# Spin-orbit coupling-enhanced valley ordering of malleable bands in twisted
bilayer graphene on WSe2

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
Youngju Park  Department of Physics, University of Seoul, Seoul 02504, Korea
Dongkyu Lee  Department of Physics, University of Seoul, Seoul 02504, Korea
Department of Smart Cities, University of Seoul, Seoul 02504, Korea Suvronil
Datta  Department of Instrumentation and Applied Physics, Indian Institute of
Science, Bangalore, 560012, India Radhika Soni  Department of Instrumentation
and Applied Physics, Indian Institute of Science, Bangalore, 560012, India 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 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 Jeil Jung<EMAIL_ADDRESS>Department of Physics,
University of Seoul, Seoul 02504, Korea Department of Smart Cities, University
of Seoul, Seoul 02504, Korea U. Chandni<EMAIL_ADDRESS>Department of
Instrumentation and Applied Physics, Indian Institute of Science, Bangalore,
560012, India

###### Abstract

New phases of matter can be stabilized by a combination of diverging
electronic density of states, strong interactions, and spin-orbit coupling.
Recent experiments in magic-angle twisted bilayer graphene (TBG) have
uncovered a wealth of novel phases as a result of interaction-driven spin-
valley flavour polarization [1, 2, 3, 4, 5, 6, 7, 8]. In this work, we explore
correlated phases appearing due to the combined effect of spin-orbit coupling-
enhanced valley polarization and large density of states below half filling
($\nu\lesssim 2$) of the moiré band in a TBG coupled to tungsten diselenide.
We observe anomalous Hall effect, accompanied by a series of Lifshitz
transitions, that are highly tunable with carrier density and magnetic field.
Strikingly, the magnetization shows an abrupt sign change in the vicinity of
half-filling, confirming its orbital nature. The coercive fields reported are
about an order of magnitude higher than previous studies in graphene-based
moiré systems [4, 5, 9, 10, 11, 12, 13], presumably aided by a Stoner
instability favoured by the van Hove singularities in the malleable bands.
While the Hall resistance is not quantized at zero magnetic fields, indicative
of a ground state with partial valley polarization, perfect quantization and
complete valley polarization are observed at finite fields. Our findings
illustrate that singularities in the flat bands in the presence of spin-orbit
coupling can stabilize ordered phases even at non-integer moiré band fillings.

Fig. 1. Ferromagnetism and valley polarization at $\boldsymbol{\nu<}~{}2$. a.
Schematic of hBN-encapsulated TBG-WSe2 heterostructure on SiO2/Si substrate.
b. Four-probe longitudinal resistance $R_{xx}$ as a function of filling $\nu$
measured at $T=0.3$ K and $B=0$. c. Hall resistance $R_{xy}$ for three
perpendicular magnetic fields $B=0.05,0.4$ and 1.2 T for density being swept
back and forth. The red and blue arrows indicate directions of density sweep.
d. $R_{xy}$ at three different fillings for $B$ swept back and forth, as
indicated by the arrows. A reversal of hysteresis is seen at $\nu=1.86$. The
coercive fields are indicated as $B^{+}$ and $B^{-}$. e. Colorplot of $\Delta
R_{xy}$, defined as the difference between the values of $R_{xy}$ for the
opposite field sweeps, as a function of $\nu$ and $B$. The change in colour
shows the reversal of magnetization that accompanies the occupation of
electrons in different $K$ and $K^{\prime}$ valleys represented with the inset
schematics. In the bottom panel, the ratio of the magnitude of negative and
positive coercive fields is plotted as a function of $\nu$. Coercive fields
are asymmetric in positive and negative $B$ and the asymmetry flips exactly
when the magnetization is reversed.

Fig. 2. Fermi surface reconstructions and malleability of bands at
$\boldsymbol{\nu<}~{}2$. a. Hall density plotted as a function of $\nu$ for
$B=0.02,0.05,0.1$ and 0.2 T at $T=2$ K. Lifshitz transitions and reset of
charge carriers are seen for $B=0.02$ and 0.05 T , whereas only reset of
charge carriers occur for $B=0.1$ and 0.2 T. b. Expected behaviour of $n_{H}$
as a function of $\nu$ and the corresponding density of states D($E$) vs $E$
profile. The purple line indicates a Lifshitz transition with a sign reversal
in Hall density while the green line shows a reset where the Hall density
reaches a minimum value without a sign change. c. $n_{H}$ vs $\nu$ at $B=0.05$
T for the shaded region in Fig. 2a. Different colorbars are used to indicate
Lifshitz transitions (purple) flanking the reset of carriers (green). d.-e.
Temperature dependence of Lifshitz transitions at $B=0.05$ T and reset at
$B=0.2$ T. f. Density of states showing a finite imbalance in occupation of
states between $K$ and $K^{\prime}$ valleys leading to orbital magnetism.

Fig. 3. Quantized orbital Chern insulator at $\boldsymbol{\nu=}~{}2$ and
possible Chern bands. a-b. Landau fan diagram in longitudinal resistance
$R_{xx}$ and transverse resistance $R_{xy}$ plotted as a function of $\nu$ for
different $B$ up to 10 T. c. Fitting of the Diophantine equation along the
minima in $R_{xx}$ and wedge like states in $R_{xy}$. The slopes of the
straight lines give the Chern numbers. d. $R_{xy}$ data at $B=10$ T plotted
for a small density range near $\nu=2$ marked by the blue colorbar on the top
axis in panel 3c. $R_{xy}$ is quantized to $h/2e^{2}$ accompanied by a minima
in $R_{xx}$ at $B=10$ T. e. Schematic of flat bands with different Chern
numbers via valley polarizing $\mathcal{T}$-symmetry breaking and proximity-
induced SOC. The dark and light colors represent each spin-valley flavor’s
lower and upper bands, respectively. In the presence of SOC, the spin-valley
flavor degeneracy can be completely lifted, leading to 8 isolated bands for
appropriate system parameters. Blue(green) and red(yellow) indicate the up and
down spin components for valley $K$($K^{\prime}$). f. Schematic of valley
Chern bands giving rise to $C_{\rm net}=+2$ at $\nu=+2$. The band degeneracy
can be lifted completely in the presence of SOC and a spin splitting exchange
field that naturally accompanies a ferromagnetic spin polarized phase. We have
used the SOC parameters for graphene on WSe2 following Ref. [14] together with
the exchange field $\lambda_{\rm ex}=0.5~{}\rm meV$ as summarized in the
methods section. The degeneracy split Chern bands lead to finite orbital
magnetism $M_{\rm(orb)}$ that changes with the filling density $\nu$, shown
here from $-4$ to $4$ in the right-most sub-panel. We note the changing signs
in the total orbital magnetization due to the relative filling of
$K\downarrow$($K^{\prime}\uparrow$) bands near $\nu=+2$, suggesting that
delicate changes in level ordering with carrier density due to Coulomb
interactions can strongly impact the net magnetization.

