††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Nanoscale electrostatic control in ultra clean van der Waals
heterostructures by local anodic oxidation of graphite gates

Liam A. Cohen  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Noah L. Samuelson  Department of
Physics, University of California at Santa Barbara, Santa Barbara CA 93106,
USA  Taige Wang  Department of Physics, University of California, Berkeley,
California 94720, USA  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Kai Klocke  Department of
Physics, University of California, Berkeley, California 94720, USA  Cian C.
Reeves  Department of Physics, University of California at Santa Barbara,
Santa Barbara CA 93106, USA  Takashi Taniguchi  International Center for
Materials Nanoarchitectonics, National Institute for Materials Science, 1-1
Namiki, Tsukuba 305-0044, Japan  Kenji Watanabe  Research Center for
Functional Materials, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  Sagar Vijay  Department of Physics, University of
California at Santa Barbara, Santa Barbara CA 93106, USA  Michael P. Zaletel
Department of Physics, University of California, Berkeley, California 94720,
USA  Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Andrea F. Young<EMAIL_ADDRESS>Department of Physics, University of California at Santa Barbara, Santa
Barbara CA 93106, USA

###### Abstract

In an all-van der Waals heterostructure, the active layer, gate dielectrics
and gate electrodes are assembled from two-dimensional crystals that have a
low density of atomic defects. This design allows two-dimensional electron
systems with very low disorder to be created, particularly in heterostructures
where the active layer also has intrinsically low disorder, such as
crystalline graphene layers or metal dichalcogenide heterobilayers. A key
missing ingredient has been nanoscale electrostatic control, with existing
methods for fabricated local gates typically introducing unwanted
contamination. Here we describe a resist-free local anodic oxidation process
for patterning sub 100nm features in graphite gates, and their subsequent
integration into an all-van der Waals heterostructure. We define a quantum
point contact in the fractional quantum Hall regime as a benchmark device and
observe signatures of chiral Luttinger liquid behaviour, indicating an absence
of extrinsic scattering centres in the vicinity of the point contact. In the
integer quantum Hall regime, we demonstrate in situ control of the edge
confinement potential, a key requirement for the precision control of chiral
edge states. This technique may enable the fabrication of devices capable of
single anyon control and coherent edge-state interferometry in the fractional
quantum Hall regime.

Van der Waals heterostructures have recently emerged as a rich platform to
study the physics of delicate correlated electronic states, including (but not
limited to) fractional quantum Hall phaseszibrov_tunable_2017; li_even_2017;
dean_fractional_2020, exciton condensatesdean_fractional_2020;
li_excitonic_2017; liu_quantum_2017; ma_strongly_2021, quantized anomalous
Hall insulatorsserlin_intrinsic_2020; li_quantum_2021, fractional Chern
insulatorsspanton_observation_2018; xie_fractional_2021, and
superconductorscao_unconventional_2018; yankowitz_tuning_2019;
zhou_superconductivity_2021; zhou_isospin_2022; zhang_spin-orbit_2022. A key
driver of continued improvement in sample quality has been the removal of
charged impurities, first with the use of high purity two-dimensional crystals
of hexagonal boron nitride (hBN) as a substratedean_boron_2010 and gate
dielectricyoung_electronic_2012; mayorov_direct_2011; wang_one-
dimensional_2013, and more recently with the use of graphite, rather than
amorphous metal, for the gate layerszibrov_tunable_2017. These ‘all-van der
Waals heterostructures’ take advantage of the fact that none of the components
in the stack host dangling bonds in their two-dimensional bulk. In addition,
numerous van der Waals interfaces appear to be self-cleaninghaigh_cross-
sectional_2012, irreversibly expelling hydrocarbon residues during processing
and leaving an atomically uniform interface.

A central feature of these platforms is electrostatic tunability, enabling a
variety of correlation-driven ground states to be accessed by field effect
gating in a single device. Electrostatic control on the nanoscale, then,
allows one- and zero-dimensional structures to be created within a correlated
two-dimensional state, opening the door to experiments that probe the
structure of interfaces between distinct phases as well as adiabatic
manipulation of individual quasi-particles and edge modes. A wide class of
these experiments require electrostatic confinement on length scales
comparable to the correlation length of superconductors or fractional quantum
Hall states, which is typically below $100\text{\,}\mathrm{n}\mathrm{m}$. The
confining potentials are also required to be energetically uniform, in the
sense that they should not introduce uncontrolled local electrical potentials
larger than the $\approx$1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ energy
gaps of the correlated states to be studied. There are two options empowered
by traditional electron beam lithography, which is capable of patterning at
the length scales required for constructing such nano-scale potentials. First,
the all-van der Waals geometry may be abandoned, patterning at least some
gates from evaporated metal. Second, heterostructures may be assembled and
then graphite gates patterned by subtractive processes. However, both
techniques lead to disorder in critical regions of the device. For example,
edge state interferometers manufactured using either technique remain limited
to the integer quantum Hall regime despite the presence of well formed
fractional quantum Hall phases in the two-dimensional sample bulk
ronen_aharonov-bohm_2021; zimmermann_tunable_2017; deprez_tunable_2021;
zhao_graphene-based_2022.

Figure 1: Local anodic oxidation and integration of graphite gates into van
der Waals heterostructures. (a) A schematic example of an exfoliated graphite
flake etched using atomic force microscope-actuated local anodic oxidation
(AFM-LAO). Here we show the x-shaped geometry used to form the quantum point
contact gates which constitute the device studied in this work. (b) A transfer
stamp with an already picked-up hBN flake is engaged with the etched graphite
flake which has some residual oxide residue from the AFM-LAO process. (c) Once
the etched graphite is laminated onto to the hBN flake the structure can be
disengaged, leaving behind the oxidized residue and resulting in a pristine
microstructure. (d) Example edge state configuration of a quantum point
contact resulting from the electrostatic potential created by an x-shaped
graphite gate. (e) Same as panel (d) but with an array of holes in the top
graphite gate used to create a superlattice of quantum dots. (f) Same as
panels (d-e) but with an edge-state interferometer structure. Here the
isolated Fabry-Pérot cavity can be realized by careful alignment of etched
graphite gates in both the top and bottom layers of the heterostructure. (g-i)
AFM topography images of patterns etched into graphite flakes using AFM-LAO
corresponding to the top gate structures in panels (d-f). (j-l) AFM topogrophy
images of etched graphite on hBN post van der Waals pick-up of the flakes in
panels (g-i) demonstrating that most of the oxidized residue (white dots) seen
in panel (g-i) is not transferred.  Figure 2: Quantum point contact operation
in the integer quantum Hall regime. (a) Optical micrograph of measured device.
Transport contacts are denoted by C1-C8. Gate contacts are labeled by $V_{i}$
with an appropriate subscript to denote which region the gate controls. b)
Diagonal conductance, $G_{\mathrm{D}}$, versus $V_{\text{EW}}-V_{\mathrm{B}}$
demonstrating integer QH edge mode partitioning at the QPC. Here the east/west
gates are swept simultaneously with the back gate such that the bulk filling
factor, $\nu_{EW}\propto V_{\text{EW}}+\alpha V_{\mathrm{B}}$, remains
constant. Additionally, $\nu_{NS}\propto V_{\text{NS}}+\alpha
V_{\mathrm{B}}=0$ is held constant.

##  I Device Fabrication

Here we show how high-quality mesoscopic devices may be created by abandoning
traditional lithographic techniques in favor of patterning graphite gates at
sub-$100\text{\,}\mathrm{n}\mathrm{m}$ length scales using a resist-free
process which takes advantage of atomic force microscope-actuated local anodic
oxidation (AFM-LAO) of graphite li_electrode-free_2018. These gates are then
integrated directly into a van der Waals heterostructure using a low-strain
variation of the standard dry-transfer process wang_one-dimensional_2013;
novoselov_2d_2016; masubuchi_fabrication_2009. Fig. 1a-c portray a schematic
description of this processs. Fabrication begins with the use of an atomic
force microscope (AFM) to locally oxidizemasubuchi_fabrication_2009;
li_electrode-free_2018 a region of the graphite flake. In this process, a
conductive AFM tip is brought close to the graphite surface in a humid
environment. Capillary forces form a nano-scale water meniscusbutt_using_2006
connecting the tip and graphite surface. When a high frequency excitation is
applied to the tip, the voltage drop across the water meniscus catalyzes
oxidation of the graphite into gaseous and amorphous byproducts. Scanning the
tip across the graphite surface while this reaction occurs allows nano-scale
subtractive lithographyli_electrode-free_2018 without introducing contaminants
to the two dimensional graphite bulk, as would occur in solvent-based resist
removal processes. The patterned graphite gate can then be integrated into a
van der Waals heterostructure through pick-up by an unpatterned van der Waals
flake (see Methods and Fig. 1b-c) to produce a wide array of geometries,
several examples of which are depicted in Figs. 1d-f.

In the anodic oxidation process, amorphous residue–likely carbon and carbon
oxides–typically remains, manifesting as features localized at critical
interfaces in the AFM topographs of Figs. 1(g-i). However, these byproducts
adhere more strongly to the original SiO2 substrate than to the hBN flake used
for pick-up. This results in a self-cleaning process that enables the transfer
of pristine microstructures into the middle layers of the heterostructure.
Figs. 1(j-l) show AFM topographs of the patterned graphite after pick-up by a
hexagonal boron nitride flake–i.e., imaged with the graphite in the
configuration shown in Fig. 1c. Etch byproducts visible in the as-cut graphite
(Fig. 1g-i) are not transferred, leaving pristine nano-scale subtractive
patterns. Notably, in this process the areas closest to the critical regions
are not exposed to additional fabrication residues, in contrast to graphite
gates patterned by plasma-etching ronen_aharonov-bohm_2021.

Remarkably, our process permits transfer of high-density patterns without
degradation or tearing. Three examples are shown in Figs. 1d-f, depicting a
quantum point contact, quantum dot array, and Fabry-Pérot interferometer, all
meant to be operated in the quantum Hall regime. We note that to preserve the
alignment of fine lithographic features during pattern transfer, the patterned
graphite flakes must remain contiguous: free-floating graphite pieces will
typically slide during transfer leading to electrical shorts between nominally
disconnected areas. In most cases, additional conventional subtractive
processing is used to electrically disconnect different gate regions and make
electrical contact to sample and gate layers. However, certain desirable
electrostatic geometries can be realized using complementary patterns in both
the graphite top- and bottom- gates and aligning the two during transfer. An
example of this strategy, used to construct a quantum Hall edge state Fabry-
Pérot interferometer, is shown in Figs. 1f, i, and l, in which a patterned
bottom gate is used to create an isolated, independently density-tunable
region in a graphene device without requiring a free-floating gate.

While the process described above produces topographically pristine gate
geometries, topography alone is not a guarantee of electronic quality. In
particular, without electrical characterization, we cannot exclude that the
anodic oxidation itself produces unacceptably large local potentials on the
graphite edge. To qualify our technique, then, we use a monolayer graphene
quantum point contact (QPC) device operating in the quantum Hall regime. A QPC
is formed when a narrow constriction in a two-dimensional device restricts the
number of quantized channels through which electrons can
flowvan_wees_quantized_1988. In the quantum Hall regime, when the constriction
is on the order of the magnetic length $\ell_{B}$, the transmission of chiral
edge modes through the QPC varies sensitively with the width. To form a QPC
using our fabrication method, we start from the cross gate geometry of Fig.
1d, and perform additional lithographic processing (see Methods) to isolate
the four quadrants of the graphite top gate producing four isolated gates we
denote North (N), South (S), East (E) and West (W). Voltages applied to these
gates and a global graphite bottom gate can be used to deplete the monolayer
into the $\nu=0$ gap in the $N$ and $S$ regions, forming a narrow
constriction, with the filling factor in the $E$ and $W$ regions held
constant. An optical micrograph of the completed device is shown in Fig. 2a.

