# Energy gap of the even-denominator fractional quantum Hall state in bilayer
graphene

Alexandre Assouline  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  Material Science
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA  Haoxin Zhou  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Liam A. Cohen  Department of Physics,
University of California at Santa Barbara, Santa Barbara CA 93106, USA
Fangyuan Yang  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Ruining Zhang  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  Roger S. K. Mong
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA
15260, USA  Michael P. Zaletel  Department of Physics, University of
California, Berkeley, California 94720, USA  Material Science 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

Bernal bilayer graphene hosts even denominator fractional quantum Hall states
thought to be described by a Pfaffian wave function with nonabelian
quasiparticle excitations. Here we report the quantitative determination of
fractional quantum Hall energy gaps in bilayer graphene using both thermally
activated transport and by direct measurement of the chemical potential. We
find a transport activation gap of $5.1\text{\,}\mathrm{K}$ at
$B=$12\text{\,}\mathrm{T}$$ for a half-filled $N=1$ Landau level, consistent
with density matrix renormalization group calculations for the Pfaffian state.
However, the measured thermodynamic gap of $11.6\text{\,}\mathrm{K}$ is
smaller than theoretical expectations for the clean limit by approximately a
factor of two. We analyze the chemical potential data near fractional filling
within a simplified model of a Wigner crystal of fractional quasiparticles
with long-wavelength disorder, explaining this discrepancy. Our results
quantitatively establish bilayer graphene as a robust platform for probing the
non-Abelian anyons expected to arise as the elementary excitations of the
even-denominator state.

Non-Abelian anyonsMoore and Read (1991) are thought to enable fault tolerant
topological quantum bits through their non-trivial braiding statisticsNayak
_et al._ (2008). In an ideal scenario, the error rate of such qubits is
limited only by the density of thermally excited quasiparticles present in the
system. Such processes—analogous to quasiparticle poisoning in superconducting
qubits—are exponentially suppressed at low temperature by an Arrhenius law,
$n_{\mathrm{qp}}\propto\exp\left(-\Delta_{\mathrm{qp}}/2k_{B}T\right)$, where
$\Delta_{\mathrm{qp}}$ is the energy gap for non-Abelian quasiparticles and
$T$ is temperature. The energy gap is thus a key figure of merit for candidate
nonabelian states. According to numerical calculationsMorf (1998); Rezayi
(2017), nonabelian ground states are the leading candidates to describe the
even denominator fractional quantum Hall (FQH) states observed in the second
orbital Landau level of single-component systems such as GaAs quantum
wellsWillett  _et al._ (1987). While these numerical results are thought to be
reliable, the small energy gaps measured for these states in GaAsKumar  _et
al._ (2010); Watson  _et al._ (2015); Chung  _et al._ (2021) have hampered
experimental efforts to directly probe nonabelian statistics via fusion and
braiding of individual quasiparticles.

Within the simplest model of bilayer graphene, the $N=0$ and $N=1$ orbital
levels are both pinned to zero energyMcCann and Fal’ko (2006). Combined with
the spin- and valley degeneracies native to graphene quantum Hall systemsDean
_et al._ (2020), this produces an eight-fold degeneracy—a seemingly
inauspicious arena for the single-component physics of nonabelian FQH states.
However, as a wealth of experimental work has shown, all of these degeneracies
are lifted by the combination of electronic interactions and the applied
displacement fieldFeldman  _et al._ (2009); Martin  _et al._ (2010); Dean  _et
al._ (2010); Lee  _et al._ (2014); Kou  _et al._ (2014); Ki  _et al._ (2014);
Maher  _et al._ (2014); Hunt  _et al._ (2017); Huang  _et al._ (2022); Zibrov
_et al._ (2017); Li  _et al._ (2017). In particular, broad domains of density
and displacement field are characterized by partial filling of a singly
degenerate $N=0$ or $N=1$ Landau level. In the $N=1$ regime, an incompressible
state is observed at half-integer fillingKi  _et al._ (2014); Zibrov  _et al._
(2017); Li  _et al._ (2017); Huang  _et al._ (2022), which calculations show
should be described by a nonabelian Pfaffian ground stateApalkov and
Chakraborty (2011); Papic and Abanin (2014); Balram (2022); Zibrov  _et al._
(2017). Prior measurements of energy gaps have found activation gaps as large
as $1.8\text{\,}\mathrm{K}$ at $B=$14\text{\,}\mathrm{T}$$; however, precise
comparisons of activation and thermodynamic gaps to theoretical expectations
have not been previously reported.

Figure 1: Chemical potential and inverse compressibility of bilayer graphene
fractional quantum Hall states. (A) Device schematic showing the hBN layers
(blue), top and bottom graphite gates (dark grey), monolayer graphene detector
layer connected to Corbino contacts, and bilayer graphene sample layer. (B)
Optical image of the Corbino contacts to the monolayer graphene detector.
White dashed lines show the trajectory of a chiral edge state along trenches
etched through the device, which ensures contact between the metal and dual
gated sample bulk. (C) The top panel shows the measured $\mu$ at
$B=$13.8\text{\,}\mathrm{T}$$ and $T=$50\text{\,}\mathrm{m}\mathrm{K}$$ in the
partially filled N=0 level spanning $0<\nu<1$. The bottom panel shows the
inverse compressibility, $d\mu/d\nu$, calculated by numerically
differentiating the data in the top panel. (D) The same as C, but for the
partially filled N=1 orbital Landau level spanning $-1<\nu<0$.

Here we report energy gaps for both odd- and even-denominator FQH states in
bilayer graphene using both transport and chemical potential measurements.
Thermally activated transport measures the energy cost of creating a
physically separated quasiparticle-quasihole pair. We measure activated
transport using a Corbino-like geometryPolshyn  _et al._ (2018); Zeng  _et
al._ (2019), which directly probes the conductivity of the gapped, insulating
bulk. Chemical potential measurements record a jump at incompressible filling
factors known as the thermodynamic gap, which—in the clean limit—measures the
difference between adding charge $\pm e$ to the gapped system. We measure the
thermodynamic gap using a direct-current charge sensing technique based on a
double-layer deviceEisenstein  _et al._ (1994); Yang  _et al._ (2021).
Combining these techniques, we find several new features, including weak FQH
states at $\nu=5/11$, $\nu=6/11$ and $\nu=5/9$ of a partially filled N=1
Landau level. Moreover, both schemes show an energy gap for a half-filled
single component Landau level that is several times larger than reported to
date for a candidate nonabelian state in any system Gammel  _et al._ (1988);
Kumar  _et al._ (2010); Watson  _et al._ (2015); Falson  _et al._ (2015);
Zibrov  _et al._ (2017); Li  _et al._ (2017); Shi  _et al._ (2020); Chung  _et
al._ (2021). Notably, these measurement schemes effectively average over
$\sim$10\text{\,}\mu\mathrm{m}^{2}$$ sized areas, a testament to the
exceptional uniformity of the electron gas in bilayer graphene.