Fig. 4. Summary of the various phases observed. a. Four different colors are
used to indicate the extent of the phases in the density and magnetic field
subspace; Ferromagnetism (orange), Chern insulator (light blue), Lifshitz
transitions (red) and reset (blue). The inset shows low field phase diagram up
to $B=0.2$ T for a density range of $\nu=1.5-2$ in the shaded region of the
main panel. The densities corresponding to Lifshitz transitions and reset are
marked by red and blue circles respectively. While ferromagnetism, Lifshitz
transitions and reset occur below $\nu=2$, the Chern insulator emanates from
exactly $\nu=2$ outside the ferromagnetic region. b. Coercive field as a
function of $\nu$ for various reports on graphene-based moiré systems, clearly
indicating the large coercive field observed in this work. TBG, TMBG, TDBG,
and TLG stand for twisted bilayer graphene, twisted monolayer bilayer
graphene, twisted double bilayer graphene, and trilayer graphene,
respectively.

Topology of the Fermi surfaces and the density of states (DOS) at the Fermi
level govern various competing orders in quantum materials [15, 16]. The
formation of a broken-symmetry phase, such as a magnet, is often treated as an
instability in the parent electron liquid phase, driven by singularities [17].
For example, van Hove singularities (vHSs) which are associated with saddle
points of energy dispersion in momentum space, feature strongly diverging DOS
and favour localization of electronic states that stabilizes phases such as
density waves, ferromagnetism and superconductivity [18, 19, 15, 16, 17].
Contrary to these ‘local’ vHSs, the whole electronic band in the magic-angle
TBG is nearly flat, with a large, ‘global’ DOS, that favor emergent correlated
phases, including correlated insulators [1, 2, 3], orbital magnets [3, 4, 5,
20, 21], non-Fermi liquids [22], and Chern insulators [7, 23, 8, 24, 25],
typically at integer fillings of the moiré unit cell. Experiments in TBG have
shown that the inversion (C2) [4, 5, 26] or time reversal ($\mathcal{T}$)
symmetry breaking [7, 8] can lift the degeneracy of the flat bands and
polarize spin-valley degrees of freedom leading to Chern insulators. Reports
of anomalous Hall effect (AHE) at zero magnetic field at moiré filling factors
$\nu=3$ [4, 5] and most recently at $\nu=1,\pm 2$ [12, 27, 28] necessitates a
non-zero difference in the occupation of electronic states of the two valleys,
that produces a finite Berry curvature. While AHE and Chern insulators were
most significantly observed in TBG samples aligned with a hexagonal boron
nitride (hBN) layer [4, 5] or with the application of a large magnetic field
[7, 8, 23], spin-orbit coupling (SOC) can also drive topological order and
symmetry-broken phases [12, 29, 6]. Proximity-induced SOC can break
C${}_{2}\mathcal{T}$ symmetry at zero magnetic fields and polarize the charge
carriers within a single valley [12, 30]. The interplay of SOC, interactions
and topology, driven by the presence of vHSs has, however, remained largely
unexplored [31]. The presence of vHSs within the nearly flat moiré bands can
lead to enhanced correlations that can be manifested in the emergence of new
instabilities over a narrower range of tunable, non-integer moiré band
fillings.

In this work, we investigate the non-integer filling regime of the moiré band
in TBG proximitized by tungsten diselenide (WSe2). We report signatures of
valley polarization along with Fermi surface reconstructions that suggest a
Stoner-like instability favoured by vHSs in the vicinity of $\nu<2$.
Significantly, the band reconstructions are malleable and can be tuned via a
combination of carrier density and magnetic field. Fig. 1a shows the schematic
of our device consisting of a multilayer WSe2 on magic-angle TBG encapsulated
by two hBN layers. The four-probe longitudinal resistance $R_{xx}$ as a
function of filling $\nu$ at a magnetic field $B=0$ shows well-defined peaks
at the charge neutrality point (CNP) $\nu=0$ and half fillings $\nu=\pm 2$
(Fig. 1b). We estimate the twist angle to be $\theta\approx 1.17^{\circ}$,
consistent with the estimation from Landau fan diagram (Fig. 3). The data
presented in Fig. 1 were taken at a temperature $T=0.3$ K. In Fig. 1c, the
Hall resistance $R_{xy}$ at low $B$-fields, most strikingly, shows a
hysteretic behavior over a wide range of fillings $\nu<2$, as the Fermi energy
is swept back and forth. The hysteresis becomes narrower with increasing
$B$-field and vanishes at $B\approx 1.2$ T. Notably, while $R_{xy}$ shows
zero-crossings at $B=50$ mT, no sign change is observed for higher fields up
to $B=1.2$ T with $R_{xy}$ remaining negative for $B>50$ mT. This feature will
be discussed in detail in Fig. 2. It is evident that $R_{xy}$ strongly depends
on the history of the sample in out-of-plane $B$-field training, and this
leads to a non-zero $R_{xy}$ at $B=0$ when the field is swept back and forth
(Fig. 1d). To our surprise, we find an abrupt sign change in the hysteresis of
$R_{xy}$ at $\nu=1.86$ (Fig. 1e). The magnitude of the coercive field, where
the hysteresis disappears, is about one order of magnitude higher than
previous reports on moiré systems [4, 5, 32, 10, 11, 12]. The large coercive
field suggests a more robust ferromagnetic phase than in previous experiments
and may also indicate domain wall pinning due to disorder and local
inhomogeneities in the twist angle. We expect that the coercive fields will
couple more strongly with the orbital magnetic moments rather than the spin
whose shifts in energy are typically of $\sim$ 0.1 meV per Tesla. The
hysteresis in $R_{xy}$ with respect to both $\nu$ and $B$ suggests that the
sample remains magnetized without any external magnetic influence. We note
that the measured $R_{xy}$ is much lower than the quantum of resistance
($h$/$e^{2}$, where $h$ is the Planck’s constant and $e$ is electronic
charge). The width of the hysteresis in $B$-field changes as the carrier
density is tuned in the vicinity of $\nu<2$, as evident from the color plot in
Fig. 1e, where we have plotted the difference in $R_{xy}$ for two opposite
directions of field sweep, $\Delta
R_{xy}=R_{xy}(\overleftarrow{B})-R_{xy}(\overrightarrow{B})$, as a function of
$B$ and $\nu$. Surprisingly, we observe a significant asymmetry between
positive ($B^{+}$) and negative coercive fields ($B^{-}$). The sudden
switching behavior of $R_{xy}$ as a function of density at $\nu=1.86$ is
accompanied by a reversal of the asymmetry between $B^{+}$ and $B^{-}$
coercive fields. We quantify the latter feature using the parameter $\alpha$,
defined as $\alpha=|B^{-}|/|B^{+}|$, obtaining $\alpha>1$ for $\nu\leq 1.85$
and $\alpha<1$ for $\nu>1.85$ (bottom panel of Fig. 1e).