To characterize the operation of our QPC we measure the four-terminal
‘diagonal conductance,’ $G_{\mathrm{D}}$ (see Methods), which in the integer
quantum Hall (IQH) regime gives a direct measure of the number of edge modes
$N_{\text{qpc}}$ transmitted across the device such that
$G_{\mathrm{D}}=N_{\text{qpc}}\frac{e^{2}}{h}$datta_electronic_1995;
zimmermann_tunable_2017. Fig. 2b shows $G_{\mathrm{D}}$ measured at $B=6$ T
and T=300 mK as the gate voltages are adjusted to tune $N_{\text{qpc}}$ from 0
to 2. In this measurement, we fix the value of $V_{\mathrm{EW}}+\alpha
V_{\mathrm{B}}$ (where $\alpha=c_{b}/c_{t}$ is the ratio of the bottom- and
top-gate capacitances to the monolayer, and $V_{\text{EW}}$ denotes a single
voltage applied to the $E$ and $W$ gates – see Fig. 5a-c). This fixes the bulk
filling factor in the $E$ and $W$ regions within the $\nu=-4$ plateau. We also
fix $V_{\mathrm{NS}}+\alpha V_{\mathrm{B}}$ to keep the $N$ and $S$ regions
within the $\nu=0$ plateau. Tuning $V_{\text{EW}}-V_{\mathrm{B}}$ then
controls the QPC width independent of the bulk filling factors in the four
quadrants (see Fig. S1 and Fig. S2 for additional info on QPC operation in the
integer quantum Hall regime). As is evident in Fig. 2b, we observe integer-
quantized conductance plateaus separated by monotonic transitions; in
particular, we do not observe non-monotonic conductance features
characteristic of disorder-mediated or resonant tunneling effects, typical of
quantum point contacts in both III-V semiconductors and graphene
milliken_indications_1996; zimmermann_tunable_2017; ronen_aharonov-bohm_2021;
radu_quasi-particle_2008; baer_interplay_2014.

##  II Chiral Luttinger liquid behavior at a fractional quantum point contact

Figure 3:  Partitioning of fractional quantum Hall edges and quasi-particle
tunneling. (a) $G_{\mathrm{D}}$ map at fixed $\nu_{EW}=-\frac{5}{3}$. As
$V_{\mathrm{B}}$ is adjusted along the y-axis, $V_{\text{EW}}$ is swept in
tandem to keep $V_{\text{EW}}+\alpha V_{\mathrm{B}}\propto\nu_{EW}$ constant
while allowing $V_{\mathrm{B}}-V_{\text{EW}}$ to vary, adjusting the saddle
point electrostatics. On the x-axis, the range of $V_{\text{NS}}$ is varied as
$V_{\mathrm{B}}$ changes in order to keep the range $V_{\text{NS}}+\alpha
V_{\mathrm{B}}$ within the $\nu=0$ plateau. (b) $G_{\mathrm{D}}$ along the
dashed white contour in panel a. Two fractional plateaus are highlighted at
$4/3$ and $5/3$ resepctively. (c) Bias dependence of the
$G_{\mathrm{D}}=\frac{4}{3}$ plateau, along the dashed blue line shown in
panel a. The tunneling conductance is suppressed at zero bias well into the
plateau, a distinct signature of chiral Luttinger liquid behavior. (d)
Extracted back-scattered conductance, $R$, versus scaled DC bias,
$V_{\text{bias}}$, at a fixed $V_{\text{NS}}=3.925V$ for the temperatures $T=$
$141\text{\,}\mathrm{m}\mathrm{K}$, $169\text{\,}\mathrm{m}\mathrm{K}$,
$199\text{\,}\mathrm{m}\mathrm{K}$, $229\text{\,}\mathrm{m}\mathrm{K}$,
$259\text{\,}\mathrm{m}\mathrm{K}$, $289\text{\,}\mathrm{m}\mathrm{K}$,
$320\text{\,}\mathrm{m}\mathrm{K}$, $350\text{\,}\mathrm{m}\mathrm{K}$,
$379\text{\,}\mathrm{m}\mathrm{K}$, $410\text{\,}\mathrm{m}\mathrm{K}$,
$440\text{\,}\mathrm{m}\mathrm{K}$, $470\text{\,}\mathrm{m}\mathrm{K}$,
$499\text{\,}\mathrm{m}\mathrm{K}$, $530\text{\,}\mathrm{m}\mathrm{K}$,
$534\text{\,}\mathrm{m}\mathrm{K}$, $540\text{\,}\mathrm{m}\mathrm{K}$,
$544\text{\,}\mathrm{m}\mathrm{K}$, $550\text{\,}\mathrm{m}\mathrm{K}$. (e)
Back-scattered conductance, $R$, at zero DC bias versus temperature. Power-law
behavior onsets above $300\text{\,}\mathrm{m}\mathrm{K}$; we find $R\propto
T^{-1.057\pm 0.008}$, yielding a Luttinger parameter of $g=0.47\pm 0.004$. The
cross-over energy scale between weak electron tunneling and weak quasi-
particle back-scattering is estimated to be
$T^{*}=64\pm$3\text{\,}\mathrm{m}\mathrm{K}$$ (notably power-law behavior with
negligible contributions from electron tunneling for quasi-particle back
scattering is not expected unless $T>>T^{*}$). (f) Scaling collapse of DC bias
dependent tunneling curves for different temperatures above power-law onset.
Here $R/T^{-1.06}$ is plotted against $eV_{\text{bias}}/k_{b}T$ for the
temperatures
$T=$299\text{\,}\mathrm{m}\mathrm{K}$,$320\text{\,}\mathrm{m}\mathrm{K}$,$340\text{\,}\mathrm{m}\mathrm{K}$,$360\text{\,}\mathrm{m}\mathrm{K}$,$379\text{\,}\mathrm{m}\mathrm{K}$,$399\text{\,}\mathrm{m}\mathrm{K}$,$420\text{\,}\mathrm{m}\mathrm{K}$,$440\text{\,}\mathrm{m}\mathrm{K}$,$460\text{\,}\mathrm{m}\mathrm{K}$,$480\text{\,}\mathrm{m}\mathrm{K}$,$499\text{\,}\mathrm{m}\mathrm{K}$,$520\text{\,}\mathrm{m}\mathrm{K}$,$540\text{\,}\mathrm{m}\mathrm{K}$$.
The black-dashed is a fit to the expected bias dependence for a chiral
Luttinger liquid with g = 0.47 (see Eq. (1)) yielding an effective quasi-
particle charge of $e^{*}=(0.55\pm 0.001)|e|$.

An even more stringent characterization of disorder at the QPC can be obtained
in the fractional quantum Hall (FQH) regime. Fig. 3a shows $G_{\mathrm{D}}$
plotted as a function of $V_{\mathrm{B}}$ and $V_{\text{NS}}+\alpha
V_{\mathrm{B}}$ at B=13T and T=300mK. Throughout this range, the filling
factor of the $E$ and $W$ region is fixed at $\nu=-5/3$, and that of the $N$
and $S$ regions is fixed within the $\nu=0$ plateau. Data taken along the
white dashed line in Fig. 3a is plotted in Fig. 3b, and shows two distinct
quantized plateaus at $G_{\mathrm{D}}=4/3$ and $5/3$ , in addition to the
integer plateau at $G_{\mathrm{D}}=1$. This observation suggests the
$\nu=-5/3$ state hosts two edge modes each with conductance
$-\frac{e^{2}}{3h}$. Notably, this is not consistent with theoretical
expectations for a sharp confining potentialjohnson_composite_1991;
kane_randomness_1994. In that limit, a pair of counter propagating edge modes
is expected where one carries a conductance of $-\frac{e^{2}}{h}$ and the
other carries a conductance of $\frac{e^{2}}{3h}$ (see Fig. 6a); evidence for
this structure was observed recently in carefully designed III-V
heterostructuresnakamura_half-integer_2022. Our observation is instead
consistent with a picture of the 5/3 edge which includes the effects of edge
state reconstruction resulting from a soft confinement potential defining the
boundary of the topological bulksabo_edge_2017. Theoretically, softly confined
edges of hole-conjugate fractional quantum Hall states are
expectedchamon_sharp_1994; jianhui_wang_edge_2013; meir_composite_1994 to host
neutral modes along with the observed fractionally quantized charged modes
(see Fig. 6b). While these neutral modes do not directly couple to an applied
electric field, they may affect thermal transport or renormalize the tunneling
spectra of nearby charged modes kane_edge-state_1996.

The edge modes at the boundary of a fractional quantum Hall state are expected
to behave like chiral Luttinger liquidswen_chiral_1990; kane_edge-state_1996.
In this state of matter, the quasi-particle wave functions are orthogonal to
the electrons from which they are microscopically constructed. This
‘orthogonality catastrophe’ results in a soft gap in the electron tunneling
density of states which vanishes like $\rho\propto(E-E_{F})^{1/g-1}$, where
the constant $g$ is known as the Luttinger parameter luttinger_exactly_1963. A
remarkable consequence of this fact is that even arbitrarily weak barriers
between edge modes will suppress tunneling at sufficiently low temperature and
bias voltage.

Fig. 3c shows $G_{\mathrm{D}}$ as a function of source-drain voltage
$V_{\text{bias}}$ and $V_{\text{NS}}$, corresponding to the dashed blue
trajectory in Fig. 3a. The data shows a single sharp zero-bias conductance dip
throughout the transmission plateau at
$G_{\mathrm{D}}\approx\frac{4}{3}\frac{e^{2}}{h}$–in contrast with the weak
bias dependence observed for an integer edges (see Fig. 7). In Fig. 3c,
$V_{\text{NS}}$ tunes the electrostatic confinement at the QPC and thus the
height of the tunnel barrier separating the two outermost edge modes incident
upon the junction. As expected, a decrease in $V_{\text{NS}}$ lowers the
potential barrier at the QPC and enhances transmission. The lack of resonant
structure in Fig. 3c, features observed in previous
experimentsmilliken_indications_1996; radu_quasi-particle_2008;
baer_interplay_2014; zimmermann_tunable_2017, even at high transmission
suggests that scattering between edge modes occurs at a single, gate-
controlled saddle point.

To compare our data with expectations from the chiral Luttinger liquid
theorywen_edge_1991; kane_edge-state_1996, Fig. 3d shows the reflection
coefficient, $R$, which measures the probability of backscattering the
outermost fractional edge mode (see also Fig. S3 for a detailed analysis,
which follows Refs. wen_edge_1991; roddaro_nonlinear_2003; radu_quasi-
particle_2008; baer_experimental_2014). For two fractional quantum Hall edges
which are weakly tunnel coupled via the exchange of quasi-particles, the
reflection coefficient $R$, as a function of $V_{\text{bias}}$, may be
computed perturbatively wen_edge_1991. Evidently, even for high transmission,
voltage bias alone can tune $R$ from over 75% to nearly zero, demonstrating
that the incident edge modes become decoupled a low energies – a manifestation
of the ‘orthogonality catastrophe’ – a hallmark of chiral Luttinger liquid
physics.