Fig. 1A shows a schematic of the experimental geometry used to measure the
chemical potential $\mu$. A graphene bilayer hosting the FQH system of
interest is separated by a $62\text{\,}\mathrm{n}\mathrm{m}$-thick hexagonal
boron nitride (hBN) dielectric from a graphene monolayer that functions as a
sensor. Both layers are encapsulated by additional hBN dielectrics and
graphite gates, creating a four plate geometry that allows independent control
of the carrier density on both the monolayer detector and bilayer sample
layer. We measure Corbino transport in the detector layer, where a FQH state
functions as a sensitive detector of the local potential. An optical image of
the Corbino contacts is shown in Fig. 1B. As described in detail in the
supplementary information, monitoring transport in the sensor layer allows us
to precisely determine $\mu$ of the bilayer sample. An advantage of our
technique is that it avoids finite-frequency modulation of the carrier
density, allowing us to accommodate charge equilibration times as large as a
second.

Figs. 1C-D show $\mu$ and $d\mu/d\nu$ measured in our bilayer graphene device
at $B=$13.8\text{\,}\mathrm{T}$$. In the $N=0$ Landau level, incompressible
spikes are observed at fillings corresponding to the two- and four-flux ‘Jain’
sequenceJain (1989), with denominators as high as $17$. In the $N=1$ orbital,
a different hierarchy is observed, including a prominent state at $\nu+1=1/2$
along with states at $8/17$ and $7/13$ filling. This sequence is consistent
with a Pfaffian state at half filling and abelian ‘daughter’ states built from
its elementary excitationsLevin and Halperin (2009); Zibrov  _et al._ (2017).
Additional peaks are observed at fillings consistent with the four-flux Jain
sequence, at $3/5$ and $2/5$, and finally several weaker states at $5/11$,
$6/11$ and $5/9$ which were not previously reported. Away from these
incompressible fillings, the compressibility is negative throughout the
partially filled Landau levelEisenstein  _et al._ (1992). Additional negative
compressibility is observed near the incompressible states, associated with
the formation of Wigner crystals of fractionally charged quasiparticles at low
quasiparticle density.

Figure 2: Comparison of activation and thermodynamic gap sin a partially
filled $N=1$ Landau level. (A) Two terminal conductance measured in a Corbino
geometry as a function of filling factor at $B=$12\text{\,}\mathrm{T}$$ for
different temperatures. The temperature spacing is 5mK. (B) Activation gap
from the Arrhenius fit for $\nu=-3+1/2$ (red) and $\nu=-3+2/3$ (blue). (C)
Chemical potential measurement near $\nu_{0}=-1+1/2$ (red dots) at
$B=$13.8\text{\,}\mathrm{T}$$. Theory fit using the Wigner crystal model in
the clean limit (light red line) and in the disordered limit (red line). (D)
Chemical potential measurement near $\nu_{0}=-1+2/3$ (blue dots) at
$B=$13.8\text{\,}\mathrm{T}$$. Theory fit using the Wigner crystal model in
the clean limit (light blue line) and in the disordered limit (blue line).

Fig. 2A shows the two terminal conductance ($G$) measured at
$B=$12\text{\,}\mathrm{T}$$ in a second sample consisting of a dual gated
bilayer with Corbino-like geometry (see supplementary). Measurements are taken
at $B=$12\text{\,}\mathrm{T}$$ in a partially filled N=1 Landau level
corresponding to filling factors $0.25\lesssim\nu+3<0.75$ (see supplementary
information). The three most prominent FQH states, at $\nu+3=1/3,1/2$, and
$2/3$, all show vanishing conductance at the lowest temperatures. Fig. 2B
shows the minimal conductance for $\nu+3=1/2$ and 2/3 as a function of
temperature, along with fits to an Arrhenius law $G\propto
n_{\mathrm{qp}}\propto e^{-\Delta_{\mathrm{qp}}/2k_{B}T}$. For the $1/2$
state, the activation gap is found to be
$\Delta_{\mathrm{qp}}^{\mathrm{act}}=$5.1\pm 0.2\text{\,}\mathrm{K}$$ at
$B=$12\text{\,}\mathrm{T}$$, considerably larger than previous measurements in
grapheneZibrov  _et al._ (2017); Li  _et al._ (2017) or other two-dimensional
electron systemsChung  _et al._ (2021); Watson  _et al._ (2015); Falson  _et
al._ (2015); Kumar  _et al._ (2010); Shi  _et al._ (2020).

We may compare the result for the activation gap with a numerical calculation
that accounts for the microscopic details of bilayer graphene, accomplished
using the density matrix renormalization group (DMRG) Zaletel  _et al._
(2013); Mong  _et al._ (2017). Following Ref. Zibrov  _et al._ (2017), these
calculations are conducted on an infinite cylinder within a 4-band model of
BLG and account for mixing between the $N=0$ and $1$ Landau levels, screening
from the gates, and—crucially–screening due to inter-Landau level transitions,
which is treated within the random phase approximation (see supplementary
information). We obtain a charge gap
$\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}=0.011E_{C}$, where the Coulomb energy
scale $E_{C}$ depends on both the magnetic field and the dielectric constant
for hBN, which we take as
$\epsilon_{\mathrm{hBN}}=\sqrt{\epsilon_{xy}\epsilon_{z}}=4.5$Yang  _et al._
(2021). The calculated gap is $5.6\text{\,}\mathrm{K}$ at
$12\text{\,}\mathrm{T}$, within 10% of the experimental value.

The jump in chemical potential at fractional filling, $\Delta\mu$, provides an
alternative measurement of the FQH energy gaps, as shown in Figs. 2C-D
measured at $B=$13.8\text{\,}\mathrm{T}$$. In the clean limit, $\Delta\mu$
corresponds to the energy cost of adding a whole electron to the gapped
system, and is expected to be $e/e^{*}$ times larger than the quasiparticle
gap, where $e^{*}$ is the quasiparticle charge. At $\nu=-1/2$, where
$e/e^{*}=4$, the quasiparticle gap
$\Delta_{\mathrm{qp}}^{\mathrm{\mu}}=\Delta\mu/4=$2.9\text{\,}\mathrm{K}$$
implied by the measured thermodynamic is significantly smaller than
$\Delta_{\mathrm{qp}}^{\mathrm{act}}\approx$5.1\text{\,}\mathrm{K}$$, even
before accounting for the small difference in $B$ between Figs. 2A and C. A
similar discrepancy is seen at $\nu+1=2/3$, where
$\Delta_{\mathrm{qp}}^{\mathrm{act}}=$7.6\pm 0.5\text{\,}\mathrm{K}$$ but the
quasiparticle gap from thermodynamic measurements is
$\rm\Delta_{qp}^{\mu}=$5.2\text{\,}\mathrm{K}$$. For comparison, at
$\nu=1+2/3$ $\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}=$11.7\text{\,}\mathrm{K}$$.

We attribute the discrepancy to the contrasting role of disorder on the
thermodynamic and activation gaps. In the simplest model for activated
transportPolyakov and Shklovskii (1995) disorder does not affect the
activation gap (though in more involved models the activation gap becomes
sensitive to the spatial correlations of the disorder potentiald’Ambrumenil
_et al._ (2011); Nuebler  _et al._ (2010)). The thermodynamic gap, on the
other hand, is sensitive to the presence of disorder-induced localized states
which lead directly to a finite compressibility. To assess this hypothesis, we
compare our data against a phenomenological model for $\mu(\nu)$ that accounts
for both the disorder and quasiparticle interactions. Our model assumes that
the compressible states adjacent to the incompressible FQH states are Wigner
crystals of fractionally charged quasiparticlesEzawa and Iwazaki (1992);
Eisenstein  _et al._ (1992). As a starting point, we compute the energy
density $\mathcal{E}(\nu)$ of this pristine Wigner crystal under the
assumption that the fractional point charges $e^{\ast}$ form a triangular
lattice and interact through an effective Coulomb potential which accounts for
screening from the gates as well as the dielectric response of the parent
gapped state. In the disorder-free limit, we obtain theoretical $\mu(\nu)$
curves in which an infinitely-sharp jump of
$\Delta\mu=\frac{e}{e^{*}}\Delta_{\mathrm{qp}}$ is flanked by the negative
compressibility of the screened Wigner crystal (see supplementary
information). As shown in Figs. 2C-D, we find this disorder-free model
provides a good fit to the data at moderate quasiparticle densities, where the
compressibility is strongly negative.