Ferromagnetism at $\nu=2$ is unexpected in TBG since a valley-polarized ground
state is energetically unfavourable due to inter-valley Hund’s coupling [26].
However, SOC in TBG leads to flat bands with non-zero Berry curvature within a
single valley [12, 30]. Presence of proximity-induced SOC is confirmed by weak
antilocalization measurements in our device (Supplementary Fig. S7). The
reversal of magnetization is a strong evidence for spontaneous switching of
valley polarization induced by tuning the carrier density. A ferromagnet can
be classified as spin or orbital, depending on whether the magnetization is
due to spontaneous spin or valley polarization. In an orbital Chern insulator,
the magnetization jumps abruptly when the chemical potential crosses the Chern
gap. The edge state contribution is sufficient to change the sign of
magnetization simply by tuning the density below the gap of an orbital Chern
insulator [33, 11, 34]. Therefore, the abrupt reversal of hysteresis indicates
the dominance of orbital magnetism over the spin magnetism, and the
energetically favourable ground state is solely determined by the gate
voltages in weak magnetic fields. The observation of a non-quantized $R_{xy}$,
however, suggests that the ground states have partially valley polarized bands
with unequal occupation of different valleys as a function of carrier density.
Partial valley polarization is not incompatible with the intervalley-coherent
phases proposed in literature [35, 36, 37] that can mitigate a fully valley
polarized phase. We note, however, that the SOC terms by themselves do not mix
electronic states from $K$ and $K^{\prime}$ and is not the microscopic origin
for the intervalley-coherent phases. The sign switching of valley
polarizations as a function of density leads to an abrupt reversal of
magnetization (Fig. 1e) indicating a clear phase transition point between
these competing phases for magnetic fields below $\sim$ 0.5 T.

Having established the orbital nature of the ferromagnet at $\nu<2$, we now
turn to the zero-crossings in $R_{xy}$ that accompany the hysteresis in $\nu$
at $B=50$ mT (Fig. 1c). The $\nu$-dependence of Hall density $n_{H}$ gives
insights into the Fermiology of a system. In Fig. 2a, $n_{H}=-(1/e)(B/R_{xy})$
is plotted as a function of $\nu$ at four different low $B$-fields, but at a
higher temperature $T=2$ K, where ferromagnetism disappears (Supplementary
Fig. S5) and the Hall data is independent of the direction of density sweep.
We observe a rich sequence of sign changes and resets in $n_{H}$ around
$\nu<2$, particularly for the lowest fields 20 mT and 50 mT. Assuming a single
particle energy band diagram for TBG, the DOS is expected to show a vHS around
$\nu=2$ (see Fig. 2b, top right panel). As the Fermi energy is swept through
the vHS, a Lifshitz transition is expected that changes the topology of the
Fermi surface, flipping the sign of $n_{H}$ with a logarithmically diverging
profile, as shown in the top left panel [38]. However, when the bands are
malleable, as the Fermi energy approaches the peak in the DOS, it can reset
the bands and produce a split DOS profile as shown in the bottom right panel
[8, 6]. This leads to a ‘reset’ of charge carriers, where $n_{H}$ drops to a
low value before rising again, without a sign change. Our experiments reveal a
completely new set of phase transitions in comparsion to previous reports on
magic-angle TBG, where a reset is typically observed near $\nu=2$ [8, 39, 6].
A closer look at our data near $\nu<2$ at 50 mT shows two Lifshitz transitions
that flank a reset, indicated by the colorbars in Fig. 2c. We note that the
vHS within the nearly flat bands can shift the density of states weights for
small changes in the twist angle [40, 41]. Surprisingly, our experiments
reveal an unprecedented tunability of the DOS with $B$-field and density,
further validating the malleability of the TBG bands. The Lifshitz transitions
disappear at $B=100$ mT, and $n_{H}$ shows a peak-like feature that decreases
to zero and increases slowly (Fig. 2a). Such a ‘reset’ of charge carriers at a
relatively higher field, with no additional Lifshitz transitions, indicates
$B$-field-driven changes in the DOS of the bands. Fig. 2d-e show that these
phase transitions become weaker and fade away with increasing temperature.
Remarkably, these distinct features in $n_{H}$ appear around the same density
$\nu<2$ where we have observed ferromagnetism at lower temperatures of $T\leq
1$ K. The flat band condition of on-site Coulomb interactions ($U$) dominating
over the kinetic energy of the carriers, and the diverging DOS around $\nu<2$
easily satisfy the Stoner criterion of ferromagnetism $UD(E_{F})>1$, where
$D(E_{F})$ is the DOS at the Fermi energy $E_{F}$ [17, 20, 42]. We speculate
that such a strong instability in the DOS at $\nu<2$ favours spontaneous
valley polarization, leading to the observed AHE along with the switching of
magnetization (Fig. 1d-e). Our theoretical calculations discussed below show
that spin polarization together with SOC assists valley polarization. Thus, a
valley polarized orbital magnet with a non-zero spin polarization should be
favored over a valley polarized magnetic phase without a net spin
polarization. In Fig. 2f we illustrate the scenario where conventional Stoner
spin polarizing ferromagnetic phase is accompanied by valley polarization
where $K$ and $K^{\prime}$ valleys are unevenly occupied.

