# Impact of bulk-edge coupling on observation of anyonic braiding statistics
in quantum Hall interferometers

J. Nakamura  Department of Physics and Astronomy, Purdue University  Birck
Nanotechnology Center, Purdue University  S. Liang  Department of Physics and
Astronomy, Purdue University  Birck Nanotechnology Center, Purdue University
G. C. Gardner  Birck Nanotechnology Center, Purdue University  Microsoft
Quantum Lab Purdue, Purdue University  M. J. Manfra<EMAIL_ADDRESS>Department of Physics and Astronomy, Purdue University  Birck Nanotechnology
Center, Purdue University  Microsoft Quantum Lab Purdue, Purdue University
School of Electrical and Computer Engineering, Purdue University  School of
Materials Engineering, Purdue University

###### Abstract

Quantum Hall interferometers have been used to probe fractional charge, and
more recently, fractional statistics of quasiparticles. Theoretical
predictions have been made regarding the effect of electrostatic coupling on
interferometer behavior and observation of anyonic phases. Here we present
measurements of a small Fabry-Perot interferometer in which these
electrostatic coupling constants can be determined experimentally,
facilitating quantitative comparison with theory. At the $\nu=1/3$ fractional
quantum Hall state, this device exhibits Aharonov-Bohm interference near the
center of the conductance plateau interrupted by a few discrete phase jumps,
and $\Phi_{0}$ oscillations at higher and lower magnetic fields, consistent
with theoretical predictions for detection of anyonic statistics. We estimate
the electrostatic parameters $K_{I}$ and $K_{IL}$ by two methods: by the ratio
of oscillation periods in compressible versus incompressible regions, and from
finite-bias conductance measurements, and these two methods yield consistent
results. We find that the extracted $K_{I}$ and $K_{IL}$ can account for the
deviation of the values of the discrete phase jumps from the theoretically
predicted anyonic phase $\theta_{a}=2\pi/3$. In the integer quantum Hall
regime, we find that the experimental values of $K_{I}$ and $K_{IL}$ can
account for the the observed Aharonov-Bohm and Coulomb dominated behavior of
different edge states.

##  I Introduction

Fractional quantum Hall states Tsui1982  are predicted to host exotic
quasiparticles exhibiting fractional charge and fractional anyonic braiding
statistics Leinaas1977 ; Wilczek1982 ; Laughlin1983 ; Halperin1984 ;
Arovas1984 . Fractional charge has been demonstrated through shot noise
measurements Heiblum1997 ; Saminadayar1997 ; Heiblum2008 , resonant tunneling
Goldman1995Science ; Kou2012 ; McClure2012 ; Heiblum2017 ; Ensslin2020-2 , and
interferometery Heiblum2010 ; Willett2009 ; Nakamura2019  for numerous
fractional quantum Hall states. Recently, experimental evidence for anyonic
statistics was demonstrated through quasiparticle collisions Bartolomei2020
and Fabry-Perot interference Nakamura2020 . Additionally, evidence of non-
Abelian braiding statistics has been seen in interferometry experiments at
$\nu=5/2$ Willett2009 ; Willett2013 ; Willett2019 .

Electronic interferometers using quantum point contacts (QPCs) to partition
edge states have been proposed as a method to probe both the fractional charge
and fractional braiding statistics of quasiparticles Kivelson1989 ;
Kivelson1990 ; Gefen1993 ; Chamon1997 , and substantial theoretical work has
been made to understand the behavior of quantum Hall interferometers Kim2006 ;
Halperin2007 ; Halperin2011 ; Rosenow2012 ; Levkivskyi2012 , including their
application to non-Abelian states Halperin2006 ; Bonderson2006 ; Bishara2008 ;
Bishara2009 ; Stern2010 . For a Fabry-Perot quantum Hall interferometer, the
phase difference determining interference will be given by Eqn. 1 Chamon1997 ;
Halperin2011 :

$\frac{\theta}{2\pi}=e^{*}_{in}\frac{AB}{\Phi_{0}}+N_{L}\frac{\theta_{a}}{2\pi}$
(1)

$B$ is the magnetic field, $A$ is the area of the interference path set by the
gates, $e^{*}_{in}$ is the quasiparticle charge on the interfering edge state,
$\Phi_{0}$ is the flux quantum, $N_{L}$ is the number of localized
quasiparticles inside the interferometer, and $\theta_{a}$ is the anyonic
phase associated with the interfering quasiparticles. With QPCs tuned to weak
backscattering, oscillations in the conductance across the interferometer will
occur with $\delta G\propto\cos(\theta)$, enabling fractional charge and
statistics to be probed through transport measurements. For the $\nu=1/3$
state $\theta_{a}$ is predicted to be $\frac{2\pi}{3}$ Arovas1984 ;
Halperin1984 ; Jain2003 ; Jain2004 ; Jain2010 , while different anyonic phases
are predicted for different fractional quantum Hall states JainBook ;
Halperin2011 .

An important consideration for operation of Fabry-Perot interferometers is the
role of Coulomb interactions. It has been shown that the Coulomb interaction
between charge in the bulk of the interferometer and charge at the edge has a
major effect on interferometer behavior as it can cause the area of the
interferometer to change when charge in the bulk changes, which modifies the
Aharonov-Bohm contribution to the interferometer phase Halperin2011 ; this has
important effects in both integer and fractional quantum Hall interference.
Strong bulk-edge coupling can result in unusual interference behavior,
including a decrease in magnetic flux through the interference path when
magnetic field is increased, resulting in positively sloped lines of constant
phase. For the integer regime this has been referred to as the Coulomb-
dominated regime, while behavior where the bulk-edge coupling is weak and
interference exhibits the conventional negatively-sloped lines of constant
phase has been referred to as the Aharonov-Bohm regime. This definition of
Aharonov-Bohm and Coulomb-dominated regimes is not as meaningful for
fractional states due to the effect of anyonic statistics Halperin2021 .
Nevertheless, a strong bulk-edge interaction still has critical effects in the
fractional regime. Most importantly, strong bulk-edge coupling can make the
anyonic phase unobservable Halperin2011 , making it important to suppress
Coulomb charging effects in interferometers Zhang2009 ; Heiblum2016-2 ;
Nakamura2019 .

