-$B$ analogue of
the integer quantum Hall effect (QHE) came from the Haldane model [15], where
non-interacting electrons hopping on a honeycomb lattice with a zero-average,
spatially inhomogeneous $B$ exhibit quantized Hall conductances. Subsequently,
interacting lattice models with nearly flat Chern bands proposed fractional
QHE in the absence of $B$ [16, 17]. The key to realizing these fractional
states is the search for materials with extremely flat bands featuring non-
trivial topological properties. TBG, where two graphene sheets are stacked
together with a rotational mismatch near the first magic-angle of
$1.1^{\circ}$, is an ideal material platform to investigate such zero-$B$
states. Recent experiments have revealed a plethora of broken-symmetry phases
in TBG [1, 2, 4, 2, 5, 18, 8, 9, 10, 11, 12, 13, 14, 19] and related moiré
materials [20, 21, 22, 23], where the external tuning of several parameters
leads to the desired symmetry breaking. For instance, $C\textsubscript{2}$
inversion symmetry can be broken by aligning the TBG layers with hexagonal
Boron Nitride (hBN) [9], while the application of a $B$-field breaks time
reversal symmetry $T$. These experimental knobs enable access to the low
energy degrees of freedom of the band structure, which hosts non-trivial
topological, as well as trivial band insulators at or near integer fillings of
the moiré unit cell. Incompressible states stabilized by electron-electron
interactions within the flat band, were found to appear at large $B$-fields,
in the form of Hall plateaus featuring Chern numbers $C$ at various band
filling index $s$ [10, 11, 12, 13, 24]. Recently, transport measurements in
TBG uncovered a high $B$-field-induced state at $s$ = 3.5, suggesting the
possibility of a new VHS and fractional Chern insulator [11]. However, states
originating from fractional fillings of the unit cell at zero or weak
$B$-field have remained elusive in TBG devices, although a recent report on
twisted monolayer-bilayer graphene indicates incompressible insulators with
$C=1$ at fractional band fillings of $s=$ 1.5 and 3.5 [20]. Notably, states at
several fractional fillings have been observed in transition metal
dichalcogenide (TMD)-based moiré systems, attributed to generalized Wigner
crystallization of the underlying lattice and charge density waves ordering
[25]. States featuring fractional $C$ and/or fractional $s$, which unify the
possibility of many exotic states such as spin/charge density wave phases
(fractional $s$), symmetry broken Chern insulators (integer $C$, fractional
$s$), and fractional Chern insulators (fractional $C$ and $s$), are thus of
great interest to our growing understanding of this fascinating material
platform. While all these states emerge from strong electron-electron
interactions within the flat bands, subtle changes in experimental
configurations can modify the anisotropy energy that decides the preferred
ordered phase when the degenerate phases are in close competition [26, 27].