At zero bias, when quasi-particle backscattering is weak (corresponding to
small $R$), $R$ is expected to follow a simple power law with respect to
temperature, $R\propto T^{2g-2}$. We plot the zero-bias $R$ in Fig. 3e. For
$T$ between 300-550mK—corresponding to $R\lesssim 0.4$—the reflection
coefficient is well fit by a power law with $g=0.47$. In this weak-
backscattering regime, the dependence of $R$ on $V_{\text{bias}}$ and $T$ is
theoretically expectedwen_edge_1991 to follow a scaling form,

$\frac{R(x)}{T^{2g-2}}=A\beta(g+i\frac{e^{*}x}{2\pi},g-i\frac{e^{*}x}{2\pi})\times\\\
\bigg{[}\pi\cosh(\frac{e^{*}x}{2})+2\sinh(\frac{e^{*}x}{2})\text{Im}[\Psi(g+i\frac{e^{*}x}{2\pi})]\bigg{]}$
(1)

where $x=\frac{eV_{\text{bias}}}{k_{b}T}$ is the scaled voltage bias, $\beta$
and $\Psi$ denote the corresponding Euler integrals, $e^{*}$ is the effective
quasi-particle charge in units of the electron charge, and A is a constant
related to the height of the tunnel barrier.

It follows from this equation that if $R(x)$ is scaled appropriately by the
temperature, tunneling spectra taken in the temperature range where the zero-
bias power law applies should follow a universal curve, with $e^{*}$ providing
a single additional adjustable parameter. Fig. 3f shows $R/T^{-1.06}$ versus
$eV_{\text{bias}}/k_{b}T$ for several temperatures in the range of
$300\text{\,}\mathrm{m}\mathrm{K}$ to $550\text{\,}\mathrm{m}\mathrm{K}$. The
curves are observed to collapse onto each other fixing $g=0.47$. The collapsed
curves are then averaged and fit to Eq. (1) extracting the quasi-particle
charge as $e^{*}=0.55\pm 0.001$.

The observed scaling collapse provides evidence of tunneling between chiral
Luttinger liquids at a single point within the QPC. However, the measured
values of $g=0.47$ and $e^{*}=0.55$ are at odds with predictions that that the
boundary of a fractional quantum Hall phase should host a chiral Luttinger
liquid with a quantized Luttinger parameter and quasi-particle charge, namely
$g=1/3$ and $e^{*}=1/3$wen_chiral_1990. We attribute this discrepancy to the
presence of neutral modes at the $\nu=-5/3$ boundary, consistent with an
electrostatically shallow edge permitting significant Coulomb-induced
reconstruction effectskane_edge-state_1996; meir_composite_1994 (See Fig. 6b).
While a universal regime is always expected to exist, for the electrostatic
configuration explored here, this regime most likely occurs significantly
below $300\text{\,}\mathrm{m}\mathrm{K}$, outside of the weak quasi-particle
backscattering limit. Notably, past experiments on a variety of fractional
Hall states in III-V semiconductor systemschang_observation_1996;
chang_plateau_2001; milliken_indications_1996; radu_quasi-particle_2008;
chang_observation_1996; chang_plateau_2001 have also shown significant
deviations from the expected universal behavior. The deviation from
universality in all of these experiments, including the one presented here, is
likely traceable to the complex edge structure arising from the shallow edge
confinement typical of gate-defined tunnel barriers.

Figure 4:  Tuning edge sharpness via electrostatic gating. (a).
$G_{\mathrm{D}}$ map at fixed $\nu_{EW}=-2$ at B = 2T. As in Fig. 3a, the
gates are swept to maintain constant filling in the E, W regions while varying
$N_{QPC}$ and the sharpness of the confining potential. (b) Data of panel a
along the marked trajectories, showing a qualitative change from individual
$1\frac{e^{2}}{h}$ steps to a single continuous transition from
$G_{\mathrm{D}}=2\frac{e^{2}}{h}$ to $G_{\mathrm{D}}=6\frac{e^{2}}{h}$ (c)
Thomas-Fermi simulation of the filling factor at the center of the QPC.

This deficiency may be addressed in our geometry using in situ control of the
confinement potential. The softness of the confinement potential is quantified
by the ratio of $E_{V}\equiv e\frac{\partial\phi_{ext}}{\partial x}\ell_{B}$
(here $x$ is the coordinate perpendicular to the confined edge, and $\ell_{B}$
is the magnetic length) and the Coulomb energy
$E_{C}\equiv\frac{e^{2}}{4\pi\epsilon\ell_{B}}$. In our geometry, independent
control of the voltages applied to the N/S, E/W, and bottom gates is
equivalent to independent control of the filling factor in the E/W regions,
the number of transmitted modes $N_{\text{qpc}}$, and $E_{V}$, allowing us to
explore the effects of the confinement energy within the QPC independently of
the barrier height.

To illustrate this, Figure 4a shows a map of $G_{\mathrm{D}}$ measured at
$B=2$ T with fixed $\nu=-6$ in the E/W regions. In this plot, moving along a
diagonal from lower left to upper right reduces $N_{\text{qpc}}$, while moving
along a diagonal from upper left to lower right increases $E_{V}$. We observe
a qualitative change in the nature of the transition between
$G_{\mathrm{D}}=2\frac{e^{2}}{h}$ to $G_{\mathrm{D}}=6\frac{e^{2}}{h}$ as a
function of $E_{V}$. For small $E_{V}$—i.e. soft confinement—we observe
distinct conductance plateaus corresponding to each integer quantum Hall state
between $\nu=-2$ and $\nu=-6$, indicating that the four distinct edge modes
corresponding to the $\nu=3$ through $\nu=6$ states can be individually
partitioned. For large $E_{V}$, in contrast, the steps merge into one
continuous jump from $G_{\mathrm{D}}=2\frac{e^{2}}{h}$ to
$G_{\mathrm{D}}=6\frac{e^{2}}{h}$, with no discernible intermediate plateaus
at partial transmission (see Fig. 4b).

This behavior can be understood in the context of the competition between
$E_{V}$ and $E_{C}$ at the QPC. In the regime of Fig. 4, in the bulk, the
four-fold spin and valley degeneracy of the Landau levels is lifted due to
quantum Hall ferromagnetismdean_fractional_2020. In the presence of a soft
confining potential, the electron system can lower its energy by forming
islands of incompressible fluid at integer filling factor within the QPC
saddle point, due to the emergence of $E_{C}$-scale integer quantum Hall
ferromagnetic gaps. However, as $E_{V}$ is increased it is no longer
energetically favorable to form such incompressible islands within the QPC,
and instead a smooth change in electron density at the QPC as function of
transmission is preferred. Finite element electrostatics calculations show
$E_{V}/E_{C}$ is approximately $1.5$ and $2.8$ for the trajectories marked AB
and CD in Fig. 4a, respectively, where we take $E_{V}$ from the maximum value
of the potential slope at the boundary between the N and W regions.

To assess the plausibility of this explanation, we perform simulations of the
electron density in our device within the Thomas-Fermi approximation (see
Supplementary Information Section 5 for details of the simulation). These
simulations take as inputs the device electrostatics as well as a
phenomenological model for the graphene chemical potential, which influences
the quantum capacitance. For $E_{V}/E_{C}=2.2$, the simulated density in the
vicinity of the QPC exhibits a “wedding cake” profile, with well defined
strips of integer filling, while for larger $E_{V}/E_{C}=4.4$ the density
drops sharply in a single step across the quantum point contact (see Fig. 8).
As a heuristic proxy for $G_{D}$, we plot the simulated filling factor at the
center of the QPC, $|\nu^{\textrm{QPC}}|$, in Fig. 4c. The soft confinement
regime shows a series of integer steps, while sharp confinement does not. This
indicates that only in the case of soft confinement do the edge-states remain
well separated at the QPC, allowing their transmission to be partioned
individually as observed in Fig. 4a.

As discussed previously, rearrangement of the electron density also causes
significant effects on the edge in the fractional quantum Hall regime. These
effects, referred to generally as “edge-reconstruction,” can nucleate
additional edge modes at soft potential boundaries with quantitative effects
on tunneling behavior. We estimate $E_{V}/E_{C}\sim 1.3$ for the fractional
quantum Hall tunneling experiment of Fig. 3, squarely in the regime where edge
reconstruction effects are expected. Adjusting $E_{V}$ such that the effects
of $E_{C}$ at the edge are negligible should allow exploration of universal
edge mode physics in both tunneling and interferometry experiments.

##  III Conclusions

In summary, we have shown that AFM-based local oxidation
lithographyli_electrode-free_2018—applied to graphite and coupled with a low-
strain dry transfer procedure—allows for nanoscale features to be fabricated
in the gating layers of a van der Waals heterostructure, while keeping the
disorder level sufficiently small to resolve the subtle physics of tunneling
between chiral Luttinger liquids at a single point. Control over $E_{V}$, in
addition to $N_{\text{qpc}}$ and the bulk filling $\nu_{EW}$, provides a new
knob to turn which likely will enable operation of quantum Hall mesoscopic
devices in regimes where universal physics is expected and the behavior is
significantly simpler to interpret than in previous measurements
chang_observation_1996; chang_plateau_2001. We expect these techniques may be
immediately applied to direct probes of topological order, as quantum point
contacts constitute the essential building block for measuring the Luttinger
liquid parameter radu_quasi-particle_2008, quantized thermal Hall
responsebanerjee_observation_2018; banerjee_observed_2017, and
interferometrynakamura_direct_2020. More generally, however, the techniques we
describe enable a new level of complexity in device design, allowing the
equivalent of monolithic integrated circuits built from gate-tunable
correlated phases in van der Waals heterostructures.

##  IV Acknowledgements

The authors thank J. Folk and A. Potts for comments on the manuscript, and
C.R. Dean and J. Swan for advice on humidity stabilization. We additionally
acknowledge Youngjoon Choi for providing AFM topographs of quantum dot arrays.
Work at UCSB was primarily supported by the Air Force Office of Scientific
Research under award FA9550-20-1-0208 and by the Gordon and Betty Moore
Foundation EPIQS program under award GBMF9471. L.C. and N.S. received
additional support from the Army Research Office under award W911NF20-1-0082.
T.W. and M.Z. were supported by the Director, Office of Science, Office of
Basic Energy Sciences, Materials Sciences and Engineering Division of the U.S.
Department of Energy under contract no. DE-AC02-05-CH11231 (van der Waals
heterostructures program, KCWF16). K.K. was supported by the U.S. Department
of Energy, Office of Science, National Quantum Information Science Research
Centers, Quantum Science Center. C.R. was supported by the National Science
Foundation through Enabling Quantum Leap: Convergent Accelerated Discovery
Foundries for Quantum Materials Science, Engineering and Information
(Q-AMASE-i) award number DMR-1906325. K.W. and T.T. acknowledge support from
JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).

##  V Author Contributions

L.A.C and A.F.Y conceived of the experiment. L.A.C and N.S. fabricated the
devices and performed the measurements. L.A.C, N.S, M.Z., T.W., K.K., and
A.F.Y analyzed the data and wrote the paper. M.Z. and S.V. proposed the
simulation methodology. T.W., K.K., and C.R. performed numerical calculations
based on the proposed theoretical modeling. T.T. and K.W. synthesized the
hexagonal boron nitride crystals

##  VI Competing Interests

The authors declare no competing interests.

