# Mapping twist-tuned multi-band topology in bilayer WSe2

Benjamin A. Foutty  Geballe Laboratory for Advanced Materials, Stanford, CA
94305, USA  Department of Physics, Stanford University, Stanford, CA 94305,
USA  Carlos R. Kometter  Geballe Laboratory for Advanced Materials, Stanford,
CA 94305, USA  Department of Physics, Stanford University, Stanford, CA 94305,
USA  Trithep Devakul  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Aidan P. Reddy  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Kenji Watanabe  Research Center for Electronic and Optical
Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba
305-0044, Japan  Takashi Taniguchi  Research Center for Materials
Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  Liang Fu  Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Benjamin E.
Feldman<EMAIL_ADDRESS>Geballe Laboratory for Advanced Materials, Stanford,
CA 94305, USA  Department of Physics, Stanford University, Stanford, CA 94305,
USA  Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA

###### Abstract

Semiconductor moiré superlattices have been shown to host a wide array of
interaction-driven ground states. However, twisted homobilayers have been
difficult to study in the limit of large moiré wavelength, where interactions
are most dominant, and despite numerous predictions of nontrivial topology in
these homobilayers, experimental evidence has remained elusive. Here, we
conduct local electronic compressibility measurements of twisted bilayer WSe2
at small twist angles. We demonstrate multiple topological bands which host a
series of Chern insulators at zero magnetic field near a ‘magic angle’ around
$1.23^{\circ}$. Using a locally applied electric field, we induce a
topological quantum phase transition at one hole per moiré unit cell.
Furthermore, by measuring at a variety of local twist angles, we characterize
how the interacting ground states of the underlying honeycomb superlattice
depend on the size of the moiré unit cell. Our work establishes the
topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe2,
demonstrating a tunable platform for strongly correlated topological phases.

##  I Introduction

The richness of interacting condensed matter systems motivates the development
of experimental platforms with tunable Hamiltonians, which can admit
systematic study over a wide range of phase space bloch_quantum_2012 ;
kennes_moire_2021 . Recently, moiré heterostructures have emerged as a venue
to engineer flat electronic bands and realize emergent correlated states. In
twisted graphene structures, interacting phases generally only form near
certain ‘magic’ angles andrei_graphene_2020 ; balents_superconductivity_2020 ;
cao_correlated_2018 ; cao_unconventional_2018 . In contrast, moiré transition
metal dichalcogenide (TMD) heterostructures exhibit strong correlations across
a wide range of twist angles (and corresponding moiré wavelengths)
mak_semiconductor_2022 . Because the underlying electronic structure can often
be captured by extended Hubbard models, these systems are promising for
quantum simulation with effective parameters tunable by moiré wavelength,
choice of TMD, and applied electromagnetic fields kennes_moire_2021 ;
wu_hubbard_2018 ; zhang_moire_2020 ; tang_simulation_2020 . In practice, the
ability to meaningfully adjust parameters has been limited by the experimental
challenges of poor electrical contact and local disorder
mak_semiconductor_2022 ; lau_reproducibility_2022 . As a result, study of
correlated states in these systems to date has focused mostly on
heterobilayers, where lattice mismatch provides a nearly constant moiré
wavelength tang_simulation_2020 ; xu_correlated_2020 ; regan_mott_2020 ;
li_continuous_2021 ; li_quantum_2021 ; li_charge-order-enhanced_2021 , or on
homobilayers with large interlayer twist, where angular variations have a
muted effect wang_correlated_2020 ; ghiotto_quantum_2021 ; xu_tunable_2022 .
While this has resulted in the discovery of many exotic ground states, the
ability to vary parameters while maintaining strong interactions relative to
electronic bandwidth in a single system has remained limited.

Of particular interest is the ability to engineer tunable topological flat-
band structures, in which interactions can lead to nontrivial insulating
states. In graphene-based moiré superlattices, quantum anomalous Hall (QAH)
insulators have been observed, but this typically requires fine-tuning of
external parameters (in addition to twist angle) serlin_intrinsic_2020 ;
chen_tunable_2020 ; pierce_unconventional_2021 ; polshyn_electrical_2020 ;
stepanov_competing_2021 , and the topological bands do not allow a local
tight-binding description andrei_graphene_2020 ; mak_semiconductor_2022 ;
kane_quantum_2005 . Theoretical calculations have predicted that certain TMD
moiré homobilayers realize a generalized version of the Kane-Mele model in
which the lowest energy pair of bands have nonzero Chern number, and that
these bands could support QAH states at odd filling due to interactions
wu_topological_2019 . However, there has been disagreement in the literature
about the exact parameters for which topological bands are expected
devakul_magic_2021 ; pan_band_2020 . While there has been no experimental
evidence of topology in TMD homobilayers to date, recent experiments in AB-
stacked MoTe2/WSe2 heterobilayers were able to mimic similar physics using a
large out of plane electric field to hybridize the electronic states on the
two layers, and realize QAH and quantum spin hall insulating states at moiré
filling factors $\nu=-1$ and $\nu=-2$, respectively li_quantum_2021 ;
tschirhart_intrinsic_2023 ; zhao_realization_2022 . The precise symmetry-
breaking which underlies these insulating phases is a subject of ongoing
experimental and theoretical studies tao_valley-coherent_2022 ; zhang_spin-
textured_2021 ; xie_valley-polarized_2022 ; pan_topological_2022 ;
devakul_quantum_2022 ; rademaker_spin-orbit_2022 ; dong_excitonic_2023 ;
mai_interaction-driven_2023 ; mai_14_2023 .