To gain further insights into the possible ground state at half-filling, we
have measured $R_{xx}$ and $R_{xy}$ simultaneously in a $B$-field up to $B=10$
T, at $T=0.3$ K. A series of symmetry-broken Chern insulators in the form of
minima in $R_{xx}$ and wedge-like features in $R_{xy}$ emerge from different
fillings (Fig. 3a-b). The Chern insulators 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, $C$ is the Chern number,
$\phi$ is the magnetic flux per moiré unit cell, $\phi_{0}=h/e$ is the flux-
quantum, and $s$ is the band filling index or the number of carriers per unit
cell at $B=0$ T. For sufficiently strong magnetic fields the four fold spin-
valley degeneracy is completely lifted near the CNP: ($C,\nu$) = ($\pm 1,0$),
($\pm 2,0$), ($\pm 3,0$), ($\pm 4,0$). In addition, we observe states
emanating from different integers $\nu$ as ($C,\nu$) = ($+3,+1$), ($\pm 2,\pm
2$), ($+4,+2$), ($\pm 1,\pm 3$) (Fig. 3c). The Chern insulator $C=2$ at
$\nu=2$ is perfectly quantized to $h/2e^{2}$ at a high $B$-field (Fig. 3d). In
TBG devices, such topological incompressible insulators have been described
within the picture of isolated eight fold bands with broken $\mathcal{T}$
symmetry, where the Chern numbers of the bands are the same in the two valleys
[7, 23, 8], but opposite for valence and conduction bands (Fig. 3e). The
valley imbalanced filling of the bands results in the net Chern number
observed, consistent with previous studies. While a large $B$-field is usually
used to break $\mathcal{T}$-symmetry, proximity-induced SOC in graphene due to
WSe2 breaks $C_{2}\mathcal{T}$ symmetry inherently at $B=0$ [30] and together
with an exchange field it can generate spin-valley degeneracy-lifted bands
(Fig. 3e). Moreover, the spin-valley flavor resolved Chern numbers can be
tuned by varying the sublattice splitting energy and Ising SOC in TBG-WSe2
systems. The values of exchange fields are expected to change with the degree
of spin polarization at the onset of magnetism that depends on specific system
parameters and Coulomb interactions, see supplementary information and Fig. S9
that shows how different initial conditions for a mean-field self-consistent
Hubbard model result in meta-stable spin polarized states at different carrier
densities. We illustrate in Fig. 3f, the spin-valley resolved band degeneracy
lifting introduced by a finite exchange field of $\lambda_{\rm ex}=0.5$ meV in
the Hamiltonian that models the spin polarization of a ferromagnetic phase,
together with a proximitized SOC term discussed in the methods section. The
associated spin-valley resolved bands develop a well defined Chern number that
will lead to a finite orbital moment when they are filled. We illustrate by
using frozen bands how the total orbital moment evolves with filling density
giving rise to a magnetization of the order of $\sim$10-2 $\mu_{\rm B}/A_{\rm
M}$ that can change its sign depending on the specific carrier density value.
Since the orbital magnetization depends sensitively on the actual exchange
field for the specific spin configuration, see Fig. A1-A2, it is expected that
its integrated magnitude as well as the local values in experiments vary
considerably with density, especially near the phase transition points.
Experimentally, the orbital magnetization maps can reach local values as large
as a few $\mu_{\rm B}/A_{\rm M}$ [43, 44]. In our experiments, an abrupt
change in orbital moment as a function of filling indicates that reordering of
levels at $\nu=1.86$ takes place due to a close competition between the
magnetic phases of opposite signs.

In Fig. 4a, we have presented a diagram with the summary of various phases
discussed throughout this report. The Lifshitz transitions and reset of
carriers at $B=20$ and 50 mT occur at the densities near the extreme left
boundary of the ferromagnetic domain as well as within the domain. At $B\geq
100$ mT the Lifshitz transitions disappear and we find a second reset near the
extreme right boundary of the ferromagnetic domain (inset in Fig. 4). As
evidenced from the diagram, ferromagnetism is accompanied by a series of Fermi
surface reconstructions at the vHSs. However, the observation of a Chern
insulator emanating from an integer value of $\nu=2$ that lies outside the
ferromagnetic domain in both density and magnetic field indicates that
$\mathcal{T}$-symmetry breaking via high $B-$fields is distinct from the
ferromagnetism observed at $\nu<2$. Finally, in the context of a very high
coercive field in our data, we have plotted coercive field reported in several
moiré graphene systems at different $\nu$ (Fig. 4b). The plot clearly
indicates the coercive field observed in our work is the highest in comparison
to other reports to date.

