# Generalized anomalous Hall crystals in twisted bilayer–trilayer graphene

Ruiheng Su1,2∗ Dacen Waters3,4∗ Boran Zhou5 Kenji Watanabe6 Takashi Taniguchi7
Ya-Hui Zhang5 Matthew Yankowitz3,8† Joshua Folk1,2† 1Quantum Matter Institute,
University of British Columbia, Vancouver, British Columbia, V6T 1Z1, Canada
2Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia, V6T 1Z1, Canada  3Department of Physics,
University of Washington, Seattle, Washington, 98195, USA  4Intelligence
Community Postdoctoral Research Fellowship Program, University of Washington,
Seattle, Washington, 98195, USA  5Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, Maryland, 21205, USA  6Research Center for
Functional Materials, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  7International Center for Materials
Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  8Department of Materials Science and Engineering,
University of Washington, Seattle, Washington, 98195, USA  ∗These authors
contributed equally to this work<EMAIL_ADDRESS>(M.Y<EMAIL_ADDRESS>(J.F.)

In a dilute two-dimensional electron gas, Coulomb interactions can stabilize
the formation of a Wigner crystal wigner1934interaction ; grimes1979wigner ;
Andrei1988 . Although Wigner crystals are topologically trivial, it has been
predicted that electrons in a partially-filled band can break continuous
translational symmetry and time-reversal symmetry spontaneously to form an
anomalous Hall crystal (AHC) zhou2023fractional ; dong2023anomalous ;
kwan2023anomalous ; sheng2024quantum ; tan2024parent ; soejima2024anomalous ;
dong2024stability . Here, we report the observation of a generalized AHC in
twisted bilayer–trilayer graphene whose formation is seeded by the moiré
potential. The crystal quadruples the moiré unit-cell area, coinciding with an
integer quantum anomalous Hall effect. The Chern number of the state is
exceptionally tunable, and can be switched reversibly between $+1$ and $-1$ by
electric and magnetic fields. Recent theories have predicted the generalized
AHC to act as a parent state of the fractional quantum anomalous Hall effect
zhou2023fractional ; dong2023anomalous , pointing to twisted bilayer–trilayer
graphene as a promising platform in the search for new correlated and
topological phenomena.

Berry curvature is the foundational ingredient required for the formation of
topological electronic states. When time-reversal symmetry is broken
spontaneously, these states give rise to dramatic experimental signatures at
zero magnetic field. Quantum anomalous Hall (QAH) insulators are a notable
example, characterized by precise quantization of the Hall conductance to
integer multiples of the square of the electron charge divided by Planck’s
constant, $Ce^{2}/h$ haldane1988model ; RevModPhys.95.011002 . This
quantization is robust against sample variations, protected by a topological
Bloch-band invariant known as the Chern number, $C$, that is equivalent to the
integrated Berry curvature of the filled bands in the Brillouin zone
thouless1982quantized ; berryphase2010effects . QAH insulators have been
observed in several van der Waals (vdW) platforms, including graphene
sharpe2019emergent ; serlin2020intrinsic ; stepanov2021competing ;
polshyn2022topological ; chen2020tunable ; lu2024fractional  and transition
metal dichalcogenide (TMD) moiré lattices li2021quantum ; cai2023signatures ;
zeng2023thermodynamic ; park2023observation ; tao2024valley , as well as
rhombohedral-stacked multilayer graphene han2023large ; sha2024observation .
In each of those examples, flat electronic bands are susceptible to the
spontaneous breaking of valley (and therefore time-reversal) symmetry by
electron–electron interactions. The finite Chern number of the filled bands
then defines the topological state of the system.

Figure 1: Generalized anomalous Hall crystal in twisted bilayer–trilayer
graphene. a, Schematic of the dual-gated twisted bilayer–trilayer graphene
device. b, Spatial distribution of carrier density, $n(\mathbf{r})$, at a
filling of $\nu=1/4$. The upper panel is from a single-particle calculation
and corresponds to a metallic state. The lower panel is from a Hartree–Fock
calculation and corresponds to a $C=1$ insulator that spontaneously quadruples
the moiré unit cell area. c, Illustration of the moiré Brillouin zone (BZ) and
zoom-in of the reduced moiré BZ with the quadrupled unit cell, labeled with
high symmetry points. d, Single-particle calculated band structure plotted
along the trajectory through the moiré BZ shown in c, for
$\theta=1.50^{\circ}$ and $\delta=90$ meV. e, Hartree–Fock band structure
calculated at $\nu=1/4$, plotted along the trajectory through the reduced
moiré BZ in c. The lowest band (shaded blue) is fully filled, with $C=+1$.
Dashed lines at zero energy in d-e denote the Fermi energy at $\nu=1/4$. f,
Single-particle (top) and Hartree–Fock (bottom) calculations of the Berry
curvature, $\Omega(\mathbf{k})$, within the moiré BZ as in illustrated panel
c. Hartree–Fock results were divided by $4$ for clarity. g, Antisymmetrized
map of $\rho_{xy}$ obtained at $T=100$ mK and $|B_{\perp}|=0.9$ T. The top
axis ($\nu$) shows the corresponding number of electrons per moiré unit cell.
The inset shows an optical micrograph of the device labeled with the
measurement configuration. The white scale bar is 2 $\mu$m. h, Antisymmetrized
$\rho_{xy}$ and symmetrized $\rho_{xx}$ maps obtained in the boxed region in g
at $|B_{\perp}|=10$ mT. i, Measurements of $\rho_{xy}$ and $\rho_{xx}$ versus
$D$ at fixed $\nu=0.251$ and $B_{\perp}=-50$ mT. j, $\rho_{xy}$ measured as
$B_{\perp}$ is swept back and forth at $\nu=0.253$ and $D=532$ mV nm-1, $T=10$
mK.

Electron–electron interactions can also induce the spontaneous breaking of
translational symmetry, creating a crystal of electrons known as a Wigner
crystal. Traditional Wigner crystals are topologically trivial, usually pinned
by disorder in two dimensions to behave as electrical insulators. However,
there are decades-old predictions that topological states resembling the
Wigner crystal can also exist tevsanovic1989hall . When these states form at
zero magnetic field due to spontaneously broken time-reversal symmetry, the
result is an anomalous Hall crystal (AHC) that exhibits both a charge gap and
a quantized anomalous Hall effect.

Although the pure AHC state is expected to require ultra-strong interactions
to emerge dong2024stability , a generalized version of the AHC may arise more
readily in moiré systems, using the commensuration energy of the moiré lattice
to form crystals with an integer-multiple enlargement of the unit cell (Figs
1a,b). These states fold the original moiré bands into a smaller Brillouin
zone (Figs 1c–e), multiplying the number of bands by the same integer that
describes the unit-cell enlargement. The Chern number of any given folded band
is ultimately determined by its total Berry curvature, which may be strongly
modified by interactions (Fig. 1f). There is no inherent connection to the
Chern number of the parent band prior to the crystal formation, although the
sum of the Chern numbers of the folded bands must equal that of the parent.

Here, we report the observation of a generalized AHC, along with an array of
topological charge density wave (TCDW) states emerging in a modest magnetic
field, all arising in twisted bilayer–trilayer graphene. The generalized AHC
forms at a doping of one electron per four moiré unit cells, corresponding to
a band filling factor of $\nu=1/4$ within the four-fold degenerate lowest
moiré conduction band whose total filling capacity is $\nu=4$. The generalized
AHC state is reminiscent of earlier observations of TCDW states in various
moiré lattices, both with wang2015evidence ; spanton2018observation ;
saito2021hofstadter ; xie2021fractional ; he2023symmetry ; xie2024strong  and
without polshyn2022topological  an external magnetic field, but is clearly
distinct in several important ways. It is the first such state to establish a
clear disconnect between the Chern number of the generalized AHC state ($C=1$
at $\nu=1/4$) and that of the parent state at integer band filling ($C=0$ at
$\nu=1$), as it is impossible to evenly partition the parent-state Berry
curvature among the folded bands generated by the four-times larger unit cell.
Interactions therefore play an essential role in determining the Chern numbers
of the folded bands, and by extension that of the resulting generalized AHC
state. Because of this, the Chern number of the state is exquisitely tunable
with external fields — the electric displacement field, $D$, and both the in-
plane, $B_{\parallel}$, and out-of-plane, $B_{\perp}$, magnetic field —
enabling unprecedented control of its topological properties.