Figure 1: Multiple topological bands and ground states in tWSe2. a, Schematic
showing the moiré pattern in twisted WSe2 (tWSe2), in which MX and XM stacking
sites (bottom illustrations) form a honeycomb lattice. b, Cartoon of lowest
energy moiré bands and their occupations at the first four integer hole
fillings, with Chern numbers $C$ of the corresponding ground states labeled.
c, Measurement of inverse electronic compressibility
$\mathrm{d}\mu/\mathrm{d}n$ and chemical potential $\mu(n)$ of tWSe2 as
function of holes per moiré unit cell $\nu$ in a location with twist angle
$\theta=1.23^{\circ}$. d-g, $\mathrm{d}\mu/\mathrm{d}n$ at low magnetic fields
$B$ around each of the first four integer moiré fillings $\nu$. The gaps at
$\nu=-1$ and $-3$ have $C=+1$, while $\nu=-2$ and $-4$ have $C=0$.

In this work, we use scanning single electron transistor (SET) microscopy to
investigate twisted bilayer WSe2 (tWSe2) at much lower relative twist angle
$\theta$ than previously studied. By measuring at different twist angles, we
map how the system depends on local moiré wavelength. In addition to
widespread correlated insulating states, we find ‘magic-angle’ topology in the
vicinity of $\theta\approx 1.23^{\circ}$. Around that twist angle, we observe
a series of QAH insulators, indicating multiple moiré bands with nonzero Chern
numbers. Using an applied displacement field, we drive a topological phase
transition of the $\nu=-1$ insulator between a QAH state and a topologically
trivial layer-polarized state. Together, our results represent systematic
quantum simulation of generalized Kane-Mele-Hubbard physics on a tunable real-
space lattice.

##  II Series of Chern insulators in tWSe2

We study a twisted WSe2 bilayer with a small relative angle
$1.1^{\circ}<\theta<1.6^{\circ}$. At these twist angles, the lowest two moiré
valence bands are well-described by a tight-binding model on a honeycomb
lattice consisting of states localized at the XM and MX sites, shown
schematically in Fig. 1a devakul_magic_2021 . We first focus on an area of the
sample with $\theta=1.23^{\circ}$. Figure 1c shows the chemical potential
$\mu$ and the inverse electronic compressibility d$\mu/$d$n$ as a function of
moiré filling factor $\nu$, where $\nu=-1$ signifies one hole per moiré unit
cell. Gapped states occur at $\nu=-1,-2,-3,-4$, and $-6$, superimposed on an
overall downslope in $\mu$, which reflects long-range electronic interactions
near the valence band edge eisenstein_negative_1992 ; foutty_tunable_2023 .

To determine the nature of the insulating states, we probe the behavior near
integer fillings $|\nu|\leq 4$ at low magnetic field $B$ and temperature
$T=330$ mK (Fig. 1d-g). The slopes of the gaps encode their topological
character (and Hall conductance) via the Streda formula streda_theory_1982 .
At $\nu=-1$ and $-3$, the gaps have slopes corresponding to Chern number
$C=+1$ all the way down to zero magnetic field, indicating that both are QAH
states (Supplementary Sec. 1). At the even integers $\nu=-2$ and $-4$, we
measure gaps with total Chern number $C=0$. Based on the observed nontrivial
topology at odd integer fillings, these gaps are likely to be quantum spin
Hall insulators with either two or four copies of fully filled $|C|=1$ bands
(Fig. 1b). We measure broadly similar behavior at $T=1.6$ K (Extended Data
Fig. 1) and across a wide spatial region (roughly 600 nm $\times$ 1 $\mu$m) in
our sample with uniform twist angle between $1.22^{\circ}$ and $1.23^{\circ}$
(Extended Data Figs. 2,3).