In this work, we report the observation of novel states at half-integer moiré
band fillings in TBG/WSe2 devices, that persist at $B\approx 0$, suggesting
the possibility of a spin/charge density wave order. We focus on
magnetotransport and thermoelectric transport measurements on two TBG/WSe2
moiré heterostructures D1 and D2, with twist angles $\theta\approx
1.14^{\circ}$ and $1.16^{\circ}$ respectively, fabricated using the ‘tear and
stack’ technique (see Methods and Supplementary Fig. S6) [28, 4]. Few-layer
WSe2 flake of thickness $\approx$ 2-3 nm was placed on the TBG, sandwiched
between two hBN layers, with the stack assembled on SiO2/Si substrate (Fig.
1a). Characterization for D1 starts in Fig. 1b with the four-probe
longitudinal resistance $R_{xx}$ as a function of charge carrier density ($n$)
in the temperature ($T$) range of 3$-$32 K for $B=0$ T. The hole side in
device D1 was inaccessible due to dielectric leakage. We observe correlated
states at all partial fillings $\nu$ = 1, 2, and 3 and a resistive peak at
$\nu$ = 4, where the conduction band is fully occupied. The twist angle
($\theta\approx$ $1.14^{\circ}$) is estimated from the density at the
superlattice gap ($\nu=4$) (see Methods). The resistance at $\nu$ = 1, 2, 3
increases with increasing temperature in the measured range, and the peaks
disappear at higher $T$. Further, we observe that linear-in-$T$ behavior of
$R_{xx}(n)$ at $\nu$ = 1, 2 and 3 persists up to $T=32$ K (see Supplementary
Fig. S8). Device D2 also exhibits similar correlated states at different $\nu$
across various contact configurations (see Supplementary Fig. S7). Fig. 1c
shows the Landau fan diagram for device D1 in a perpendicular $B$-field, where
the linearly dispersing features emanating from the charge neutrality point
(CNP) and other band fillings can be characterized by fitting the Diophantine
equation, $n/n_{0}=C\phi/\phi_{0}+s$, where $n_{0}$ is the density
corresponding to one carrier per moiré unit cell, $\phi$ is the magnetic flux
per moiré unit cell, $\phi_{0}=h/e$ is the flux-quantum, $h$ is Planck’s
constant, $e$ is the charge of electron, and $s$ is the band filling index or
the number of carriers per unit cell at $B=0$ T. The slope gives the Chern
number via the Streda formula, $C$ ($=h/e$ $\partial n/\partial B$). The
Landau-levels (LLs) diverging from the CNP have filling factors of 4, 8, 12,
and 16 (black lines) that increase in steps of 4 due to the four-fold
spin/valley degeneracy. For $B\gtrsim$ 5 T, we observe an additional state
that can be extrapolated to $\nu$ = 1 at $B=0$, with a slope of 2 (green
line). We now focus on the low $B$-field Hall measurements. Fig. 1d shows Hall
conductivity $\sigma_{xy}$($n,B$) as a function of $n$ at a perpendicular $B$
= 1 T. We find that, while zero-crossings of $\sigma_{xy}$ are observed, as
expected, at the CNP and the band insulator at $\nu=$ 4, a ‘reset’ behaviour
is observed at $\nu$ = 2, 3, namely $\sigma_{xy}$ decreases to zero and rises
again through $\nu$ = 2 and 3 without any sign change. These features have
been reported in earlier studies of TBG and related systems [11, 29], and have
been attributed to enhanced Coulomb interactions within the malleable bands
that favor a gap opening by splitting the band into a low-energy filled and
high-energy empty sub-band when the Fermi energy $E_{F}$ crosses the VHS at
$\nu=2$. A similar splitting of the bands that occur at the consequently
developed VHS at $\nu=3$, leads to yet another reset [11]. Surprisingly, some
novel observations with respect to earlier work are the additional zero-
crossings in $\sigma_{xy}(n)$ at $\nu=1$ and half-integer $\nu=0.5$ and 3.5.
Intriguingly, a minor peak in $R_{xx}(n)$ around $\nu$ = 3.5 was seen even at
$B=0$ in several later thermal cycles as shown in the inset to Fig. 1d. It is
often illustrative to plot the Hall density $n_{H}$, calculated from Hall
resistance $R_{xy}$ as $n_{H}$ = ${-\frac{1}{e}}{\frac{B}{R_{xy}}}$, which
provides insights into the VHSs and Fermi surface topology. In Fig. 1e,
$n_{H}$ exhibits features consistent with $\sigma_{xy}$, diverging with
opposite signs on both low and high-density sides of $\nu$ = 0.5, 1 and 3.5.
Furthermore, no sign-reversal in $n_{H}$ is observed at $\nu$ = 2, 3.
Remarkably, we find robust states at half-integer $\nu$ even at $B$-fields as
low as $\sim$0.3 T in device D1 (see Supplementary Fig. S9).

To further investigate the presence and robustness of states at fractional
fillings we report in Fig. 2 detailed magnetotransport measurements in a
second device D2. We observe similarities and differences, the latter possibly
due to specific device details. Consistent with device D1, sign changes are
observed in $n_{H}$ at $\nu=\pm 3.5$, down to $B=0.2$ T (see Fig. 2a for $B=2$
T and supplementary Fig. S10), which remain robust at high $B$ as well (Fig.
2d-e). Further, $n_{H}$ shows reset of charge carriers at $|\nu|=2$ similar to
device D1. Fig. 2b and 2c show the Landau fan diagrams in $R_{xx}$ and
$R_{xy}$ respectively. The filling factors of $\pm 2$ emanating from the CNP
indicate that the four-fold spin/valley symmetry is broken. Additionally,
multiple linearly dispersing minima in $R_{xx}$ are observed at high $B$,
originating from various $s$. We also note that these minima in $R_{xx}$ are
accompanied by the wedge-like resistive features in $R_{xy}$ (Fig. 2c). Our
observation of possible Chern insulators, labelled by the two indices $C$ and
integer $s$ with ($C$, $s$) = ($\pm 2,\pm 2$), ($1,3$) and ($3,1$) are in a
good agreement with recent reports on TBG systems [10, 11, 12, 13]. Chern
insulators at integer $s$ can be understood within the two spin/valley
degenerate flat band model in TBG where $C_{2}T$ symmetry breaking opens a gap
at the CNP and produces isolated flat Chern bands with opposite Chern numbers
$C=+1$ and $C=-1$ [30, 12, 11, 10, 13]. The most striking feature in our data
is a new state at half-integer $s\approx-0.5$ for $B\gtrsim 6.5$ T (denoted by
red arrow in Fig. 2b and red line in the lower panel of Fig. 2c) for which we
obtain $C=-4$ from Streda formula in both $R_{xx}$ and $R_{xy}$. The magnetic
field evolution of the state along the same red line in Fig. 2c is shown in
Fig. 2f, where we observe that $R_{xy}$ approaches $h/4e^{2}$ within the
experimental range of $B$-field, providing further evidence for the Chern
insulating state.