##  VII Tables

##  VIII Figure Legends/Captions

##  IX References

## References

  * (1) Zibrov, A. A. _et al._ Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level.  _Nature_ 549, 360–364 (2017).
  * (2) Li, J. I. A. _et al._ Even denominator fractional quantum Hall states in bilayer graphene.  _Science_ eaao2521 (2017).
  * (3) Dean, C., Kim, P., Li, J. I. A. & Young, A.  Fractional Quantum Hall Effects in Graphene.  In _Fractional Quantum Hall Effects: New Developments_ , 317–375 (World Scientific, Singapore, 2020).
  * (4) Li, J. I. A., Taniguchi, T., Watanabe, K., Hone, J. & Dean, C. R.  Excitonic superfluid phase in double bilayer graphene.  _Nature Physics_ 13, 751–755 (2017).
  * (5) Liu, X., Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P.  Quantum Hall drag of exciton condensate in graphene.  _Nature Physics_ 13, 746–750 (2017).
  * (6) Ma, L. _et al._ Strongly correlated excitonic insulator in atomic double layers.  _Nature_ 598, 585–589 (2021).
  * (7) Serlin, M. _et al._ Intrinsic quantized anomalous Hall effect in a moiré heterostructure.  _Science_ 367, 900–903 (2020).
  * (8) Li, T. _et al._ Quantum anomalous Hall effect from intertwined moiré bands.  _Nature_ 600, 641–646 (2021).
  * (9) Spanton, E. M. _et al._ Observation of fractional Chern insulators in a van der Waals heterostructure.  _Science_ 360, 62–66 (2018).
  * (10) Xie, Y. _et al._ Fractional Chern insulators in magic-angle twisted bilayer graphene.  _Nature_ 600, 439–443 (2021).
  * (11) Cao, Y. _et al._ Unconventional superconductivity in magic-angle graphene superlattices.  _Nature_ 556, 43–50 (2018).
  * (12) Yankowitz, M. _et al._ Tuning superconductivity in twisted bilayer graphene.  _Science_ 363, 1059–1064 (2019).
  * (13) Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F.  Superconductivity in rhombohedral trilayer graphene.  _Nature_ 598, 434–438 (2021).
  * (14) Zhou, H. _et al._ Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene.  _Science_ 375, 774–778 (2022).
  * (15) Zhang, Y. _et al._ Spin-Orbit Enhanced Superconductivity in Bernal Bilayer Graphene (2022).  URL http://arxiv.org/abs/2205.05087.  Number: arXiv:2205.05087 arXiv:2205.05087 [cond-mat].
  * (16) Dean, C. R. _et al._ Boron nitride substrates for high-quality graphene electronics.  _Nature Nanotechnology_ 5, 722–726 (2010).
  * (17) Young, A. F. _et al._ Electronic compressibility of layer-polarized bilayer graphene.  _Physical Review B_ 85, 235458 (2012).
  * (18) Mayorov, A. S. _et al._ Direct evidence for micron-scale ballistic transport in encapsulated graphene at room temperature.  _1103.4510_ (2011).
  * (19) Wang, L. _et al._ One-Dimensional Electrical Contact to a Two-Dimensional Material.  _Science_ 342, 614–617 (2013).
  * (20) Haigh, S. J. _et al._ Cross-sectional imaging of individual layers and buried interfaces of graphene-based heterostructures and superlattices.  _Nat Mater_ 11, 764–767 (2012).
  * (21) Ronen, Y. _et al._ Aharonov-Bohm effect in graphene-based Fabry-Pérot quantum Hall interferometers.  _Nature Nanotechnology_ 16, 563–569 (2021).
  * (22) Zimmermann, K. _et al._ Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices.  _Nature Communications_ 8, 14983 (2017).
  * (23) Déprez, C. _et al._ A tunable Fabry-Pérot quantum Hall interferometer in graphene.  _Nature Nanotechnology_ 16, 555–562 (2021).
  * (24) Zhao, L. _et al._ Graphene-based quantum Hall interferometer with self-aligned side gates (2022).  URL http://arxiv.org/abs/2206.05623.  ArXiv:2206.05623 [cond-mat, physics:quant-ph].
  * (25) Li, H. _et al._ Electrode-Free Anodic Oxidation Nanolithography of Low-Dimensional Materials.  _Nano Letters_ 18, 8011–8015 (2018).
  * (26) Novoselov, K. S., Mishchenko, A., Carvalho, A. & Neto, A. H. C.  2D materials and van der Waals heterostructures.  _Science_ 353, aac9439 (2016).
  * (27) Masubuchi, S., Ono, M., Yoshida, K., Hirakawa, K. & Machida, T.  Fabrication of graphene nanoribbon by local anodic oxidation lithography using atomic force microscope.  _Applied Physics Letters_ 94 (2009).
  * (28) Butt, H.-J., Farshchi-Tabrizi, M. & Kappl, M.  Using capillary forces to determine the geometry of nanocontacts.  _Journal of Applied Physics_ 100, 024312 (2006).
  * (29) van Wees, B. J. _et al._ Quantized conductance of point contacts in a two-dimensional electron gas.  _Phys. Rev. Lett._ 60, 848–850 (1988).
  * (30) Datta, S.  _Electronic Transport in Mesoscopic Systems_.  Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Cambridge University Press, Cambridge, 1995).  URL https://www.cambridge.org/core/books/electronic-transport-in-mesoscopic-systems/1E55DEF5978AA7B843FF70337C220D8B.
  * (31) Milliken, F. P., Umbach, C. P. & Webb, R. A.  Indications of a Luttinger liquid in the fractional quantum Hall regime.  _Solid State Communications_ 97, 309–313 (1996).
  * (32) Radu, I. P. _et al._ Quasi-Particle Properties from Tunneling in the $\nu$=5/2 Fractional Quantum Hall State.  _Science_ 320, 899–902 (2008).
  * (33) Baer, S. _et al._ Interplay of fractional quantum Hall states and localization in quantum point contacts.  _Physical Review B_ 89, 085424 (2014).
  * (34) Johnson, M. D. & MacDonald, A. H.  Composite edges in the $\nu$=2/3 fractional quantum Hall effect.  _Physical Review Letters_ 67, 2060–2063 (1991).
  * (35) Kane, C. L., Fisher, M. P. A. & Polchinski, J.  Randomness at the edge: Theory of quantum Hall transport at filling $\nu$=2/3.  _Phys. Rev. Lett._ 72, 4129–4132 (1994).
  * (36) Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J.  Half-integer conductance plateau at the $\nu = 2/3$ fractional quantum Hall state in a quantum point contact (2022).  URL http://arxiv.org/abs/2211.17240.  ArXiv:2211.17240 [cond-mat].
  * (37) Sabo, R. _et al._ Edge reconstruction in fractional quantum Hall states.  _Nature Physics_ 13, 491–496 (2017).
  * (38) Chamon, C. d. C. & Wen, X. G.  Sharp and smooth boundaries of quantum Hall liquids.  _Physical Review B_ 49, 8227–8241 (1994).
  * (39) Jianhui Wang, Yigal Meir & Yuval Gefen.  Edge Reconstruction in the $\nu = 2/3$ Fractional Quantum Hall State.  _Phys. Rev. Lett._ 111 (2013).
  * (40) Meir, Y.  Composite edge states in the $\nu$ = 2/3 fractional quantum Hall regime.  _Physical Review Letters_ 72, 2624–2627 (1994).
  * (41) Kane, C. L. & Fisher, M. P. A.  Edge-State Transport.  In _Perspectives in Quantum Hall Effects_ , 109–159 (John Wiley & Sons, Ltd, 1996).  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527617258.ch4.
  * (42) Wen, X. G.  Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states.  _Physical Review B_ 41, 12838–12844 (1990).
  * (43) Luttinger, J. M.  An Exactly Soluble Model of a Many‐Fermion System.  _Journal of Mathematical Physics_ 4, 1154–1162 (1963).
  * (44) Wen, X.-G.  Edge transport properties of the fractional quantum Hall states and weak-impurity scattering of a one-dimensional charge-density wave.  _Physical Review B_ 44, 5708–5719 (1991).
  * (45) Roddaro, S. _et al._ Nonlinear Quasiparticle Tunneling between Fractional Quantum Hall Edges.  _Physical Review Letters_ 90, 046805 (2003).
  * (46) Baer, S. _et al._ Experimental probe of topological orders and edge excitations in the second Landau level.  _Physical Review B_ 90, 075403 (2014).
  * (47) Chang, A. M., Pfeiffer, L. N. & West, K. W.  Observation of Chiral Luttinger Behavior in Electron Tunneling into Fractional Quantum Hall Edges.  _Physical Review Letters_ 77, 2538–2541 (1996).
  * (48) Chang, A. M., Wu, M. K., Chi, C. C., Pfeiffer, L. N. & West, K. W.  Plateau Behavior in the Chiral Luttinger Liquid Exponent.  _Physical Review Letters_ 86, 143–146 (2001).
  * (49) Banerjee, M. _et al._ Observation of half-integer thermal Hall conductance.  _Nature_ 559, 205–210 (2018).
  * (50) Banerjee, M. _et al._ Observed quantization of anyonic heat flow.  _Nature_ 545, 75–79 (2017).
  * (51) Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J.  Direct observation of anyonic braiding statistics.  _Nature Physics_ 16, 931–936 (2020).
  * (52) Abbas, A.  _Nanofabrication Using Electron Beam Lithography: Novel Resist and Applications_.  Master of Applied Science, University of Waterloo, Waterloo, Ontario CA (2013).
  * (53) Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F.  Independent superconductors and correlated insulators in twisted bilayer graphene.  _Nature Physics_ 1–5 (2020).
  * (54) Akash Laturia, Maarten L. Van de Put & William G. Vandenberghe.  Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk.  _npj 2D Mater Appl_ 2 (2018).
  * (55) We first use an unbounded range for $n(\mathbf{r})$ to determine the external potential that fix bulk filling $\nu_{EW}=1$ and $\nu_{NS}=0$. Then we constrain $n(\mathbf{r})$ to fall in between $\nu=0$ and $\nu=1$ such that any transition from $\tilde{\nu}=0$ to $\tilde{\nu}=1$ in $\tilde{\nu}(\mathbf{r})$ has a finite width of scale $\ell_{B}$.
  * (56) Yang, F. _et al._ Experimental Determination of the Energy per Particle in Partially Filled Landau Levels.  _Physical Review Letters_ 126, 156802 (2021).

##  X Methods

###  X.1 van der Waals Heterostructure Assembly

Graphene and hBN were mechanically exfoliated from bulk crystals using a
combination of thermal release tape and scotch magic tape. The initial mother-
tape is prepared using 3M Scotch-brand magic tape for graphite or 3M Scotch-
brand greener magic tape for hBN. For hBN, thermal release tape is adhered to
the mother-tape to generate a daughter tape. For graphite, the daughter-tape
is made with 3M Scotch-brand magic tape. The daughter-tape is removed from the
mother-tape, cleaving the bulk crystals along the c-axis, then transferred
onto a 1cm x 1cm doped Si chip with 285nm of thermally-grown SiO2 on the
surface. For graphite, the substrate is heated to $110^{\circ}$ C for 60
seconds, before removing the tape quickly to reduce glue residue remaining on
the SiO2 surface. For hBN, the daughter-tape is transferred onto the SiO2
surface at room temperature and is removed from the SiO2 slowly. The Si/SiO2
substrates are cleaned by a standard solvent process: the chip is cleaned in
acetone for 5 minutes in a high power ultrasonic bath, followed by an IPA
wash, and finished with an N2 blow-dry. Additionally, for graphite the SiO2
surface is treated in O2 plasma at 100W and 300mTorr for 60 seconds in order
to promote adhesion of large multilayers. The resulting exfoliated crystals
are then characterized by optical microscopy.