Figure 2: Magnetic field dependence at $\theta=1.23^{\circ}$. a Magnetic field
dependence of d$\mu$/d$n$ up to $B=11$ T. b, Thermodynamic gaps
$\Delta_{(C,s)}$ for states with intercept $s$ and slope $C$ as identified by
the Diophantine equation, as a function of $B$, for the pair of competing
states with intercept $s=-1$. c, $\Delta_{(C,s)}$ as a function of $B$, for
the insulating gaps that persist to zero magnetic field at $s=-1,-2,-3$, and
$-4$. d, Theoretically calculated single-particle Hofstadter spectrum for a
single-spin species as a function of $B$ near a topological band inversion,
with single-particle energy $E$ measured relative to the WSe2 band edge. The
prominent $C=+1$ and $C=0$ gaps with $s=-1$ are labelled. e, Wannier plot of
the density of states of the Hofstadter spectrum in d as a function of $\nu$
and $B$, with $0.5$ meV Gaussian broadening.

To the best of our knowledge, this is the only moiré system reported to have
multiple topological bands that are not related by degenerate quantum degrees
of freedom. Our experiment is most consistent with the pair of lowest energy
moiré bands having identical Chern number in a single spin-valley sector (see
Supplementary Sec. 2 for further discussion). This is beyond any two-band
tight binding model (which must have zero total Chern number), highlighting
that the topological structure in tWSe2 extends beyond the Kane-Mele model.

In Fig. 2a, we present the full magnetic field dependence at this twist angle
up to $B=11$ T. A feature of both Chern insulating states at $\nu=-1$ and
$\nu=-3$ is the coexistence of topologically trivial $C=0$ states emerging at
relatively low magnetic fields $B\approx 0.6$ T, where no other Landau levels
(Hofstadter states) are seen. The thermodynamic charge gap (Methods) of the
trivial state is smaller, and both gaps shrink in moderate magnetic fields
(Fig. 2b). This behavior is consistent with theoretical expectations for Chern
bands close to a topological band inversion (Fig. 2d-e), for which a Landau
level is pinned to the top of the second moiré band, leading to two dominant
Hofstadter gaps originating from $\nu=-1$.

At moderate magnetic fields, multiple additional Hofstadter states emerge.
Each of these can be described by the Diophantine equation
$\nu=C(\Phi/\Phi_{0})+s$, where $C$ is the Chern number of the gap,
$\Phi_{0}=h/e$ is the flux quantum, and $s$ is the intercept. Some features of
the observed Hofstadter pattern are not readily understood from the expected
degeneracies of the bands. For example, the $(C,s)=(+2,-3)$ state is
comparatively weak and is not resolved at high fields, inconsistent with a
series of singly degenerate Hofstadter states emanating from an interaction-
driven spin/valley polarized gap at $\nu=-3$. In addition to the Hofstadter
states, we observe phase transitions as a function of magnetic field (and
density), signified by sharp negative compressibility and changes in the
pattern and/or strength of gaps such as that near $\nu=-3$ at $B=2.6$ T. This
likely reflects changing energetics between distinct moiré bands in a magnetic
field cao_correlated_2018 ; kometter_hofstadter_2022 .

The magnitudes of the dominant gaps emanating from integer fillings exhibit
nontrivial dependence at low magnetic fields (Fig. 2c). All of these gaps are
non-monotonic below $B=1$ T. This is particularly stark at $\nu=-1$ and $-2$,
where the gaps sharply increase up to $B\approx 0.25$ T, before decreasing
beyond that field. One explanation for this gap dependence might be changes in
the spin ordering of the ground states at very low magnetic fields. However,
given the topology of the $\nu=-1$ gap and the theoretically predicted spin-
valley polarization of the QAH state, this is unlikely. Another possibility is
a change in the nature of the lowest energy excitations, such as their spin
configuration, as a field is applied. We note that the non-monotonic
dependence of the gaps at $\nu=-1$ and $\nu=-2$ persists to angles where all
states are topologically trivial (Extended Data Fig. 4).

##  III Twist angle dependence

Measurements at other locations with local angles ranging from $\theta\sim
1.1-1.6^{\circ}$ enable us to systematically investigate how these states
depend on moiré wavelength. We present d$\mu$/d$n$ as a function of $\nu$ up
to $B=4$ T at a selection of different twist angles in Fig. 3a, with the
observed incompressible states schematically depicted in the adjacent Wannier
diagrams. Outside of a small range of angles
($\theta=1.2^{\circ}-1.25^{\circ}$) we do not observe any zero-field Chern
insulators. At $\theta=1.20^{\circ}$, we find that a QAH state is favored at
$\nu=-1$, but not $\nu=-3$, indicating a slight difference in the twist angle
dependence of these two states (Extended Data Fig. 5). Qualitatively, below
this range, the integer gaps become much stronger, and fewer Hofstadter states
are observed. Additionally there is a reversal of the relative gap strengths:
at low twist angle, $\Delta_{-3}$ becomes larger than the $\Delta_{-2}$,
likely due to details of the interaction strength relative to the single-
particle gap between moiré bands. The overall level of disorder also
consistently appears worse, which may reflect an enhanced susceptibility to
local atomic reconstruction at low angles. For $\theta>1.3^{\circ}$, no zero-
field gaps are observed at $|\nu|\geq 3$, and the $\nu=-2$ gap vanishes at
lower magnetic fields with increasing angle. Additionally, we observe a larger
number of (weaker) Hofstadter states tracing back to all integers.