FIG. 4. Degeneracy lifting of the folded bands. a. Spin-resolved band
structure and corresponding DOS of TBG under a 1D periodic potential given in
Eq. 2 with $C_{0}=0$, $C_{z}=30$ meV, $\phi=0$, $\Delta=300$ meV, $t_{2}=15$
meV and $\lambda_{\rm R}=10.5$ meV. The red and blue shades indicate the
strength of the respective spin orientations induced by the SOC terms. b.
Schematic that combines the colormap of the interlayer distances from the
relaxed TBG system (AA stackings appear as bright dots) together with the 1D
sine potential whose zero is at AA-stacking to illustrate the rectangular cell
whose area is twice as large as that of TBG moiré unit cell. The inset
represents the high symmetry points and the $k$-path corresponding to the band
structure in the main panel. c. The sketch illustrates how the zone-folding
doubles the number of flat bands and halves the Brillouin zone area, while
splitting of the spin and valley degrees of freedom bring the total number of
bands to 16. The high/low intensity of colors for the lines distinguish the
$K$ and $K^{\prime}$ valleys and the red and blue colors refer to the spin
degree of freedom. The valley-degeneracy lifting is illustrated by a small
shift in the K′ valley levels with respect to the bands in the K valley.

While Hall measurements demonstrate the existence of states at fractional
fillings for low $B$, the technique precludes direct observation of these
states at $B=0$. With this aim, we have performed thermoelectric transport
measurements at $B=0$ for device D1. The experimental technique used for
measurement of the thermo-voltage is briefly discussed below and in methods
section. A temperature gradient ($\Delta T$) over the entire TBG/WSe2 channel
is created by passing a sinusoidal current ($I_{h}$) through SLG contacts
outside the twisted region, shown in the inset of Fig. 3a. We measure the
second-harmonic component of thermo-voltage $V_{2\omega}$ for different
$I_{h}$ (described in detail elsewhere [31]) in a density range from the CNP
to $\nu$ = 4 (Fig. 3a). The linear variation of $V_{2\omega}$ with
${I_{h}}^{2}$ ensures a linear response regime for the range of heating
currents used. The $n$-dependence of $V_{2\omega}$ shows a series of zero-
crossings at $\nu$ = 0, 1, 2, 3. Most importantly, we observe that the sign-
reversal of $V_{2\omega}$ also occurs at half-integer $\nu$ = 3.5 at $B=0$. In
Fermionic systems at very low temperatures ( $T\ll T\textsubscript{F}$, where
$T_{F}$ is the Fermi temperature), where scattering of charge carriers is
diffusive, semiclassical Mott relation (SMR) establishes a connection between
thermoelectric power ($S=V_{2\omega}/\Delta T$) and electrical resistance
($R_{xx}$) [32, 31] as,

$S_{\mathrm{Mott}}=\frac{\pi^{2}k_{\mathrm{B}}^{2}T}{3|e|}\frac{1}{R_{xx}}\frac{\mathrm{d}R_{xx}}{\mathrm{d}n}\frac{\mathrm{d}n}{\mathrm{d}E}\bigg{|}_{E_{\mathrm{F}}},$
(1)

where $\frac{\mathrm{d}n}{\mathrm{d}E}$ represents the DOS. In deriving the
SMR, a non-interacting scenario is assumed with the electrical conductance
varying smoothly near $E_{F}$. SMR holds true in a large number of van der
Waals materials, including monolayer and bilayer graphene [33, 34] and
disordered metals/insulators. We compare $V_{2\omega}$ with derivative of
$R_{xx}$ with respect to $n$, shown in the bottom panel of Fig. 3a. We observe
exact mapping of zero-crossings at $\nu=1,2$ and 3, with peaks in $R_{xx}$.
While, $\frac{dR_{xx}}{dn}$ shows a weak peak at $\nu=3.5$, a clear sign
change is observed in $V_{2\omega}$. Intriguingly, the state at $\nu=0.5$ does
not appear at $B=0$ in either $R_{xx}$ or $V_{2\omega}$. This we believe is
because the sign changes in $V_{2\omega}$ occur in accordance with the
resistive peaks in $R_{xx}$, while the sign changes in $\sigma_{xy}$ and
$n_{H}$ are intricately connected to the Fermi surface topology via $R_{xy}$.
The temperature dependence of $V_{2\omega}$ is illustrated in Fig. 3b. The
sign reversals are seen as a series of vertical red and blue bands throughout
the measured $T$ range. While the transitions at the CNP and $\nu$ = 1, 2 are
robust and persist up to the measured temperature of $T=40$ K, the feature at
$\nu$ = 3.5 vanishes for $T\gtrsim$ 18 K.