In our previous experiment probing the anyonic phase at $\nu=1/3$ Nakamura2020
, the interferometer was in a regime in which the Coulomb charging effects
leading to the bulk-edge interaction were highly suppressed, allowing the
anyonic phase to be extracted without being greatly obscured. However, this
suppression of Coulomb effects also likely results in thermal smearing of
quasiparticle transitions in high and low field regions where the bulk becomes
compressible.

Here we report on measurements and analysis of interference in a small gate-
defined Fabry-Perot interferometer with lithographic dimensions 800 nm
$\times$ 800 nm specifically designed to investigate the impact of increased
bulk edge coupling. Two quantum point contacts are used to partially
backscatter the edge states with interference occurring between the two
different backscattered paths. The effective area based on the magnetic field
oscillation period at $\nu=1$ of approximately 21 mT is $\approx 0.2$ $\mu$m2,
implying a lateral depletion of approximately 180 nm, similar to previous
results for these types of interferometers Nakamura2019 ; Nakamura2020 .

Figure 1: Interferometer at the $\nu=1/3$ fractional quantum Hall state. a)
SEM image of interferometer with false color. Yellow regions are the gates
which define the interference path. Blue regions corrsepond to the 2DES. There
is a center gate in the middle which is kept grounded and does not affect the
2DES density. Red lines indicate edge state trajectories, with backscattered
paths denoted by dashed lines. b) Bulk $R_{xy}$ measurement showing the
$\nu=1/3$ conductance plateau. The approximate positions in magnetic field
where the interferometer transitions from negatively sloped Aharonov-Bohm
behavior to flat lines of constant phase with $\Phi_{0}$ modulations are
marked with dashed lines. c) Interference at $\nu=1/3$ at a mixing chamber
temperature of 10 mK. Near the center there are several discrete jumps in
phase which are associated with removal of quasiparticles localized by
disorder in the interior of the interferometer. At high and low field the
lines of constant phase become nearly independent of magnetic field, but
modulations with period approximately $\Phi_{0}$ can be seen. These
modulations are more prominent in the high-field region, particularly close to
the transition point at approximately 7.7 T.

##  II $\Phi_{0}$ period modulations in the interference pattern at $\nu=1/3$

A key finding from the Rosenow & Stern model for Fabry-Perot interferometers
Rosenow2020  is that due to the finite energy gap for the creation of
quasiparticle and quasihole excitations, there will be a finite range of
magnetic field near the center of the state where filling factor
$\nu\equiv\frac{n\Phi_{0}}{B}$ remains fixed and the bulk is incompressible,
and in this regime no quasiparticles are created and $\theta$ will evolve
primarily due to the Aharonov-Bohm phase. Once the magnetic field has been
varied sufficiently far from the center of the state that the chemical
potential is outside of the spectral gap, quasiparticles or quasiholes should
start to enter the device with the expected period $\Phi_{0}$. At $\nu=1/3$
the behavior expected for this regime where the bulk is compressible that is
upon the addition of magnetic flux $\Phi_{0}$, a quasiparticle will be removed
from the device (or a quasihole will be added), giving a shift in the anyonic
phase of $-2\pi/3$ which cancels out the Aharonov-Bohm contribution to the
phase. This results in the leading order interference having no magnetic field
dependence, but oscillations still occur as a function of gate voltage
Halperin2011 ; Halperin2021 . Higher-order contributions to interference are
also expected due $\Phi_{0}$ periodic changes in quasiparticle number
Rosenow2020  but are thermally suppressed. In a previous work at $\nu=1/3$
Nakamura2020  the lines of constant phase were observed to shift from a
negative slope near the center to zero slope at high and low field, consistent
with a shift from constant $\nu$ with an incompressible bulk to constant $n$
with a compressible bulk. $\Phi_{0}$ modulations from quasiparticle
transitions were not observed in the high and low field regions, which was
attributed to thermal smearing of the quasiparticle number. Significant
thermal smearing is expected due to the small quasiparticle charge $e^{*}=e/3$
and large screening needed to suppress bulk-edge coupling Rosenow2020 .

The device we have measured in this work has an area smaller by approximately
a factor of 2 compared to the device in Nakamura2020 . An SEM image is shown
in Fig. 1a. Bulk magnetoresistance $R_{xy}$ is shown in Fig. 1b indicating the
$\nu=1/3$ state and resistance plateau. Conductance data measured across the
interferometer as a function of $B$ and gate voltage variation $\delta V_{g}$
at $\nu=1/3$ is shown in Fig. 1c ($\delta V_{g}$ is applied to both side
gates, and is relative to -0.8 V). A smooth background is subtracted to
emphasize the interference oscillations. For this measurement the QPCs were
individually tuned to approximately 90% transmission at the center of the
$\nu=1/3$ state to achieve the weak backscattering regime. The overall
behavior is similar to Nakamura2020 : near the center of the plateau the lines
of constant phase have a negative slope, which is interrupted by several
discrete jumps in phase. At low field and high field the lines of constant
phase flatten out, consistent with transitions to a compressible bulk with
populations of quasiparticles (at low field) or quasiholes (at high field).
Unlike Nakamura2020 , however, there are additional modulations in the
interference pattern in the low and high field regions which have a period of
$\approx 22$ mT in the low field region and $\approx$ 20 mT in the high field
region (Fourier transforms illustrating these periods are shown in
Supplemental Fig. 2 and discussed in Supplemental Section 2). This period is
close to the Aharonov-Bohm period of $\approx 21$ mT at the integer state
$\nu=1$, indicating that the modulations in the low and high field regions
have close to $\Phi_{0}$ period as predicted in Rosenow2020 . This suggests
that the reduction in device size has enabled quasiparticle transitions and
associated higher-order interference terms to be partially resolved in the
compressible regions, giving additional experimental evidence of anyonic
statistics. Repeatability of the data is discussed in Supplemental Section 1
and Supplemental Fig. 1.