To summarize, our experiments have revealed a phase diagram of competing
phases in TBG near its first magic angle which is indicative of vHSs within
the quasi-flat bands, where proximity SOC plays a role in favoring the valley
polarized solutions. This diverging DOS within the nearly flat bands of TBG
reveals a finer internal structure of the bands that is manifested through
multiple phase transitions as a function of temperature, magnetic fields and
carrier densities in the vicinity of $\nu\sim 1.8$. Uncovering the underlying
physics of the various quasi-degenerate, competing ground states will require
an overarching theoretical analysis of strongly interacting many-body physics.
Our primary findings of AHE and Fermi surface reconstructions are reported
away from the usual commensurate filling of $\nu=2$. The various features in
our data including abrupt reversal of magnetization and non-quantized $R_{xy}$
are clear signs of orbital magnetism and partially valley polarized ground
state, where both bulk and edge modes are expected to contribute to the
transport. The bulk transport may be affected by percolating conduction
channels between topological domains of closely competing phases where
external electric or magnetic fields can be used as control knobs to favor
certain phases over the other. Varying the twist angle between the graphene
sheet and the WSe2 can modify the proximity SOC strength that would in turn
modify the phase diagram of the expected ground states. The high sensitivity
of the electronic structure to experimental conditions makes it both a
challenge to perform experiments reproducibly and at the same time provides an
opportunity to explore the physics near multi-phase transition points where
the electronic response functions will be particularly sensitive to external
perturbations. Whether the observed singularities within the flat bands fall
under the category of higher-order vHSs, recently proposed in TBG and related
systems [45, 46], would need to be addressed in future works. Another
interesting direction would be to identify the connection, if any, between the
valley-polarized Stoner magnet found in our work and the superconducting phase
in the vicinity of $\nu=\pm 2$ [2, 3, 39, 47, 48].

## Acknowledgements

We gratefully acknowledge the usage of the MNCF and NNFC facilities at CeNSE,
IISc. U.C. acknowledges funding from SERB via SPG/2020/000164 and
WEA/2021/000005. Y.J.P. was supported by the Korean National Research
Foundation grant NRF-2020R1A2C3009142 and D.L. was supported by grant
NRF-2020R1A5A1016518, as well as the Korean Ministry of Land, Infrastructure
and Transport (MOLIT) from the Innovative Talent Education Program for Smart
Cities. J.J. 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 and the network support from
KREONET. K.W. and T.T. acknowledge support from JSPS KAKENHI (Grant Numbers
19H05790, 20H00354 and 21H05233).

## Author Contributions

S.B. fabricated the device, performed the measurements and analysed the data.
B.G. contributed to measurements and analysis of data. Y.J.P., D.L. and J.J.
performed the theoretical calculations. S.D. and R.S. assisted in
measurements. K.W. and T.T. grew the hBN crystals. A.G. advised on
experiments. U.C. supervised the project. S.B., J.J. and U.C. wrote the
manuscript, with inputs from all authors.

## Competing interests

The authors declare no competing interests.

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## METHODS

### Device fabrication

The well known ‘tear and stack’ method was used to assemble the
heterostructure in this work. Polypropylene carbonate (PPC) film coated on a
polydimethylsiloxane (PDMS) stamp was used for picking up individual layers of
hBN, WSe2 and graphene. The final device was etched into a multi-terminal Hall
bar by reactive ion etching using CHF3/O2 followed by electron-beam
lithography, and thermal evaporation of Ohmic edge contacts and top gate using
Cr/Au (5 nm/60 nm). WSe2 layer with a thickness of $\sim$ 3 nm was exfoliated
from bulk crystals procured from 2D Semiconductors.

### Transport Measurements

Electrical transport measurements were performed in a He3 cryostat with a 10 T
magnetic field and a cryogen-free, pumped He4 cryostat with a 9 T magnetic
field. Magnetotransport measurements were carried out with a bias current of
$10$ nA, using an SR830 low-frequency lock-in amplifier at $17.81$ Hz. The
carrier density in the system was tuned by the top gate. The twist angle was
estimated using 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.
For the measurements of hysteresis in $R_{xy}$, Onsager reciprocity theorem
[49, 4] was used, details of which are given in the supplementary information.

### Proximity SOC in G/WSe2

The interlayer coupling, in particular the proximity SOC induced in the
graphene sheet on top of a WSe2 can be modeled by combining sublattice
dependent site potential differences together with Rashba and intrinsic SOC
terms [50]. In the following we briefly outline how the bands of graphene can
be altered under the proximity SOC effects of WSe2.

Fig. A1 Exchange-field induced valley split bands of graphene on WSe2 and TBG
on WSe2 calculated within the continuum model including SOC. For TBG we assume
the magic twist angle $\theta=1.05^{\circ}$ with a Fermi velocity of
$\upsilon_{F}=1\times 10^{6}~{}\rm m/s$. The SOC parameters $\lambda_{I}^{A}$,
$\lambda_{I}^{B}$, and $\lambda_{R}$, $\lambda_{PIA}^{A}$, $\lambda_{PIA}^{B}$
use the default values proposed in Ref. [14] except when they are defined
explicitly. The proximity SOC induced by the WSe2 layer lifts the degeneracy
of the spin up and down bands within each valley but the intervalley
degeneracy between opposite spins of $K\uparrow$ ($K\downarrow$) and
$K^{\prime}\downarrow$ ($K^{\prime}\uparrow$) bands is maintained. An exchange
field $\lambda_{\rm ex}$ that reflects the magnitude and sense of the spin
polarization breaks the remaining degeneracy and can easily alter the ordering
of the spin-valley flavours.

Fig. A2 Orbital magnetization of the TBG-WSe2 bands in Fig. A1 using continuum
model. In the absence of the exchange field ($\lambda_{\rm ex}=0$), the
orbital magnetization from the $K$ and $K^{\prime}$ valleys cancel each other
leading to the zero total orbital magnetization (black). For the finite
$\lambda_{\rm ex}=\pm 0.5,\ 2,\ 4$ meV, the orbital magnetization of each
valley remains finite where the sign flip of the slope along with the band
filling $v$ indicates the topological phase transition from positive to
negative Chern number.

