Direct observation of anyonic braiding statistics at the ν=1/3 fractional quantum Hallstate

J. Nakamura,1, 2 S. Liang,1, 2 G. C. Gardner,2, 3 and M. J. Manfra1, 2, 4, 3, 5, ∗

1Department of Physics and Astronomy, Purdue University2Birck Nanotechnology Center, Purdue University3Microsoft Quantum Purdue, Purdue University

4School of Electrical and Computer Engineering, Purdue University5School of Materials Engineering, Purdue University

(Dated: June 26, 2020)

Utilizing an electronic Fabry-Perot interferometer in which Coulomb charging effects are sup-pressed, we report experimental observation of anyonic braiding statistics for the ν = 1/3 fractionalquantum Hall state. Strong Aharonov-Bohm interference of the ν = 1/3 edge mode is punctuatedby discrete phase slips consistent with an anyonic phase of θanyon = 2π

3. Our results are consistent

with a recent theory of a Fabry-Perot interferometer operated in a regime in which device chargingenergy is small compared to the energy of formation of charged quasiparticles [17]. Close correspon-dence between device operation and theoretical predictions substantiates our claim of observationof anyonic braiding.

BACKGROUND

Quantum theory requires that all fundamental par-ticles must be fermions or bosons, which has profoundimplications for particles’ statistical behavior. However,theoretical works have shown that in two dimensions itis possible for particles to violate this principle and obeyso-called anyonic statistics, in which exchange of particleposition results in a quantum mechanical phase changethat is not π or 2π (as for fermions or bosons), but a ra-tional fraction of π [1, 2]. While anyons cannot exist asfundamental particles in nature, certain condensed mat-ter systems are predicted to host exotic quasiparticleswhich obey a certain form of anyonic statistics.

The quantum Hall effect is a remarkable example ofa topological phase of matter occurring when a two-dimensional electron system (2DES) is cooled to lowtemperature and placed in a strong magnetic field. Inthe quantum Hall regime the bulk forms an insulator,and charge flows in edge currents which are topologicallyprotected from backscattering and exhibit quantized con-ductance. The elementary excitations of fractional quan-tum Hall states [3] are not simply electrons, which obeyfermionic statistics, but instead are emergent quasipar-ticles which are predicted to have highly exotic prop-erties including fractional charge and anyonic statistics[4]. In two dimensions, two exchanges of particle po-sitions are topologically equivalent to one quasiparticleencircling the other in a closed path [5], referred to as abraid; this is illustrated in Fig. 1a. The anyonic char-acter of these quasiparticles is reflected in the fractionalphase the system obtains from braiding; thus they aresaid to obey anyonic braiding statistics. The statistics offractional quantum Hall states have been studied in the-oretical [6, 7] and numerical [8–12] works. The anyonicphase does not depend on the trajectory taken but onlyon the number of quasiparticles encircled, making braid-

ing another manifestation of topology in quantum Hallphysics; this topological robustness has motivated aggres-sive pursuit of fault-tolerant quantum computation basedon braiding operations in various condensed matter sys-tems [5, 13–15]. In a recent experimental work anyonicstatistics were inferred from noise correlation measure-ments [16]; however, direct observation of the anyonicphase in braiding experiments will further our under-standing of the exotic behavior of quantum Hall quasi-particles and is a necessary step to towards quasiparticlemanipulation.

Electronic interferometry has been used to study edgephysics in previous theoretical [17–25] and experimen-tal [26–45] works, and has been proposed as an ex-perimental means to observe anyonic braiding statistics[18, 20, 46, 47] including the highly exotic non-Abelianform of anyonic statistics [48–55]. An electronic Fabry-Perot interferometer consists of a confined 2DES usingquantum point contacts (QPCs) to partition edge cur-rents, as shown in Fig. 1b. Quasiparticles backscatteredby the QPCs will braid around quasiparticles localizedinside the interferometer; therefore changes in Nqp, thenumber of quasiparticles localized inside the interferom-eter, will result in a shift in the interference phase dueto the anyonic contribution θanyon [18, 20, 46, 47], withθanyon = 2π

2p+1 for a Laughlin fractional quantum Hall

state ν = 12p+1 [6, 7]. The interferometer phase differ-

ence θ is a combination of the Aharonov-Bohm phasescaled by the quasiparticle charge e∗ and the anyoniccontribution, written in Eqn. 1 [18, 20, 46]:

θ = 2πe∗

e

AIB

Φ0+Nqpθanyon (1)

The total current backscatterd by the interferometerwill depend on cos (θ), so the interference phase can beprobed by measuring the conductanceG across the device

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[58].