To account for disorder, we make the assumption that the disorder potential
varies slowly in comparison with both the inter-quasiparticle distance and the
distance to the gates. As described in the supplementary material, this allows
us to make a local density approximation; $\mu(\nu)$ can then be solved for
explicitly given the interaction energy density $\mathcal{E}(\nu)$ and the
disorder distribution $P[V_{D}]$, which we assume to be a Gaussian of width
$\Gamma$. We note that these assumptions may not be correct. For example, it
will not be the case if the disorder arises from dilute Poisson-distributed
charge impurities in the hBN. Nevertheless, it results in a tractable model
that accounts for the competition between disorder and interactions.

Fits to this model are shown in Fig.2C-D near $\nu=-1+1/2$ and $\nu=-1+2/3$.
The fit is parameterized by the quasiparticle gap $\Delta^{\rm
fit}_{\mathrm{qp}}$, a phenomenological parameter $\chi$ which accounts for
the dielectric response of the parent state, and the disorder broadening
$\Gamma$ (see supplementary information). We find quantitative agreement
between the Wigner crystal model and experiment, providing strong evidence for
a Wigner crystal of fractionalized quasiparticles. From the fit we infer
$\Delta^{\mathrm{fit}}_{\mathrm{qp}}=$7\text{\,}\mathrm{K}$$ for the 1/2
state, within 20% of
$\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}=$6.0\text{\,}\mathrm{K}$$. The same
analysis for the $\nu_{0}=-1+2/3$ gives
$\Delta^{\mathrm{fit}}_{\mathrm{qp}}=$11.6\text{\,}\mathrm{K}$$, again within
20% of the $\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}=$11.7\text{\,}\mathrm{K}$$.
For both fillings, we find $\Gamma=$1.0\pm
0.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, consistent with previous
estimates for the Landau level broadening Polshyn  _et al._ (2018); Zeng  _et
al._ (2019). The comparison between experimental and theoretical gaps is
summarized in Table 1.

Filling $\nu+1$ |  B |  $\Delta_{\mathrm{qp}}^{\mathrm{\mathrm{act}}}$ |  $\Delta_{\mathrm{qp}}^{\mathrm{\mu}}$ |  $\Delta_{\mathrm{qp}}^{\mathrm{fit}}$ |  $\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}$
---|---|---|---|---|---
$\displaystyle\frac{1}{2}$ |  $12\text{\,}\mathrm{T}$ |  $5.1\text{\,}\mathrm{K}$ |  $-$ |  $-$ |  $5.6\text{\,}\mathrm{K}$
|  $13.8\text{\,}\mathrm{T}$ |  $-$ |  $2.9\text{\,}\mathrm{K}$ |  $7.0\text{\,}\mathrm{K}$ |  $6.0\text{\,}\mathrm{K}$
$\displaystyle\frac{2}{3}$ |  $12\text{\,}\mathrm{T}$ |  $7.6\text{\,}\mathrm{K}$ |  $-$ |  $-$ |  $10.8\text{\,}\mathrm{K}$
|  $13.8\text{\,}\mathrm{T}$ |  $-$ |  $5.2\text{\,}\mathrm{K}$ |  $11.6\text{\,}\mathrm{K}$ |  $11.7\text{\,}\mathrm{K}$
Table 1: Comparison of the quasiparticle gaps at $1/2$ and $2/3$ filling in
the $N=1$ Landau level as determined by DMRG calculations
$\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}$, thermally activated transport
$\Delta_{\mathrm{qp}}^{\mathrm{act}}$, the chemical potential jump
$\Delta_{\mathrm{qp}}^{\mu}$, and from the fit to the Wigner crystal model
$\Delta_{\mathrm{qp}}^{\mathrm{fit}}$.

Given the rather large discrepancies between experiment and numerics in GaAsMa
_et al._ (2022)—particularly at half filling—the level of agreement we find
for both activated and thermodynamic gaps with numerical modeling is
encouraging. We note that several sources may account for the remaining
quantitative discrepancies in our work. These including differences in inter-
Landau level screening strength at $\nu\sim-3$ relative to $\nu\sim-1$Shizuya
(2007), as well as possible spin textures in the excitation spectrum, which
can lower the activation gap but are not accounted for in our modeling. For
$\rm\Delta_{qp}^{fit}$, moreover, the phenomenological nature of our model for
disorder may not capture the microscopic physics at a quantitative level.

Fig. 3A shows the $\mu$ measured at different temperatures near the
$\nu+1=1/2$ gap. We focus on the strong temperature dependence of $\Delta\mu$,
plotted for several incompressible filling factors in Fig. 3B (see also the
supplementary information). We fit the low temperature limit of $\Delta\mu(T)$
using the Sommerfeld expansion $\Delta\mu(T)=\Delta_{0}-bT^{2}+\cdots$, which
is justified so long as the quasiparticles experience short-range repulsion.
The fitted values $\Delta_{0}$ and $b$ are reported in Figs. 3C and D,
respectively.

Notably, the $\nu=-1+1/2$ state shows anomalously strong temperature
dependence, manifesting as a large value of the $b$ parameter. According to
the Maxwell relation $|\frac{d\mu}{dT}|_{n}=-|\frac{ds}{dn}|_{T}$, this
suggests an anomalous contribution to the entropy in the dilute quasiparticle
limit. Anomalous entropy is expected in the vicinity of non-Abelian
statesCooper and Stern (2009) owing to the topological degeneracy of a dilute
gas of nonabelian anyons. However, this contribution is considerably smaller
than the anomalous entropy we observe. To see this, we assume the
quasiparticles are Ising anyons, endowing the ground state with an anomalous
entropy of $S_{\mathrm{topo}}=k_{B}N_{\mathrm{qp}}\ln(\sqrt{2})$ Cooper and
Stern (2009). Accounting for this contribution adds a linear-in-$T$ term to
the low-temperature expression,
$\Delta\mu(T)=\Delta_{0}-4\ln(2)T-bT^{2}+\cdots$. Refitting the $1/2$ state to
account for this modification reduces the best-fit $b$ to $36\pm
3\text{\,}\mathrm{K}^{-1}$–still larger than $b$ for all odd-denominator
states.

This analysis implies that the anomalous entropy near $\nu=1/2$—at least at
the filling factors corresponding to the extrema in $\mu$—does not arise
solely from the topological degeneracy. Notably, these extrema occur at a
density of quasiparticles where the average inter-quasiparticle distance is
larger than the distance to the gate. Disorder is expected to dominate this
regime, as inter-quasiparticle interactions are screened. Crudely, if disorder
is more important than the long-range Coulomb interaction, we expect
$b\propto(e/e^{\ast})^{2}/\Gamma$, where $\Gamma$ is the strength of the
disorder. However, determining the prefactor requires understanding the
thermodynamics of a Coulomb glass of fractionalized particles in an unknown
disorder distribution, a challenge we leave to future work.