### Continuum model bands

We model the single-layer graphene Hamiltonian contacting a TMD layer through
the staggered potential (U), exchange field (ex), intrinsic (I) and Rashba (R)
SOC, and pseudospin inversion asymmetry (PIA). [14]

$h_{i}\rightarrow h_{i}+h_{U}+h_{I}+h_{\rm ex}+h_{R}+h_{PIA}$ (1)

with

$\displaystyle h_{U}$ $\displaystyle=\Delta{\bf\sigma}_{z}\mathbf{1}_{\bf
s}\mathbf{1}_{\bf\tau}$ (2) $\displaystyle h_{I}$
$\displaystyle=\frac{1}{2}\left[\lambda_{I}^{A}(\sigma_{z}+\sigma_{0})+\lambda_{I}^{B}(\sigma_{z}-\sigma_{0})\right]\eta{\bf
s}_{z}$ $\displaystyle h_{\rm ex}$ $\displaystyle=\lambda_{\rm
ex}\mathbf{1}_{\bf\sigma}{\bf s}_{z}\mathbf{1}_{\bf\tau}$ $\displaystyle
h_{R}$ $\displaystyle=\frac{\lambda_{R}}{2}(\eta{\bf\sigma}_{x}{\bf
s}_{y}-{\bf\sigma}_{y}{\bf s}_{x})\mathbf{1}_{\tau}$ $\displaystyle h_{PIA}$
$\displaystyle=\frac{a}{2}\left[\lambda_{PIA}^{A}(\sigma_{z}+\sigma_{0})+\lambda_{PIA}^{B}(\sigma_{z}-\sigma_{0})\right]\,\left(k_{x}{\bf
s}_{y}-k_{y}{\bf s}_{x}\right)\mathbf{1}_{\bf\tau}$

where ${\bf\sigma}$ and ${\bf s}$ are Pauli matrices that represent the A/B
sublattice and $\uparrow/\downarrow$ spin, and $\mathbf{1}$ is a $2\times 2$
identity matrix. The simultaneous presence of a spin-splitting exchange field
and a SOC term is known to introduce valley polarization. For instance, an
intrinsic SOC captured through the Kane-Mele model together with sublattice
staggering potential introduces unequal gaps at the $K$ and $K^{\prime}$
valleys that leads to an anomalous Hall effect by populating a given spin-
valley band when the system is carrier doped. Similarly, a bilayer graphene
system subject to a Rashba SOC and spin-splitting exchange field is known to
give rise to an anomalous Hall effect for appropriate system parameters [51].
Both examples illustrate different mechanisms for the onset of an anomalous
Hall effect when SOC is accompanied by an exchange field that separates the
spin up and down bands in a ferromagnetic phase. The Rashba SOC mixes spins
but does not mix valleys. The band structure of G/WSe2 in Fig. A1 illustrates
how the two-fold low energy nearly flat bands are split into eight different
bands due to spin-valley degeneracy lifting using a model system, whereas
explicit mean-field calculations for the Hubbard model are presented in the
supplementary information to illustrate the sensitivity of the calculated
results to different initial conditions and carrier densities. The high
sensitivity of the orbital level ordering to spin configuration also leads to
sensitive changes in the orbital moments with exchange field parameters. We
illustrate in Fig. A2, the small changes in $\lambda_{\rm ex}$ up to $\sim$4
meV in magnitude which is sufficient to change the number of nodes crossing
zero and therefore resulting in sign flips of the orbital magnetization slopes
as a function of filling, in turn related with the spin-valley Chern numbers.

Below, we write the pristine TBG continuum model Hamiltonian for one spin-
valley flavor as 8$\times$8 matrix to emphasize the three dominant interlayer
tunneling [52]

$H_{0}^{\eta}({\bf k})=\begin{pmatrix}h_{1}({\bf k})&t_{12}({\bf
k}_{0})&t_{12}({\bf k}_{+})&t_{12}({\bf k}_{-})\\\ t_{12}^{\dagger}({\bf
k}_{0})&h_{2}({\bf k}_{0})&{\bf 0}&{\bf 0}\\\ t_{12}^{\dagger}({\bf
k}_{+})&{\bf 0}&h_{2}({\bf k}_{+})&{\bf 0}\\\ t_{12}^{\dagger}({\bf
k}_{-})&{\bf 0}&{\bf 0}&h_{2}({\bf k}_{-})\end{pmatrix}$ (3)

with

$\displaystyle h_{1}({\bf k})$
$\displaystyle=\hbar\upsilon_{F}\begin{pmatrix}0&(\eta
k_{x}-ik_{y})e^{i(+\frac{\theta}{2})}\\\ (\eta
k_{x}+ik_{y})e^{i(-\frac{\theta}{2})}&0\end{pmatrix},$ (4) $\displaystyle
h_{2}({\bf k})$ $\displaystyle=\hbar\upsilon_{F}\begin{pmatrix}0&(\eta
k_{x}-ik_{y})e^{i(-\frac{\theta}{2})}\\\ (\eta
k_{x}+ik_{y})e^{i(+\frac{\theta}{2})}&0\end{pmatrix},$ $\displaystyle
t_{12}({\bf k}_{j})$ $\displaystyle=\begin{pmatrix}\omega^{\prime}&\omega
e^{i(2\pi/3)j}\\\ \omega e^{i(2\pi/3)j}&\omega^{\prime}\end{pmatrix}$

where ${\bf k}$ and ${\bf k}_{j}$ are the wave vectors measured from the
$K^{(\eta)}$ of the layer 1 and 2 with the valley index $\eta=\pm 1$, and
${\bf k}_{j}={\bf k}+{\bf G}_{j}$ with $j=0,+,-$ are connected with the three
moire reciprocal lattice vectors, which adjusts the momentum difference
between two different Dirac points from layer 1 and 2. The $h_{1}$ and $h_{2}$
describe the Dirac bands of each layer where we use the Fermi velocity
$\upsilon_{F}=1\times 10^{6}$ m/s and the twist angle $\theta$. The $t_{12}$
term is the tunneling matrix between the two graphene layers with tunneling
constants $\omega=0.12$ eV and $\omega^{\prime}=0.0939$ eV. For the actual
calculation, we use 392$\times$392 Hamiltonian matrices for each valley and
each ${\bf k}$ point, which includes the hopping terms between the two spins,
and use four sublattices, and 49 reciprocal lattice points. [53] The valley
mixing terms are not included in our spin-orbit coupling models.