AFM-LAO was performed on a Bruker Dimension Icon AFM. Exfoliated graphite
flakes, prepared as described above, are loaded into the AFM. The humidity is
controlled using a bang-bang style humidity controller. The plant is formed by
a beaker filled with 250mL of deionized water placed on a hot plate at
$120^{\circ}$ C. Once the humidity sensor measures higher than 50% RH, the
hot-plate is switched off. We pattern sub 100 nm crosses into a 3 nm graphite
flake to form the top gate of the quantum point contact. This is accomplished
using a Pt/Ir coated Arrow-NcPt AFM probe from nanoandmore. A topographical
map of the graphite flake is obtained and the cross pattern is placed in an
area with no visible defects. The lithography is performed in Bruker’s Nanoman
software package which allows for precise control of the direction, speed, and
deflection of the conducting AFM probe. Graphite can be etched with AFM-LAO
under a variety of conditions, however we have found that lithography
performed in contact mode with an 18V peak-to-peak excitation at 150kHz
provided the smallest line-widths achievable in our system. For new AFM
probes, typical line-widths are on the order of 60-70 nm . This leads to a QPC
critical dimension (tip-to-tip distance) between 90 and 100 nm . However, due
most likely to hydrocarbon build up or natural wear on the AFM probe, the cut-
width broadens to 100 nm after $\sim$150 $\mu$m of cutting.

Before assembling the van der Waals heterostructure we fabricate a ‘transfer-
slide:’ a PDMS stamp adhered to a glass slide with a polycarbonate laminate
transferred on top used to ‘pick-up’ the first layer of hBN. Initially, 8g of
Sylgard 184 PDMS is mixed in a 10:1 ratio by weight with a curing agent and
poured into a standard 100 mm plastic petri dish. The PDMS is left to cure at
room temperature for 24 hours in a vacuum chamber in order to remove any
bubbles that formed during mixing. Additionally, another 3g of Sylgard 184
PDMS solution is mixed with a curing agent and left to cure partially at room
temperature under vacuum for 2 hours. A PDMS cylinder is cut out using a 2 mm
hole punch from the 8g PDMS batch and is then adhered to a glass slide using
the partially cured PDMS. An additional droplet of partially cured PDMS is
pipetted onto the cylinder in order to form a dome. Slides are left to finish
curing for another 24 hours at room temperature. The resulting slides are then
inspected under an optical microscope for dirt particulates.

The polycarbonate laminate is made from a 13.3% wt/vol ratio of Bisphenol-A
polycarbonate (PC) in cyclopentanone. Unaided, cyclopentanone will only
dissolve a 4% wt/vol solution of PC, however by using the QSonica 500
ultrasonic wand, solutions up to 15% wt/vol can be made. This recipe was
derived from previous work by Abbas abbas_nanofabrication_2013. Due to its
significantly lower vapor pressure than chloroform, PC dissolved in
cyclopentanone can be effectively spun onto a Si substrate. Thin films of PC
are prepared by spinning the mixture onto a Si substrate at 1400 rpm for 60
seconds. The thin film is then subsequently transferred onto a domed PDMS
stamp. The transfer slide is heated between $160^{\circ}$C to $180^{\circ}$C
for 5 minutes in order to increase the adhesion between the PC film and the
PDMS stamp as well as to remove any air bubbles that may have formed during
transfer of the film.

We begin the assembly of the van der Waals heterostructure by picking up an
hBN flake larger than 100um in both width and length. Ensuring the first hBN
has the largest area aids in assembling the remaining layers since subsequent
crystals will adhere entirely to another van der Waals material. The transfer
slide is engaged at a $0^{\circ}$ tilt angle, with the substrate heated to
$70^{\circ}$C, and with the touch-down point 100-150um away from the center of
the first hBN layer. Once the transfer slide is brought into near contact the
substrate is heated to $105^{\circ}$C. This causes the transfer slide to fully
engage and laminate over the first hBN layer. The substrate is then cooled
naturally back down to $70^{\circ}$ which retracts the transfer slide and
removes the target hBN flake from the SiO2. Next, the AFM-LAO etched graphite
top gate is picked up while entirely encapsulated by the large initial hBN
layer. There are a number of advantages to using a domed PDMS transfer slide,
one of which is that the engage point is in the center of the PDMS stamp, as
opposed to the edge, which has fewer dust particulates that can interfere with
assembly. Moreover, domed PDMS transfer slides are known to reduce strain
during vDH assembly saito_independent_2020; it turns out this technique is
critical for picking up AFM-LAO nano-structures without inducing tears or
folds.

The PC laminate is extremely uniform as a result of being spun onto a Si
substrate. This also minimizes a common issue in assembling van der Waals
heterostructures where the polycarbonate exhibits stochastic adherence to the
SiO2 substrate. This often causes a ‘jerk’ like motion during stacking which
easily can cause graphite nano-structures to tear. The remaining layers are
assembled in the same manner; the full resulting device stack can be seen in
Fig. 1c. The stack is then deposited onto a doped silicon substrate with 285nm
of thermally grown oxide, which forms the basis for a global bottom gate used
to dope the graphene contacts. This is accomplished by engaging the transfer
slide and heating the substrate to $180^{\circ}$C (above the glass transition
temperature of PC) to detach the PC from the PDMS stamp. The laminate is then
dissolved in chloroform for a minimum of 30 min. Afterwards, the sample is
rinsed in acetone and IPA, and then blown-dry with N2.

###  X.2 Device Post Processing

Heterostructures are post-processed using standard electron beam lithography,
vacuum deposition, and dry-etching processes. A device-defining plasma etch is
used to separate the nanotextured graphite into four quadrants that we label
North (N), South (S), East (E), and West (W): this is accomplished by
inductively coupled plasma (ICP) etching in 40 sccm of CHF3 and 4 sccm of O2.
This etch also separates the graphene contacts C1-8 such that any two contacts
are only connected through the dual-gated region. Additionally, the conducting
Si substrate is used to dope each contact and prevent the formation of p-n
junctions at the boundary between the contacts and the dual-gated region.
Finally, several trenches are etched across the boundary to the dual-gated
region to introduce local scattering sites that improve equilibration between
the contacts and quantum Hall edge modes in the device region.

The etch mask is patterned by lifting off 40nm of Al using a polymethyl
methacrylate (PMMA) A4 495K / A2 950K bilayer resist. The PMMA is exposed
using a 30kV electron-beam in an FEI SEM at 4004uC/cm2 and developed in a
mixture of DI:IPA 1:3 kept at $10^{\circ}$C. Al is deposited at 0.3A/s and the
lift off is done for over 12 hours in N-Methyl-2-Pyrrolidone (NMP). Post
etching, the Al is dissolved in AZMIF300 photoresist developer which contains
$<3$% by weight of Tetramethylammonium hydroxide (TMAH). Edge contacts are
deposited onto the exposed graphene contacts and graphite gates using the same
bilayer PMMA mask. Before vacuum deposition, a brief contact cleaning etch in
an ICP with 40 sccm of CHF3 and 9 sccm of O2 is performed. Subsequently, a
metal stack of Cr/Pd/Au 3/15/185 nm is deposited and lifted off. Special care
is taken to deposit the Cr layer at 0.3A/s in order to improve coating
uniformity.

###  X.3 Measurement

Experiments were performed in a dry dilution refrigerator with a base
temperature of 20mK. Electronic filters are used in line with transport and
gate contacts in order to lower the effective electron temperature. To improve
edge mode equilibration to the contacts most measurements are performed at
300mK unless otherwise noted. Electronic measurements were performed using
standard lock-in amplifier techniques. For the diagonal conductance
measurements an AC voltage bias at 17.77Hz is applied via a 1000x resistor
divider to (see Fig. 2a for contact references) C3-4 and the resulting current
is measured using an Ithaco 1211 trans-impedance amplifier on C5-6 with a gain
of $10^{-7}$ A/V. We use transport contacts in pairs to decrease the contact
resistance and improve edge-state equilibration, exciting an AC voltage on
contacts C3/C4, and the measuring the current $I_{out}$ on contacts C5/C6. The
diagonal voltage drop, $V_{D}$, is measured between C1/C2 and C7/C8. We then
define $G_{\mathrm{D}}\equiv I_{out}/V_{D}$. Note that the sign convention
used here is such that $G_{D}$ is positive for data measured on the hole side
of the device, $\nu<0$. The voltage is measured between contacts C1-2 and C7-8
with an SR560 voltage pre-amplifier with a gain of 1000. For two terminal
measurements the same AC bias is applied to contacts C1-4 and the current is
measured via C5-8. DC bias was added on top of the AC bias using a passive
summer.

###  X.4 Thomas-Fermi calculation

We consider a classical effective model wherein the electron density
$n(\mathbf{r})$ of the two-dimensional electron gas adjusts according to the
local electrostatic potential and compressibility of an interacting Landau-
level (LL). The classical energy functional can be decomposed into the Hartree
energy, the interaction with an externally applied potential
$\Phi(\mathbf{r})$, and the remainder,

$\displaystyle\mathcal{E}[n(\mathbf{r})]$
$\displaystyle=\frac{e^{2}}{2}\int_{{\bf{r_{1}},\bf{r_{2}}}}n({\bf{r_{1}}})V({\bf{r_{1}},\bf{r_{2}}})n({\bf{r_{2}}})$
(2)
$\displaystyle\qquad\qquad-e\int_{\mathbf{r}}\Phi(\mathbf{r})n(\mathbf{r})+\mathcal{E}_{xc}[n(\mathbf{r})],$

$\mathcal{E}_{xc}[n]$ contains not only the exchange-correlation energy but
also single-particle contributions (e.g. inter-LL interactions, , Zeeman
energies, etc.). If $n(\mathbf{r})$ varies slowly compared with $\ell_{B}$, we
may neglect the dependence of the functional $\mathcal{E}_{xc}[n(\mathbf{r})]$
on the gradient $\bm{\nabla}n$ and employ the local density approximation
(LDA),

$\mathcal{E}_{xc}[n(\mathbf{r})]=\int_{\mathbf{r}}E_{xc}(n(\mathbf{r}))$ (3)

where $E_{xc}(n)$ is determined for a system at constant density $n$. Our aim
here is to find the density configuration $n(\mathbf{r})$ corresponding to the
global minimum of the free energy $\mathcal{E}[n(\mathbf{r})]$.

We follow the geometry shown in Fig. 1g. There are four top gates a distance
$d_{t}$ above the sample and one back gate a distance $d_{b}$ below, between
which the space is filled by hBN with dielectric constant $\epsilon_{\perp}=3$
and $\epsilon_{\|}=6.6$ laturia_dielectric_2018. The N/S gates and the E/W
gates are shifted by $w_{NS}$ and $w_{EW}$, respectively, from the center. We
make an approximation by treating the cut-out “X”-shaped region to define the
gates as a metal held at fixed voltage $V=0$ (rather than as a vacuum). This
allows us to analytically solve for the electrostatic Green’s function and
gate-induced potentials without resorting to e.g. COMSOL simulations.