In closing, we note that a related manuscript reports scanning tunneling
microscopy to study the same bilayer graphene FQH states studied here111A.
Yazdani, Private communication. In that work, the gate voltage $\delta V_{g}$
over which the FQH gaps appear provides a local measurement of the
thermodynamic gap. Those authors find $4\Delta^{\textrm{STM}}_{\rm
qp}=$30\text{\,}\mathrm{K}$$ for the 1/2 state at $B=$14\text{\,}\mathrm{T}$$.
This result is consistent with the intrinsic gap inferred from our WC model,
$4\Delta^{\textrm{WC}}_{\rm qp}\sim$28\text{\,}\mathrm{K}$$, as expected for a
local measurement that probes the chemical potential at length scales smaller
than the disorder correlation length. The large intrinsic gaps manifesting
across several experimental techniques show that bilayer graphene is an ideal
platform to explore the intrinsic physics of nonabelian anyons in the solid
state.

Figure 3: Temperature dependent $\mu$ near fractional filling at B=13.8T. (A)
Chemical potential near half filling of an $N=1$ Landau level at several
different temperatures. (B) Chemical potential jump across the incompressible
states as a function of temperature for different filling factors in an $N=1$
LL (dots). The solid lines are a low temperature fit,
$\Delta\mu(T)=\Delta_{0}-bT^{2}$. (C) Chemical potential jump $\Delta_{0}$
extracted from the fit for different fractional states in the N=0
($\tilde{\nu}=\nu$, red dots) and (N=0 orbital) and $N=1$
($\tilde{\nu}=\nu+1$, orange dots) orbital Landau levels. (D) Temperature
decay parameter $b$ extracted from the fit same states.

###### Acknowledgements.

The authors would like to acknowledge discussions with A. Stern, and A.
Yazdani for a related collaboration and sharing unpublished results, and
Evgeny Redekop for providing the device image shown in Fig. 1A. Experimental
work at UCSB was primarily supported by the Office of Naval Research under
award N00014-23-1-2066 to AFY. AFY acknowledges additional support by Gordon
and Betty Moore Foundation EPIQS program under award GBMF9471. MZ and TW were
supported by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division, under Contract
No. DE-AC02-05CH11231, within the van der Waals Heterostructures Program
(KCWF16). RM is supported by the National Science Foundation under Award No.
DMR-1848336. KW and TT acknowledge support from the Elemental Strategy
Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) and
JSPSKAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233). This research
used the Lawrencium computational cluster provided by the Lawrence Berkeley
National Laboratory (Supported by the U.S. Department of Energy, Office of
Basic Energy Sciences under Contract No. DE-AC02-05CH11231)

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Supplementary information

##  I Experimental methods

###  I.1 Sample preparation

The van der Waals heterostructure is assembled using a polycarbonate based dry
stacking technique. An image of the final heterostructure is shown in Fig S1A
together with all the different hBN thicknesses, measured by atomic force
microscopy. Two subsequent lithography steps are performed: first, we define
holes in the top gate (Fig. S1B); then we define smaller holes in a ‘Pac-Man’
shape within the top gate holes to expose the graphene edge (Fig. S1C). The
etching is realized with a RIE using CHF3/O2 gas. Another etch is done to
define trenches in the monolayer, so that the quantum Hall edge states are
connected to the bulk of the device (Fig. S1D). Metal deposition of Cr/Pd/Au
(3nm/15nm/120nm) makes electrical contact to the bilayer graphene, monolayer
graphene, and gate layers (Fig. S1E). In order to connect the isolated
graphene contacts without shorting them to the exposed top gate edges, we use
overdosed PMMA bridges (Fig. S1F). Finally the graphene contacts are connected
to macroscopic leads. The finished device shown in Fig. S1G.

For the Corbino transport measurements, the sample consists of a dual-graphite
gated, hBN encapsulated Bernal bilayer graphene layer, shown in Fig. S1H. The
top hBN thickness is 48.9nm and the bottom hBN thickness is 37.2nm. A
transport phase diagram from this sample is shown in Fig. S4B.

Figure S1: Sample fabrication chronological steps. (A) Optical microscope
image of the final heterostructures with the layer borders highlighted with
colors (B) Holes are defined in the top gates. (C) Smaller hole in a Pac-Man
shape to the monolayer are etched within the top gate holes. (D) Contact
trenches are defined together with openings to contact all the layers. (E)
Metal deposition to make contact to the different layers. (F) Overdose PMMA is
used to define bridges on top of the exposed top gate edges. (G) Final image
of the chemical potential device. (H) Final image of the Corbino transport
device.

###  I.2 Chemical potential measurements

A schematic of the chemical potential measurement scheme is shown in Fig. S2A.
A graphene monolayer transducer is placed on top of the bilayer sample in a
dual graphite gated device. We tune the monolayer density to a sharp
conductance minima defined at fractional filling $\rm\nu=-2+7/9$ with the top
gate voltage, as shown in Fig. S2B. While the bilayer is grounded, its density
is adjusted using the back gate voltage. The chemical potential shift of the
bilayer can then be detected via the shift of the monolayer conductance minima
on the top gate axis.

Figure S2: Measurement principle. (A) Device schematic showing the dual
graphite gated sample with a graphene monolayer detector capacitively coupled
to the bilayer graphene sample of interest. (B) Monolayer detector conductance
as a function of top gate voltage at B=13.5T. The four flux fractional state
conductance minima $\nu=-2+7/9$, highlighted with the red arrow, is used as a
sharp detector of the chemical potential change in the bilayer graphene
sample.

We model our system as a four plate capacitor model which accounts for the
chemical potential of the bilayer and of the monolayer detector. The system
electrostatics are described by the relation:

$\hat{C}(\vec{V}+\vec{\mu}/e)=-e\vec{n},$ (S1)

Where $\hat{C}$ is the geometric capacitance matrix, $\vec{V}$ are the applied
voltages, $\vec{n}$ is the vector of charge carrier densities, and $\vec{\mu}$
is the chemical potential of the layers which we take to be fixed for the top
and bottom gates but is a (density dependent) quantity for the sample and
detector layers.

$\hat{C}=\left({\begin{array}[]{ccccc}-c_{td}&c_{td}&0&0\\\
c_{td}&-c_{td}-c_{ds}&c_{ds}&0\\\ 0&c_{ds}&-c_{ds}-c_{sb}&c_{sb}\\\
0&0&c_{sb}&-c_{sb}\\\
\end{array}}\right),\quad\vec{V}=\left({\begin{array}[]{ccccc}V_{t}\\\
V_{d}\\\ V_{s}\\\ V_{b}\\\
\end{array}}\right),\quad\vec{\mu}=\left({\begin{array}[]{ccccc}0\\\
\mu_{d}(n_{d})\\\ \mu_{s}(n_{s})\\\ 0\\\
\end{array}}\right),\quad\vec{n}=\left({\begin{array}[]{ccccc}n_{t}\\\
n_{d}\\\ n_{s}\\\ n_{b}\\\ \end{array}}\right)$ (S2)

Here, $c_{ij}$ is the geometric capacitance between the layers $i$ and $j$,
with $i,j=t,d,s,b$, indicating the top gate, detector layer, sample layer, and
bottom gate, respectively. Note that $c_{ij}=c_{ji}$.