A major obstacle towards the observation of any-onic phases through interferometry has been been theCoulomb interaction of the interfering edge state withcharge located in the bulk of the interferometer [19]. Astrong bulk-edge interaction causes the area AI of the in-terferometer to change when charge in the bulk changes[19, 20]. As a consequence, for so-called Coulomb-dominated devices with strong bulk-edge interaction, thechange in Aharonov-Bohm phase due to the change inAI when Nqp is changed cancels out the anyonic phaseθanyon, making quasiparticle braiding statistics unob-servable [20]. While novel physics has been exploredin Coulomb-dominated devices [30, 32, 34, 45, 56, 57],this bulk-edge interaction must be reduced to make any-onic braiding observable. Various techniques have beenimplemented to reduce this Coulomb bulk-edge interac-tion, including the use of metal screening gates [30, 34],low-temperature illumination to enhance screening bythe doping layer [52, 54, 55], addition of an Ohmic con-tact inside the interferometer [39], and incorporation ofauxilliary screening layers inside the semiconductor het-erostructure [58]. The screening layer technique has en-abled the use of small highly coherent interferometersthat exhibit robust Aharonov-Bohm interference, includ-ing at fractional quantum Hall states [58].

DEVICE DESIGN

The device used for these experiments utilizes a uniquehigh-mobility GaAs/AlGaAs heterostructure [59, 60]with screening layers to minimize the bulk-edge inter-action (see the layer stack in Supp. Fig. 1) [58]. Theinterferometer is defined using metal surface gates whichare negatively biased to deplete the 2DES underneath.Two narrow constrictions define QPCs to backscatteredge currents, and wider side gates define the rest of theinterference path. An SEM image of the device is shownin Fig. 1b; the device has a nominal area of 1.0µm ×1.0 µm, and measurements suggest that lateral depletionof the 2DES makes the interferometer area smaller byapproximately 200nm on each side, similar to the experi-mental and numerical results in [58] (see also [61]). Notethat the length scale of the interferometer is much greater

than the magnetic length lB ≡√

hceB in the regime inves-

tigated, with lB ≈9nm at ν = 1/3, so the condition thatthe interfering quasiparticles be well separated from thelocalized quasiparticles inside the interferometer whichthey may braid around should hold [10, 11]. Compared tothe device used in [58], the device used in this work has alower electron density n, which improves device stabilitybecause smaller gate voltages can be used. The devicealso has a somewhat smaller area, which may increasecoherence and visibility of interference. Experiments are

performed in a dilution refrigerator with a base mixingchamber temperature of T ≈10mK; Coulomb blockademeasurements of different quantum dot devices suggesta somewhat higher electron temperature of T ≈ 22mK.Negative voltages of ≈-1V are applied to the QPC gatesand ≈-0.8V on the side gates; conductance is measuredas a function of the side gate voltage variation δVg, whichis relative to -0.8V and applied to both side gates. An ad-ditional metal gate in the center of the device (not shownin Fig. 1b for clarity) is held at ground potential, so itdoes not affect the 2DES density; this gate is intendedto make the confining potential from the gates sharper.Measurements are performed using standard 4-terminaland 2-terminal lock-in amplifier techniques.

DISCRETE PHASE SLIPS

We operated the device at high magnetic field B at thefilling factor ν = 1/3 quantum Hall state. In Fig. 2 weshow the conductance variation δG measured across theinterferometer versus B and δVg near the center of theν = 1/3 conductance plateau. The QPCs remain in theregime of weak backscattering across this region with ap-proximately 90% transmission, and a smooth backgroundconductance is subtracted so that the interference oscil-lations can be seen clearly. As can be seen in the figure,the predominant behavior observed is conductance os-cillations with negatively-sloped lines of constant phase;however, quite conspicuously there are also a small num-ber of discrete phase jumps in the data; dotted linesare guides to the eye for these features. The jumps inphase were found to be repeatable in subsequent scans;see Supp. Fig. 5.

Eqn. 1 provides a straightforward explanation forour observations. Continuous phase evolution withnegatively-sloped lines of constant phase is the signatureof the Aharonov-Bohm effect [20]. This can be seen bytaking the derivative of Eqn. 1 with θ and Nqp held fixed,

which yields∂Vg

∂B = − 1βAI

B (here β ≡ ∂AI

∂Vgparameterizes

the change in interferometer area with side gate voltage).The negative sign implies negative slope to lines of con-stant phase as has been observed in previous experimentsin the integer [26, 30, 34] and fractional [58] quantum Hallregimes. The second term in Eqn. 1 predicts a discretechange in phase when the number of localized quasipar-ticles changes; therefore, it is natural to associate thediscrete phase jumps with the anyonic phase contribu-tion θanyon. It is noteworthy that the discrete jumps inphase occur across lines with positive slope in the B-Vgplane. This can be understood from the fact that in-creasing B is expected to remove quasiparticles from thebulk (or create quasiholes) [4, 18], while increasing gatevoltage would make it electrostatically favorable to in-crease the number of localized quasiparticles. Thus, the