Figure 2: Interference at elevated $T$. a) Interference at $\nu=1/3$ at a
mixing chamber temperature of 50 mK. Interference is still of reasonable
amplitude, and a few of the discrete jumps in the center are still visible,
but the $\Phi_{0}$ modulations are only weakly visible. b) Interference at 90
mK. The $\Phi_{0}$ modulations are completely washed out, and the interference
amplitude in the low and high field regions are greatly reduced.

Interference measurements at elevated temperatures 50 mK and 90 mK are shown
in Fig. 2a and b. At 50 mK the $\Phi_{0}$ modulations are greatly suppressed,
and at 90 mK they are completely washed out, similar to the observation in
Nakamura2020 . This is consistent with the prediction that the dominant
behavior at $\nu=1/3$ when the bulk is compressible should be oscillations
with zero magnetic field frequency Halperin2011 ; Halperin2021 , while the
$\Phi_{0}$ modulations are a higher order effect that is more easily thermally
smeared. The discrete jumps in the central region also become noticeably less
sharp and more smeared out at the higher temperatures.

It is noteworthy in Fig. 1c that the $\Phi_{0}$ modulations are more prominent
in the high-field region than in the low-field region, and we have found that
this is usually the case in multiple data sets. This may suggest an effect of
particle-hole asymmetry in terms of confining quasiparticles inside the
interferometer, although we do not have a clear explanation for the effect.
Though they carry equal and opposite total charge, quasiparticles and
quasiholes have different charge distributions. This can be understood from
the composite-fermion picture Jain1989  from the fact that quasiparticle
states involve addition of charge to excited lambda levels Jain2003A . How the
difference in behavior between quasiparticles and quasiholes might contribute
to the more clear $\Phi_{0}$ modulations in the high field region requires
further investigation.

##  III Effect of bulk-edge coupling on interference

A wide range of interferometer behavior beyond the negatively sloped pure
Aharonov-Bohm regime has been observed in previous experiments Goldman2009 ;
Zhang2009 ; Heiblum2010 ; McClure2012 ; Nakamura2019 ; Nakamura2020 ;
Ensslin2020 . Theoretical analyses Halperin2007 ; Halperin2011 ;
VonKeyserlingk2015 ; Rosenow2020  have established that electrostatic
interaction parameters are crucial in determining the observed behavior, with
key parameters being the edge stiffness $K_{I}$ which describes the energy
cost to vary the area of the interfering edge state, and $K_{IL}$ which
parameterizes the coupling of the bulk to the edge. Previous experiments have
investigated the case of interference when multiple Landau levels are present
and inferred the electrostatic parameters governing interference Heiblum2016-2
; Ensslin2020 . Device behavior has been modeled by defining an energy
function for the electrostatic energy involving these parameters (Eqn. 2)
Halperin2011 :

$E=\frac{K_{I}}{2}(\delta n_{I}^{2})+\frac{K_{L}}{2}(\delta
n_{L})^{2}+K_{IL}\delta n_{L}\delta n_{I}$ (2)

In this equation $\delta n_{L}$ is the variation of the charge in the bulk
from the background charge (which includes the quantum Hall condensate density
and the contribution from localized charges) and $\delta n_{I}$ is the
variation in charge at the edge from the ideal value. Minimizing the
electrostatic energy will result in variations in the area, $\delta A=-\delta
n_{L}\frac{K_{IL}}{K_{I}}\frac{\Phi_{0}}{\Delta\nu B}$. Including this
variation in area in Eqn. 1 yields Eqn. 3 VonKeyserlingk2015 :

$\frac{\theta}{2\pi}=e^{*}_{in}\frac{\bar{A}B}{\Phi_{0}}-\frac{K_{IL}}{K_{I}}\frac{e^{*}_{in}}{\Delta\nu}(e^{*}_{in}N_{L}+\nu_{in}\frac{\bar{A}B}{\Phi_{0}}-\bar{q})+N_{L}\frac{\theta_{a}}{2\pi}$
(3)

In Eqn. 3 $\bar{A}$ is the average area not including the variations $\delta
A$ due to the bulk edge coupling, $\Delta\nu$ is the difference between the
filling factor corresponding to the interfering edge state and the filling
factor of the next-outer edge state which is fully transmitted (for integer
states $\Delta\nu=1$, while for $\nu=1/3$ $\Delta\nu=1/3$). $\nu_{in}$ is the
filling factor corresponding to the interfering edge state, and $\bar{q}$ is
the background charge (which is primarily determined by the ionized donors and
may also be changed by the gate voltage).