Figure 3: Twist angle dependence of correlated states. a, Left:
$\mathrm{d}\mu/\mathrm{d}n$ as a function of $\nu$ and $B$ at a selection of
twist angles. From top to bottom, $\theta=1.12^{\circ}$, $1.19^{\circ}$,
$1.23^{\circ}$, $1.31^{\circ}$, $1.42^{\circ}$, and $1.50^{\circ}$. Right:
corresponding Wannier plot for each angle. Black lines show states with $C=0$,
gray lines show (Hofstadter) states with $C\neq 0$ that do not persist to
$B=0$, red lines show states with $C\neq 0$ that persist to $B=0$. b-c,
Thermodynamic gaps $\Delta_{\nu}$ at $B=0$ for fillings $\nu=-1$ (b) and
$\nu=-3$ (c) as a function of twist angle (bottom axis) and moiré wavelength
$\lambda_{m}$ (top axis). Red dots and shading indicate where the gap is
topological.

These data, along with more measurements at intermediate twists (Extended Data
Figs. 3,6), allow us to quantify gap size dependence on twist angle at zero
magnetic field, and to correlate it with the nature of the ground states that
we observe. At $\nu=-1$, topological gaps appear at an overall minimum of a
non-monotonic dependence of $\Delta_{-1}$ as a function of twist angle (Fig.
3b). This is consistent with a pair of twist-tuned band inversions. In
contrast, the trend is generally monotonic as a function of angle at $\nu=-3$,
and the gap only becomes topological just as it closes (Fig. 3c). At higher
twist angles, interactions are likely not strong enough to induce an
insulating state, so we realize a (trivial) insulator to QAH insulator to
metal transition as a function of increasing twist angle at $\nu=-3$.

##  IV Topological phase transition at $\nu=-1$

Figure 4: Electric field tuning and topological phase diagram at $\nu=-1$. a,
Schematic of the single-electron transistor (SET) tip above a tWSe2 sample
with a global back gate. The effective top gate is set by the voltage applied
to the sample $V_{2D}$ relative to the grounded tip, while the effective back
gate voltage is given by the difference between the back gate voltage $V_{BG}$
and $V_{2D}$. b, Cartoon of spatial and layer-resolved density within a moiré
unit cell. At low $|D|$, states are evenly spread across the XM and MX sites,
forming a honeycomb lattice. At high $D$, holes preferentially occupy a single
layer at one of these two sites, chosen by the sign of $D$. c-d, d$\mu$/d$n$
as a function of $D_{\rm{eff}}$ and $\nu$ at $B=0$ T (c) and $B=1.6$ T (d) in
a location with $\theta=1.2^{\circ}$. The black dashed line marks the critical
field $D_{c}$ of a topological transition. e, d$\mu$/d$n$ as a function of
$\nu$ and $B$ around $\nu=-1$ at $V_{2D}=3.2$ V in which the tip induces a
reduced effective displacement field $D_{\rm{eff},\nu=-1}\approx 30$ mV/nm.
The resulting gap is topological with Chern number $C=+1$. f, d$\mu$/d$n$ as a
function of $\nu$ and $B$ around $\nu=-1$ at $V_{2D}=-2$ V in which the tip
induces a higher effective displacement field $D_{\rm{eff},\nu=-1}\approx 80$
mV/nm. The resulting gap is trivial with Chern number $C=0$. g, Experimental
phase diagram of the topological character of the $\nu=-1$ gap as a function
of twist angle (top axis) or moiré wavelength (bottom axis) and effective
displacement field. Red indicates topological ($C=+1$) while black indicates
trivial. Dots represent locations/conditions where we have conducted field
sweeps (as in e-f), while lines indicate measurements as a function of $n$ and
$D_{\rm{eff}}$ at fixed $B$ (as in c-d). The blue dashed line indicates the
condition where the tip does not dope the sample.