In our experiment, we vary the bottom gate voltage by $\Delta V_{b}$, while
leaving the sample at constant voltage so that $\Delta V_{s}=0$. We then find
the top gate voltage $\Delta V_{t}$ such that the detector density $\delta
n_{t}=0$, which in turn implies $\Delta\mu_{t}=0$. Under these conditions, we
have the relation

$\Delta V+\Delta\mu/e=\left({\begin{array}[]{ccccc}\Delta V_{t}\\\ 0\\\
\Delta\mu_{s}/e\\\ \Delta V_{b}\\\ \end{array}}\right).$ (S3)

Using Eq. (S1), the second row of the matrix gives

$\displaystyle c_{td}\Delta V_{t}+c_{ds}\Delta\mu_{s}/e$
$\displaystyle=-e\Delta n_{d}=0$ (S4) $\displaystyle\Delta\mu_{s}$
$\displaystyle=-e\Delta V_{t}\frac{c_{td}}{c_{ds}}$ (S5)

Giving the bilayer chemical potential shift from the measured shift on the top
gate. The third row of Eq. (S1) gives an expression for the density of the
sample layer $\Delta n_{s}$,

$\Delta n_{s}=-(c_{sd}+c_{sb})\Delta\mu_{s}/e+c_{sb}\Delta V_{b}$ (S6)

Using the relations above requires accurate knowledge of the geometric
capacitances. Particularly the ratios $\rm\frac{c_{d,t}}{c_{d,s}}$ and
$\rm\frac{c_{s,d}}{c_{s,b}}$. This is done by tracking the conductance minima
as a function of various gate voltage as shown in Fig. S3. In these
experiments, a fixed density feature in the detector layer is tracked through
a region where the sample layer is incompressible (for example, in the $\nu=4$
integer quantum Hall gap) or in the middle of the compressible Landau level.
This allows the relevant capacitance ratios to be measured directly with high
accuracy.

Figure S3: Capacitance ratio. (A) Monolayer conductance map as a function of
bilayer gate and top gate. (B) Derivative of the top gate shift as a function
of bilayer gate. The black dashed line shows the value of the capacitance
ratio $c_{ds}/c_{td}=0.58$. (C) Monolayer conductance map as a function of
back gate and top gate across $\nu=4$. (D) Derivative of the top gate shift as
a function of back gate. The black dashed line shows the value of the
capacitance ratio $\frac{c_{sb}c_{ds}}{(c_{sb}+c_{ds})c_{td}}=0.26$.

A final electrostatic consideration for bilayer graphene is the displacement
field $D$, which modifies the precise nature of the filled Landau level
orbitals, favoring valley polarization at large $D$. We obtain the
displacement field from the relation
$D=\frac{1}{2\epsilon_{0}}(c_{sd}(V_{d}-V_{s})-c_{sb}(V_{b}-V_{s}))$. While we
do not map the entire phase diagram of our bilayer sample as a function of
$D$, this phase diagram is highly reproducible across devicesZibrov  _et al._
(2017). In Fig. S4A, we show the trajectories in this phase diagram acquired
on another device (Device B of reference Zibrov  _et al._ (2017))
corresponding to the data shown in the main text. Also, the transport phase
diagram showing where the activation gaps have been measured is shown in Fig.
S4B.

Figure S4:  Capacitance and transport phase diagram of bilayer graphene. (A)
Capacitance map from Device B of Ref. Zibrov  _et al._ (2017) at B=14T. The
green (N=0 orbital) and red (N=1 orbital) dashed lines correspond to
trajectories shown in the main text Figs 1C and 1D, respectively. (B)
Conductance map from the bilayer Corbino device at B=12T. The red dashed line
correspond to the line cut where the main text data shown in Fig 2A has been
measured.

##  II Computational methods

###  II.1 DMRG calculation of the charge gaps

To find the charge gaps at the incompressible fractional quantum Hall (FQH)
states in bilayer graphene (BLG), we first use infinite density matrix
renormalization group (iDMRG) on a cylinder to obtain the ground state at the
corresponding fractional fillings $\nu$, following the formulation developed
in Ref. Zibrov  _et al._ , 2017. In the DMRG calculation, we always assume
full isospin polarization, which is justified experimentally in Ref. Zibrov
_et al._ , 2017; Hunt  _et al._ , 2017. Compared to the conventional lowest
Landau level (LLL) in GaAs, the “zeroth” Landau level (ZLL) in bilayer
graphene has an additional $N=1$ orbital, consisting of a mixture of
conventional $n=0$ and $n=1$ LLs. We keep both the $N=0$ and $N=1$ orbitals
within the ZLL and write the ZLL-projected density operator as

$n_{\mathrm{ZLL}}(q)=\sum_{N,N^{\prime}=0}^{1}\mathcal{F}_{N,N^{\prime}}(q)\bar{\rho}_{NN^{\prime}}(q)$
(S7)

where $\bar{\rho}_{NN^{\prime}}(q)$ is the usual density operator projected
into $N,N^{\prime}$ orbitals and $\mathcal{F}$ are the BLG “form factors”
defined in Ref. Zibrov  _et al._ , 2017. The remaining $|N|\geq 2$ LLs are
separated from the ZLL by at least the cyclotron gap $\hbar\omega_{c}$, whose
main effect is to screen the interaction via inter-LL virtual excitations
which can be captured by the effective interaction discussed below.

The bare Coulomb interaction is screened by the encapsulating hBN, the
graphite gates, and inter-Landau level (LL) transition. Screening from the
$\mathrm{hBN}$ and the graphite gates can be captured by

$V^{0}(\mathbf{q})=E_{C}\ell_{B}\frac{4\pi\sinh\left(\beta
d_{t}|\mathbf{q}|\right)\sinh\left(\beta
d_{b}|\mathbf{q}|\right)}{\sinh\left(\beta\left(d_{t}+d_{b}\right)|\mathbf{q}|\right)|\mathbf{q}|}$
(S8)

where $E_{C}=e^{2}/4\pi\epsilon_{0}\epsilon_{\mathrm{hBN}}\ell_{B}$ is the
Coulomb scale, $d_{t}$ ($d_{b}$) are the distance from the sample to the the
top (bottom) graphite gate. Here
$\epsilon_{\mathrm{hBN}}=\sqrt{\epsilon_{\perp}\epsilon_{\|}}$ and
$\beta=\sqrt{\epsilon_{\|}/\epsilon_{\perp}}$ are defined with the out-of-
plane (in-plane) dielectric constant $\epsilon_{\perp}$ ($\epsilon_{\|}$) of
hBN. In all DMRG calculations we stick to
$d_{t}=d_{b}=$60\text{\,}\mathrm{n}\mathrm{m}$$ mimicking the device geometry.
The precise dielectric constants of hBN have been subject to debate, but we
pick $\epsilon_{\perp}=3.15$ and $\epsilon_{\|}=6.6$ from Ref. Geick  _et al._
, 1966 and our own measurement.

The static dielectric response due to inter-LL virtual excitations can be
obtained within the random phase approximation (RPA),

$V_{\mathrm{RPA}}(\mathbf{q})=\frac{V_{0}(\mathbf{q})}{1-V_{0}(\mathbf{q})\Pi_{\nu}(\mathbf{q},\omega=0)}$
(S9)

where the polarizability $\Pi_{\nu}(\mathbf{q})$ sums over all inter-LL
transitions $m\to n$ except for $0\leftrightarrow 1$.