##  IV Inferring $\frac{K_{IL}}{K_{I}}$ from magnetic field periods

Eqn. 3 implies that in the presence of finite $K_{IL}$, in ranges of magnetic
field where the localized quasiparticle number is fixed the the magnetic field
oscillation period will be modified from the base value of
$\frac{\Phi_{0}}{e^{*}\bar{A}}$. In the presence of bulk-edge coupling this
period will change to Eqn. 4 Halperin2021 :

$\Delta
B=\frac{\Phi_{0}}{e^{*}_{in}\bar{A}}(1-\frac{K_{IL}}{K_{I}}\frac{\nu_{in}}{\Delta\nu})^{-1}$
(4)

At $\nu=1$ we observe a region in magnetic field towards the center of the
state where the magnetic field period is larger (Fig. 3), suggesting that in
this region $\mu$ is in the gap and the localized electron number is fixed, as
predicted by the model in Rosenow2020 . At higher and lower field the period
becomes smaller, suggesting that localized holes and electrons are being added
and bringing the period back to the base value of
$\frac{\Phi_{0}}{e^{*}\bar{A}}$. This allows $\frac{K_{IL}}{K_{I}}$ to be
extracted based on the ratio of the periods, yielding a value of 0.25 (see
Supplemental Section 3 and Supplemental Fig. 3). Similar analysis at $\nu=1/3$
yields $\frac{K_{IL}}{K_{I}}=0.24$. These values imply a moderate effect of
bulk-edge coupling at $\nu=1$ and $\nu=1/3$.

Figure 3:  Interferometer conductance oscillations versus magnetic field and
gate voltage at $\nu=1$. Near the center the magnetic field oscillation period
is larger, suggesting an incompressible bulk, while at high and low fields the
period becomes smaller (while still maintaining an overall negative slope)
suggesting a transition to a compressible bulk.

##  V Extracting $K_{I}$ and $K_{IL}$ from finite-bias measurements

Estimating $\frac{K_{IL}}{K_{I}}$ from Eqn. 4 gives only the ratio of the two
parameters, and gives limited insight into the relevant factors which
contribute to each term. To extend this analysis and estimate the magnitude of
$K_{I}$ and $K_{IL}$ in our device, we adopt a simple picture in which the
energy is a combination of the single-particle energy $E_{sp}$ (which is
determined by the external electrostatic confining potential) and the electron
interaction energy $E_{int}$ which we estimate by approximating the device as
a quantum dot (and extract from zero-field Coulomb blockade measurements). The
total energy is a combination of the two terms, $E=E_{sp}+E_{int}$.

In this approximation we estimate the interaction energy by assuming that it
can be treated as zero-dimensional quantum dot-like object, so that
$E_{int}=\frac{\delta q_{total}^{2}}{2C}$. $\delta q_{total}$ is the combined
bulk and edge excess charge, $\delta q_{total}=e\delta n_{I}+e\delta n_{L}$,
and $C$ is the electrostatic self capacitance of the device. This yields :

$E_{int}=\frac{\delta q_{total}^{2}}{2C}=\frac{e^{2}}{C}(\frac{\delta
n_{I}^{2}}{2}+\frac{\delta n_{L}^{2}}{2}+\delta n_{L}\delta n_{I})$ (5)

From this we can see that the quantum dot-like charging energy
$\frac{e^{2}}{2C}$ contributes to $K_{I}$, $K_{IL}$, and $K_{L}$.
$\frac{e^{2}}{2C}$ can be extracted from the height of Coulomb blockade
diamonds. For our device the $B=0$ Coulomb blockade measurements Beenakker1991
yield $\frac{e^{2}}{C_{total}}\approx 90$ $\mu$eV (Fig. 4a). This can be
refined by subtracting the contribution from the single-particle level spacing
due to the finite Density of States (DOS) per unit area
$\frac{m^{*}}{\pi\hbar^{2}}$ in 2D at $B$ = 0 which gives a quantum
contribution $\frac{e^{2}}{C_{quantum}}\approx 18$ $\mu e$V for a device with
area $\approx 0.2$ $\mu$m2. Then
$\frac{e^{2}}{C}=\frac{e^{2}}{C_{total}}-\frac{e^{2}}{C_{quantum}}\approx$ 90
$\mu$eV-$18$ $\mu$eV$=72$ $\mu$eV.

$E_{sp}$ is set by the confining potential which increases the system energy
when area is changed. This external potential can be approximated as a
constant electric field $\mathcal{E}$ assuming the variations in area are
small. Then, the additional electrostatic potential the charge added to the
edge experiences will be $\mathcal{E}\delta l$, where $\mathcal{E}$ is the
electric field and $\delta l=\frac{\delta A}{L}$ is the increase in the radius
of the interferometer ($L$ is the perimeter of the interference path). $\delta
A$ will depend on the amount of charge added to the edge and the sheet density
of the interfering edge state, $\delta A=\frac{\delta n_{I}}{\rho}$ and
$\rho=\frac{\Delta\nu B}{\Phi_{0}}$. The total change in energy will be the
average change in potential times the amount of charge added,
$E_{sp}=\frac{e\delta n_{I}\mathcal{E}\delta l}{2}=\frac{e\delta
n_{I}^{2}\mathcal{E}\Phi_{0}}{2LB\Delta\nu}$. To find the contribution of
$E_{sp}$ to $K_{I}$ we need to determine the value of
$\frac{e\mathcal{E}\Phi_{0}}{LB\Delta\nu}$

The electric field $\mathcal{E}$ also drives the edge velocity, via
$\vec{v}_{edge}=\frac{\vec{\mathcal{E}}\times\vec{B}}{B^{2}}$, so a measure of
velocity can be used to extract the electric field and get $E_{sp}$ Chamon1997
. For integer states (where $e^{*}=e$ and $\Delta\nu=1$) and weak
backscattering, when the experiment of measuring differential conductance as a
function of gate voltage or magnetic field and source drain bias $V_{SD}$ is
performed, it exhibits a checkerboard pattern with $\delta
G\propto\cos(\frac{2\pi AB}{\Phi_{0}})\cos(\frac{LeV_{SD}}{2\hbar v_{edge}})$
McClure2009 ; Nakamura2019 ; Deprez2021 ; Ronen2021  (note that this assumes a
symmetric potential drop; the symmetry of the potential drop is discussed in
Supplemental Section 4). The product of cosines will result in nodes in the
oscillation pattern at $\frac{LeV_{SD}}{2\hbar v_{edge}}=\pi(n+1/2)$, so that
the voltage spacing between nodes $\Delta V_{SD}=\frac{2\pi\hbar
v_{edge}}{eL}=\frac{hv_{edge}}{eL}=\frac{\Phi_{0}\mathcal{E}}{LB}$. This gives
$E_{sp}=\frac{\delta n_{I}^{2}}{2}\Delta V_{SD}$.