###  .1 Symmetry breaking in twisted bilayer–trilayer graphene

Figure 1a shows the sample structure, consisting of Bernal-stacked bilayer and
trilayer flakes twisted by $1.50^{\circ}$ with respect to each other (see
Extended Data Fig. 1), as well as top and bottom graphite gates that can be
used to tune the carrier density $n$, and displacement field, $D$. The band
structure depends both on the magnitude and sign of $D$, owing to the
difference in the number of graphene layers above and below the moiré
interface. In this work, we focus on electronic states that emerge for $D>0$,
such that the conduction band is polarized towards the trilayer graphene
waters2024topological . Figure 1c shows a representative single-particle band-
structure calculation for the system (see Methods), including a potential
difference between the top and bottom layers, $\delta$, to capture the effect
of an applied $D$. The flat, isolated moiré conduction band (blue curve) is
conducive to the spontaneous symmetry-breaking that is essential for the
generalized AHC state to emerge. Indeed, this band has previously been
observed to host multiple states that break spin and/or valley (together
called isospin) degeneracy waters2024topological .

Figure 2: Competing topological states at fractional band filling. a, Landau
fan of antisymmetrized $\rho_{xy}$ (left) and symmetrized $\rho_{xx}$ (right),
obtained at fixed $D=531$ mV nm-1, $T=10$ mK, with $\nu$ as the fast
measurement axis. Dashed lines show the expected evolution of $|C|=1$,
$\nu=1/4$ states ($C_{1/4}^{+1}$, $C_{1/4}^{-1}$) based on the Streda formula.
b, Schematic diagram depicting the primary features in a. The total
magnetization of the state is assumed to be aligned with $B_{\perp}$. c,
Antisymmetrized $\rho_{xy}$ maps obtained at several values of $B_{\perp}$.
The dashed (dotted) line indicates $D=531$ mV nm-1 ($526$ mV nm-1). d, Landau
fan obtained at fixed $D=526$ mV nm-1. The dashed line shows the expected
evolution of a $C=-1$, $\nu=1/3$ state ($C_{1/3}^{-1}$). e, Schematic diagram
depicting the primary features in d.

Signatures of isospin symmetry-breaking can be easily seen in a map of the
antisymmetrized Hall resistivity, $\rho_{xy}$, shown in Fig. 1g. There are
multiple instances in which $\rho_{xy}$ diverges and changes its sign over a
narrow range of $n$ with essentially no dependence on $D$, corresponding to
the chemical potential traversing an energy gap. For example, these features
occur at the charge neutrality point ($n=\nu=0$), and at $n=5.17\times
10^{12}$ cm-2 where the lowest conduction band is fully filled with four
electrons per moiré unit cell ($\nu=4$, see Methods). Both are consistent with
the expectation of single-particle gaps from the calculated band structure
(Fig. 1d). Similar features at quarter– and half–fillings ($\nu=1,2$, and also
weakly at $\nu=3$) reflect correlation-induced energy gaps. Our data are
broadly similar to previous reports across the family of small–angle twisted
multilayer graphene structures Shen2020 ; Liu2020 ; Cao2019b ; Burg2019 ;
He2021tdbg ; chen2020tMBG ; polshyn2020electrical ; Xu2021tmbg ;
waters2024topological .

###  .2 Evidence for the generalized anomalous Hall crystal

We focus first on one symmetry-broken state that appears very close to charge
neutrality and around $D=0.53$ V nm-1, within the boxed region in Fig. 1g.
Figure 1h maps out $\rho_{xy}$ and $\rho_{xx}$ over this region near zero
field. Remarkably, a stripe of large $\rho_{xy}$ close to $h/e^{2}\approx
25.81$ k$\Omega$ emerges from a comparably featureless background. The carrier
density where the stripe appears, $n=0.323\times 10^{12}$ cm-2, corresponds to
adding one electron for every four moiré unit cells ($\nu=1/4$). This
quantized Hall resistance, concomitant with a vanishing $\rho_{xx}$, is a
hallmark of an integer QAH state (Fig. 1i). Accordingly, the quantization of
$\rho_{xy}$ persists to zero external magnetic field and up to a temperature
of $T\approx 400$ mK (see Extended Data Figs. 2 and 3), and is sharply
hysteretic with a coercive field $B_{c}\approx 50$ mT (Fig. 1j). The sign of
$\rho_{xy}$ flips abruptly across $B_{c}$ to form a single square loop. The
formation of a gapped topological state with integer quantization of
$\rho_{xy}$ at a doping less than $\nu=1$ necessitates the simultaneous
breaking of time-reversal and translational symmetry, corresponding to a
generalized AHC.

Figure 3: Topological charge density wave states in a magnetic field. a,
Antisymmetrized $\rho_{xy}$ and symmetrized $\rho_{xx}$ maps obtained at
$|B_{\perp}|=2.5$ T, $T=10$ mK. b, Schematic phase diagram of the primary
features in a. TCDW states (blue and orange) are labeled with their respective
Chern numbers. Regions of normal–, half–, and quarter–metal are assigned based
on the degeneracy of low-field quantum oscillations waters2024topological .
Trivial insulating states ($C=0$) are denoted by vertical gray stripes.
Contours of $R_{xy}=0$, corresponding to either van Hove singularities or
spontaneous isospin-symmetry breaking, are denoted by red curves. The purple
curve denotes the same first-order phase transition (PT) shown in Figs. 2b and
e. c, Landau fans of antisymmetrized $\rho_{xy}$ and symmetrized $\rho_{xx}$
obtained by sweeping $V_{\text{tg}}$ with fixed $V_{\text{bg}}=4.89$ V (dashed
line in right panel of a), $T=10$ mK. The dashed black lines on the
$\rho_{xx}$ fan show the trajectories of TCDW states with $|C|=1$ tracing back
to $\nu=1/4$, $1/2$, and $2/3$. d, Schematic diagram depicting the primary
features in c. In addition to states that follow the conventions in b, gray
lines indicate trajectories of quantum Hall states tracing back to $\nu=0$ and
$\nu=1$, with thicker lines denoting states that are observed experimentally.
The inset shows a line-cut of $\rho_{xy}$ and $\rho_{xx}$ from c, taken along
the linear trajectory expected for a $C=1$, $\nu=1/2$ state.

The quantized value $\rho_{xy}=h/e^{2}$ indicates that the state has a Chern
number of $|C|=1$, which is confirmed by its evolution in $n$ with out-of-
plane magnetic field (Fig. 2a). As an applied $B_{\perp}$ modifies the
degeneracy of filled Chern bands, the charge density required to reach a
topological gap changes with $B_{\perp}$. This is described by the Streda
formula, $\frac{dn}{dB}=C\frac{e}{h}$, which relates the slope of the
trajectory of a gapped state in the $(n,B)$ plane to its Chern number
streda1982quantised ; streda1982theory . The locations of the
$\rho_{xy}=+h/e^{2}$ plateau and $\rho_{xx}$ minima indeed disperse to larger
values of $n$ as $B_{\perp}$ is increased from zero, in excellent agreement
with the expected slope of a $C=+1$ state.

Analogous measurements made at $\nu=1$ show that the state corresponding to a
fully filled first moiré band is a gapless metal at zero magnetic field,
although it exhibits an anomalous Hall effect upon doping (see Extended Data
Figs. 4 and 5). The state transitions to an insulator with $C=0$ upon
application of a small $B_{\perp}$, with diverging $\rho_{xx}$ and an abrupt
sign flip in $\rho_{xy}$, neither of which disperse in doping as $B_{\perp}$
is raised (Extended Data Fig. 5). Thus, the $C=1$ QAH state at $\nu=1/4$
appears to emerge directly out of a quarter–metal phase associated with full
isospin degeneracy lifting at $\nu=1$, as identified by the frequency of
quantum oscillations formed in a small magnetic field waters2024topological .
The disconnect between $C=1$ for the state at $\nu=1/4$ versus $C=0$ at
$\nu=1$ is an identifying characteristic of the anomalous Hall crystal.