Theoretical predictions assert that the topological phase at $\nu=-1$ is
highly sensitive to an applied electric field, which favors polarization onto
one of the layers and breaks the honeycomb sublattice symmetry of the emergent
generalized Kane-Mele bands (Fig. 4b) devakul_magic_2021 . Though the scanning
SET tip is generally noninvasive, if the d.c. sample voltage $V_{2D}$ is
adjusted, the tip can locally dope the region underneath it. We find that
within certain bounds of $V_{2D}$, the tip acts as an effective gate that
tunes the local displacement field $D_{\rm{eff}}$ applied to the sample (Fig.
4a, Methods). Remarkably, we observe a transition at $\nu=-1$ as a function of
$D_{\rm{eff}}$ at certain twist angles.

We focus on a location with $\theta=1.20^{\circ}$. At $B=0$, the gap at
$\nu=-1$ first decreases before re-emerging and strengthening as
$D_{\rm{eff}}$ is increased (Fig. 4c). At $B=1.6$ T, states with different $C$
occur at different density. At lower $D_{\rm{eff}}$, the $(C,s)=(+1,-1)$ state
is dominant, while the $(0,-1)$ state strengthens at high $D_{\rm{eff}}$ (Fig.
4d). At both magnetic fields, the change in behavior occurs at a similar
critical displacement field around $D_{c}=65$ mV/nm (Fig. 4c-d, black dashed
line). This suggests a phase transition from a topological state below $D_{c}$
to a trivial $\nu=-1$ gap at higher displacement fields. The observations at
both magnetic fields are consistent with the thermodynamic gap closing and
reopening across $D_{c}$, as expected from a topological quantum phase
transition hasan_colloquium_2010 ; li_quantum_2021 .

To confirm this interpretation, we measure d$\mu$/d$n$ as a function of
magnetic field at different effective displacement fields. In Fig. 4e, we
present a measurement at the same location with a small displacement field
$D_{\rm{eff}}\approx 30$ mV/nm. The resulting $\nu=-1$ state is topological,
with the $C=+1$ gap continuing down to $B=0$. However, when the displacement
field is increased to $D_{\rm{eff}}\approx 80$ mV/nm, a field sweep in the
same location shows a topologically trivial gap (Fig. 4f).

By repeating similar measurements at other locations, we build up a phase
diagram of the topological behavior at $\nu=-1$ as a function of twist angle
and applied displacement field (Fig. 4g). In this plot, vertical lines signify
measurements as a function of $D_{\rm{eff}}$, while points signify field
sweeps to confirm the topological character of each state. In addition to
being able to tune the topological gaps to trivial at
$\theta=1.20^{\circ}-1.25^{\circ}$, we are also able to induce topological
states where trivial states had been favored (at $\theta=1.19^{\circ}$) by
decreasing the applied displacement field (Supplementary Sec. 3). The range of
twist angles over which we observe topological behavior is quite sharp, likely
due to other competitive states at $\nu=-1$.

We perform Hartree-Fock calculations for the continuum model (Supplementary
Sec. 2) which indicate a rich landscape of correlated ground states at integer
$\nu$. Within a certain range of twist angles at $\nu=-1$, a spin-valley
polarized Chern insulator is expected to be favored, with transitions to a
topologically trivial layer-polarized state at sufficiently high $D$ or low
$\theta$. At higher twist angle, our calculations predict that the Chern state
will give way to an intervalley-coherent trivial state. Additionally, we find
a quantum spin Hall insulator at $\nu=-2$, and a Chern insulator at $\nu=-3$
with the same sign as at $\nu=-1$, as observed in experiment.

##  V Outlook

In conclusion, we have established twisted WSe2 as a platform to realize
multiple topologically nontrivial bands which support symmetry-broken QAH gaps
at half-filling. Comparing data across a range of moiré wavelengths from
approximately 12 to 17 nm, we reveal widely varying correlated ground states
of the honeycomb lattice model in twisted WSe2, both in magnitudes of the gaps
and their topological character. This highlights the importance of moiré
wavelength as a tuning parameter, which is only accessible in low-twist
lattice-matched systems. Our work motivates further studies to confirm the
specific spin-valley symmetry breaking of the odd-integer gaps, as well as
transport studies to confirm that quantum spin Hall insulating states are
realized at even integers, including a potential double quantum spin Hall
insulator at $\nu=-4$ mai_14_2023 . More generally, this work opens a new
platform for studying tunable topological bands in the limit of larger moiré
wavelengths where interacting phases such as fractional Chern insulators may
be more readily stabilized crepel_anomalous_2022 ; morales-
duran_pressure–enhanced_2023 .

Note added: While finalizing this manuscript, we became aware of a preprint
that reports a topological moiré band and evidence of (fractional) quantum
anomalous Hall states in twisted homobilayer MoTe2 cai_signatures_2023 .