Figure 4: Finite-bias measurements. a) Differential conductance measurements
at zero magnetic field in the Coulomb blockade regime. The height of the
diamond pattern gives the characteristic charging energy
$\frac{e^{2}}{C_{total}}\approx 90$ $\mu eV$. Note that this energy will
include the contribution from the finite DOS, which should be subtracted to
yield the electrostatic component that contributes the $K_{IL}$. This yields
$\frac{e^{2}}{2C}\approx 72$ $\mu e$V. b) Interference data for the innermost
edge mode at $\nu=3$. The QPCs are tuned to partially reflect the inner mode
and fully transmit the two outer modes. The positive slope indicates that
under these conditions the device is in the Coulomb-dominated regime where the
bulk-edge interaction is strong. c) Differential conductance measurements at
$B=2.5$ T, the $\nu=1$ state. The checkerboard pattern suggests that the bias
is close to symmetric, although the fact that there is some tilt to the
pattern suggests that there is some asymmetry. d) Oscillation amplitude from a
FFT versus $V_{SD}$ for $\nu=1$. The minima in the pattern correspond to the
nodes in the checkerboard pattern, allowing extraction of $\Delta
V_{SD}\approx 162$ $\mu$V. e) Differential conductance measurement for the
inner mode at $\nu=3$. f) FFT amplitude versus $V_{SD}$. The spacing between
minima gives $\Delta V_{SD}=45$ $\mu$V, indicating a relatively low velocity
which is to be expected for an inner mode.

Differential conductance measurements at $\nu=1$ are shown in Fig. 4c, and the
amplitude versus $V_{SD}$ is plotted in Fig. 4d. The data exhibits the
expected checkerboard pattern, and the spacing between minima in the amplitude
(corresponding to the nodes in the oscillation pattern) gives $\Delta
V_{SD}\approx 162$ $\mu$V, and $\frac{e\mathcal{E}\Phi_{0}}{LB}\approx 162$
$\mu e$V. Combining Eqn. 2 with the relationships for $E_{sp}$ and $E_{int}$
gives $K_{I}=\frac{e^{2}}{C}+\frac{e\mathcal{E}\Phi_{0}}{LB}\approx 72$
$\mu$eV$+162$ $\mu$eV$=234$ $\mu e$V, while $K_{IL}=\frac{e^{2}}{C}\approx 72$
$\mu e$V. This gives $\frac{K_{IL}}{K_{I}}=0.31$. The fact that
$\frac{K_{IL}}{K_{I}}<0.5$ should place the interferometer in the Aharonov-
Bohm regime, which is consistent with the observation of predominantly
negatively-sloped behavior at $\nu=1$ (Fig. 3). Additionally, this value of
$\frac{K_{IL}}{K_{I}}$ is close to the value of 0.25 extracted from Eqn. 4,
giving further validation for the model.

##  VI Strong bulk-edge coupling at $\nu=3$

Strong bulk-edge coupling causes the area of the interference path to decrease
when the magnetic field is increased (or when localized quasiparticles are
added to the interior of the device), resulting in a positive slope to
constant phase lines when $\frac{K_{IL}}{K_{I}}>0.5$, which has been observed
in some previous experiments Goldman2009 ; Zhang2009 ; Heiblum2010 ;
McClure2012 . With $B$ set to $\nu=3$, interference data is shown for the
innermost mode in Fig. 4b, where the QPCs are tuned to backscatter the
innermost mode and fully transmit the outer modes. There is an overall
positive slope to the data, indicating that unlike at $\nu=1$, the device is
in the Coulomb dominated regime and the bulk-edge interaction is strong. This
is also supported by the fact that the magnetic field oscillation period is
approximately $\frac{\Phi_{0}}{2}$, similar to previous experiments Zhang2009
; Heiblum2010  and theory Halperin2007 ; Halperin2011  for interference of an
inner mode at $\nu=3$.

Differential conductance measurements at $\nu=3$ are shown in Fig. 4e, and the
oscillation amplitude versus $V_{SD}$ shown in Fig. 4f. From this data $\Delta
V_{SD}=45$ $\mu$V, implying a lower velocity and smaller edge stiffness than
at $\nu=1$. This low velocity can be understood from the fact that an inner
edge state is being interfered; the inner edge state will be positioned at a
region with a more shallow confining potential, resulting in a lower electric
field and thus lower velocity Chklovskii1992 ; Chklovskii1993 ; Harshad2018 ;
Nakamura2019 , making it easier for the area enclosed by this edge state to
change. The measured value yields $K_{I}=\frac{e^{2}}{C}+e\Delta V_{SD}=72$
$\mu e$V$+45$ $\mu e$V$=117$ $\mu e$V and $K_{IL}=\frac{e^{2}}{C}\approx 72$
$\mu e$V. This gives $\frac{K_{IL}}{K_{I}}$=0.62. Since
$\frac{K_{IL}}{K_{I}}>0.5$, the device is predicted to be in the Coulomb-
dominated regime, consistent with the observed interference behavior in Fig.
4b. This gives additional validation for this method of calculating $K_{I}$
and $K_{IL}$ since this model is able to correctly predict Coulomb-dominated
behavior. On the other hand the outer modes at $\nu=3$, corresponding to the
two spin configurations of the $N=0$ Landau level, exhibit negatively sloped
behavior, indicating that a higher velocity due to a steeper confining
potential farther out at the edge gives a stronger $K_{I}$ (see Supplemental
Fig. 5). The outermost edge also exhibits the period-halving effect which has
been observed previously Heiblum2015 ; Heiblum2018 ; Ashoori2020  and
attributed to inter-edge interaction Frigeri2019  or electron pairing
Frigeri2020 .