###  .3 Competing topological states

Following the trajectory of the $\nu=1/4$ state in Fig. 2a to increasing
$B_{\perp}$ reveals another remarkable feature of the data: the
$\rho_{xy}=+h/e^{2}$ plateau disappears above $\approx 0.7$ T, simultaneous
with the emergence of a $\rho_{xy}=-h/e^{2}$ plateau and $\rho_{xx}$ minimum.
This state arises at a lower charge density ($\nu<1/4$) than its zero-field
counterpart, consistent with a $C=-1$ state also originating from $\nu=1/4$.
For clarity, we label these states by their Chern number and band filling upon
extrapolation to $B_{\perp}=0$: $(C=1,\nu=1/4)\equiv C^{+1}_{1/4}$ and
$(C=-1,\nu=1/4)\equiv C^{-1}_{1/4}$. Apparently, the system undergoes a phase
transition above a critical threshold field, $B_{\text{PT}}$, from
$C^{+1}_{1/4}$ into a new ground state that also has a quadrupled unit cell
but the opposite sign of the Chern number ($C^{-1}_{1/4}$ ). This transition
is first-order in nature, as evidenced by hysteretic jumps in resistance near
$B_{\text{PT}}$ (see Extended Data Fig. 6). The topological phases represented
by the data in Fig. 2a are summarized in Fig. 2b. The sign reversal of $C$
across $B_{\text{PT}}$ is qualitatively distinct from the flip between $C=\pm
1$ states across the coercive field nearer to $B_{\perp}=0$. Whereas the
latter occurs between time-reversed counterparts whose free energies either
increase or decrease with $B_{\perp}$, the former cannot be explained in the
same way.

Figure 4: Magnetic and electric field–driven transitions. a, b, Landau fans of
antisymmetrized $\rho_{xy}$ (a) and symmetrized $\rho_{xx}$ (b), collected at
several values of in-plane magnetic field with $D=531$ mV nm-1, $T=10$ mK. c,
$\rho_{xy}$ measured as $B_{\perp}$ is swept back and forth for several values
of $B_{\parallel}$ ($\nu=0.253$, $D=531$ mV nm-1), $T=300$ mK. d, Blue markers
denote the values of $B_{\perp}$ and $B_{\parallel}$ corresponding to the
phase boundary separating $C=\pm 1$ states, inferred from a and b. The error
bars reflect the extent to which the opposite-signed quantized plateaux
overlap. The gray line traces a contour of fixed total applied magnetic field
$B_{\text{tot}}=0.71$ T. e, $\rho_{xy}$ measured as $D$ is swept back and
forth at $\nu=0.248$, $B_{\perp}=-0.05$ T, and $B_{\parallel}=0.35$ T. Each
sweep is offset by $3h/e^{2}$ for clarity. The respective values $\pm h/e^{2}$
for each are denoted by the narrow gray lines.

Signatures of the first-order phase transition persist across the entire range
of $\nu$ shown in Fig. 2a, indicating that it is not limited to commensurate
filling. The transition is also tuned by $D$, as seen in maps of $\rho_{xy}$
acquired at different fixed values of $B_{\perp}$ (Fig. 2c). The primary
features of each of these maps are the contiguous regions of red and blue
coloring, which correspond to gapped states with $C=+1$ and $-1$,
respectively. The Landau fans shown in Fig. 2a correspond to the value of $D$
denoted by the black dashed line. The first-order phase transition seen in
Fig. 2a appears as an arc in the $(\nu,D)$ plane in Fig. 2c, mapping out the
location of this phase transition in both $\nu$ and $D$. At all values of
$B_{\perp}$, the arc sharply separates regions of the maps with opposite signs
of $\rho_{xy}$ in the topological gapped states, and with slightly different
values of $\rho_{xy}$ in the metallic states.

###  .4 Additional magnetic-field–induced states

The map acquired at $B_{\perp}=0.5$ T in Fig. 2c reveals an additional pocket
of quantized Hall resistance close to $\nu=0.33$, where $\rho_{xy}=-h/e^{2}$.
Figure 2d shows a Landau fan acquired at $D=526$ mV nm-1, corresponding to the
black dotted line in Fig. 2c, which cuts through this additional blue pocket
at $B_{\perp}=0.5$ T. The sequence of phase transitions observed at this value
of $D$ is more intricate, as summarized schematically in Fig. 2e. The extra
blue pocket in the $B_{\perp}=0.5$ T map from Fig. 2c can now be seen as a
$\rho\sim-h/e^{2}$ plateau with concomitant $\rho_{xx}$ minimum that falls
along the $C=-1$ line coming from $\nu=1/3$ (i.e., $C^{-1}_{1/3}$ ). The
appearance of the $C^{-1}_{1/3}$ state is clearly connected with the same
first-order phase transition that results in a flip between the $C^{+1}_{1/4}$
and $C^{-1}_{1/4}$ states. The integer Chern number and fractional filling
factor of $C^{-1}_{1/3}$ suggest a spontaneously tripled moiré unit cell,
contrasting with the quadrupled unit cells of $C^{+1}_{1/4}$ and
$C^{-1}_{1/4}$ . Further raising $B_{\perp}$ suppresses $C^{-1}_{1/3}$ , and
$C^{-1}_{1/4}$ returns (Fig. 2d). Taken together, the data in Fig. 2 indicate
that the system switches sharply between a low-field state where the Chern
number is $+1$ and the unit cell is quadrupled, into a higher-field state
where the Chern number is $-1$ and the unit cell may be either tripled or
quadrupled, depending on $B_{\perp}$ and other parameters.

Several other TCDW states arise at even higher perpendicular magnetic fields.
Figure 3a shows maps of $\rho_{xy}$ and $\rho_{xx}$ at $B_{\perp}=\pm 2.5$ T.
A large portion of these maps is covered by extended vertical stripes that
correspond to quantum Hall states at fixed band filling. In addition, very
small pockets of suppressed $\rho_{xx}$ and enhanced $\rho_{xy}$ appear over
narrow ranges of $D$, denoted by the $|C|=1$ states in the schematic diagram
(Fig. 3b). With the exception of the state near $\nu=3/2$, all of these
features appear along the sharp diagonal boundary that separates the isospin-
unpolarized phase at large $D$ from the quarter-metal phase at smaller $D$.
Notably, this phase boundary is aligned precisely to the axis corresponding to
the top gate voltage, $V_{\rm{tg}}$, such that the phase transition is
controlled entirely by the back gate voltage, $V_{\rm{bg}}$ (see Extended Data
Fig. 7 for the same maps displayed against $V_{\rm{tg}}$ and $V_{\rm{bg}}$).
Although we do not fully understand the origin of this effect, similar
behavior is seen across a wide family of symmetry-breaking transitions in
twisted Bernal graphene multilayer structures Shen2020 ; Liu2020 ; Cao2019b ;
Burg2019 ; He2021tdbg ; chen2020tMBG ; polshyn2020electrical ; Xu2021tmbg ;
waters2024topological . We speculate that it may arise from inhomogeneous
electrostatic charging of the five layers of graphene rickhaus2019gap ;
kolavr2023electrostatic .

In order to better visualize the field-induced TCDW states, we examine a
Landau fan acquired along a trajectory parallel to the quarter-metal phase
boundary (Fig. 3c, taken along the path denoted by the black dashed line in
Fig. 3a), with salient features highlighted in Fig. 3d. The first-order phase
transition between the $C^{+1}_{1/4}$ and $C^{-1}_{1/4}$ states that was
analyzed earlier appears first at $\nu\approx 1/4$ and $B_{\perp}=0.7$ T,
identified in the data by a sharp jump in $\rho_{xx}$ and an inversion in
$\rho_{xy}$. The phase transition shifts rapidly towards larger values of
$\nu$ as the field is raised and then eventually reverses direction back
towards $\nu=0$. Notably, the $C^{+1}_{1/4}$ state briefly recurs for
$B_{\perp}\approx 6$ T, when the phase transition crosses back across the
value of $\nu$ corresponding to the projected trajectory of the $C^{+1}_{1/4}$
state. We speculate later about the possible nature of this first-order phase
transition.