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##  VI Methods

###  VI.1 Sample fabrication

The tWSe2 device was fabricated using standard dry transfer techniques. Using
a poly(bisphenol A carbonate) (PC)/polydimethylsiloxane (PDMS) stamp, we pick
up a thin (15 nm thick) hexagonal boron nitride (hBN) flake, followed by the
first half of the monolayer WSe2 flake (exfoliated from HQ Graphene source),
using the hBN to tear the flake in two, and then the second half rotated to a
controlled angle of $1.5^{\circ}$. Separately, we prepare a stack with a
bottom hBN (13 nm) and a graphite (5 nm) back gate, on which we deposit pre-
patterned Cr/Pt contacts (2 nm / 8 nm). This is annealed at $\approx 300\
^{\circ}$C for 8 hours to clean polymer and resist residues both before and
after pre-patterning of contacts. The tWSe2 stack is then dropped onto the
pre-patterned contacts. We used standard e-beam lithography techniques to
fabricate metallic contacts, as well as local “contact-gates” over the Pt
contacts, while leaving the rest of the sample ungated on top for access to
the SET movva_high-mobility_2015 ; foutty_tunable_2023 .

###  VI.2 SET Measurements

The SET sensor was fabricated by evaporating aluminum onto a pulled quartz
rod, with an estimated diameter at the apex of $50-100$ nm. The SET “tip” is
brought to about $50$ nm above the sample surface. Scanning SET measurements
were performed in a Unisoku USM 1300 scanning probe microscope with a
customized microscope head. a.c. excitations (2-5 mV peak-to-peak amplitude)
were applied to both sample and back gate at distinct frequencies between 200
and 400 Hz. We then measure inverse compressibility
$\textrm{d}\mu/\textrm{d}n\propto I_{\textrm{BG}}/I_{\textrm{2D}}$ where
$I_{\textrm{BG}}$ and $I_{\textrm{2D}}$ are measurements of the SET current
demodulated at respective frequencies of the back gate and sample excitations
yu_correlated_2022 . Except where otherwise noted (Fig. 4), a d.c. offset
voltage $V_{\textrm{2D}}$ is applied to the sample to maintain the working
point of the SET at its maximum sensitivity point within a Coulomb blockade
oscillation fringe chosen to be near the “flat-band” condition where the tip
does not gate the sample. This minimizes tip-induced doping and provides a
direct measurement of $\mu(n)$ at d.c. timescales. Depending on measurement
location, we measure a small ($<1\times 10^{-11}$ meV cm2) difference in
d$\mu$/d$n$ between a.c. and d.c. measurements, which we subtract from the
a.c. in all data presented in the main text (see Supplementary Sec. 4 for
further discussion). The contact gates are held at a large, negative voltage
throughout the measurement to maintain good electrical contact across variable
hole doping. All SET measurements are taken at $T=330$ mK unless otherwise
noted.

###  VI.3 Density and twist angle determination

The approximate twist angle of the sample ($\sim 1.5^{\circ}$) is determined
by the control over rotation in the stackng process. To ensure that the tWSe2
is not overly strained or relaxed after picking it up, we perform
piezoelectric force microscopy (PFM) using a Bruker Icon atomic force
microscope during the fabrication process on the PC slide before setting the
stack onto the prepatterned Pt contacts mcgilly_visualization_2020 ;
bai_excitons_2020 . We present this in Supplementary Sec. 5. We find a
relatively low-strain triangular moiré superlattice with lattice constant
around $13$ nm, consistent with what we extract from SET measurements.

As described in the main text, there is some variation in local twist angle
across the device. We use SET measurements to precisely determine local angle
in a similar manner to standard techniques in magic-angle twisted bilayer
graphene zondiner_cascade_2020 ; uri_mapping_2020 . From the slopes of the
Hofstadter states, we can accurately measure the sample capacitance and
convert between back gate voltage $\rm{V}_{\rm{BG}}$ and carrier density $n$.
We then use the integer gaps we measure (e.g. at $\nu=-1,-2$, and $-3$) and/or
the corresponding Hofstadter intercepts (in cases where there is no zero-field
gap) to determine the density corresponding to filling one hole per moiré unit
cell. From this density $n_{\rm{muc}}$, we convert to twist angle $\theta$ via
$1/n_{\rm{muc}}=\frac{\sqrt{3}a_{\rm{WSe}_{2}}^{2}}{4-4\cos\theta}$ and
subsequently moiré wavelength via
$\lambda_{m}=\frac{a_{\rm{WSe}_{2}}}{2\sin(\theta/2)}$, where
$a_{\rm{WSe}_{2}}=0.328$ nm.