The emergence of distinct features in magnetotransport and thermoelectricity
at half-integer band fillings for two different devices, despite the
inevitable variability typically seen in TBG samples, suggest a robust
mechanism behind our observations. We speculate that strong correlations
between electrons may generate a spin/charge density order that breaks
translation symmetry in the moiré leading to Brillouin zone (BZ) folding with
electron wave functions extending over a unit cell with doubled area. In the
non-interacting, single-particle band structure of TBG, a VHS is expected to
appear at $|\nu|=2$, which is the half-filling of the first low-energy non-
dispersive conduction or valence band, separating a two-pocket Fermi surface
around the $K-$points from a higher-energy single-pocket Fermi surface around
the $\Gamma$-point of the moiré BZ. This has been consistently verified in
transport experiments [35, 11, 4]. Broken translational symmetry and
appearance of folded BZ in the doubled moiré unit cell may favor VHS at
$\nu=1$ and other partial fillings instead, as observed in our devices. While
higher order translational symmetry breaking (tripling, quadrupling) is a
possibility, we can not rule out the presence of an inherent, accidental VHS
at $\nu=0.5$ in the TBG/WSe2 band structure either. As noted in the
theoretical model below, additional spin splitting of bands can lead to a
complex DOS hierarchy, which can explain the VHS features at half integer
fillings. Furthermore, the emergence of a Chern insulator at $\nu\approx-0.5$
in device D2, strengthens the possibility of a doubled moiré unit cell. We
also remark that the observation of a state with a slope of $C=2$ as obtained
from Streda formula from a band filling of $s=1$ for device D1, is reminiscent
of an odd parity ($p=C+s$) Chern insulator, recently reported in
compressibility measurements of TBG due to reconstruction of Chern bands
driven by the modulation of charge density waves over the real space moiré
unit cell [36].

In the following we illustrate in Fig. 4a the band structure of TBG subject to
a 1D density wave potential that doubles the moiré unit cell area (Fig. 4b)
and hence the number of bands, similar to the proposal in twisted monolayer-
bilayer graphene [20]. The four-fold degenerate conduction and valence bands
would turn into two eight-fold degenerate bands upon zone-folding when the
supercell area is doubled. In Fig. 4c, we schematically illustrate the
degeneracy lifting of these bands resulting in a total of sixteen split bands
in the presence of spin/charge density potential patterns. The states at half-
integer fillings observed in experiments are therefore compatible with an odd
integer filling of the zone-folded split bands. Our proposal of 1D patterned
phases whose periods are commensurate with the moiré patterns can give rise to
fractionally filled flat bands, analogous to the Bloch band filling index
$s=\pm 1/3,1/2$ states seen in Landau levels [37] that can be justified in the
presence of Wigner crystal phases [38]. Our tight-binding Hamiltonian in the
carbon $\pi$-orbitals basis incorporates this 1D pattern through a sinusoidal
function and further incorporates the sublattice staggering as well as the
intrinsic and Rashba spin-orbit coupling potentials that are often introduced
when describing proximity spin-orbit coupling of graphene on transition metal
dichalcogenide (TMD).

$\begin{split}H=\sum_{l,m,,\sigma}t_{lm}c^{\dagger}_{l\sigma}c_{m\sigma}+\sum_{l,\sigma}(C_{0}+\xi_{l}C_{z})\sin(\frac{\pi
x_{l}}{\lambda_{M}^{\rm TBG}}+\phi)\,c^{\dagger}_{l\sigma}c_{l\sigma}\\\
+\sum_{l_{t},\sigma}\xi_{l_{t}}\Delta
c^{\dagger}_{l_{t}\sigma}c_{l_{t}\sigma}+\sum_{\langle\langle
l_{t},m_{t}\rangle\rangle\sigma}\nu_{l_{t}m_{t}}t_{2}e^{i\pi/2}c^{\dagger}_{l_{t}\sigma}c_{m_{t}\sigma}\\\
+\frac{2i}{3}\sum_{\langle
l_{t},m_{t}\rangle}\sum_{\sigma,\sigma^{\prime}}c^{\dagger}_{l_{t}\sigma}c_{m_{t}\sigma^{\prime}}\left[\lambda_{R}(\hat{\textbf{s}}\times\textbf{d}_{l_{t}m_{t}})_{z}\right]_{\sigma\sigma^{\prime}}.\end{split}$
(2)