For a coarse-grained system with a finite resolution grid, the classical
energy functional becomes

$\displaystyle\mathcal{E}[\\{n(\mathbf{r})\\}]$
$\displaystyle=\mathcal{E}_{C}+\mathcal{E}_{xc}+\mathcal{E}_{\Phi}$ (4)
$\displaystyle\mathcal{E}_{C}$
$\displaystyle=\frac{1}{2A}\sum_{\mathbf{q}}V(\mathbf{q})n(\mathbf{q})n(-\mathbf{q})$
$\displaystyle\mathcal{E}_{xc}$
$\displaystyle=\sum_{\mathbf{r}}E_{xc}(n(\mathbf{r}))\mathop{}\\!\mathrm{d}A,$
$\displaystyle\mathcal{E}_{\Phi}$
$\displaystyle=\sum_{\mathbf{r}}\Phi(\mathbf{r})n(\mathbf{r})\mathop{}\\!\mathrm{d}A$

where $\mathop{}\\!\mathrm{d}A=\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
is the grid area, $A$ is the total area, and
$n(\mathbf{q})=\sum_{\mathbf{r}}e^{-i\mathbf{q}\cdot\mathbf{r}}n(\mathbf{r})\mathop{}\\!\mathrm{d}A$.
$\mathcal{E}_{C}$ is set by the gate-screened Coulomb interaction,

$V(\mathbf{q})=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{\mathrm{hBN}}}\frac{4\pi\sinh\left(\beta
d_{t}|\mathbf{q}|\right)\sinh\left(\beta
d_{b}|\mathbf{q}|\right)}{\sinh\left(\beta(d_{t}+d_{b})|\mathbf{q}|\right)|\mathbf{q}|},$
(5)

$\epsilon_{\mathrm{hBN}}=\sqrt{\epsilon_{\perp}\epsilon_{\|}}$ and
$\beta=\sqrt{\epsilon_{\|}/\epsilon_{\perp}}$. $\mathcal{E}_{\Phi}$ is the one
body potential term arising from the potential $\Phi(\mathbf{r})$ on the
sample due to the adjacent gates,

$\Phi(\mathbf{q})=-eV_{t}(\mathbf{q})\frac{\sinh(\beta
d_{b}|\mathbf{q}|)}{\sinh(\beta(d_{t}+d_{b})|\mathbf{q}|)}-eV_{B}\frac{d_{t}}{d_{t}+d_{b}}$
(6)

where $V_{t}(\mathbf{q})$ is the top gate potential and $V_{B}$ is the back
gate potential.

The remaining energy $E_{xc}$ is defined by integrating the chemical potential
$\mu$,

$E_{xc}(n)=\int_{0}^{n}\mathop{}\\!\mathrm{d}n^{\prime}\mu(n^{\prime}).$ (7)

which encodes information about the IQH and FQH gaps and the electron
compressibility. For $\mu$ we use a phenomenological model detailed in SM.

Physically, $n(\mathbf{r})$ can only vary on the scale of the magnetic length
$\ell_{B}$ due to the underlying quantum Hall wavefunction. To capture this
feature, we implement a square grid with periodic boundary conditions and
meshing much finer than the scale of $\ell_{B}$, and then evaluate
$\mathcal{E}$ with respect to the Gaussian convoluted density profile 111We
first use an unbounded range for $n(\mathbf{r})$ to determine the external
potential that fix bulk filling $\nu_{EW}=1$ and $\nu_{NS}=0$. Then we
constrain $n(\mathbf{r})$ to fall in between $\nu=0$ and $\nu=1$ such that any
transition from $\tilde{\nu}=0$ to $\tilde{\nu}=1$ in
$\tilde{\nu}(\mathbf{r})$ has a finite width of scale $\ell_{B}$.

$\tilde{n}(\mathbf{r})=\mathcal{N}^{-1}\sum_{\mathbf{r}^{\prime}}n(\mathbf{r}^{\prime})e^{-\left\lvert\mathbf{r}-\mathbf{r}^{\prime}\right\rvert^{2}/2\ell_{B}^{2}},$
(8)

where $\mathcal{N}$ is the corresponding normalization factor. We use a basin-
hopping global optimizer with local L-BFGS-B minimization to vary
$\\{n(\mathbf{r})\\}$ and find the lowest energy configuration.

##  XI Data Availability

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

##  XII Extended Data Figures

Figure 5:  Precise Determination of $\alpha$ and Hall conductance at B = 6T
and B = 13T: (a) Longitudinal and Hall conductance in the W region of the
device for two values of the bottom gate, $V_{\mathrm{B}}=1.0V$ and
$V_{\mathrm{B}}=2.0V$, at B = 10T. All other regions are set to $\nu=0$. (b)
Calculated shift $\Delta V_{W}$ applied to the $V_{\mathrm{B}}=1.0V$ trace
shows overlap of identical features between the two traces. (c) The region of
vanishing longitudinal conductance in $\nu=-2/3$ was used to numerically
determine the shift $\Delta V_{W}$ by minimizing the sum of the norm-squared
differences between the two traces over a region around $\nu=-2/3$. (d)
$\sigma_{xx}$ and $\sigma_{xy}$ versus $V_{\text{W}}$ at $B=13T$ and
$V_{\text{B}}=1V$. (e) $1/R_{xy}$ measured on the west side of the device
versus $V_{W}$ while $\nu_{N}=\nu_{S}=\nu_{E}=0$ is kept fixed, and
$V_{\mathrm{B}}=-0.33V$. Inset: current measured during $1/R_{xy}$ sweep
showing $\nu=0$ gap. Figure 6: Possible Edge Structures in $\nu=-5/3$ (a)
Hole-conjugate FQH states such as $|\nu|=2/3,5/3$ states can be modeled by a
Laughlin-like FQH state of holes within a bulk integer quantum Hall state,
leading to a small strip of increased filling factor around the edge of the
sample. This is shown schematically in panels a and c by plotting the filling
factor of holes $\nu_{h}\equiv-\nu$ at the boundary between a $\nu=-5/3$ and
$\nu=-1$ state, where the relevant fractional edges measured in the experiment
occur. The MacDonald model johnson_composite_1991 of the resulting edge
structure posits a downstream integer mode at the outermost edge of the
sample, as well as an upstream (counter-propagating) fractional mode. (b) In
real experiments, the two counter-propagating charged modes are rarely
observed, but rather mix through the presence of inter-edge interactions,
yielding a single effective charge-2e/3 mode propagating downstream, as well
as an upstream charge-neutral mode, as explained by the Kane-Fisher-Polchinski
model kane_randomness_1994. (c) A sufficiently soft confining potential may
make it energetically favorable to redistribute the charge in the system and
create an additional strip of density $\nu_{h}=4/3$, introducing a set of two
additional counter-propagating fractional edge modes: the Meir model.
meir_composite_1994 (d) In a real system, where these modes can also mix, the
resulting mode structure may contain two downstream fractional-conductance
modes as well as two upstream neutral modes. This scenario is consistent with
the observation of multiple fractional conductance steps within the $\nu=-5/3$
state.  Figure 7: Tunneling conductance in an integer vs. fractional edge (a)
Plot of the tunneling conductance across an integer conductance step, in the
$\nu$=1 state. In the fully reflecting and fully transmitting limits, the
conductance is constant for $V_{DC}$ less than about —1 mV—, and smoothly
varies as the edge is transmitted. (b) For a fractional edge state, the
conductance remains highly suppressed even when the edge state is partially
transmitted, with a sharply nonlinear $dI/dV$ near $V_{DC}$=0. Even when the
edge state is fully transmitted, and $dI/dV(V_{DC}=0)=4/3$, the tunneling
conductance remains nonlinear. (c) and (d) present linecuts of the data in (a)
and (b) respectively for comparison.  Figure 8:  Thomas-Fermi simulations of
the QPC pinch-off (a) The filling at the center of the QPC
$\nu^{\mathrm{QPC}}$ as a function of $V_{B}$ and $V_{\text{NS}}+\alpha
V_{B}$, with the bulk filling of the east/west regions fixed at $\nu_{EW}=-6$.
The result qualitatively mimics the measured $G_{D}$ shown in Fig. 4a since
$\nu^{\mathrm{QPC}}$ determines the number of transmitted modes and therefore
the diagonal conductance. Two line cuts which correspond to $E_{V}/E_{C}=2.2$
and $4.4$ are shown in Fig. 4c. (b-c) Calculated filling factor $\nu$ for a
realistic device geometry at $B=$2\text{\,}\mathrm{T}$$ for (b)
$E_{V}/E_{C}=4.4$ and (c) $E_{V}/E_{C}=2.2$. When $E_{V}/E_{C}=2.2$, there
exists an incompressible island with $\nu^{\textrm{QPC}}=-4$ at the center of
the QPC. Contours of $\nu=n+1/2$ are shown as white dotted lines, indicating
the location of chiral edge modes, two-pairs of which are transmitted through
the QPC. This illustrates the rule $N^{\textrm{QPC}}=\nu^{\textrm{QPC}}+2$.

Supplementary Information: Nanoscale electrostatic control in ultra clean van
der Waals heterostructures by local anodic oxidation of graphite gates

##  XIII Partitioning of integer quantum Hall edge modes

Fig. S1a shows a map of the $G_{\mathrm{D}}$ at B=6 T and T=300 mK as we vary
the $V_{\text{NS}}\equiv V_{N}=V_{S}$ and $V_{\text{EW}}\equiv V_{E}=V_{W}$.
As expected for IQH transport, the conductance maps in Fig. S1a-c are
dominated by regions of fixed conductance at integer multiples of
$\frac{e^{2}}{h}$, corresponding to transmission of an integer number of
chiral edge states across the device. For each conductance map in Fig. S1a-c,
the graphite bottom gate, $V_{\mathrm{B}}$, is fixed to a different voltage.
Using the capacitive lever arm $\alpha\equiv C_{B}/C_{T}=1.773$ (see Fig. 5),
the ranges of $V_{\text{EW}}$ and $V_{\text{NS}}$ are chosen such that the
electronic density in each region of the monolayer is kept in the same range
in all three panels. The precise mapping of gate voltages to $\nu$ is
determined by measuring the Hall effect in the W quadrant (see Fig. 5).

Figure S1: Quantum point contact operation in the integer quantum Hall regime.
(a) Diagonal conductance $G_{\mathrm{D}}$ plotted as a function of
$V_{\text{NS}}$ and $V_{\text{EW}}$ at B=6 T and T=300 mK for
$V_{\mathrm{B}}=0.56$, (b) $V_{\mathrm{B}}=-0.33$, and (c)
$V_{\mathrm{B}}=-1.22$. In panels a-c, $V_{\text{NS}}$ and $V_{\text{EW}}$
denote the voltages applied to north and south or east and west gates,
respectively. The ranges of $V_{\text{NS}}$ and $V_{\text{EW}}$ are chosen
such that the filling factor $\nu_{EW}=\nu_{E}=\nu_{W}\in[-6,0]$ and
$\nu_{NS}=\nu_{S}=\nu_{N}\in[-2,1]$ for each conductance map in a-c. The range
over which $\nu_{NS}=0$ is marked in each plot by the white bar. (d) The same
trace as in the main text Fig. 2b. The trace in the multidimensional parameter
space intersects the $G_{\mathrm{D}}$ maps of panels a-c at the position
marked I, II, and III (e) Schematic depiction of the filling factors within
the QPC at point I in panel a, (f) point II in panel b, and (g) point III in
panel c. For points I-III, the filling factor in the N/S/E/W regions is
constant but the fringe fields vary with $V_{\mathrm{B}}$, fully modulating
transmission of the two outermost edge modes through the QPC.

Tracing the behavior of the transitions between conductance plateaus reveals
two distinct regimes. In the first, plateau transitions are controlled by only
$V_{\text{EW}}$ or $V_{\text{NS}}$, producing steps in $G_{\mathrm{D}}$ along
either horizontal or vertical lines in the $(V_{\text{NS}},V_{\text{EW}})$
plane. In this regime, the diagonal conductance is primarily determined by the
number of edge states transmitted along the physical edge of the device, far
from the QPC. For example at point I in Fig. S1a, the filling factor of the
north and south regions, $\nu_{NS}=\nu_{N}=\nu_{S}$, is fixed to $\nu_{NS}=0$,
while the E/W regions are fixed to $\nu_{EW}=\nu_{E}=\nu_{W}=-4$. Point I sits
to the right of a vertical transition, solely controlled by $V_{\text{NS}}$,
where $G_{\mathrm{D}}$ goes from 0 to 1. Decreasing $V_{\text{NS}}$ starting
at point I changes the filling $\nu_{NS}$ from 0 to -1, adding an edge mode in
the N/S regions at the physical device boundary and increasing
$G_{\mathrm{D}}$.