To understand the topological phase transition along with the band filling
$\nu$, we obtain the orbital magnetization $M_{\rm(orb)}(\mu)$ as a function
of chemical potential $\mu$ [33] using

$\displaystyle M_{\rm(orb)}(\mu)$ $\displaystyle=\sum_{n}\int\frac{d^{2}{\bf
k}}{(2\pi)^{2}}\,f(\mu-\varepsilon_{n}({\bf
k}))\,(\varepsilon_{n}+\varepsilon_{n^{\prime}}-2\mu)$ (5) $\displaystyle\
\times\left[\frac{e}{\hbar}{\rm Im}\sum_{n^{\prime}\neq n}\frac{\langle
n|\partial_{k_{x}}H|n^{\prime}\rangle\langle
n^{\prime}|\partial_{k_{y}}H|n\rangle}{(\varepsilon_{n}-\varepsilon_{n^{\prime}})^{2}}\right],$

where $f(E)$ is the Fermi-Dirac distribution, $\varepsilon_{n}$ and
$|n\rangle$ are the eigen energy and vector of the $n$-th band. The Chern
number $C=(2\pi\hbar/e)dM_{\rm(orb)}/d\mu$ can be estimated by the slope of
the $M_{\rm(orb)}$ v.s. $\mu$ graphs. (See Fig. A2.)

## 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.

## SUPPLEMENTARY INFORMATION

##  I Experimental Results

Supplementary Fig. S1. Hall resistance $R_{xy}$ measured as a function of
magnetic field $B$ using Onsager reciprocity theorem [54, 4]. a. Optical image
of the device where A, B, C, D, and E are the contacts used throughout the
measurements. The scale bar denotes 10 $\mu$m. b.-c. Measurement of $R_{xy}$
for $B$-field sweep at $\nu=1.8$ by driving current from contact A to D and
measuring voltage between contact C and B. Orange and green colours are used
for current and voltage probes. Blue and red colours in the data denote the
direction of $B$-field sweep. d.-e. First, $R_{xy}$ is measured as a function
of $B$-field sweep by passing current from contact A to D and measuring
voltage between contact C and B. Second, $R_{xy}$ is measured as a function of
$B$-field sweep by passing current from contact C to B and measuring voltage
between contact A and D. The antisymmetrized $R_{xy}$ is calculated using
$R_{xy}=(R_{ADCB}-R_{CBAD})/2$. The width and shape of the hysteresis obtained
by measuring $R_{xy}$ using the above discussed two methods do not match
perfectly. The transverse voltage in an out-of-plane $B$ relies on the fact
that $R_{xy}$ is antisymmetric in $B$ and it is accurately determined using
$R_{xy}=\\{R_{xy}(B)-R_{xy}(-B)\\}/2$. However, this method does not perfectly
hold true for a ferromagnet where the dynamics of magnetic domains throughout
the device may have different responses to the magnetic field. For the precise
determination of $R_{xy}$ below coercive field we have used Onsager
reciprocity theorem and the ferromagnetism data presented in the paper are
obtained using the reciprocity theorem, as detailed in (b).

Supplementary Fig. S2. Comparison of $R_{xy}$ using Onsager antisymmetrization
and field antisymmetrization. a. $R_{xy}$ as a function of $B$ measured by
interchanging current and voltage probes as discussed in Fig. S1. b. To
compare $R_{xy}$ data resulting from different measurement configurations, we
show the hysteresis at $\nu=1.85$ obtained from field antisymmetrized
$R_{xy}(\overleftarrow{B})=\\{R_{xy}(\overleftarrow{B})-R_{xy}(\overrightarrow{-B})\\}/2$,
$R_{xy}(\overrightarrow{B})=\\{R_{xy}(\overrightarrow{B})-R_{xy}(\overleftarrow{-B})\\}/2$
where $\overleftarrow{B}$ and $\overrightarrow{B}$ denote two opposite
direction of $B$ sweep. In this method, a significant reduction in the width
of the hysteresis in $R_{xy}$ for $0.44\lesssim|B|\lesssim 0.67$ (shown by
dashed green lines) indicates the asymmetry between positive and negative
coercive fields. However, the method discussed in a. better captures the
dynamics of magnetic domains, which are asymmetric in magnetic field.
Therefore, we have used the Onsager anti-symmetrization method for the
characterization of hysteresis in our device.

Supplementary Fig. S3. Low-field Hall data for two opposite sweep directions
of carrier density at $T=0.3$ K. $R_{xy}$ plotted as a function of $\nu$ at
$B=50$ mT shows a hysteresis loop with respect to density only in the vicinity
of $\nu=2$. Here, $R_{xy}$ is calculated using Onsager reciprocity theorem as
discussed in Fig. S1.

Supplementary Fig. S4. The evolution of hysteresis in density and magnetic
field. a. $R_{yx}$ shows hysteretic behaviour when the density is swept back
and forth at low magnetic fields up to 1.2 T. The hysteresis is strongly
suppressed with increasing $B$. $R_{yx}$ shows zero crossings only at $B=0.05$
T and remains positive at other fields. a. The hysteresis in $R_{xy}$ with
repect to out of plane $B$-field in plotted for different $\nu$ below $\nu=2$.
The width of the hysteresis in $B$ is highly tunable with density and the
reversal of the hysteresis in clearly seen.

Supplementary Fig. S5. Temperature dependence of the hysteresis a.-d. $R_{xy}$
as a function of $B$ for two opposite sweep directions is plotted for
different temperatures. The hysteresis becomes weaker with increasing
temperature and disappears at $T=2$ K.

Supplementary Fig. S6. Fermi surface reconstructions at $T=0.3$ K. Hall
density $n_{H}$ plotted as a function of $\nu$ for $B=0.05,0.2,0.4$ and 0.6 T
at $T=0.3$ K. Lifshitz transitions and reset of charge carriers are observed
for $B=0.05T$ , whereas only reset of charge carriers occur for $B=0.2-0.6$ T.