Here, the first term describes the carbon interatomic hopping terms that are
tuned to generate the magic angle flat bands at $1.08^{\circ}$ [39], see the
supplementary information for the expression of the distance-dependent
$t_{lm}$ terms. The $l,\,m$ indices are lattice labels $l_{t},\,m_{t}$ with
the $t$ subscript referring to top layer sites that are contacting the TMD.
The second 1D sine term describes a density wave potential repeating with a
period of $2\lambda_{M}^{\rm TBG}$, twice the TBG moiré period,which defines a
supercell that doubles the moiré area and allows to obtain folded bands that
split at half-integers of the unperturbed system’s filling factors. The $\phi$
phase term is a control knob that shifts the potential origin position. The
remaining three terms capture the WSe2 interface effect in the contacting
graphene layer through a constant mass term with unequal sublattice resolved
potentials, and through Kane-Mele and Rashba SOC models, where
$\nu_{l_{t}m_{t}}=\pm 1$ following the conventions in Ref. [17] where
$\hat{\textbf{s}}$ are the spin Pauli matrices and $\textbf{d}_{l_{t}m_{t}}$
is the displacement vector between lattice sites $l_{t}$ and $m_{t}$. These
terms introduce an energy anisotropy that favors a certain spin/valley isospin
polarization of the ground-state [40]. Because the lattice constant mismatch
between TBG and WSe2 is already large, the proximity spin-orbit coupling terms
[41] in graphene do not change drastically when we modify the twist angle from
alignment. The system parameters for the band structures are ad hoc choices
that illustrate the degeneracy lifting of the folded flat bands compatible
with the observed half-integer filling of $\nu=0.5$ and $3.5$, and the gaps at
the integer filling of $\nu=2$ and $3$ (partially). The simultaneous presence
of mass with opposite $\xi_{l}=\pm 1$ signs for $A,\,B$ sublattice resolved
potentials and SOC terms can cause valley and spin splitting that lifts the
degeneracy between the flat bands which the Coulomb interactions can further
amplify. We refer to the supplementary information for the bands associated
with alternative $\phi=\pi/2$ cosine-like 1D potential that illustrates the
overall robustness to shifts in its origin.

Our work shows that generation of flat bands by means of moiré superlattices
is enriched by the possibility of superposing periodic potentials that are
commensurate with the original moiré pattern [42]. The favored ground state
among several quasi-degenerate phases will depend on the anisotropy energy for
the different spin/valley isospin polarization that can change with details in
sample preparation and device assembly methods. Here, we have only considered
the $s=1/2$ Bloch band filling index scenario, while $s=\pm 1/3$ for Kekule
patterns with three times larger superlattice area seen in experiments with
Graphene/hBN [37, 38] also deserve attention in future research. Different
specific material choices for the TBG dielectric environment are expected to
subtly influence the ground state configuration. In particular, the way in
which the spin and valley isospins order for various fillings will impact
system properties like the spin or orbital magnetism stemming from the
topological nature of the bands, the superconducting pairing symmetry as well
as the instabilities in the electron-electron and electron-hole channels. Our
work suggests that using TMD dielectrics is a promising pathway to further
understand and explore the nature of fractional-filling correlated states in
the limit of vanishing magnetic fields. This prospect is ever brighter, thanks
to the large variety of TMD materials that could be used to engineer moiré
heterostructures.

Note added in proof : While preparing our manuscript, we became aware of
related observations of fractional Chern insulators in hBN-interfaced magic-
angle TBG reported in [43] at filling fractions different than in our work.

## METHODS

### Device fabrication

The devices in this work were fabricated using a modified ‘tear and stack’
method (see Supplementary Fig. S6). A hBN flake ($10-22$ nm) was picked up
using a polypropylene carbonate (PPC) film that was coated on a
polydimethylsiloxane (PDMS) stamp. The hBN flake was then used to pick up a
multilayer WSe2 ($\sim 2-3$ nm). Next, hBN/WSe2 was slowly placed on a
monolayer graphene flake. After picking up the first half of the graphene,
torn by hBN, the transfer stage was rotated approximately by $1.1^{\circ}$,
and the second half of the graphene was picked up. In the final step, a thick
hBN flake ($25-30$ nm) was picked up, and the encapsulated stack was released
on a 285 nm SiO2/Si wafer at a temperature of $85^{\circ}$C. We set the
temperature at $63^{\circ}$C for picking up hBN and $55^{\circ}$C for picking
up graphene and WSe2. The PPC film was then cleaned with anisole and organic
solvents. We defined the Hall bar geometry of the final device using electron-
beam lithography, reactive ion etching using CHF3/O2, and thermal evaporation
of ohmic edge contacts using Cr/Au (5 nm/60 nm). WSe2 crystals used in this
work were obtained from 2D Semiconductors.

### Transport Measurements

Electrical and thermoelectric transport measurements were carried out in a
pumped Helium cryostat with 9 T magnetic field. Magnetotransport measurements
were performed using a bias current of $10-100$ nA using an SR830 low-
frequency lock-in amplifier at $17.81$ Hz. Local Joule heating was employed
for thermoelectric measurements to create a temperature gradient across the
TBG/WSe2 region. A range of sinusoidal currents ($300-400$ nA) at an
excitation frequency of $\omega$ $=$ $17.81$ Hz was allowed to pass through
the SLG region to create a temperature difference $\Delta T$ between two ends
of the TBG/WSe2 channel and the resulting $2^{\mathrm{nd}}$ harmonic thermo-
voltage ($V_{2\omega}$) was recorded using the same lock-in amplifier. To
estimate the twist angle, we used the relation,
$n_{s}=8\theta^{2}/\sqrt{3}a^{2}$ where $a=0.246$ nm is the lattice constant
of graphene and $n_{s}$ ($\nu=4$) is the charge carrier density corresponding
to a fully filled superlattice unit cell.