In the second regime, the plateau transitions are influenced by both
$V_{\text{EW}}$ and $V_{\text{NS}}$, producing a step in $G_{\mathrm{D}}$
along lines of slope $\approx-1$ in the $(V_{\text{NS}},V_{\text{EW}})$ plane.
This behavior is expected when edge modes are transmitted through the center
of the QPC, where they are equally sensitive to the fringe electric fields of
each of the N/S/E/W gates. Consider again point I which sits just above a
diagonal transition, where $G_{\mathrm{D}}$ goes from 0 to 1. Near the
transition, $\nu_{NS}=0$ and no conduction across the device can occur along
the etched boundary – all current must be carried via edge modes through the
QPC. However, at the transition, $G_{\mathrm{D}}$ may change sharply by either
an equal perturbation in $V_{\text{NS}}$ or $V_{\text{EW}}$ while maintaining
$\nu_{NS}=0$. The existence of such transitions in $G_{\mathrm{D}}$ implies
the filling factor in the QPC center can be held fixed via equal and opposite
modulations of $V_{\text{NS}}$ and $V_{\text{EW}}$: lines separating differing
values of $G_{\mathrm{D}}$ which are parallel to
$V_{\text{NS}}+V_{\text{EW}}=0$ demarcate sharp boundaries between regions of
different filling factor in the center of the QPC itself.

The location in density of the diagonal steps in $G_{\mathrm{D}}$ shift as a
function of $V_{\mathrm{B}}$ (Figs. S1b-c), in contrast to the horizontal and
vertical transitions whose locations in density are unaffected. This behavior
follows from the device electrostatics: Near the device boundary, transport is
determined directly by the bulk filling factor in the N/S/E/W regions. Since
the graphite bottom gate uniformly modulates the density of the whole
monolayer, the role of $V_{\mathrm{B}}$ is merely to induce a chemical
potential shift in the whole device which is compensated by offsetting the
applied gate voltages. This is not true in the central region, however, which
is doped by the fringe fields of N/S/E/W gates.

It follows that tuning the bottom gate while keeping the densities in the N,
S, E, and W regions constant changes the electrostatics of the QPC. Points
I-III in Figs. S1a-c correspond to identical carrier densities away from the
QPC, with $\nu_{EW}=-4$ and $\nu_{NS}=0$. At point I, all the edge modes are
pinched off and $G_{\mathrm{D}}=0$. As $V_{\mathrm{B}}$ is decreased, the
filling factor in the QPC changes, leading to the transmission of one
additional edge modes at point II and and two additional modes at point III.
Fig. S1d shows the continuous evolution between points I, II, and III as a
function of $V_{\text{EW}}-V_{\mathrm{B}}$, corresponding to tuning the QPC
electrostatics. A schematic depiction of the corresponding filling factor maps
in real space are shown in Figs. S1e-g.

##  XIV Extended Integer QPC Operation

Index | $G_{\mathrm{D}}[\frac{e^{2}}{h}]$ | $\nu_{EW}$ | $\nu_{NS}$ | $\nu_{qpc}$
---|---|---|---|---
I | 1 | -4 | 0 | -1
II | 1 | -2 | -1 | -1
III | 1 | -1 | -2 | 0
IV | 1 | -1 | -2 | -1
V | 2 | -2 | -2 | -2
VI | 0.7 | -1 | -4 | -2

The full parameter space that determines the value of $G_{\mathrm{D}}$ when
tuned with $V_{\text{NS}}$ and $V_{\text{EW}}$ has features which depend on
either $V_{\text{NS}}$ or $V_{\text{EW}}$ individually or features
perpendicular to $V_{\text{NS}}+V_{\text{EW}}=0$ (indicated as dashed white
lines in Fig. S2a). The former are dominated by physics at the etched edge of
the device where the edge modes of a $pp^{\prime}p$ junction are fully
equilibrated zimmermann_tunable_2017. The latter are interpreted as boundaries
between operating points that have differing values of $\nu_{qpc}$ – the
filling factor in the center of the device, determined by the fringe fields of
the four quadrant top gates (for a fixed $V_{\mathrm{B}}$). Fig. S2a has
several points denoted by roman numerals which correspond either to QPC
operation or conductance through the edge of the device. Fig. S2b shows an
illustration of the inferred filling factor in each region of the device for
each point marked in Fig. S2a. This information, along with the associated
$G_{\mathrm{D}}$ for each point is repeated in the table.

Figure S2: Extended Integer QPC Operation: (a) $G_{\mathrm{D}}$ plotted versus
$V_{\text{NS}}$ and $V_{\text{EW}}$ with $V_{\mathrm{B}}=-0.33V$. Lines
parallel to $V_{\text{NS}}+V_{\text{EW}}=0$ demarcate transitions between
integer $G_{\mathrm{D}}$ values indicating a rapid change in the density at
the QPC. Roman numerals I-VI correspond to various unique combinations of
$\nu_{NS}$, $\nu_{EW}$, and $\nu_{qpc}$, where $\nu_{qpc}$ is the filling
factor at center of the QPC. (b) Illustrations of the inferred filling factor
in each region of the device corresponding to the operating points in panel a.

Point VI is of particular note since it falls to the left of a boundary in
$G_{\mathrm{D}}$, where $G_{\mathrm{D}}$ is a fractional value below 1, that
is intersected by a dashed line. This indicates the transition is sensitive to
the potential at the QPC, but is in a region where at least one edge mode is
transmitted along the edge of the device since $\nu_{NS}<-1$. The
corresponding filling factors at VI are $\nu_{NS}=-4$, $\nu_{EW}=-1$, and
$\nu_{qpc}=-2$. Starting from point IV, where $\nu_{qpc}=-1$, as
$V_{\text{NS}}$ becomes more negative, the filling factor in the QPC is more
strongly doped towards $\nu_{NS}$. Eventually, at point VI, the filling factor
in the QPC increases to $\nu_{qpc}=-2$, and an edge mode bridges the north and
south regions through the $\nu_{EW}=-1$ bulk. This creates a scattering
channel across the device that reduces the conductance to the expected value
of $G_{\mathrm{D}}\sim 2/3$ for a $pp^{\prime}p$ junction of the given filling
factors (-1, -2, -1) zimmermann_tunable_2017; deprez_tunable_2021. This shows
it is possible to see QPC behavior even with a background conductance through
the etched edge of the device, however the value of $G_{\mathrm{D}}$ requires
some interpretation. Consequently, for most experiments presented in the main
text we focus exclusively on regimes where there is no conduction along the
edge of the device, simplifying possible interpretations of fractional values
of $G_{\mathrm{D}}$.

##  XV DC Voltage Bias Correction

We used a set of cryogenic filters at the mixing chamber in order to help
equilibrate the electrons in the graphene 2DEG to the temperature of the
dilution refrigerator (at the mixing chamber). These filters contain a set of
RF pi-filters with small component values which at low meausrement frequencies
are negligible. In addition, there are a set of RC filters with a roll-off
frequency of 6.6kHz; these filters have a series resistance of
$R=$3\text{\,}\mathrm{k}\Omega$$ each. In order to investigate the DC bias
directly across the QPC it is necessary to take into account known filter
resistances since they inherently form a voltage divider between themselves
and the two-terminal resistance across the QPC defined as $R_{\text{DUT}}$.
Determining the relationship between the applied DC voltage at the input of
the filters (a known value defined as $V_{\text{DC}}$) and the voltage bias
across the sample, defined as $V_{\text{DUT}}$, is straightforward. At low
frequencies, the current drawn to any floating contacts is negligible so the
DC voltage across the sample is well approximated by

$V_{\text{DUT}}=V_{\text{DC}}-(2(R/2)+50\Omega)\cdot I$ (9)

where $I$ is the DC current through $R_{\text{DUT}}$. In all tunneling
measurements performed in this work pairs of contacts are tied together, so
the effective series filter resistance is $R/2$. The $50\text{\,}\Omega$ is
the lock-in output impedance. While the DC current is not known a priori,
differentiating with respect to $V_{\text{DUT}}$ the above equation gives:

$1=\frac{\mathrm{d}V_{\text{DC}}}{\mathrm{d}V_{\text{DUT}}}-(R+50\Omega)\frac{\mathrm{d}I}{\mathrm{d}V_{\text{DUT}}}$
(10)

This equation can be rearranged to isolate $\mathrm{d}V_{\text{DUT}}$ and
$\mathrm{d}V_{\text{DC}}$, given below.

$\mathrm{d}V_{\text{DUT}}=\frac{\mathrm{d}V_{\text{DC}}}{1+(R+50\Omega)\frac{\mathrm{d}I}{\mathrm{d}V_{\text{DUT}}}}$
(11)

The differential conductance across the QPC,
$G_{\mathrm{D}}=\mathrm{d}I/\mathrm{d}V_{\text{DUT}}$, appears in the
denominator. Integrating this equation leaves the final correction formula,
which can be expanded for $3050\Omega\cdot G_{\mathrm{D}}\ll 1$ to give:

$V_{\text{DUT}}=\int_{0}^{V_{\text{DC}}}\frac{\mathrm{d}V_{\text{DC}}^{\prime}}{1+(R+50\Omega)G_{\mathrm{D}}(V_{\text{DC}}^{\prime})}\approx
V_{\text{DC}}-(3050\Omega)\int_{0}^{V_{\text{DC}}}\mathrm{d}V_{\text{DC}}^{\prime}G_{\mathrm{D}}(V_{\text{DC}}^{\prime})$
(12)

This correction is applied for all the tunneling curves in the main text Fig.
3d and the collapsed curves in Fig. 3f.

##  XVI Luttinger Tunneling Data Analysis

Figure S3: Extended analysis of temperature dependent tunneling data (a)
Diagonal resistance, defined as $1/G_{\mathrm{D}}$, versus $V_{\text{bias}}$
for the same gate configuration as Fig. 3c at $V_{\text{NS}}=-3.925V$ and same
temperature values as Fig. 3d. (b) The standard deviation of $R_{D}$ defined
as $\sigma R_{D}$ versus bias taken across all temperatures between
$T=$141\text{\,}\mathrm{m}\mathrm{K}$$ and
$T=$550\text{\,}\mathrm{m}\mathrm{K}$$. Additional temperature values at finer
increments are included which are not shown in panel (a). Dashed lines
indicate threshold in $V_{\text{bias}}$ where $\sigma R_{D}<0.0012$. (c)
$R_{D}$ versus $V_{\text{bias}}$ averaged over all temperatures. (d) Portion
of the temperature averaged $R_{D}$ in panel (c) plotted only where the
standard deviation $\sigma R_{D}$ falls below $0.0012$. The black dashed line
is a fit to a quartic polynomial which then defines the temperature
independent background. (e) $R_{D}$ plotted versus $V_{\text{bias}}$ for
various temperatures with the quartic background extracted in panel d
subtracted (except for the constant term). A value for $R_{\text{sat}}$, i.e,
the resistance which panel (e) saturates to can be defined by averaging
$R_{D}$ over a window of $250\text{\,}\mu\mathrm{V}$ on either side of
$V_{\text{bias}}=0$ at high bias, and then averaging together the results. (f)
$R$ plotted versus $V_{\text{bias}}$ for the same temperatures in Fig. 3d.