$\Pi_{\nu}(\mathbf{q})=\sum_{-\Lambda<m,n<\Lambda}\nu_{m}\left(1-\nu_{n}\right)\Pi_{m,n}(\mathbf{q})$
(S10)

where $m,n$ label LLs, $\nu_{m}$ is the filling of $m$th LL, and $\Lambda\gg
1$ is a high energy cutoff. For simplicity, we follow Ref. Misumi and Shizuya,
2008 by calculating $\Pi_{m,n}(\mathrm{q})$ within a two-band model of BLG
assuming only $\gamma_{0},\gamma_{1}\neq 0$. We use inter-layer bias
$U=$8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, with orbital filling in the
order of $\ket{K^{\prime}\downarrow 0},\ket{K^{\prime}\downarrow
1},\ket{K\downarrow 0},\ket{K\downarrow 1},\ket{K^{\prime}\uparrow
0},\ket{K^{\prime}\uparrow 1},\ket{K\uparrow 0},\ket{K\uparrow 1}$ at
$B>$12\text{\,}\mathrm{T}$$ where $\ket{\tau,s,N}$ labels the valley, spin,
and LL Li  _et al._ (2017); Hunt  _et al._ (2017). We note that any transition
$0\leftrightarrow 1$ within the ZLL is disregarded since the DMRG calculation
automatically accounts for such transitions by keeping both the $N=0$ and
$N=1$ orbitals. The sum in Eqn. S10 converges slowly with the cutoff
$\Lambda$, so we scale the cutoff from $\Lambda=110$ to $170$ and extrapolate
to $\Lambda\to\infty$ Yang  _et al._ (2021). We show the resulting
polarization function $\Pi_{\nu}(\mathbf{q})$ in Fig. S5, which agree with the
commonly used phenomenological form $\Pi_{\mathrm{tanh}}$ at large $q$ and
small $q$ limit but deviate at intermediate $q$ Hunt  _et al._ (2017); Papic
and Abanin (2014); Zibrov  _et al._ (2017). The deviation, which depends on
both the filling and displacement field, has a quantitative effect on the
charge gaps.

Figure S5: Dielectric function in BLG at $B=$14\text{\,}\mathrm{T}$$ with
$V^{0}(\mathrm{q})$ shown in Eqn. S8. We show the one obtained with the
calculated polarization function $\Pi_{\nu}(\mathrm{q})$ in Eqn S10 with
$U=$8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and $\nu=0$ and the one
obatined with the phenomenological polarization function
$\Pi_{\mathrm{tanh}}(\mathrm{q})$ in Eqn. S13.

After obtaining the FQH ground states in BLG, we use the “defect” DMRG method
to compute the quasiparticle gaps Zaletel  _et al._ (2013), which is
implemented for multicomponent quantum Hall systems in Ref. Mong  _et al._ ,
2017. The key idea is to enforce a single anyonic excitation $a$ to exist in
some large central region on an infinite cylinder using a “topological
boundary condition” $[l,r]$, which means fixing the ground state sector to the
left and right of the central region. We determine the energy of anyon $a$ by
comparing the energy of this particular configuration to that of the vacuum.
The quasiparticle gap is the energy required to create and separate a pair of
lowest-lying quasiparticle and quasihole $\Delta=E_{e^{*}}+E_{-e^{*}}$. In
Table S1, we show the lowest-lying quasiparticles and quasiholes at varioud
FQH states and the corresponding “topological boundary conditions” required to
trap them in “defect” DMRG.

|  lowest-lying charge excitation |  topological boundary condition $[l,r]$
---|---|---
Jain sequence $\nu=\frac{p}{2p+1}$ |  $e^{*}=\frac{e}{2p+1}$ quasiparticle |  [“$01\cdots 010$”, “$01\cdots 001$”]
|  $e^{*}=-\frac{e}{2p+1}$ quasihole |  [“$01\cdots 010$”, “$1001\cdots 0$”]
Pfaffian-like state $\nu=\frac{1}{2}$ |  $e^{*}=\frac{e}{4}$ quasiparticle |  [“$0110$”, “$1010$”]
|  $e^{*}=-\frac{e}{4}$ quasihole |  [“$0110$”, “$0101$”]
Table S1: Lowest lying charge excitations and the corresponding topological
boundary conditions at various FQH states. Here we label different ground
state sectors by their root configurations. “$01\cdots$” means “$01$”
repeating $p-1$ times. Figure S6: Quasiparticle gap at various FQH states in
BLG calculated by DMRG. We subtract off $E_{\mathrm{madelung}}$ for the dark
dots, which exhibit vanishing $L_{y}$ dependence except for at $\nu=3/7$ and
$\nu=1/2$.

The accuracy of the calculation is primarily controlled by three parameters.
The most important parameter is the cylinder circumference $L_{y}$ used in
iDMRG. Due to the absence of local curvature on the cylinder, there are no
geometric corrections typically seen in FQH numerics on the sphere Mong  _et
al._ (2017). The Coulomb interaction is screened by the gates on two sides
such that at sufficiently large $L_{y}$, we expect the interaction energy
$E_{\mathrm{madelung}}$ between the anyon and its images around the cylinder
to vanish exponentially. In practice, however, the gate distance
$d=$60\text{\,}\mathrm{nm}$$ is comparable to the accessible range of
$12\ell_{B}\leq L_{y}\leq 24\ell_{B}$, so we still need to manually subtract
off $E_{\mathrm{madelung}}$. For simplicity, we approximate the anyon on the
cylinder by a point charge at $y=0$, then $E_{\mathrm{madelung}}$ can be
computed by

$E_{\mathrm{madelung}}(L_{y})=\frac{(e^{*})^{2}}{2}\sum_{n\in\mathbb{Z},n\neq
0}V_{\mathrm{RPA}}(nL_{y})$ (S11)

where $e^{*}$ is the charge of the anyon, and the mirror images locate at
$y=nL$ with $n$ being a non-zero integer due to the periodic boundary
condition along the $y$ axis. In the calculation of the charge gaps we
consider a pair of quasiparticle and quasihole, each contributing to one copy
of the interaction energy $E_{\mathrm{madelung}}$. In Fig. S6, we show the
quasiparticle gaps computed by DMRG. After subtracting of two copies of
$E_{\mathrm{madelung}}$, the resulting gaps exhibit vanishing $L_{y}$
dependence, except for at $\nu=3/7$ and $\nu=1/2$ which requires a larger
$L_{y}$ to stablize the ground states. At these two states, we perform an
exponential extrapolation in $L_{y}$ by
$\Delta\left(L_{y}\right)=\Delta+\delta e^{-L_{y}/L_{y0}}$ and report
$\Delta$.

Another obvious parameter is the bond dimension $\chi$ within DMRG, which
determines the maximum allowed entanglement in the system. In practice, the
charge gaps change negligibly with bond dimension $\chi\geq 1600$ up to the
largest cylinder circumference $L_{y}=24\ell_{B}$ for all the FQH states we
examine. A more subtle but equally important parameter is the number of
orbitals $N_{\mathrm{defect}}$ in the central region in “defect” DMRG. The
“defect” DMRG algorithm functions by introducing $N_{\mathrm{defect}}$ tensors
in the middle of two infinite matrix product state (MPS) ground states on two
sides, and then optimizing this configuration with a finite DMRG algorithm. It
is crucial for the corresponding physical span $\frac{2\pi}{L_{y}}\times
N_{\mathrm{defect}}$ to exceed the anyon radius $\xi_{a}$, which is typically
a few $\ell_{B}$. We stick to $N_{\mathrm{defect}}=128$ which turns out to be
sufficient up to the largest cylinder circumference $L_{y}=24\ell_{B}$ for all
FQH states except for at $\nu=1/2$ where Ising anyons have a much longer tail.
At $\nu=1/2$, we extrapolate the span $N_{\mathrm{defect}}=64$ to $128$ and
restrict ourselves to $L_{y}\leq 18\ell_{B}$.