### Real-space lattice calculations

The atomic positions were relaxed using LAMMPS molecular dynamics simulations
[44] based on EXX-RPA-informed [45] force-fields, and the electronic structure
was obtained through real-space tight-binding calculations as outlined in the
supplementary information. Both the band structure and DOS were calculated by
a shift-invert diagonalization method [46] with a broadening of $0.8$ meV,
while the band structures in the absence of SOC terms are calculated using
exact diagonalization. We used a 4x3x1 Monkhorst-Pack k-point grid for the DOS
calculation in a rectangular supercell containing 22328 atoms. The graphene
itself is modeled using the F2G2 model [47]. More details on the Hamiltonian
implementation, analysis of the different terms in the Hamiltonian and phase
shifted patterns can be found in the supplementary information.

## Acknowledgements

We gratefully acknowledge the usage of the MNCF and NNFC facilities at CeNSE,
IISc. UC acknowledges funding from SERB via grants ECR/2017/001566 and
SPG/2020/000164. NL was supported by the Korean National Research Foundation
grant NRF-2020R1A2C3009142 and AS by grant NRF-2020R1A5A1016518. DL was
supported by the Korean Ministry of Land, Infrastructure and Transport(MOLIT)
from the Innovative Talent Education Program for Smart Cities. JJ was
supported by the Samsung Science and Technology Foundation under project SSTF-
BAA1802-06. We acknowledge computational support from KISTI through grant
KSC-2021-CRE-0389 and the resources of Urban Big data and AI Institute (UBAI)
at the University of Seoul. KW and TT acknowledge support from the Elemental
Strategy Initiative conducted by the MEXT, Japan (Grant Number
JPMXP0112101001) and JSPS KAKENHI (Grant Numbers JP19H05790 and JP20H00354).

## Author Contributions

SB fabricated the device, performed the measurements, and analyzed the data.
BG, MP, and PSM assisted in measurements/analysis. KW and TT grew the hBN
crystals. AG and UC advised on experiments. NL, SA, DL, and JJ performed the
theoretical calculations. SB, NL, JJ, and UC wrote the paper with inputs from
other authors.

## Competing interests

The authors declare no competing interests.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Code availability

The code that support the findings of this study are available from the
corresponding author upon reasonable request.

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SUPPLEMENTARY INFORMATION

##  I Simulation

Here, we outline the details of the tight-binding hopping terms, in particular
the interlayer coupling that we have used in our simulations. We then show the
band structures corresponding to a shifted origin 1D potential that shows the
relative robustness of band folding features. We further present band
structures for different Hamiltonian model parameters without and with spin-
orbit coupling (SOC) terms to better understand the role of the different
terms in the Hamiltonian.

###  I.1 Twisted bilayer graphene’s tight-binding Hamiltonian

The twisted bilayer graphene (TBG) interlayer hopping terms between atoms at
lattice points $\textbf{R}_{i}$ and $\textbf{R}_{j}$ are given by [39]

$-t_{lm}=-t(\textbf{R}_{l},\textbf{R}_{m})=S_{0}\exp[(\textbf{d}\cdot\textbf{e}_{z}-p)/q]\times\left[V_{pp\sigma}(d)\left(\frac{\textbf{d}\cdot\textbf{e}_{z}}{d}\right)^{2}\right].$
(3)

with $\textbf{d}=\textbf{R}_{l}-\textbf{R}_{m}$, $d=\left|\mathbf{d}\right|$
and $\textbf{e}_{z}$ the unit vector parallel to the $z$ axis and where [48]

$V_{pp\sigma}(d)=V_{pp\sigma}^{0}\exp\left(-\frac{(d-d_{0})}{r_{0}}\right)$
(4)

with $d_{0}$ the interlayer distance, $a_{0}$ the interatomic carbon distance,
$r_{0}=0.184a$ and $V_{pp\sigma}^{0}=-0.48$ eV is the transfer integral
between two vertically aligned atoms. This form in Eq. 4 is widely used in the
literature [49] and is good at reproducing the largest magic angle in TBG,
even though the corresponding effective hopping term, obtained by summating
intralayer intra-sublattice terms [47], of $-2.45$ eV (when using
$V_{pp\pi}^{0}=2.7$ eV for the transfer integral between nearest-neighbor
atoms) corresponds to a Fermi velocity which is much smaller than experiment
or even local density approximation (LDA). To correct for this approximate
model in square brackets, we introduced the variables $p=1.34$ and $q=3.25$ to
better capture the inter-layer distance modulations that enter the fitting
procedure to the DFT tunneling at the K-point for all slidings as well as the
magic angle renormalization factor $S_{0}$ equal to $0.895$ for these
structures that we relaxed using LAMMPS MD relaxations [44] using EXX-RPA-
informed [45] force fields. This choice of $S_{0}$ value leads to the maximum
flatness of bands at $1.08^{\circ}$ when all terms but the first are equal to
zero in the Hamiltonian from Eq. (2) in the manuscript. For the intralayer
terms we simply use the F2G2 model [47].