###  VI.4 Extraction of gap sizes

The thermodynamic gap sizes shown in the main text are given by the size of
the step in the chemical potential $\mu(n)$. Practically, this is extracted by
numerically integrating $\textrm{d}\mu/\textrm{d}n$ across the gap. To
accurately measure the gap on top of the widespread negative compressibility
coming from long-range interactions at low density, we subtract a small
background before integrating, analogous to Refs.
eisenstein_compressibility_1994 ; kometter_hofstadter_2022 . Generally, this
background is taken from averaging the value of $\textrm{d}\mu/\textrm{d}n$ on
either side of the gap. This background is measured sufficiently far (in
density) from the gap so that chemical potential behavior immediately adjacent
to the gap, for example enhanced negative compressibility, is not included in
the background. In a few cases in which gaps are particularly close together
(for example, $C=+1$ gaps and $C=0$ gaps with $s=-1$ between $B=0.5$ and
$B=1.5$ T), the background is just taken from a single side of the gap, so
that the behavior of one gap does not affect the extracted value of the other.
In general, disorder can reduce the size of measured gaps. In the main text
(Figs. 2-3), we present gaps from data measured at optimized locations that
have low twist angle variability in order to minimize the effects of disorder
in comparisons.

###  VI.5 Electric field tuning with SET

As mentioned in the main text, if the d.c. sample voltage $V_{\textrm{2D}}$ is
tuned away from the “flat-band” condition compensating for the work function
difference between sample and tip, the tip will locally gate the sample. We
can model this doping by treating the tip as one side of a parallel plate
capacitor:
$D_{\textrm{eff}}=\frac{1}{2\epsilon_{0}}(C_{t}(0-(V_{2D}-V_{fb}))-C_{b}(V_{BG}-V_{2D}-V_{0}))$,
where $C_{b(t)}$ is the back (top) gate capacitance, $V_{2D}$ and $V_{BG}$ are
the d.c. voltages applied to sample and back gate, $V_{fb}$ is the “flat-band”
voltage at which the tip and sample are work-function-compensated, and $V_{0}$
is the voltage at which back gate and sample are work-function-compensated
(equivalent to the voltage of the WSe2 band edge). While $C_{t}$ will depend
on the height of the tip, we experimentally extract $C_{t}\approx 0.045C_{b}$
for the measurements shown in the main text based on the shifts of constant-
density features in the $V_{BG}-V_{2D}$ plane. Data shown in Fig. 4c-d is
taken by initially fixing $V_{2D}$ and then sweeping $V_{BG}$, feeding back on
the value of $V_{2D}$ while data is taken to maintain the same position along
the SET Coulomb blockade oscillation. From this data, we apply the
transformation to convert to $n$ and $D_{\textrm{eff}}$ from $V_{2D}$ and
$V_{BG}$.

###  VI.6 Hofstadter spectrum calculation

We compute the Hofstadter spectrum for the continuum model for TMD
homobilayers wu_topological_2019 , with continuum model parameters extracted
from ab initio calculations of Ref devakul_magic_2021 :
$(V,w,\psi)=(9\mathrm{\ meV},18\mathrm{\ meV},128^{\circ})$. We use an
effective mass $m=0.3m_{e}$, where $m_{e}$ is the electron mass, and a WSe2
lattice constant of $a_{0}=3.317$Å. The calculation is performed at a twist
angle $\theta=1.2^{\circ}$ and displacement field is modeled as an interlayer
potential $\mathrm{diag}(\frac{\Delta}{2},-\frac{\Delta}{2})$ with $\Delta=4$
meV. The continuum model is described in more detail in the Supplementary Sec.
2. The spectrum for the $K$ valley bands features a topological first band
with $C=1$ that is close to a displacement field tuned band inversion with the
second band.

The finite magnetic field is incorporated by minimal substitution
$\vec{p}\rightarrow\vec{\pi}=\vec{p}-\vec{A}$ with the symmetric gauge vector
potential $A=\frac{B}{2}(x\hat{y}-y\hat{x})$. At flux per unit cell
$B=\frac{p}{2qA_{uc}}$, where $A_{uc}$ is the moiré unit cell area, the
Hamiltonian is diagonalized in the Landau level basis with a large cutoff
$N_{\mathrm{max}}=\lfloor 100q/p\rfloor$, using the method described in detail
in Ref kometter_hofstadter_2022 . The resulting Hofstadter spectrum shown in
Fig. 2d is computed for the $K$ valley bands.

To compute the Wannier plot in Fig. 2e, we first apply a Gaussian broadening
of width $\sigma=0.5$ meV to the calculated density of states. The filling
factor $\nu$ is obtained by integrating the broadened density of states from
the charge neutrality point, assuming full valley polarization.