###  II.2 Quasiparticle Wigner crystal model

We model the compressible states adjacent to the incompressible fractional
quantum Hall states as Wigner crystals of fractionally charged quasiparticles.
We start by thinking of electron Wigner crystals, whose energy can be
evaluated as

$E_{\mathrm{WC}}=\sum_{i\neq
j}\sum_{\left\\{{\mathbf{R}}_{i}\right\\}}\frac{1}{2}\frac{\left\langle\Psi\left|V\left(\left|{\mathbf{R}}_{i}-{\mathbf{R}}_{j}\right|\right)\right|\Psi\right\rangle}{\langle\Psi\mid\Psi\rangle}\approx\frac{N_{e}}{2}\sum_{{\mathbf{R}}_{i}\neq
0}V_{H}\left(\left|{\mathbf{R}}_{i}\right|\right)$ (S12)

where $N_{e}$ is the number of the electrons, $V({\mathbf{R}})$ is the
effective interaction in Eqn. (S9), and ${\mathbf{R}}_{i}$ spans the
triangular lattice of the Wigner crystal. Here we plug in the zeroth Landau
level (ZLL) wavefunctions (which include both the $N=0,1$ LLs) and expand up
to terms that contain two of the ${\mathbf{R}}$’s, neglecting all Fock terms
Maki and Zotos (1983). There is a simple relation between
$V_{H}({\mathbf{R}})$ and $V({\mathbf{R}})$ in Fourier space, i.e.
$V_{H}({\mathbf{q}})=V({\mathbf{q}})\left|F({\mathbf{q}})\right|^{4}$ with
$F({\mathbf{q}})=e^{-{\mathbf{q}}^{2}\ell_{B}^{2}/4}$ being the LLL form
factor.

The classical treatment of the WC is valid as long as the electrons are
sufficiently far away from each other $R\gg\ell_{B}$, then we can neglect
quantum statistics, which always holds in the filling range of interest.

The relevant energy that determines the chemical potential in the experiment
is the WC energy relative to the charging energy of a classical capacitor,

$E_{\mathrm{int}}=E_{\mathrm{WC}}-\frac{e^{2}N^{2}}{2C_{g}}$ (S13)

where $C_{g}$ is the geometric self capacitance of the bilayer graphene plate.
Then the desired chemical potential $\mu(\nu)$ in the clean limit is given by
$\mu_{\mathrm{int}}(\nu)=\partial_{\nu}(E_{\mathrm{int}}/N_{\phi})$ with
$N_{\phi}$ being the number of fluxes.

The formulation discussed so far has primarily focused on Wigner crystals of
electrons, which have predicted a chemical potential in excellent agreement
with experimental results near integer quantum Hall states in monolayer
graphene Zibrov  _et al._ (2017). However, our main focus is on Wigner
crystals of quasiparticles. These fractional quasiparticles carry a charge of
$e^{*}$ and possess their own Landau levels, where the magnetic length is
effectively increased to $\tilde{\ell}_{B}=\sqrt{\hbar/e^{*}B}$. Within the
composite-fermion picture, the quasiholes, such as those in the
$\nu=\frac{1}{3}$ state, even exhibit identical wavefunctions to the lowest
Landau level but with $\ell_{B}$ replaced by $\tilde{\ell}_{B}$. In the
filling range of interest in this study, these quasiparticles are sufficiently
far apart from each other that their interactions are expected to be be
reasonably well approximated as those of electric monopoles, therefore
ignoring details of the charge distribution. Therefore, a rough estimate of
the chemical potential adjacent to the fractional quantum Hall states can be
made by substituting $e$ with $e^{*}$ and $\ell_{B}$ with $\tilde{\ell}_{B}$.
One additional complication comes from the screening from composite Fermion
$\Lambda$ levels. To capture this effect, we use a phenomenological model of
the RPA polarizability $\Pi_{\nu}(\mathbf{q})$ Hunt  _et al._ (2017); Papic
and Abanin (2014); Zibrov  _et al._ (2017),

$\Pi^{\chi}(\mathbf{q})=\frac{4\log(4)}{2\pi}\tanh\left(\chi|{\mathbf{q}}|^{2}\ell_{B}^{2}\right)\frac{1}{\hbar\omega_{c}}$
(S14)

where the phenomenological RPA parameter $\chi$ controls the strength of the
screening. If we only consider virtual transitions between LLs, matching the
small ${\mathbf{q}}$ and large ${\mathbf{q}}$ behavior of
$\Pi_{\nu=0}(\mathbf{q})$ in Eqn. S10 gives $\chi=0.62$. Including transitions
between $\Lambda$ levels will result in a larger $\chi$, which is turned into
a fitting parameter for quasiparticle WC. We note that in DMRG this treatment
is unnecessary since it keeps all microscopic electron degrees of freedom,
while here we are working directly with quasiparticles. Using the effective
interaction which include both the gate and RPA-screening, the WC energy of
Eq.(S12) is then evaluated numerically as described further in Yang  _et al._
(2021). We note that the chemical potential of the quasiparticle
$\mu^{\mathrm{qp}}$ computed here is related to the electron chemical
potential $\mu$ measured in the experiment by
$\mu=\frac{e}{e^{*}}\mu^{\mathrm{qp}}$.

###  II.3 Slow-varying disorder model

One tractable limit to account for disorder is when the disorder potential
varies slowly in comparison with both the inter-quasiparticle distance in the
Wigner crystal and the gate-sample distance. In this limit we can take the
local density approximation (LDA) such that the total energy can be expressed
as a local functional of the slowly-varying charge density $n(\mathbf{r})$,

$E[n]=\int\mathrm{d}^{2}\mathcal{E}(n(\mathbf{r}))=\int\mathrm{d}^{2}\mathbf{r}\left[\mathcal{E}_{\mathrm{int}}(n(\mathbf{r}))+\frac{e^{2}n(\mathbf{r})^{2}}{2c_{g}}+n(\mathbf{r})eV_{D}(\mathbf{r})-n(\mathbf{r})eV_{g}\right]$
(S15)

where
$\mathcal{E}_{\mathrm{int}}\equiv\frac{E_{\mathrm{int}}}{2\pi\ell_{B}^{2}N_{\phi}}$
is the interaction energy density, $c_{g}=C_{g}/A$ is the geometric self
capacitance per unit area, $V_{D}(r)$ is the disorder potential, and $V_{g}$
is the bias on the gate. We note that the disorder model does not care about
the microscopic degrees of freedom in the system, which is fully characterized
by $\mathcal{E}_{\mathrm{int}}$, so we pick the convention such that
$n(\mathbf{r})$ is the electron number density. The key approximation here is
the LDA which neglects the gradient $\bm{\nabla}n$ dependence of the energy
functional $E[n]$. Optimizing the energy functional $E_{\mathrm{int}}[n]$
determines the local charge density $n(V_{g}-V_{D})$ at each point,