Recent experiments have extended quantum Hall interferometery to graphene
Deprez2021 ; Ronen2021 , and provide another opportunity to apply this method
for using $K_{I}$ and $K_{IL}$ to understand Aharonov-Bohm versus Coulomb
dominated behavior. In Ref. Deprez2021  the smallest device with area 3.1
$\mu$m2 had estimated charging energy $E_{c}=18$ $\mu e$V and $\Delta
V_{SD}\approx 70$ $\mu$V from differential conductance measurements when
interfering the outer edge state at $\nu=2$. This yields
$\frac{K_{IL}}{K_{I}}\approx\frac{18\mu V}{18\mu V+70\mu V}\approx 0.2$,
placing the device in the Aharonov-Bohm regime, which is consistent with the
negatively sloped lines of constant phase observed. Similarly, the 3 $\mu$m2
device in Ref. Ronen2021  had an estimated $E_{c}=16$ $\mu e$V and $\Delta
V_{SD}=50$ $\mu$V for the inner mode at $\nu=2$, yielding
$\frac{K_{IL}}{K_{I}}\approx 0.24$, also in concordance with the observed
negatively-sloped Aharonov-Bohm behavior. This suggests that the method for
analyzing $K_{I}$ and $K_{IL}$ can also be applied to graphene devices, which
are promising for probing exotic statistics.

##  VII Finite bias measurements at $\nu=1/3$

The finite-bias behavior of interference at the $\nu=1/3$ state is expected to
be modified by Luttinger liquid effects Chamon1997 ; Smits2013 . Ref.
Chamon1997  analyzes the current through interferometers as a function of
$V_{SD}$ and $T$, and finds that while integer states should have uniform
spacing of nodes (as discussed in the previous section), for fractional states
the innermost nodes will have a narrower spacing than the outer nodes. At low
temperature, the positions of the nodes will approximately be given by Eqn. 6:

$V_{SD}=\frac{hv_{edge}}{e^{*}L}(n+\frac{1+g}{2}),n=0,1,2,...$ (6)

Here $g$ is the tunneling exponent, expected to be 1/3 for the $\nu=1/3$
state. This implies that the innermost nodes will have a narrower spacing than
the outer ones, which will have a spacing $\Delta
V_{SD}=\frac{hv_{edge}}{e^{*}L}$. At high temperatures, Ref. Chamon1997
predicts that the innermost nodes will move outward and reach same spacing as
the outer nodes, so that node spacing is uniform as in the integer case. Since
this theory calculates the total current it is most convenient to work with
the DC current oscillation amplitude $\delta I$ rather than the differential
conductance; therefore, we have measured both $\delta G$ and $\delta I$ as a
function of $V_{SD}$ at $\nu=1/3$.

Figure 5: Finite-bias measurements at $\nu=1/3$. a) Differential conductance
measurements at $\nu=1/3$ $B=7.4$ T. b) Differential conductance oscillation
amplitude vs. $V_{SD}$. The spacing between nodes (which appear as minima in
the plot) are indicated. It is noteworthy that the spacing between the central
nodes is somewhat more narrow than the outer nodes, which may be an indication
of Luttinger liquid behavior. c) DC current versus $\delta V_{g}$ and
$V_{SD}$. d) DC current oscillation amplitude versus $V_{SD}$. The minima in
the amplitude correspond to nodes in the interference pattern. As predicted by
Chamon1997 , the central nodes have narrower spacing than the outer ones.

The differential conductance for $\nu=1/3$ is plotted in Fig. 5a, with the
amplitude shown in b. The central separation between the innermost nodes is
$\approx 120$ $\mu$V, while the separation between the outer nodes is $\approx
190$ $\mu$V. More direct comparison to Chamon1997  can be made by to
measurements of oscillations in the DC current, shown in Fig. 5c and 5d. The
outer nodes have a separation of $\approx 197$ $\mu$V, while the inner ones
have a separation of $\approx 167$ $\mu$V. This is consistent with the
expectation that the inner nodes should have a narrower spacing; however, the
ratio of the inner to the outer node spacing is 0.85, which is somewhat
smaller than the value of 2/3 predicted by Eqn. 6 from Chamon1997 . A possible
explanation for this discrepancy is that the large biases applied in these
measurements cause significant heating of the electrons in the device,
shifting behavior towards the high-temperature limit of uniform node spacing.
Additionally, at elevated mixing chamber temperature the innermost node moves
to higher $V_{SD}$, approaching the spacing of the outer nodes as anticipated
by Chamon1997 ; see Supplemental Fig. 6 and Supplemental Section 5. The
qualitative agreement suggests that this theory can be used to extract
$v_{edge}$ from the outer node spacing of $\Delta V_{SD}\approx 197$ $\mu$V
(the differential conductance measurement shows similar values for the outer
node spacing). Then, $E_{sp}=\frac{e\mathcal{E}\Phi_{0}}{LB\Delta\nu}=e\Delta
V_{SD}\frac{e^{*}}{e\Delta\nu}$; with $e^{*}=e/3$ and $\Delta\nu=1/3$ for the
$\nu=1/3$ state, $E_{sp}=e\Delta V_{SD}$ as for the integer case. Using
$\Delta V_{SD}\approx 197$ $\mu$V gives $K_{I}\approx 72$ $\mu e$V$+197$ $\mu
e$V$=269$ $\mu e$V, while as before $K_{IL}=\frac{e^{2}}{C}\approx 72$ $\mu
e$V. This gives $\frac{K_{IL}}{K_{I}}=0.27$, close to the value of 0.24
extracted from Eqn. 4. A possible limitation to this approach is that the
equations for current in Ref. Chamon1997  were developed without including
bulk-edge coupling. Future theoretical work could refine this analysis by
analysing bulk-edge coupling corrections to the positions of nodes and improve
the accuracy of $K_{I}$.