At larger values of $\nu$, additional states arise over small ranges of
$B_{\perp}$ that project back to $\nu=1/2$ and $2/3$. The state projecting to
$\nu=1/2$ has a slope corresponding to a Chern number of $+1$ (i.e.,
$C^{+1}_{1/2}$), consistent with its quantized value of $\rho_{xy}=h/e^{2}$
(see the inset of Fig 3d). Similar behavior is seen for a more weakly
developed $C=1$ state projecting to $\nu=2/3$, although the Landau fan also
shows faint signs of an incipient $C=2$ state at slightly smaller $B_{\perp}$
(i.e., $C^{+1}_{2/3}$ and $C^{+2}_{2/3}$). Another Landau fan acquired at
fixed $D=0.57$ V/nm, shown in Extended Data Fig. 5a,b, also shows evidence of
a weakly developed $C=1$ state projecting to $\nu=3/2$ (i.e., $C^{+1}_{3/2}$).
We speculate that these high-field states are essentially the same as the
generalized AHC at $\nu=1/4$ but only emerge in a modest external magnetic
field (as a matter of terminology, we only refer to states directly connected
to $B=0$ as ‘generalized AHC’ states, otherwise we use the more general ‘TCDW’
nomenclature). We also see weak signatures of high-field translational
symmetry breaking tracing to $\nu=3/4$, but without any associated gapped TCDW
states (see Landau fans in Extended Data Fig. 5c,d). The state projecting to
$\nu=2/3$ most likely corresponds to a filling of the lowest two folded
subbands of a TCDW state with a tripled moiré unit cell area. The states
projecting to $\nu=1/2$ and $3/2$ could correspond either to filling the
lowest of two folded subbands of a TCDW state that doubles the moiré unit
cell, or to the filling of multiple subbands of a quadrupled unit cell state.
Spatially resolved measurements will be required to distinguish these
possibilities.

###  .5 Chern number control of the generalized AHC

We now return to the nature of the sign reversal of the Chern number of the
generalized AHC at $\nu=1/4$. In addition to the first-order flip with
$B_{\perp}$, first seen in Fig. 2, we find that an in-plane magnetic field,
$B_{\parallel}$, can drive a similar transition. Repeating the scans from Fig.
2a with an additional $B_{\parallel}$ applied, the transition from the $C=1$
to $C=-1$ state shifts to lower $B_{\perp}$ for larger $B_{\parallel}$ (Fig.
4a,b), until $C^{-1}_{1/4}$ becomes the ground state at $B_{\perp}=0$ for
sufficiently large $B_{\parallel}$ (right-most panel of Fig. 4a). This leads
to a reversal in the orientation of the primary $\rho_{xy}$ hysteresis loop
centered around $B_{\perp}=0$, as seen in the high-$B_{\parallel}$ traces of
Fig. 4c. For intermediate values of $B_{\parallel}$, secondary hysteresis
loops are visible around $\pm B_{\text{PT}}$, corresponding to the flip of the
Chern number in the transition between $C^{+1}_{1/4}$ and $C^{-1}_{1/4}$ . The
primary and secondary hysteresis loops merge as $B_{\parallel}$ is raised
above a crossover value. A quantitative analysis of the $C^{+1}_{1/4}$ to
$C^{-1}_{1/4}$ crossover indicates that this transition occurs when the total
magnetic field, $B_{\text{tot}}\equiv\sqrt{B_{\perp}^{2}+B_{\parallel}^{2}}$,
exceeds $\approx 0.71$ T (Fig. 4d). Additionally, we find that $B_{\parallel}$
suppresses the $C^{-1}_{1/3}$ state in the narrow ranges of $B_{\perp}$ where
it appears (Extended Data Fig. 8).

At finite magnetic field, the reversal of the Chern state can also be driven
electrically, as first indicated in Fig. 2c. The purple and yellow curves in
Fig. 4e show a measurement of $\rho_{xy}$ as $D$ is swept back and forth
across the region of the generalized AHC state, sitting at $B_{\perp}=-0.05$ T
and $B_{\parallel}=0.35$ T. The hysteresis loop is now actuated by gate
voltages, with quantization to $\pm h/e^{2}$ depending on the sweeping
direction of $D$. Extended Data Fig. 9 shows that this electrical switching is
highly robust and repeatable, arising from the apparent lower energy of
$C^{-1}_{1/4}$ compared to $C^{+1}_{1/4}$ at smaller values of $D$.

###  .6 Discussion

Taken together, our results reveal that topological states driven by
simultaneous time-reversal and translational symmetry breaking are ubiquitous
in twisted bilayer–trilayer graphene. In this context, the generalized AHC is
simply the most robust of these states, persisting all the way down to zero
magnetic field. Despite their clear experimental signatures, many fundamental
questions remain about the underlying nature of these interaction-induced
topological states. One is why this particular moiré lattice acts a favorable
platform for hosting zero-field electron-crystal states, where the wide array
of other moiré structures, under intense investigation by groups around the
world, apparently do not. A closely related system is twisted
monolayer–bilayer graphene, which has been reported to exhibit incipient
integer QAH effects at $\nu=3/2$ and $7/2$. These effects indicate an
interaction-induced doubling of the moiré unit cell polshyn2022topological ,
and perhaps the formation of a generalized AHC. However, the Chern number of
the density-wave states ($C=1$) in that system is half that of the states at
integer band filling ($C=2$ at $\nu=1$ and $3$), and thus does not clearly
demonstrate the interaction-induced modification of Berry curvature that is
one hallmark of the AHC.

The nature of the first-order phase transition separating the $C=+1$ and $-1$
states also remains mysterious. One possibility is that this phase transition
represents a recondensation of carriers from one valley of the graphene
Brillouin zone into the other. Such a scenario has been previously suggested
to explain the charge-doping–induced reversal of the sign of $C$ observed in
orbital magnetic states at integer $\nu$ in magic-angle twisted bilayer
graphene and twisted monolayer-bilayer graphene zhu2020voltage ;
polshyn2020electrical ; grover2022chern . This effect is understood to result
from a flip in the sign of the total magnetization of the state upon doping
across the topological gap, reversing the valley corresponding to the ground
state. However, the simplest version of this scenario is not consistent with
our data. First, we do not find a first-order reversal in $C$ upon doping
across the $\nu=1/4$ state. More generally, a cursory overview of the data in
Fig. 2 indicates that the $C=1$ and $-1$ phases across the finite-field
transition are not a time-reversed pair. In contrast with the near-perfect
(anti-)symmetry of the transition across $B=0$, the states above and below the
$B_{\perp}=0.1-0.3$ T transition are clearly different, with $C^{-1}_{1/4}$
much narrower than $C^{+1}_{1/4}$ across the boundary, and no $C^{+1}_{1/3}$
at the equivalent location to $C^{-1}_{1/3}$ despite the fact that the
transition passes diagonally through $C^{-1}_{1/3}$ .

An alternative possibility is that the two states represent qualitatively
different electronic orders, whether within the same valley or in opposite
ones. There are many degrees of freedom that could vary between these two
orders, including both isospin textures and the real-space geometry of the
electronic crystal. Given the sensitivity of the first-order phase transition
to $B_{\parallel}$, which typically does not couple strongly to orbital
degrees of freedom in 2D materials, it is possible that spin physics might
play a role in distinguishing the two states. Extended Data Fig. 10 shows
schematic energy diagrams of a possible two-state model consistent with our
observations, agnostic to the specific ordering of each state but with the
constraint that the two states have opposite Chern numbers for a given sign of
their total magnetization in order to explain the flip between $C^{-1}_{1/4}$
and $C^{+1}_{1/4}$ across $B_{\text{PT}}$.