###  I.2 Doubling of the bands due to band-folding

In Fig. S1, we illustrate how the existence of a potential that doubles the
initial TBG moiré period by a factor two effectively leads to band-folding and
associated doubling of the initial 4-fold degenerate valence and conduction
bands. We calculate the band structures in the smallest moire triangular unit
cell and compare with the folded bands with twice the original unit cell area
where we can observe how the initial bands are being doubled in number. We
neglect any spin/valley degeneracy lifting terms in the Hamiltonian and will
discuss their effect further in Sect. I.4.

Supplementary Fig. S1. Band folding and doubling of bands. a. The top panel
illustrates the bandstructure for the unperturbed original triangular moiré
cell illustrated by the parallelogram linking 4 closest AA-stacking bright
dots in b., whereas the bottom panel shows the band structure for the doubled
area supercell. The spin is neglected in this figure, as such we see a total
of 4 bands (two valleys for the conduction band and valence band) in the top
panel becoming 8 bands by band-folding in the bottom panel. b. Real-space
schematic of the single and double area supercells indicated by the dashed red
and black solid parallelograms. c. The associated BZ for the band structures
for the single and double area supercells indicated also with dashed red and
black solid lines.

###  I.3 Alternative 1D potential modulation

We have chosen a specific form of the 1D sine potentials in the main text to
obtain the folded bands in the larger period cell. Here, we provide an
additional 1D potential that leads to a doubled supercell area compared to the
moiré pattern unit cell area that leads to quantitatively different types of
folded bands. In Fig. S2, we apply a cosine instead of a sine 1D modulation in
Eq. 2 in the manuscript (set $\phi=\pi/2$). In this case, the AA stacking
regions are located where the 1D potential is maximal or minimal.

Supplementary Fig. S2. Cosine modulation for 1D potential in the rectangular
unit cell. a. Bandstructure of TBG under a 1D cosine periodic potential in Eq.
2 by setting $\phi=\pi/2$, and using the same parameters $C_{z}=30$ meV,
$t_{2}=15$ meV, $\lambda_{R}=10.5$ meV and $\Delta=300$ meV. b. Colormap
illustrating the cosine modulation in the rectangular unit cell using the same
procedure as in the main text. The high-symmetry path in the Brillouin zone
(BZ) is the same as in the manuscript.

Supplementary Fig. S3. Impact of the different Hamiltonian terms. We gradually
include the different terms from the Hamiltonian in Eq. 2 from the manuscript,
while ignoring the Kane-Mele term by setting $t_{2}=0$ as well as the Rashba
term by setting $\lambda_{R}=0$ meV. Comparison of these figures with the
final band structure in the manuscript illustrates that each term of this
heuristic Hamiltonian can play an important role.

Supplementary Fig. S4. Effect of the constant mass-term in the band
separation. We gradually increase the constant mass-term, with $t_{2}=3$ meV
and $\lambda_{R}=0$ meV, we only show the spin up contribution here, and
$C_{z}=0.03$ meV in Eq. 2 from manuscript. The constant mass term clearly
increases the separation between the conduction and valence band groups that
is expected to reduce interband screening and therefore enhance the chances of
opening a band gap due to Coulomb interactions when the subbands are
completely filled.