The narrower spacing of the innermost nodes, being a signature of Luttinger-
liquid behavior, contrasts with the nearly uniform node spacing observed for
the integer states at $\nu=1$ and $\nu=3$ in Fig. 4a and c, which is expected
for Fermi liquids. Previous experimental evidence for Luttinger-liquid
behavior of fractional quantum Hall edge states has been seen in tunneling
experiments Chang1996 ; Chang2003 , while here we have shown evidence through
interferometry.

##  VIII Phase jumps at $\nu=1/3$

Several discrete phase jumps can be seen in Fig. 1a, similar to previous
observations Nakamura2020 , which may be caused by the anyonic phase when the
number of localized quasiparticles inside the interferometer changes. To
extract the values of these phase jumps, we have calculated the phase at each
value of the magnetic field by taking Fourier transforms along cuts parallel
to the lines corresponding to discrete jumps. Then we subtract off the
Aharonov-Bohm contribution to the phase (which simply results in continuous
phase evolution and a constant linear slope in phase vs. $B$). The process for
extracting the phases in this way is discussed in detail in the Supplemental
Section 5 and illustrated in Supplemental Fig. 7; this method should enable a
more accurate phase extraction than the fitting method in Nakamura2020  and
has the additional advantage of not needing the position of each jump to be
specified. The resulting phase after subtracting the Aharonov-Bohm
contribution should be due to the anyonic contribtuion, and is plotted in Fig.
6a. As can be seen in Fig. 1a, some of the phase jumps are very close to each
other, so that the individual jumps in phase are not readily resolvable; in
particular there appear to be two very close jumps at $\approx 7.28$ T and
three close jumps at $\approx 7.37$ T. While the individual phase jumps cannot
be isolated, the combined phase jump can be extracted from the data. At low
fields (below approximately 7.2 T) and high fields (above approximately 7.7 T)
the phase exhibits a staircase pattern due to $\approx\Phi_{0}$ periodic
additions of quasiparticles, although since there is still significant
smearing this staircase pattern is not sharp.

The values of the phase jumps (both the individual ones from isolated jumps
and the combined ones when multiple are very close) are listed in Fig. 6a, and
the corresponding part of the data where the jumps occur is indicated in b.
These values are calculated by taking the average value of the phase on each
plateau and subtracting the adjacent values to get the jump in phase.
Averaging all the jumps (and taking into account the fact that some of the
changes in phase are most likely due to multiple discrete jumps) yields an
average change in phase $\frac{\overline{\Delta\theta}}{2\pi}=-0.24\pm 0.04$
(uncertainty is estimated from the standard deviation of the phase jumps).
This change is smaller than the value of $\theta_{a}=\frac{2\pi}{3}$ expected
from theory (note however that the difference in sign is accounted for the
fact that quasiparticles are expected to be removed as field is increased).
However, theoretical works Halperin2011 ; VonKeyserlingk2015 ; Halperin2021
predict a modification to the value of the phase jump that occurs when a
quasiparticle is added due to bulk edge coupling (Eqn. 7):

$\frac{\Delta\theta}{2\pi}=\frac{\theta_{a}}{2\pi}-\frac{K_{IL}}{K_{I}}\frac{e^{*2}}{\Delta\nu}$
(7)

This modification comes about because when a quasiparticle enters the bulk,
it’s electric charge will cause the area of the interferometer to change,
leading to a change in the Aharonov-Bohm phase in addition to the anyonic
phase. This correction can be included into the extraction of the anyonic
phase at $\nu=1/3$ by
$\frac{\theta_{a}}{2\pi}=-\frac{\overline{\Delta\theta}}{2\pi}-\frac{1}{3}\frac{K_{IL}}{K_{I}}$.
Using the value of $\frac{K_{IL}}{K_{I}}=0.24$ extracted from Eqn. 4 gives
$\frac{\theta_{a}}{2\pi}=0.32\pm 0.05$, while using
$\frac{K_{IL}}{K_{I}}=0.27$ from finite bias measurements yields
$\frac{\theta_{a}}{2\pi}=0.33\pm 0.05$; these values are close to the value of
$\theta_{a}=\frac{2\pi}{3}$ from theoretical and numerical studies Arovas1984
; Jain2003 ; Jain2004 ; Jain2010 . Thus, although the bulk-edge interaction
partially obscures the anyonic phase, accounting for this effect indicates
that the anyonic phase is close to the theoretically predicted value, giving
strong support to the theoretical works Halperin2011 ; VonKeyserlingk2015 ;
Halperin2021  and consistent with previous experiments at $\nu=1/3$
Bartolomei2020 ; Nakamura2020 .

Figure 6: Phase versus magnetic field. a) Phases extracted from Fourier
transforms of the data in Fig. 1a. The FFTs are performed along cuts of the
conductance parallel to discrete phase jumps. The phase is evaluated at the
peak frequency which corresponds to the Aharonov-Bohm oscillation frequency.
The Aharonov-Bohm effect gives a a constant linear change of phase with $B$
which has been subtracted off to yield the contribution from localized
quasiparticles, and plateaus in phase occur which correspond to the regions
between phase jumps. The change in phase for the discrete jumps are indicated;
the leftmost discrete jump appears to correspond to two closeby jumps, while
the second from the left appears to consist of three closeby jumps; in these
cases the individual phase jumps are not readily resolved, but the total phase
change can be calculated and divided into average individual phase changes. b)
Raw data (repeated from Fig. 1a) indicating where the discrete jumps in a)
occur in the data.