Finally, we discuss possible future directions enabled by our work. First,
scanning probe measurements to directly visualize the geometry of the
generalized AHC will be critical for unraveling its precise nature. Second,
recent theories have predicted that fractional quantum anomalous Hall (FQAH)
states can emerge upon partially filling a generalized AHC band
zhou2023fractional ; dong2023anomalous , which host emergent quasiparticles
with fractional electric charge and anyonic quantum exchange statistics.
Indeed, the generalized AHC has been proposed as an explanation for the
integer QAH insulator seen at one electron per moiré unit cell ($\nu=1$) in
rhombohedral pentalayer graphene aligned with hBN zhou2023fractional ;
dong2023anomalous ; tan2024parent ; soejima2024anomalous ; dong2024stability ,
out of which FQAH states emerge at $\nu<1$ lu2024fractional . However, in
pentalayer graphene, the state at $\nu=1$ does not need to break any
translation symmetry beyond that of the hBN–graphene moiré lattice, making it
nearly impossible to distinguish from a standard Chern insulator. The
generalized AHC in twisted bilayer–trilayer graphene, which corresponds to a
spontaneous enlargement of the moiré unit cell, provides a more direct
opportunity to investigate the possible relationship between topological
electron crystals and the FQAH effect. In particular, further studies of this
system can help to clarify whether FQAH daughter states can emerge upon doping
the generalized AHC, or if the two topological many-body orders directly
compete. Finally, future experiments based on more complex device geometries —
for example, efforts aiming to proximitize the integer (or potential
fractional) QAH states with a superconductor to create non-Abelian Majorana
(or parafermion) modes — will benefit enormously from the experimental
simplicity of the twisted bilayer–trilayer structure, which only contains the
favored Bernal stacking configuration rather than the metastable rhombohedral
order.

##  I Methods

Device fabrication. Mechanically exfoliated graphene flakes with connected
bilayer and trilayer regions were first identified using optical microscopy.
The bilayer and trilayer regions were then separated using polymer-free anodic
oxidation nanolithography Li2018 . The vdW heterostructure was sequentially
assembled using a polycarbonate (PC)/polydimethylsiloxane (PDMS) stamp
wang2013one  in the following order: graphite, hBN, bilayer graphene, trilayer
graphene, hBN, graphite. The graphene flakes were rotationally misaligned by
$\theta\approx 1.5^{\circ}$ by rotating the stage after picking up the bilayer
graphene. In principle, this should create a twisted AB-ABA stacking geometry.
However, we note that AB-BA stacking faults are known to exist in bilayer
graphene flakes Ju2015  and cannot be detected optically. Therefore, it is
possible that the device is in a twisted BA-ABA stacking geometry, rather than
the assumed AB-ABA configuration. Future work will be needed to resolve
whether such a difference in stacking configuration is experimentally
meaningful.

After assembly, the completed vdW stack is dropped onto a Si/SiO2 wafer. We
used standard electron beam lithography and CHF3/O2 plasma etching to define
vdW stacks into a Hall bar geometry and standard metal deposition techniques
(Cr/Au) wang2013one  to make electrical contact.

Transport measurements. The transport measurements were carried out across two
thermal cycles, first in a Bluefors LD dilution refrigerator equipped with a
3-axis superconducting vector magnet and then in a Bluefors XLD dilution
refrigerator with a one-axis superconducting magnet. In both systems, the
nominal base mixing chamber temperature was $T=10$ mK, as measured by a
factory-supplied RuOx sensor. Unless otherwise specified, measurements were
carried out at $T=100$ mK. Four-terminal lock-in measurements were performed
by sourcing a small alternating current of $I<1.5$ nA at a frequency $<20$ Hz,
chosen to accurately capture sensitive transport features while minimizing
electronic noise. In addition, a global bottom gate voltage of between $3.0$ V
and $55.5$ V was applied to the Si substrate to improve the contact
resistance.

The charge carrier density, $n$, and the out-of-plane electric displacement
field, $D$, were defined according to
$n=\left(C_{\text{bg}}V_{\text{bg}}+C_{\text{tg}}V_{\text{tg}}\right)/e$ and
$D=\left(C_{\text{t
g}}V_{\text{tg}}-C_{\text{bg}}V_{\text{bg}}\right)/2\epsilon_{0}$, where
$C_{\text{tg}}$ and $C_{\text{bg}}$ are the top and bottom gate capacitance
per unit area, $e$ is the elementary charge and $\epsilon_{0}$ is the vacuum
permittivity. $C_{\rm{tg}}$ and $C_{\rm{bg}}$ were estimated using the slope
of the gate-voltage–dependent Hall density, $n_{H}=1/(eR_{H})$, where
$R_{H}=\rho_{xy}/B$ is the antisymmetrized Hall coefficient. The carrier
density, $n_{s}$, required to fully fill a four-fold degenerate moiré band was
estimated using the same method. Using $n_{s}$, the filling factor $\nu$ was
defined according to $\nu=n/(n_{s}/4)$. A twist angle of $\theta=1.50^{\circ}$
was estimated from the doping density corresponding to adding four electrons
per moiré unit cell $n_{s}$ = 8$\theta^{2}/\sqrt{3}a^{2}$, where $a$ = 0.246
nm is the lattice constant of graphene.

To reduce geometric mixing between measured longitudinal and transverse
voltage when $B_{\perp}>0$, we plot the field-symmetrized values of
resistance. $\rho_{xx}$ was symmetrized according to
$\left(\rho_{xx}(B_{\perp}>0)+\rho_{xx}(B_{\perp}<0)\right)/2$, and
$\rho_{xy}$ was antisymmetrized according to
$\left(\rho_{xy}(B_{\perp}>0)-\rho_{xy}(B_{\perp}<0)\right)/2$.

The sample had multiple contacts that could be used as current or voltage
probes; these are labelled in the schematic in Extended Data Fig. 1, with
differences highlighted in Extended Data Fig. 3. Extended Data Fig. 1
characterizes the sample homogeneity with maps of $\rho_{xx}$ obtained from
several adjacent voltage probes. The maps are nearly identical over the
accessible range of gate voltages, providing evidence for a highly homogeneous
moiré lattice and minimal twist angle disorder. Except where noted below,
$\rho_{xx}$ measurements used contacts 1 and 3 and $\rho_{xy}$ measurements
used contacts 2 and 4. However, in the following figure panels $\rho_{xx}$ and
$\rho_{xy}$ measurements were based on the diagonal contact pair 1 and 4 after
field symmetrization/antisymmetrization: Fig. 1h; Fig. 2a,c,d; Fig. 3a (left,
$\rho_{xy}$); Fig. 4a; Extended Data Fig. 5. Extended Data Fig. 8c used
contact pair 2-3 for the $\rho_{xx}$ measurement. The source and drain
contacts are (B,C) throughout.

Band structure calculation. We use the Bistritzer-MacDonald continuum model to
calculate the band structure. The Hamiltonian for each valley and spin is:

$H=\begin{pmatrix}H_{AB}&H_{M}^{\dagger}\\\ H_{M}&H_{ABA}\end{pmatrix},$ (1)

where $H_{AB}$ and $H_{ABA}$ are the Hamiltonian of the bilayer and the
trilayer respectively. $H_{M}$ represents the moiré tunneling between the
bilayer and the trilayer. The bilayer Hamiltonian $H_{AB}$ is:

$\displaystyle
H_{AB}=\begin{pmatrix}-\frac{\delta}{2}&\gamma_{0}\pi&\gamma_{4}\pi&\gamma_{3}\pi^{\dagger}\\\
\gamma_{0}\pi^{\dagger}&-\frac{\delta}{2}&\gamma_{1}&\gamma_{4}\pi\\\
\gamma_{4}\pi^{\dagger}&\gamma_{1}&-\frac{\delta}{4}&\gamma_{0}\pi\\\
\gamma_{3}\pi&\gamma_{4}\pi^{\dagger}&\gamma_{0}\pi&-\frac{\delta}{4}\end{pmatrix},$
(2)

and the trilayer Hamiltonian $H_{ABA}$ is:

$\displaystyle
H_{ABA}=\begin{pmatrix}0&\gamma_{0}\pi&\gamma_{4}\pi&\gamma_{3}\pi^{\dagger}&\frac{\gamma_{2}}{2}&0\\\
\gamma_{0}\pi^{\dagger}&0&\gamma_{1}&\gamma_{4}\pi&0&\frac{\gamma_{5}}{2}\\\
\gamma_{4}\pi^{\dagger}&\gamma_{1}&\frac{\delta}{4}&\gamma_{0}\pi&\gamma_{4}\pi^{\dagger}&\gamma_{1}\\\
\gamma_{3}\pi&\gamma_{4}\pi^{\dagger}&\gamma_{0}\pi^{\dagger}&\frac{\delta}{4}&\gamma_{3}\pi&\gamma_{4}\pi^{\dagger}\\\
\frac{\gamma_{2}}{2}&0&\gamma_{4}\pi&\gamma_{3}\pi^{\dagger}&\frac{\delta}{2}&\gamma_{0}\pi\\\
0&\frac{\gamma_{5}}{2}&\gamma_{1}&\gamma_{4}\pi&\gamma_{0}\pi^{\dagger}&\frac{\delta}{2}\end{pmatrix},$
(3)

where $\pi=(k_{x}-\mathrm{i}k_{y})e^{\pm\mathrm{i}\frac{\theta}{2}}$, with
$\theta$ being the twist angle. The $\pm$ sign corresponds to the
bilayer/trilayer respectively. $\delta$ is the potential difference between
the top and bottom graphene layers. The hopping parameters are
$(\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5})=(2610\frac{\sqrt{3}}{2},361,-20,-283\frac{\sqrt{3}}{2},-140\frac{\sqrt{3}}{2},20)~{}\mathrm{meV}$.
The moiré tunneling is:

$\displaystyle H_{M}=t_{M}\begin{pmatrix}\alpha&e^{-\mathrm{i}\frac{2\pi
j}{3}}\\\ e^{\mathrm{i}\frac{2\pi j}{3}}&\alpha\end{pmatrix},$ (4)

where $j=0,1,2$ represents the tunneling between the bottom layer of the
bilayer and the top layer of the trilayer, with momentum differences given by
$\mathbf{Q_{j}}$. The vectors
$\mathbf{Q_{0}}=(0,0)^{T},\mathbf{Q_{1}}=(-\frac{2\pi}{\sqrt{3}a_{M}},-\frac{2\pi}{a_{M}})^{T},\mathbf{Q_{2}}=(\frac{2\pi}{\sqrt{3}a_{M}},-\frac{2\pi}{a_{M}})^{T}$.
We use $t_{M}=110~{}\mathrm{meV}$ and $\alpha=0.3$ in our calculation.

Hartree–Fock calculations. We performed the Hartree–Fock calculation in the
reduced Brillouin zone at $\nu=1/4$. The Coulomb interaction term is given by:

$\displaystyle
H_{\mathrm{int}}=\frac{1}{2A}\sum_{\mathbf{q}}\sum_{l,l^{\prime}}V_{l;l^{\prime}}(\mathbf{q}):\rho(\mathbf{q})\rho(-\mathbf{q}):,$
(5)

where $\rho_{l}(\mathbf{q})$ is the density operator at layer $l$, and $A$ is
the area of system. The interaction potential is defined as:

$V_{l,l^{\prime}}(\mathbf{q})=\frac{e^{2}e^{-q|l-l^{\prime}|d_{\mathrm{layer}}}\tanh{(q\lambda)}}{2\epsilon\epsilon_{0}|q|},$
(6)

where $d_{\mathrm{layer}}=0.34~{}\mathrm{nm}$ is the distance between adjacent
layers, $\epsilon=6$ is the dielectric constant and $\lambda=30~{}\mathrm{nm}$
is the screening length. We include 4 conduction bands in the reduced BZ at
single-particle level and perform the self-consistent calculation. We test
$100$ random initial ansatzes and select the one with the lowest energy.

Spatial charge distribution calculation. The spatial distribution
$n(\mathbf{r})$ in Fig. 1(b) is defined as $\langle
c^{\dagger}(\mathbf{r})c(\mathbf{r)\rangle}$, where $c(\mathbf{r})$ is
electron annihilation operator in real-space. It can be calculated as:

$\displaystyle n(\mathbf{r})=\sum_{\mathbf{k}}\sum_{\mathbf{q}}\langle
u(\mathbf{k+q})\lvert u({\mathbf{k}})\rangle e^{-\mathrm{i}\mathbf{q\cdot
r}},$ (7)

where $\mathbf{k}$ is defined in the BZ, $\lvert u(\mathbf{k+q})\rangle$ is
the periodic part of the Bloch wavefunction.
$\mathbf{q}=n_{1}\mathbf{G_{1}}+n_{2}\mathbf{G_{2}}$ is any integer linear
combination of reciprocal lattice vectors. For the metallic state, we use the
single-particle Bloch wavefunction and sum over $\mathbf{k}$ below the Fermi
energy at $\nu=1/4$. The reciprocal lattice vectors are moiré reciprocal
lattice vectors. For the generalized AHC state, we use the Hartree–Fock Bloch
wavefunction and sum over $\mathbf{k}$ in the reduced moiré BZ. The first
conduction band is fully filled at $\nu=1/4$. The reciprocal lattice vectors
are half the moiré reciprocal lattice vectors.

Berry curvature and Chern number calculation. We use the method from Ref.
fukui2005chern,  to calculate Berry curvature. The BZ is discretized into an
$N_{1}\times N_{2}$ grid. For each $\mathbf{k}$ in the BZ, we calculate the
$U(1)$ link:

$\displaystyle U_{\mu}(\mathbf{k})=\frac{\langle u(\mathbf{k}+\delta
k_{\mu}\hat{k}_{\mu})\lvert u(\mathbf{k})\rangle}{N_{\mu}(\mathbf{k})},$ (8)

where $N_{\mu}(\mathbf{k})=|\langle u(\mathbf{k}+\delta
k_{\mu}\hat{k}_{\mu})\lvert u(\mathbf{k})\rangle|$ and $\mu=1,2$. The Berry
curvature is then given by:

$\displaystyle\Omega(\mathbf{k})=-\frac{\mathrm{i}\ln W(\mathbf{k})}{\delta
k_{1}\delta k_{2}},$ (9)

where $W(\mathbf{k})$ is $U_{1}(\mathbf{k})U_{2}(\mathbf{k}+\delta
k_{1}\hat{k}_{1})U^{-1}_{1}(\mathbf{k}+\delta
k_{2}\hat{k}_{2})U_{2}^{-1}(\mathbf{k})$. We ensure that $-\pi<-\mathrm{i}\ln
W(\mathbf{k})\leq\pi$. For the metallic state, the BZ is the moiré BZ. For the
generalized AHC state at $\nu=1/4$, the BZ is the reduced moiré BZ. The Chern
number is calculated as:

$\displaystyle
C=\frac{1}{2\pi}\int_{\mathrm{BZ}}d^{2}\mathbf{k}\Omega(\mathbf{k}).$ (10)

Extended Data Fig. 11 shows the calculated Chern number of the generalized AHC
state at $\nu=1/4$ as various parameters ($\theta$, $\delta$, $\alpha$, and
$\epsilon$) are tuned. Our Hartree–Fock calculations predict a $C=1$ state for
realistic parameters, consistent with our experiment. However, we caution that
we do not know the precise values of any of these parameters experimentally
except for $\theta$, and the calculations also find other Chern number ground
states within relatively narrow parameter ranges. Furthermore, Hartree–Fock
predicts a generalized AHC state over a substantially wider range of $\delta$
than is observed experimentally; this is consistent with the tendency of
Hartree–Fock calculations to underestimate the formation energy of symmetry-
broken states. Despite these uncertainties, the $C=1$ states found
experimentally and theoretically motivate a plausible connection between the
two. The sensitivity of the Chern number in Hartree–Fock calculations to the
dielectric screening, $\epsilon$, highlights the importance of interactions in
determining the topological properties of the electronic crystal.