##  VII Data availability

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

##  VIII Code availability

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

##  IX Acknowledgements

Experimental work was primarily supported by the Department of Energy, Office
of Basic Energy Sciences, award number DE-SC0023109. B.E.F. acknowledges an
Alfred P. Sloan Foundation Fellowship and a Cottrell Scholar Award. The work
at Massachusetts Institute of Technology was supported by the Air Force Office
of Scientific Research (AFOSR) under award FA9550-22-1-0432. K.W. and T.T.
acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354 and
23H02052) and World Premier International Research Center Initiative (WPI),
MEXT, Japan. B.A.F. acknowledges a Stanford Graduate Fellowship. Part of this
work was performed at the Stanford Nano Shared Facilities (SNSF), supported by
the National Science Foundation under award ECCS-2026822.

##  X Author contribution

B.A.F. and C.R.K. conducted the scanning SET measurements. B.A.F. and B.E.F.
designed the experiment. T.D., A.P.R., and L.F. conducted theoretical
calculations. B.A.F. fabricated the sample. K.W. and T.T. provided the hBN
crystals. B.E.F. and L.F. supervised the project. All authors participated in
analysis of the data and writing of the manuscript.

##  XI Competing interests

The authors declare no competing interest.

##  XII Extended Data Figures

Extended Data Figure 1: d$\mu$/d$n$ at $T=1.6$ K. d$\mu$/d$n$ as a function of
$\nu$ and $B$ at temperature $T=1.6$ K. All other parameters (including
location in the sample) are identical to the data shown Fig. 2a. Extended Data
Figure 2: Device micrograph and spatial characterization. a, Optical
micrograph of the device, with individual points marking measurements. Points
1-6 mark the locations of the measurements presented in Fig. 3a, from the top
to bottom. Point 3 is the location where the data in Figs. 1-2 is taken.
Points 7-8 mark the locations of the measurements in Extended Data Fig. 3.
Point 9 marks the location of Fig. 4c-f and Extended Data Fig. 5. Points 10-13
are clean locations twisted to $\theta=1.25^{\circ}$, $1.35^{\circ}$
$1.45^{\circ}$, and $1.60^{\circ}$ respectively that are included in Figs.
3b-c and 4g and shown in Extended Data Fig. 6. Orange and green lines show the
locations of data plotted in c-d, where the dot indicates $x=0$ for each line.
b,c d$\mu$/d$n$ as a function of back gate voltage $V_{BG}$ and spatial
position $x$ along two roughly orthogonal lines in real space (shown in a)
spanning a uniform region with $\theta\approx 1.23^{\circ}$, at a constant
magnetic field of $B=1.6$ T. Spatial variability in the location of features
is mostly due to potential disorder in the sample rather than twist angle
disorder. Features related to $\nu=-1,-2,$ and $-3$ are clustered around
$V_{BG}=-1.1,-1.5$, and $-1.9$ V, respectively. The dominance of the $C=+1$
state (relative to $C=0$ at $\nu=-1$) is consistent spatially, and coexistence
of both is also visible near $\nu=-3$. $C=+1$ gaps are highlighted by red
arrows, while $C=0$ gaps are highlighted by black arrows. While these spatial
line cuts respectively span approximately 500 nm and 800 nm, spot checks in
other locations indicate qualitatively uniform behavior throughout a 600 nm
$\times$ 1 $\mu$m region. Extended Data Figure 3: Topological gaps at other
independent locations. a,b d$\mu$/d$n$ as a function of $\nu$ and $B$ in two
independent locations in the sample. These locations are separated from the
data shown in Fig. 1 and from each other by $>500$ nm. The twist angle is
roughly the same (within $\pm 0.005^{\circ}$) across the three locations.
Extended Data Figure 4: Field dependence of thermodynamic gaps at non-
topological twist angles. a, Gap sizes of the $(C,s)=(0,-1)$ gaps at
$\theta=1.31^{\circ}$ and $\theta=1.35^{\circ}$. b, Gap sizes of the
$(C,s)=(0,-2)$ gaps at $\theta=1.45^{\circ}$ and $\theta=1.50^{\circ}$. Both
gaps show non-monotonic gap size dependence on magnetic field, and there is no
evidence for topological bands at these twist angles. Extended Data Figure 5:
Low-field measurement at $\theta=1.20^{\circ}$. d$\mu$/d$n$ as a function of
$\nu$ and $B$ at low magnetic fields, in the same spot as the measurements in
Fig. 4c-f, but with no tip doping. This is the only twist angle where at
$B=0$, we resolve a clear $C=+1$ gap at $\nu=-1$ but a $C=0$ state at $\nu=-3$
without changing the displacement field. Extended Data Figure 6: Inverse
compressibility data at other twist angles. a-d, d$\mu$/d$n$ as a function of
$\nu$ and $B$ at four locations at different twist angles from those presented
in Fig. 3a, $\theta=1.60^{\circ}$ (a), $\theta=1.45^{\circ}$ (b),
$\theta=1.35^{\circ}$ (c), and $\theta=1.25^{\circ}$ (d).