$\mu_{\mathrm{int}}(n(V_{g}-V_{D}))+\frac{e^{2}n(V_{g}-V_{D})}{c_{g}}=eV_{g}-eV_{D}$
(S16)

If we consider a disorder distribution $P(V_{D})$ with zero mean $\langle
V_{D}\rangle_{P}=0$, we find

$\bar{\mu}_{\mathrm{int}}(V_{g})+\frac{e^{2}\bar{n}(V_{g})}{c_{g}}=eV_{g}$
(S17)

where
$\bar{\mu}_{\mathrm{int}}(V_{g})=\int\mathrm{d}V_{D}P(V_{D})\mu_{\mathrm{int}}(n(V_{D}))$
is the disorder averaged chemical potential and
$\bar{n}(V_{g})=\int\mathrm{d}V_{D}P(V_{D})n(V_{g}-V_{D})$ is the average
charge density. In the following we will show
$\bar{\mu}_{\mathrm{int}}(V_{g})$ is precisely the chemical potential measured
in the experiment.

As before, the experimentally relevant energy $\bar{E}_{\mathrm{int}}$ is
defined relative to a classical capacitor

$\bar{E}_{\mathrm{int}}(N_{e})=E[n]-\frac{e^{2}N_{e}^{2}}{2C_{g}}+N_{e}eV_{g}$
(S18)

Here the number of electrons $N_{e}=\bar{n}A$ is determined by the average
charge density $\bar{n}$. Then the chemical potential measured in the
experiment is

$\mu_{\mathrm{expt}}(N_{e})=\partialderivative{\bar{E}_{\mathrm{int}}}{N_{e}}=\frac{1}{A}\int\mathrm{d}^{2}\mathbf{r}\partialderivative{n}{\bar{n}}\partialderivative{n}\mathcal{E}(n(\mathbf{r}))-\frac{e^{2}N_{e}}{C_{g}}+eV_{g}=-\frac{e^{2}\bar{n}}{c_{g}}+eV_{g}=\bar{\mu}_{\mathrm{int}}(V_{g})$
(S19)

where the integral vanishes because $\partial_{n}\mathcal{E}(n)=0$. In
practice, we first numerically find the gate bias $V_{g}$ required to
reproduce the electron filling in the experiment using Eq. (S17), and then
compute the disorder averaged chemical potential $\bar{\mu}_{\mathrm{int}}$
assuming a Gaussian disorder distribution
$P(V_{D})=\frac{1}{\sqrt{2\pi}\Gamma}\exp(-V_{D}^{2}/2\Gamma^{2})$.

|  $\Delta$ ($\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$) |  $\Gamma$ ($\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$) |  $\chi$
---|---|---|---
$\displaystyle\nu=-1+1/2$ |  $2.4\pm 0.1$ |  $1.0\pm 0.5$ |  $0.85$
$\displaystyle\nu=-1+2/3$ |  $3.0\pm 0.1$ |  $1.2\pm 0.3$ |  $0.92$
Table S2: Fitting parameters of the disordered Wigner crystal model at $1/2$
and $2/3$ filling in the $N=1$ LL.

The chemical potential $\mu_{\mathrm{int}}$ we have discussed so far sorely
comes from the quasiparticle Wigner crystal, where at the FQH fillings
$\nu_{0}$ there is an additional contribution from the correlated FQH gaps,

$\mu_{\mathrm{int}}\to\mu_{\mathrm{int}}^{\mathrm{FQH}}=\begin{cases}\frac{1}{2}\Delta+\mu_{\mathrm{int}}(\nu-\nu_{0}),\quad&\nu>\nu_{0}\\\
-\frac{1}{2}\Delta+\mu_{\mathrm{int}}(\nu-\nu_{0}),\quad&\nu<\nu_{0}\end{cases}$
(S20)

where $\Delta=\frac{e}{e^{*}}\Delta_{\mathrm{qp}}$ is the thermodynamic gap at
$\nu_{0}$. The quasiparticle Wigner crystal model does not break the particle
hole symmetry, so we can obtain the chemical potential at $\nu<\nu_{0}$ by
$\mu_{\mathrm{int}}(\Delta\nu)=-\mu_{\mathrm{int}}(-\Delta\nu)$. As shown in
Fig. 2, we fit the disorder-averaged chemical potential
$\bar{\mu}_{\mathrm{int}}^{\mathrm{FQH}}$ to the experiment data with three
free parameters: the thermodynamic gap $\Delta$, the disorder scale $\Gamma$
and the screening parameter $\chi$. We summarize these fitting parameters at
$\nu=-1+1/2$ and $-1+2/3$ in the N=1 LL in Table S2.

##  III Extended data and figures

Figure S7: Chemical potential jump as a function of temperature for the Jain states in N=0 at $B=$13.8\text{\,}\mathrm{T}$$. The data (dots) are fitted using the low temperature Sommerfeld expansion (solid lines). Filling $\nu$ |  $\Delta_{\mathrm{qp}}^{\mathrm{\mu}}$ (K) |  $\Delta_{\mathrm{qp}}^{\mathrm{fit}}$ (K) |  $\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}$ (K)
---|---|---|---
$1/3$ |  $7.6\text{\,}\mathrm{K}$ |  $17.7\text{\,}\mathrm{K}$ |  $13.5\text{\,}\mathrm{K}$
$2/5$ |  $3.6\text{\,}\mathrm{K}$ |  $7.7\text{\,}\mathrm{K}$ |  $6.3\text{\,}\mathrm{K}$
$3/7$ |  $1.9\text{\,}\mathrm{K}$ |  $5.7\text{\,}\mathrm{K}$ |  $4.5\text{\,}\mathrm{K}$
$4/9$ |  $0.9\text{\,}\mathrm{K}$ |  $3.4\text{\,}\mathrm{K}$ |  $-$
$5/11$ |  $0.4\text{\,}\mathrm{K}$ |  $2.1\text{\,}\mathrm{K}$ |  $-$
$6/11$ |  $0.4\text{\,}\mathrm{K}$ |  $2.1\text{\,}\mathrm{K}$ |  $-$
$5/9$ |  $1.0\text{\,}\mathrm{K}$ |  $3.5\text{\,}\mathrm{K}$ |  $-$
$4/7$ |  $2.1\text{\,}\mathrm{K}$ |  $6.0\text{\,}\mathrm{K}$ |  $-$
$3/5$ |  $4.4\text{\,}\mathrm{K}$ |  $10.5\text{\,}\mathrm{K}$ |  $-$
$2/3$ |  $10.5\text{\,}\mathrm{K}$ |  $21.2\text{\,}\mathrm{K}$ |  $14.3\text{\,}\mathrm{K}$
$2/7$ |  $0.9\text{\,}\mathrm{K}$ |  $4.3\text{\,}\mathrm{K}$ |  $-$
Table S3: Comparison of the quasiparticle gaps at the Jain sequence in the
$N=0$ LL at $B=$13.8\text{\,}\mathrm{T}$$ as determined by DMRG calculations
$\Delta_{\mathrm{qp}}^{\mathrm{DMRG}}$, the chemical potential jump
$\Delta_{\mathrm{qp}}^{\mu}$, and from the fit to the Wigner crystal model
$\Delta_{\mathrm{qp}}^{\mathrm{fit}}$.