The valley Chern number of the single-particle moiré conduction band is
calculated to be $+2$. Based on this, a spin-valley–polarized state at
$\nu=-1$ is expected to have $|C|=2$ since it corresponds to a single filled
valley. Because the state at $\nu=1$ in our device is metallic, it is not
possible to extract its Chern number. However, the vanishing AHE at $\nu=1$
(Extended Data Fig. 4) is inconsistent with a gapless but valley-imbalanced
$|C|=2$ state. Our observation of a $C=0$ correlated insulator at finite field
(Extended Data Fig. 5) is also inconsistent with the prediction from the
single-particle band structure calculation, although the Chern number of a
correlated state at finite $B_{\perp}$ need not be the same as its value at
$B_{\perp}=0$. Future work will be necessary to identify the origin of this
apparent inconsistency. In any case, the Chern number of $1$ for the $\nu=1/4$
state is not equal to one-quarter of either of the possible Chern numbers of
the parent state at $\nu=1$ (i.e., $0$ or $2$). Thus, the ambiguity in
identifying the latter does not affect our primary conclusion that a
combination of band folding and interactions determine the Chern number of the
generalized AHC state.

##  II Acknowledgements

The authors thank Liang Fu, Leon Balents, Marcel Franz, Michael Zaletel, and
Andrea Young for helpful discussions. Experiments at UBC were undertaken with
support from the Natural Sciences and Engineering Research Council of Canada;
the Canada Foundation for Innovation; the Canadian Institute for Advanced
Research; the Max Planck-UBC-UTokyo Centre for Quantum Materials and the
Canada First Research Excellence Fund, Quantum Materials and Future
Technologies Program; and the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program, Grant Agreement
No. 951541. Work at UW was supported by National Science Foundation (NSF)
CAREER award no. DMR-2041972 and NSF MRSEC 1719797. The development of twisted
graphene samples was partially supported by the Department of Energy, Basic
Energy Science Programs under award DE-SC0023062. D.W. was supported by an
appointment to the Intelligence Community Postdoctoral Research Fellowship
Program at University of Washington administered by Oak Ridge Institute for
Science and Education through an interagency agreement between the US
Department of Energy and the Office of the Director of National Intelligence.
K.W. and T.T. acknowledge support from the Elemental Strategy Initiative
conducted by the MEXT, Japan (grant no. JPMXP0112101001) and JSPS KAKENHI
(grant nos. 19H05790, 20H00354 and 21H05233). Y.-H.Z. was supported by the
National Science Foundation under Grant No. DMR-2237031. This work made use of
shared fabrication facilities at UW provided by NSF MRSEC 1719797.

##  III Author Contributions

R.S. performed the measurements in the Folk lab at UBC, and analyzed the data;
D.W. made the sample and performed initial transport measurements in the
Yankowitz lab at UW; M.Y. and J.F. supervised the measurements; R.S., M.Y. and
J.F. wrote the manuscript with B.Z. and Y.Z. providing theory support; K.W.
and T.T. provided the hBN crystals.

## Competing interests

The authors declare no competing interests.

## Additional Information

Correspondence and requests for materials should be addressed to Matthew
Yankowitz or Joshua Folk.

## Data Availability

Source data are available for this paper. All other data that support the
findings of this study are available from the corresponding author upon
request.

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## Extended Data

Extended Data Fig. 1:  Characterization of twist angle homogeneity. a, Line-
cut of $\rho_{xx}(n)$ for three adjacent sets of voltage probes along the Hall
bar, at fixed $D=0$ and $T=4.2$ K. Inset: optical micrograph of the device.
Electrical contacts are labeled A-C and 1-4. The scale bar is 5 $\mu$m. b, c,
d, Gate voltage dependence of $\rho_{xx}$ measured from contact pairs B-1,
1-2, and 2-3, respectively. The dashed line shows the contour of $D=0$, where
line-cuts in a were obtained. The close alignment of $\rho_{xx}$ features
demonstrates the high degree of twist angle homogeneity across the length of
the sample. The source and drain contacts used for b were A and C. All other
measurements used B and C as source and drain contacts.

Extended Data Fig. 2:  Temperature dependence of the $\nu=1/4$ state. a,
Deviation of the Hall plateau from $h/e^{2}$ quantization, as a function of
displacement field and temperature, obtained with a small $B_{\perp}=-50$ mT.
Since $\rho_{xy}<0$, $\delta\rho_{xy}$ is shown as $h/e^{2}+\rho_{xy}$ for
clarity. Numerical subscripts (1-4 in this case) indicate the contact pairs
used for the measurement, as labeled in the inset of Extended Data Fig. 1a. b,
$\rho_{xx}$ measured in the same way as a. c, Percent deviation of the Hall
resistivity,
$\overline{\delta\rho_{xy}}=100\times\delta\rho_{xy}\times(h/e^{2})^{-1}$. d,
Longitudinal resistivity averaged using data points between $D=526$ and $533$
mV nm-1, $\overline{\rho_{xx}}$. Error bars represent one standard deviation.
e, f, $-\rho_{xy}(T)$ and $\rho_{xx}(T)$, respectively, at fixed $D=532$ mV
nm-1 and $B_{\perp}=-50$ mT.

Extended Data Fig. 3:  Filling factor and temperature dependence of the
$\nu=1/4$ generalize AHC state. a, Temperature dependence of $\rho_{xx,1-3}$
as $\nu$ is tuned across the generalized anomalous Hall crystal state, at
fixed $D=531$ mV nm-1 with a small magnetic field of $B_{\perp}=-50$ mT
applied. b, Line-cuts from a at selected values of $T$ between 0.65 K and 10
mK. c-f, Measurements obtained under the same conditions as a-b but for c-d
$\rho_{xy,2-4}$ and e-f $\rho_{1-4}$ (corresponding to a mixture of
longitudinal and Hall geometries since contacts 1 and 4 are separated by
$\approx 1$ $\mu$m along the length of the channel and located by opposite
sides of the Hall bar). Contact 2 fails severely for $\nu\lessapprox 0.15$,
leading to extremely large and rapidly oscillating resistance. Similar contact
issues in the generalized AHC state at $\nu=1/4$ likely also explain the large
fluctuating resistance of $\rho_{xy,2-4}$ near base temperature. $\rho_{1-4}$
exhibits a mixture of $\rho_{xx}$-like and $\rho_{xy}$-like properties when
probing metallic states but faithfully reflects the quantization of the
anomalous Hall plateau in the topological gap ($\nu\approx 0.25$) since the
$\rho_{xx}$ contribution vanishes.

Extended Data Fig. 4:  Doping dependence of the anomalous Hall effect. a,
Antisymmetrized $\rho_{xy}(B_{\perp})$ hysteresis loops at increasing values
of $\nu$ away from $\nu=1/4$. Data obtained at fixed $D=532$ mV nm-1 and
$T=300$ mK. The solid/dashed traces correspond to forward ($\uparrow$) and
reverse ($\downarrow$) scans of $B_{\perp}$, respectively. b, Map of
$\Delta\rho_{xy}/2=(\rho_{xy}(B_{\uparrow})-\rho_{xy}(B_{\downarrow}))/2$,
showing the AHE over a broad range of $\nu$. The color scale is saturated to
show the weakest AHE features near $\nu=1$. c, Line-cut of $\Delta\rho_{xy}/2$
at $B_{\perp}=0$, showing that the small anomalous Hall effect vanishes near
$\nu=1$.

Extended Data Fig. 5:  Additional Landau fans at fixed $D$. a, b, Landau fans
of antisymmetrized $\rho_{xy}$ and symmetrized $\rho_{xx}$, respectively,
obtained at $D=570$ mV nm-1. Dashed lines correspond to the trajectories of
states with Chern number $C=+1$ as described by the Streda formula, tracing
back to $\nu=1/2$, $2/3$ and $3/2$ at $B_{\perp}=0$. Deviations in both
$\rho_{xy}$ and $\rho_{xx}$ along these trajectories indicate the formation of
TCDW states. Additionally, both electron- and hole-like quantum oscillations
emerge from the correlated insulators at $\nu=1$ and $\nu=2$ and develop into
quantum Hall states at high field. At fixed $\nu=1$ and $2$, $\rho_{xx}$ is
diverging whereas $\rho_{xy}=0$ and abruptly changes sign with $\nu$.
Together, these features indicate that the correlated insulating states have
$C=0$. Notably, the $\nu=1$ state only becomes insulating in a small
$B_{\perp}$ and is metallic at $B_{\perp}\approx 0$. This metal-to-insulator
transition with $B_{\perp}$ is seen irrespective of the precise value of $D$.
c, d, Similar Landau fans obtained at $D=531$ mV nm-1. Dashed lines correspond
to the trajectories of states with Chern numbers $C=+1$ and $+2$ tracing back
to $\nu=3/4$ at $B_{\perp}=0$. There are weak deviations in both $\rho_{xy}$
and $\rho_{xx}$ along these trajectories, suggesting the possibility of
gapless TCDW states with a four-fold enlargement of the moiré unit cell.