###  I.4 Role of the different Hamiltonian terms

In Figs. S3, S4, and S5, we revisit the band-structure shown in the manuscript
main text consisting of a rectangular unit cell and sine 1D modulation
potential to illustrate the effect of the different terms in our Hamiltonian
in more detail. In Fig. S3, we divide in different panels the band structures
and start by showing in Fig. S3a the band structure for the TBG, nearly flat
bands in the rectangular unit cell. The area of this rectangular unit cell is
twice as large as the area of the single TBG system (smallest parallelogram
delimited by 4 AA-stacking regions); hence we observe a doubling in the number
of the flat bands. In the absence of spin, but including the valley degeneracy
(inherent to our real-space approach), we have 8 flat bands in the system,
distributed evenly among conduction and valence bands. In Fig. S3b, we include
the 1D sine modulation pattern and observe a clear energy separation of the
conduction and valence bands. Alternatively, in Fig. S3c, we check the effect
of adding a constant mass-term in the bottom graphene layer using the
rectangular unit cell in the absence of a sine modulation. We see that in this
case, the conduction bands and valence bands are being separated from each
other too, but for our choice for the value of $\Delta$, the doubled
conduction bands remain very close to each other and cannot be disentangled at
the meV energy level (same observation for the doubled valence bands). We
finally combine in Fig. S3d the effect of both the sine potential and the
constant mass-term. The 4 valence and 4 conduction bands get separated into 2
sets of 2 bands each. We note that the weak band degeneracy lifting at the
high symmetry points originates in the $\xi_{c_{i}}$ factor, which
differentiates between sublattices that we have included in our 1D sine
modulation. We have checked that in the absence of this factor, this
degeneracy lifting is nonexistent. Finally, for the case in Fig. S3d, only the
Rashba and the Kane-Mele term are missing when compared to the band structure
from the paper; this shows that for this choice of parameters, these terms are
crucial in their ability to resolve the degeneracy lifted spins that are
causing the features at $\nu=0.5$ and $3.5$. In Fig. S4, we further discuss
qualitatively the effect of either the mass-term in the presence of a constant
Kane-Mele term, and in Fig. S5, we vary the magnitude of the Kane-Mele term
that can arise due to proximity effects induced by the substrate. We omit the
Rashba term for simplicity here and only show the spin-up component. We notice
that while the mass-term tends to push the highest set of valence bands above
the lowest set of the conduction band, the Kane-Mele term shifts them in the
opposite directions. The combination of these two terms recovers the original
ordering of the bands, and both terms tend to widen the energy spectrum range
where the nearly flat bands distribute.

Supplementary Fig. S5. Effect of the the value of $t_{2}$ controlling the
strength of the Kane-Mele term. We gradually increase the value of the Haldane
pre-factor $t_{2}$ entering the Kane-Mele term, with $C_{z}=30$ eV,
$\Delta=300$ meV and $\lambda_{R}=0$ eV, in Eq. 2 from manuscript (we only
show the spin-up component). Increasing the value of $t_{2}$ resolves the
degeneracy of the spin-up and spin-down bands in momentum space although it
does not lead to an overall splitting in the DOS.

##  II Experimental Results

Supplementary Fig. S6. Device fabrication. a-e. Schematic of the various steps
in the ‘tear and stack’ method used for fabrication of the moiré
heterostructure. A layer of PDMS/PPC on a glass slide is used to sequentially
pick the various heterostructure layers. A twist angle of $\approx
1.1^{\circ}$ is used to create the magic-angle twisted bilayer graphene. f.
Optical image of the final stack for device D2. The top and bottom graphene
(red and yellow), WSe2 (black) and two hBN (white) layers are shown using
dashed lines. The scale bar is $12$ $\mu$m.

Supplementary Fig. S7. Characterization of device D2. a. Optical image of
device D2 where the scalebar denotes 3 $\mu$m. b. Four-probe resistance
measurements at $T=$ 6 K for different pair of contacts show twist angle
uniformity throughout the TBG/WSe2 channel. Different colors have been used
for different pairs of voltage probes. Cr/Au top gate (see (a)) was used to
tune the carrier density in the Hallbar.

Supplementary Fig. S8. Linear-in-$T$ resistance in device D1, $\theta\approx
1.14^{\circ}$ The longitudinal resistivity $\rho_{xx}$ at various filling
factors $\nu=1,2$ and 3 vary linearly with $T$ in the measured range of
temperature (3 - 32 K). The slopes for $\nu=1$, 2, and 3 were found to be 16,
13, and 14 $\Omega/K$ respectively.

Supplementary Fig. S9. Low-field Hall data in device D1, $\theta\approx
1.14^{\circ}$. a. $n_{H}-\nu$ plot for $B=0.3-1.2$ T. b. $\sigma_{xy}$ as a
function of $\nu$ for the same range of $B$. The zero crossings at $\nu=0.5,1$
and 3.5 and reset at $\nu=2$ and 3 are robust with $B$.

Supplementary Fig. S10. Low field Hall data in device D2, $\theta\approx
1.16^{\circ}$. a. $n_{H}-\nu$ plot for $B=0.2-2$ T. The sign-change at
$\nu=3.5$ appears at the lowest $B$-field of 0.2 T. In addition, the reset at
charge carriers is observed at $\nu=\pm 2$. b. $\sigma_{xy}$ as a function of
$\nu$ for the same range of $B$.

Supplementary Fig. S11. Hall conductivity at $B=9$ T in device D2. a.
$\sigma_{xy}$ shows plateaus as $\sigma_{xy}=Ce^{2}/h$ associated with the
minima in $\sigma_{xx}$. Different color bars (blue for the CNP, green for
$\nu=1,\pm 2,3$ and red for $\nu=-0.5$) have been used to show the sequence of
symmetry-broken quantized states nucleating from several partial fillings of
flat bands.

Supplementary Fig. S12. Magneto-thermoelectricity measurements in device D1.
Thermoelectric voltage $V_{2\omega}$ measured at $B=0,1,1.5$, and 2 T and
$T=3$ K for a heating current of 300 nA. The feature at $\nu=3.5$ is clearly
visible at all $B$.