Supplementary Fig. S7. Weak antilocalization data at $T=0.3$ K. The change in
longitudinal magneto-conductivity ($-\Delta\rho/\rho^{2}$) with respect to
$B=0$ mT data, is plotted as a function of $B$ from -20 mT to 20 mT for a gate
voltage $V_{G}=-0.8$ V (shown by the dashed red line in the inset). A peak in
$-\Delta\rho/\rho^{2}$ at $B=0$ mT indicates weak antilocalization and hence,
a finite spin-orbit coupling in graphene layers proximity coupled to WSe2.

Supplementary Fig. S8. Hall conductivity $\sigma_{xy}$ and longitudinal
conductivity $\sigma_{xx}$ at $B=10$ T. $\sigma_{xy}$ shows plateaus as
$\sigma_{xy}=Ce^{2}/h$ associated with the minima in $\sigma_{xx}$. Different
color bars (yellow for the CNP, green for $\nu=1,2,3$) have been used to show
the quantized states emanating from several partial fillings of flat bands.

##  II Spin-polarized phase in mean-field Hubbard model calculation

Supplementary Fig. S9. Spin magnetization of converged states of Hubbard model
calculation. The total spin magnetic moments of the moiré unit-cell was
obtained from self-consistent spin density. From $M_{init}\leq 0.45$ (Light
green) and $M_{init}\geq 1.35$ (Red), the non-magnetic phase and the
ferromagnetic phases are clearly identified. It is confirmed from the trend of
magnetization change that the two phases become closer as the filling
increases. In $M_{init}=0.81,0.9,0.99$ case, the states are separated between
$\nu=1.6$ and $\nu=1.9$ and converged to the two different phases manifested
in lower and higher initial magnetization cases.

A mean field Hubbard model of the system introduces the spin-polarizes phase
which further enhances the degeneracy lifting and therefore the valley
polarization. The initial spin densities were assumed as uniformly distributed
spin-ordered states of total magnetization in a unit moiré cell
$M_{init}=0\sim 4.5[\mu_{B}/A_{Moire}]$. Interestingly, two stable convergence
states of the non-magnetic and ferromagnetic phases were revealed according to
the initial spin density. In addition, it was shown that the convergence phase
transition at $\nu=1.7$ under the initial magnetic condition of
$M_{init}=0.9$.

The tight-binding Hamiltonian of graphene on WSe2 for the Hubbard model
calculation inherits the proximity spin orbit coupling effect and sublattice
dependent potentials due to the contacting layer. Following the formulations
presented in Refs. [14] we consider the Hamiltonian

$\begin{split}H=&H_{0}+\sum_{i\in
Bot,\sigma}\Delta\xi^{i}c^{\dagger}_{i,\sigma}c_{i,\sigma}\\\
&+\frac{2i}{3}\sum_{\langle
i,j\rangle}^{Bot}\sum_{\sigma,\sigma^{\prime}}\lambda_{R}c^{\dagger}_{i,\sigma}c_{j,\sigma^{\prime}}(s^{x}_{\sigma,\sigma`}d^{y}_{i,j}-s^{y}_{\sigma,\sigma^{\prime}}d^{x}_{i,j})\\\
&+\frac{i}{3\sqrt{3}}\sum_{\langle\langle
i,j\rangle\rangle}^{Bot}\sum_{\sigma,\sigma^{\prime}}\lambda^{i}_{I}c^{\dagger}_{i,\sigma}c_{j,\sigma^{\prime}}\nu_{i,j}s^{z}_{\sigma,\sigma^{\prime}}\\\
&+\frac{2i}{3}\sum_{\langle\langle
i,j\rangle\rangle}^{Bot}\sum_{\sigma,\sigma^{\prime}}\lambda^{i}_{PIA}c^{\dagger}_{i,\sigma}c_{j,\sigma^{\prime}}(s^{x}_{\sigma,\sigma`}d^{y}_{i,j}-s^{y}_{\sigma,\sigma^{\prime}}d^{x}_{i,j})\\\
&+\sum_{i,\sigma,\sigma^{\prime}\neq\sigma}U\rho_{i,\sigma^{\prime}}c^{\dagger}_{i,\sigma}c_{i,\sigma}\end{split}$
(6)

Here, H0 is non-interacting hopping model for magic-angle twisted bi-layer
graphene with relaxation effects [40], and the second to fifth terms express
spin-orbit interactions applied to the bottom layer graphene to effectively
reflect the WSe2 layers[14]. The 1.7∘ commensurate tBG structure is used for
this calculation with rescaling parameter $S^{\prime}=1.608$ corresponding
with the magic angle. $\Delta$ is the induced orbital staggered potential,
$\lambda_{I}$ is the intrinsic spin-orbit couplings, $\lambda_{R}$ is the
Rashba spin-orbit coupling and $\lambda_{PIA}$ is the pseudospin-inversion-
asymmetry(PIA) spin-orbit terms. We used $\Delta$ = 0.54 meV and $\lambda_{R}$
= 0.56 meV. In the case of $\lambda_{I}$ and $\lambda_{PIA}$, the values are
different for each sublattice and are $\lambda^{A}_{I}$=$-$1.22 meV,
$\lambda^{B}_{I}$=1.16 meV, $\lambda^{A}_{PIA}$ = $-$2.69 meV,
$\lambda^{B}_{PIA}$ = 2.54 meV. The last term considers the mean-field Hubbard
model. The spin density $\rho_{i,\sigma}$, considering the neutral density
$n^{0}$ obtained from the Non-interacting Hamiltonian, is defined as
$\rho_{i,\sigma}\equiv\langle
c^{\dagger}_{i,\sigma}c_{i,\sigma}\rangle-n^{0}_{i,\sigma}$. The self-
consistent calculations were performed on $5\times 5$ Monkhorst-Pack grids for
U = 5 eV.