##  IX Can the phase jumps at $\nu=1/3$ be explained by bulk-edge coupling
alone?

Since bulk-edge coupling can cause discrete jumps in phase even for electrons
Halperin2011 ; Heiblum2016-2 ; Ensslin2020 , it should be considered whether
the discrete jumps in phase observed at $\nu=1/3$ can be explained by bulk-
edge coupling alone rather than anyonic statistics Halperin2021 .

From Eqn. 3 and Ref. VonKeyserlingk2015  the change in phase for adding a
quasiparticle is $-\frac{K_{IL}}{K_{I}}\frac{e^{*2}_{in}}{\Delta\nu}$ if
$\theta_{a}$ is assumed to be zero. For $\nu=1/3$ where $\Delta\nu=1/3$ and
$e^{*}_{in}=1/3$, this would result in phase jumps of
$\frac{\Delta\theta}{2\pi}=\frac{1}{3}\frac{K_{IL}}{K_{I}}$ (note the change
in sign due to the fact that quasiparticles are being removed by increasing
the magnetic field rather than added). Using the value of
$\frac{K_{IL}}{K_{I}}$ of 0.27 extracted from the period measurements (with
the value from differential conductance measurements being similar, assuming
that the interpretation at $\nu=1/3$ is correct) gives an expected
$\frac{\Delta\theta}{2\pi}=0.09$. This value is significantly different from
the phase jumps observed in the data in Fig. 6, and is of opposite sign. This
suggests that while bulk-edge coupling does reduce the value of the phase
jumps observed in this device, the phase jumps cannot be explained by bulk
edge coupling alone without anyonic statistics. An assumption made in this
analysis is that the charge of the localized quasiparticles is equal to the
theoretically predicted value $e^{*}=e/3$, whereas a larger localized charge
would result in a greater phase jump contribution from bulk-edge coupling.
Scanning probe experiments have observed $e/3$ localized charge at the
$\nu=1/3$ state Yacoby2004 , supporting the assumption of fractional charge.
Additionally, in previous measurements of a larger device with weak bulk edge
coupling discrete phase jumps close to the expected anyonic phase of $2\pi/3$
were observed Nakamura2020 , slightly larger than the jumps measured in the
present device. If the phase jumps were caused only by bulk edge coupling they
would be expected to be significantly larger in the smaller device with
greater $K_{IL}$. The fact that they are instead slightly smaller is
consistent with the anyonic phase being partially obscured by bulk-edge
coupling, but not with being caused exclusively by it.

##  X Conclusions

In conclusion, we have demonstrated experimental evidence for multiple
theoretical predictions of quantum Hall interferometers. We have observed
$\Phi_{0}$ period modulations in interference at the $\nu=1/3$ state, which
are a signature of anyonic statistics when the bulk is compressible. We have
demonstrated two approaches for estimating the impact of bulk edge coupling:
using the ratio of the magnetic field periods, and extracting the
electrostatic coupling constants $K_{I}$ and $K_{IL}$ directly from finite
bias measurements. Uneven node spacing observed at $\nu=1/3$ in finite-bias
measurements indicates Luttinger liquid behavior. Although our model makes
several simplifications, we find that this approach validates theoretical
predictions for distinguishing between the Aharonov-Bohm and Coulomb dominated
regimes in the integer quantum Hall regime. Accounting for the correction to
$\Delta\theta$ from finite $\frac{K_{IL}}{K_{I}}$ yields values of
$\theta_{a}$ in agreement with the theoretically predicted value at $\nu=1/3$,
supporting previous experiments. An important finding is that the parameter
$\frac{K_{IL}}{K_{I}}$ can vary between different edge states in the same
device, which makes inner edge states more likely to be Coulomb dominated.
This work will inform future experimental and theoretical analysis of quantum
Hall interferometry.

##  XI Methods

This interferometer utilizes a high mobility GaAs/AlGaAs heterostructure grown
by molecular beam epitaxy Manfra2014 ; Gardner2016 . The bulk electron density
is approximately $0.6\times 10^{11}$cm-2 and mobility is $3.2\times 10^{6}$
cm2V${}^{-1}s^{-1}$. The structure also includes additional screening wells
with a setback of 25 nm from the main quantum well to reduce the charging
energy and bulk-edge coupling so that anyonic statistics can be observed. The
screening well design may also enhance the steepness of the confining
potential Harshad2018 ; Nakamura2019 , which may be important for preventing
edge reconstruction that may lead to dephasing by neutral modes Heiblum2019 ;
Hu2009 ; Gefen2016 . The screening well heterostructure design has been
described in detail in Ref. Nakamura2019 . Though the structure has the same
layer stack as the one in Ref. Nakamura2020 , the wafer is different and was
grown at a different time.

Optical lithography and wet etching to define the mesa. Ni/Au/Ge Ohmic
contacts were deposited and annealed to make electrical contact to the 2DEG.
Electron beam lithography and electron beam evaporation (5nm Ti/10nm Au) were
used to define the interferometer gates. Optical lithography and electron beam
evaporation (20nm Ti/150nm Au) were used to define bondpads and the surface
gates around the Ohmic contacts. The substrate was mechanically polished to to
make it thin enough to define metal backgates to deplete electrons in the
bottom screening well, which were patterned by optical lithography and
deposited by electron beam evaporation (100nm Ti/150nm Au).

Measurements are performed using standard voltage-biased low frequency lock-in
amplifier techniques with a typical excitation of 5 $\mu$V and frequency of 37
Hz in a dilution refrigerator with a base mixing chamber temperature of 10 mK.

##  XII Acknowledgements

This work is supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, under award number DE-SC0020138. G. C.
Gardner and M. J. Manfra acknowledge support from Microsoft Quantum. The
authors thank Bertrand Halperin for valuable suggestions to improve the
manuscript. M. J. Manfra thanks Dima Feldman and Steve Simon for fruitful
discussions during the early phases of this project.

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