The theoretical expectation given by Eq. (1) is that the reflection
coefficient $R$ has no terms which depend solely on voltage bias and not
temperature. Consequently, to compare our raw data with the predictions from
chrial Luttinger liquid theory we need to remove a temperature independent
background. Fig. S3a shows the unprocessed dependence of $1/G_{\mathrm{D}}$
versus $V_{\text{bias}}$ for the same series of temperatures listed in Fig.
3d. At large voltage bias, the temperature dependent part of $1/G_{D}$ is
expected to flatten out, however while we do observe negligible temperature
dependence for $|V_{\text{bias}}|>$500\text{\,}\mu\mathrm{V}$$, there is a
smooth variation in the bias dependence which is the same for all
temperatures. Fig. S3b shows the standard deviation of $1/G_{D}$ over
temperature, denoted as $\sigma_{T}(1/G_{D})$, versus $V_{\text{bias}}$. We
define the voltage bias value for which there is no longer any discernible
temperature variation between the data sets by choosing $V_{\text{bias}}$ such
that $\sigma_{T}(1/G_{D})<0.0012$. This range includes all values of
$V_{\text{bias}}$ which fall outside the bounds of the black dashed lines in
Fig. S3b. Fig. S3c shows $1/G_{D}$ versus $V_{\text{bias}}$ averaged over all
data sets at varying temperatures between $141\text{\,}\mathrm{m}\mathrm{K}$
and $550\text{\,}\mathrm{m}\mathrm{K}$, defined as $\overline{1/G_{D}}$
(additional data sets not shown in Fig. S3a for clarity are included in this
average). The background, plotted in Fig. S3d, is then defined by
$\overline{1/G_{D}}$ where ever $\sigma_{T}(1/G_{D})<0.0012$; the black dashed
line in the same panel is the best-fit fourth order polynomial to the
extracted background. Fig. S3e shows $1/G_{D}$, for the same temperatures in
Fig. S3a, with the background polynomial subtracted (not including the
constant offset). Here we define the saturation resistance $R_{\text{sat}}$ as
the value $\overline{1/G_{D}}$ saturates to at high bias for all temperatures;
practically this is computed by averaging $\overline{1/G_{D}}$ over
$V_{\text{bias}}\in[-$1\text{\,}\mathrm{m}\mathrm{V}$,$-0.75\text{\,}\mathrm{m}\mathrm{V}$]$
and
$V_{\text{bias}}\in[$0.75\text{\,}\mathrm{m}\mathrm{V}$,$1\text{\,}\mathrm{m}\mathrm{V}$]$,
yielding an $R_{\text{sat}}=0.775\;h/e^{2}$. Knowing $R_{\text{sat}}$, the
value of the reflection coefficient can be computed from $R_{D}\equiv
1/G_{D}$, given by the formula inset into Fig. S3f, following the procedure
outlined in Refs. wen_edge_1991; roddaro_nonlinear_2003; radu_quasi-
particle_2008; baer_experimental_2014.

##  XVII Thomas-Fermi calculation of the local density across the edge

Figure S4:  Effective model considered within the Thomas-Fermi framework. (a)
Top-view of the gate setup, which involves four square gates labeled N,S,E,
and W respectively, each of which is held at a potential denoted by
$V_{N,S,E,W}$. The back-gate (gray) is held at constant $V_{B}$. We allow for
the gate displacement from the center to be different for E/W and N/S gates.
(b) Side-view of the gates reveals two distances $d_{t}$ and $d_{b}$ for the
separation between the 2DEG and the top gate or bottom gate, respectively. (c)
A phenomenological model of the chemical potential and the corresponding
internal energy $E_{\textrm{xc}}$ considered in SM XVII.1.

###  XVII.1 Phenomenological model of the chemical potential

Here we provide details of the Thomas-Fermi calculations for the QPC with
$\nu_{EW}=-6,\nu_{NS}=-2$, $B=$2\text{\,}\mathrm{T}$$. Thomas-Fermi
calculations require as input the filling-dependent chemical potential
$\mu(\nu)$, as has been measured recently yang_experimental_2021. However,
those measurements were at high-field as opposed to
$B=$2\text{\,}\mathrm{T}$$. We thus construct a phenomenological model for
$\mu(\nu)$ which captures the dominant low-field features: jumps in the
chemical potential at the IQH gaps, and negative compressibility in between.
In monolayer graphene (MLG), the single-particle integer quantum Hall (IQH)
gaps appear at $\nu=\pm 2,\pm 6,\cdots$. Within the parameter regimes in Fig.
4a, we focus on the Landau-level (LL) $N=-1$ between $\nu=-6$ and $-2$ and
assume the IQH gap below $\nu=-6$ and above $\nu=-2$ is effectively infinite.
As we will apply our model at low field ($B=$2\text{\,}\mathrm{T}$$) we
neglect the fine structure coming from fractional quantum Hall (FQH) gaps in
between. The dominant features in $\mu(\nu)$ are then (1) jumps in the
chemical potential $\Delta\mu$ at integer fillings $\nu=-5$, $-4$ and $-3$ due
to quantum Hall ferromagnetism (QHF) in the isospin space
$(K,\uparrow),(K,\downarrow),(K^{\prime},\uparrow),(K^{\prime},\downarrow)$;
(2) a compensating negative-compressibility
$\frac{\partial\mu}{\partial\nu}\approx-\Delta\mu$ at intermediate fillings.

The thermodynamic gaps $\Delta\mu$ of the QHFs contain a single particle
contribution from the valley and spin splittings $E_{V},E_{Z}$, and an
interacting contribution from Coulomb exchange. In an $N\neq 0$ LL, we expect
$E_{V}\approx 0$ (this is in contrast to the $N=0$ LL, where valley is locked
to sublattice, and a sublattice splitting is generated by the hBN substrate).
The Zeeman energy $E_{Z}=$0.23\text{\,}\mathrm{meV}$$ is also very small at
$B=$2\text{\,}\mathrm{T}$$. The quantum Hall ferromagnet gap $\Delta\mu$ is
thus dominated by the Coulomb exchange contribution yang_experimental_2021

$\Delta\mu=\int\frac{d\mathbf{q}^{2}}{(2\pi)^{2}}V(\mathbf{q})|F(\mathbf{q})|^{2}=$12.7\text{\,}\mathrm{meV}$$
(13)

where $V(\mathbf{q})$ is the gated screened Coulomb interaction as in Eq. (5)
and $F(\mathbf{q})=(F_{00}(\mathbf{q})+F_{11}(\mathbf{q}))/2$ is the MLG form
factor in the $N=-1$ LL.

$F_{m,n}(\mathbf{q})=e^{-\frac{1}{4}|\mathbf{q}|^{2}\ell_{B}^{2}}\left(\frac{i|\mathbf{q}|\ell_{B}}{\sqrt{2}}\right)^{m-n}L_{n}^{m-n}\left(\frac{|\mathbf{q}|^{2}}{2}\right)\sqrt{\frac{n!}{m!}}$
(14)

is the usual GaAs form factor with $L_{b}^{a}(|\mathbf{q}|)$ being the
generalized Laguerre polynomial.

Based on $E_{V}=E_{Z}\approx$0\text{\,}\mathrm{meV}$$ and
$\Delta\mu=$12.7\text{\,}\mathrm{meV}$$, we consider a phenomenological model
of the chemical potential as shown in Fig. S4c. We phenomenologically include
the effect of disorder broadening such that $\mu$ increases by $\Delta\mu$
within a small window $\delta\nu=\pm 0.005$ close to integer filling.

###  XVII.2 Evolution of the QPC pinch-off

To approximate the experimental device, we simulate a system with magnetic
field $B=$2\text{\,}\mathrm{T}$$ (magnetic length
$\ell_{B}=$18.14\text{\,}\mathrm{n}\mathrm{m}$$), gate distances
$d_{t}=$60\text{\,}\mathrm{n}\mathrm{m}$,d_{b}=$30\text{\,}\mathrm{n}\mathrm{m}$$
and channel widths $w_{EW}=w_{NS}=$31\text{\,}\mathrm{n}\mathrm{m}$$ (see Fig.
8a for definition) on a $30\ell_{B}\,\times\,30\ell_{B}$ system with grid size
$\mathop{}\\!\mathrm{d}x=\mathop{}\\!\mathrm{d}y=\ell_{B}/2$. We numerically
checked that the charge density has converged in both system size and grid
size.

As explained in Fig. S4a, the quantity we are most interested in from the
Thomas-Fermi calculation is the charge density $\nu^{\mathrm{QPC}}$ at the
center of the QPC, which determines the number of edge modes
$N^{\mathrm{QPC}}$ transmitted through the QPC. $N^{\mathrm{QPC}}$ can be
directly related to the total number of edge modes $N$ across the device and
therefore the diagonal conductance $G_{D}$,

$N^{\text{QPC}}=\nu^{\mathrm{QPC}}-(-2),\quad N=-2+N^{\text{QPC}},\quad
G_{\mathrm{D}}\sim|N|\frac{e^{2}}{h}\sim\left|\nu^{\mathrm{QPC}}\right|\frac{e^{2}}{h},$
(15)

since there are two outer chiral modes outside the N/S region. We refer
readers to Fig. S1 and Fig. S2 for a more comprehensive illustration. Because
the edge modes straddle the density contours $\nu\sim n+1/2$, when
$\nu^{\mathrm{QPC}}$ is not an integer certain edge modes approach very close
to the saddle point, leading to partial transmission and therefore a non-
quantized diagonal conductance. Thus away from integer $N^{\text{QPC}}$ the
identification $G_{\mathrm{D}}\sim N^{\text{QPC}}\frac{e^{2}}{h}$ is purely
heuristic, as the Thomas-Fermi calculations cannot be expected to capture the
transmission coefficient through the QPC.

In Fig. 8b, we show $\nu^{\mathrm{QPC}}$ as a function of gate voltages. As we
pinch off the QPC, there is a clear transition from $\nu^{\mathrm{QPC}}$
dropping in a single sharp step to taking multiple steps due to the
competition between the confinement energy $E_{V}$ and the Coulomb energy
$E_{C}$. This transition indicates two different charge density configurations
within the QPC. When the confining potential is soft, i.e. $E_{V}/E_{C}$ is
small, it is energetically more favorable to form an incompressible island at
the QPC, so that $\nu^{\mathrm{QPC}}$ will quantize at integer fillings. When
the confining potential is sharp, i.e. $E_{V}/E_{C}$ is large, it is
energetically more favorable for the charge density $\nu^{\mathrm{QPC}}$ at
QPC to change continuously. We note that the $E_{V}/E_{C}$ obtained the
simulation is slightly larger than obtained in the experiment, which could
originate from the screening from virtual inter-LL transitions that had been
neglected in the simulation.

We show two prototypical charge density configurations in Fig. 8b-c
corresponding to these two cases. When $E_{V}/E_{C}$ is large, charge density
within the the QPC changes continuously as shown in Fig. 8b. When
$E_{V}/E_{C}$ is small, a small incompressible island with $\nu=-4$ forms at
the QPC as shown in Fig. 8c. Whether this island extends to incompressible
strips along the electrostatically defined edge depends on $E_{V}/E_{C}$
across the edge, which is generally expected to be larger since the confining
potential is sharper at the edge than at the junction. In this IQHE setting,
the edge structure far from the QPC is not so relevant to the diagonal
conductance $G_{D}$ measured in the experiment.