Extended Data Fig. 6:  First-order phase transition and additional
characterization of the $\nu=1/4$ and $1/3$ states. a, b, Landau fans of
antisymmetrized $\rho_{xy}$ and symmetrized $\rho_{xx}$ obtained at fixed
$D=527$ mV nm-1, respectively. (Which is larger than the $D=526$ mV nm-1 fans
shown in Fig. 2d of the main text.) Panel c is overlaid with dashed/dotted
lines, indicating $\nu=0.322$ and $B_{\perp}=0.5$ T. Panel d is overlaid with
dashed lines that show the expected trajectories of $|C|=1$ states originating
from $\nu=1/4$ and $\nu=1/3$. c, $\rho_{xy}$ for forward and reverse scans of
$\nu$, at fixed $B_{\perp}=0.5$ T and $D=526$ mV nm-1, forming a hysteresis
loop as $\nu$ is swept past the phase boundary. d, Antisymmetrized $\rho_{xy}$
measured as $B_{\perp}$ is swept back and forth at fixed $\nu=0.322$, $D=526$
mV nm-1. There is hysteresis when the sample enters/exits the TCDW state,
consistent with a first-order phase transition.

Extended Data Fig. 7:  Gate–dependent resistance maps. a, b, Antisymmetrized
$\rho_{xy}$ and symmetrized $\rho_{xx}$ maps, respectively, obtained at
$B_{\perp}=2.5$ T and plotted against $V_{\rm{tg}}$ and $V_{\rm{bg}}$. The
horizontal dashed lines mark fixed $V_{\rm{bg}}$ $=4.95$ V. The arrows show
the directions of fixed $n$ and $D$, related to $V_{\rm{tg}}$ and
$V_{\rm{bg}}$ via the linear transformation described in the Methods. The
dashed lines separate a region of normal metal with no broken isospin
degeneracies at large negative $V_{\rm{bg}}$ from a region of full isospin
degeneracy lifting (sometimes coexisting with translational symmetry breaking)
at smaller values of $V_{\rm{bg}}$.

Extended Data Fig. 8:  Effect of an in-plane magnetic field on the states at
$\nu=1/4$ and $1/3$. a, b, $\rho_{xx}$ and $\rho_{xy}$ maps, respectively,
obtained at several values of $B_{\parallel}$ with fixed $B_{\perp}=0.5$ T,
$T=10$ mK. c, Maps of $\rho_{xx}$ as a function of $\nu$ and $B_{\parallel}$
measured at $D=527$ mV nm-1 (denoted by the red dashed lines in a). An in-
plane magnetic field stabilizes the $C^{-1}_{1/4}$ state, whereas the
$C^{-1}_{1/3}$ state is suppressed. d, Measurements of $-\rho_{xy}$ and
$\rho_{xx}$ versus $B_{\parallel}$ at fixed $\nu=0.319$, obtained under the
same conditions as c. Measurements are obtained at $T=10$ mK, without
(anti)symmetrization.

Extended Data Fig. 9:  Time-series response of the Hall resistance with
$B_{\perp}$ and $D$ sweeps. a, $\rho_{xy}$ (bottom) recorded at $\nu=0.251$ in
response to a sequence of forward and reverse sweeps of $B_{\perp}$ (top) and
$D$ (middle), with a fixed $B_{\parallel}=0.35$ T. The sign of $\rho_{xy}$
switches abruptly to align with the sign of $B_{\perp}$, consistent with a
$C=+1$ state. Excursions of $D$ about $531$ mV nm-1 can also switch the sign
of $\rho_{xy}$ (obtained with fixed $B_{\perp}=-50$ mT). b, c, Representative
traces of $\rho_{xx}$ and $\rho_{xy}$ measured as $D$ is swept back and forth
with $B_{\perp}=-50$ mT. d-f, Analogous measurement to panels a-c, acquired
with $B_{\parallel}=0$. In this case, excursions of $D$ about $531$ mV nm-1 do
not switch the sign of $\rho_{xy}$. g-i, Analogous measurements with
$B_{\parallel}=2$ T. In contrast to a-f, the sign of $\rho_{xy}$ in response
to $B_{\perp}$ indicates a $C=-1$ state. Excursions of $D$ about $531$ mV nm-1
also do not change the sign of $\rho_{xy}$. All data was obtained at $T=300$
mK.

Extended Data Fig. 10: Possible two-state model explaining the Chern number
reversal of the generalized AHC state at $\nu=1/4$. Schematic diagrams showing
a possible mechanism for the reversal in the sign of $C$ across
$B_{\text{PT}}$, as analyzed in Fig. 2 of the main text, as well as its
dependence on $B_{\parallel}$ and $D$ as analyzed in Fig. 4. The diagram on
the left posits the existence of two closely competing generalized AHC states
at $B_{\perp}=B_{\parallel}=0$, denoted by the blue and yellow markers. The
blue circle represents the ground state since it has the lowest energy.

The model is agnostic to the precise details of the two different states, but
assumes that the magnetization of the state marked by the yellow square is
larger than that of the state marked by the blue dot, and also that the
direction of magnetization for a given sign of Chern number is opposite for
the two states. The second assumption is plausible because there is, in
general, no fixed relationship between the sign of $C$ and the sign of the
total magnetization for orbital magnetic states. The energies of the $C=+1$
and $-1$ branches of each state evolve oppositely in an applied out-of-plane
field, and their magnetization is proportional to the slope of these lines.
Above a certain value of $B_{\perp}$, one branch of the state with larger
magnetization (yellow square) has a lower energy than either branch of the
original ground state (blue circle). The crossing points are denoted as
“$\pm$PT”, indicating the value of $B_{\perp}$ at which a first-order phase
transition between the two states is anticipated. For each state, the sign of
the Chern number of the branch with lower energy in an applied $B_{\perp}$
corresponds to which has its total magnetization aligned with the external
field.

These are a minimal set of assumptions needed to explain the reversal in the
sign of the Chern number across $B_{\text{PT}}$. In this model, the additional
reversal with $B_{\parallel}$ could result from its coupling to the energetic
hierarchy of the two states at fixed $B_{\perp}$. For instance,
$B_{\parallel}$ could lower the energy of the state marked by the yellow
square to a value lower than that of the state marked by the blue circle, thus
making the former the ground state. A similar ground state reversal may also
result from changing $D$, explaining the gate-induced hysteresis at fixed
$B_{\perp}$ and $B_{\parallel}$. Although these models are consistent with all
of our measurements, we cannot rule out alternative explanations not
considered here.

Extended Data Fig. 11: Hartree–Fock calculations of the Chern number of the
generalized AHC state at $\nu=1/4$. a, Calculated Chern number as a function
of twist angle, $\theta$, and potential difference, $\delta$, with fixed
tunneling parameter, $\alpha=0.3$, and dielectric constant, $\epsilon=6$ (see
Methods for definitions of parameters). Parameters that do not yield a gap at
$\nu=1/4$ are uncolored. At the twist angle appropriate for our sample,
$\theta=1.50^{\circ}$, the model predicts a $C=1$ state for $\delta=90$ meV.
We do not attempt to convert $\delta$ directly to the experimental electric
displacement field, $D$, as this would require knowledge of the self-
consistent electrostatics of the five graphene layers. We note that to a first
approximation, the $\delta$ between layers depends on the applied displacement
field $D$ and the spacing between layers, $\delta\approx
4eDd_{\mathrm{layer}}$, where $4d_{\mathrm{layer}}$ is the spacing between the
top of the bilayer and bottom of the trilayer. However, this conversion
between $\delta$ and $D$ is not to be taken to be precise: it will be modified
by any charges that build up on different layers, which we know to be a
significant effect. b, Calculated Chern number as a function of $\alpha$ and
$\epsilon$ with fixed $\theta=1.50^{\circ}$ and $\delta=90$ meV.