3

Isource

200nm

a)

b)

=

FIG. 1. Quasiparticle braiding experiment. a)Schematic representation of quasiparticle exchange; quasi-particles are represented by red vortices, and trajectoriesare shown in dashed lines. Two quasiparticle exchanges(left) which bring the particles back to their original positionare topologically equivalent to one quasiparticle executing aclosed loop around the the other, and in each case the systemgains a quantum mechanical phase θanyon due to the quasi-particle’s anyonic braiding statistics. b) False-color SEM im-age of interferometer. Blue regions indicate the GaAs wherethe 2DES resides, and metal gates under which the 2DESis depleted are highlighted in yellow. When operated at theν = 1/3 fractional quantum Hall state, the current is carriedby quasiparticles traveling in chiral edge states (red arrows),and dotted arrows indicate the backscattered quasiparticlepaths which may interfere. Quasiparticles may be localizedinside the chamber of the interferometer, as represented bythe red vortices, and the backscattered paths enclose a looparound these quasiparticles, making the interferometer sensi-tive to θanyon. The lithographic area is 1.0µm × 1.0µm. Thedevice used in the experiments also has a metal gate coveringthe top of the interferometer not shown in b), which is keptat ground potential and does not affect the 2DES density un-derneath.

magnetic field at which it becomes favorable to removea quasiparticle should increase when gate voltage is in-creased, and a positive slope to the quasiparticle transi-tions is expected, as observed in resonant tunneling ex-periments [32, 41, 56, 57, 62]. The fact that we do in-deed observe a positive slope strongly suggests that thesediscrete phase jumps are associated with changes in lo-calized quasiparticle number, and the magnitude of the

slope is also consistent with this; see Supp. Discussion1 for additional analysis. Furthermore, a central prin-ciple of quantum Hall theory is that quasiparticles arelocalized in the hills and valleys of the disorder potential[63], and the fact that the discrete phase jumps are ir-regularly spaced indicates that their positions are in factdetermined by disorder as expected.

To determine the value of the change in phase as-sociated with each phase jump in the data, we per-formed a least-squares fit in the regions between thephase jumps, fitting the conductance data to the formδG = δG0cos(2π

13AIBΦ0

+ θ0), with the fitting parameterbeing θ0. This expression for the conductance assumesthat between the discrete phase jumps, the phase evolvesonly by the change in Aharonov-Bohm phase with chang-ing B and changing AI (via the change in Vg), and θ0

is the excess phase which cannot be attributed to theAharonov-Bohm effect. We determine the value of thephase jump by computing ∆θ, the difference in the fit-ted values of θ0 in adjacent regions. The fitted data areshown highlighted in Fig. 2, and the extracted valuesof ∆θ

2π are shown above each jump. Taking an averageand assuming that each phase jump corresponds to theremoval of a quasiparticle (or equivalently addition of aquasihole), we obtain θanyon = 2π × (0.31 ± 0.04); thisis consistent with the theoretical value of θanyon = 2π

3for the ν = 1/3 state [6, 7]. Our work thus provides ex-perimental confirmation for the prediction of fractionalbraiding statistics at the ν = 1/3 quantum Hall state.

TRANSITION FROM CONSTANT FILLING TOCONSTANT DENSITY

A recent theoretical work analyzed the case of a Fabry-Perot interferometer operated at the ν = 1/3 state inwhich strong screening is utilized to reduce the charac-teristic Coulomb charging energy and thus suppress thebulk-edge interaction [17]. A key prediction is that thedevice will transition from a regime of constant filling fac-tor to regimes of constant electron density when the mag-netic field is varied away from the center of the state andthe chemical potential moves away from the center of thegap in the density of states. The authors find that over awide range of magnetic field the bulk 2DES stays at fixedν = 1/3 filling. In this regime of constant ν the predomi-nant contributor to the phase will be the Aharonov-Bohmphase, but a small number of well-separated quasiparti-cle transitions should occur from which θanyon may beextracted, consistent with our results described above.Once the magnetic field is varied away from the center,the authors predict that the electrostatic energy cost ofvarying density to maintain fixed ν will cause a transi-tion from constant filling factor to constant density. Inthe regimes of constant density, many quasiparticles (atlow field) or quasiholes (at high magnetic field) will be

4

8.75 8.80 8.85 8.90 8.95 9.00

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dV

g (

mV

)

B (T)

-4

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-1

0

1

2

3

4dG (x10-2e2/h)

Δ𝜃

2𝜋= −0.32

Δ𝜃

2𝜋= −0.38

Δ𝜃

2𝜋= −0.28

Δ𝜃

2𝜋= −0.29

FIG. 2. Conductance oscillations versus magnetic field and side gate voltage. The predominant behavior is negativelysloped Aharonov-Bohm interference, but a small number of discrete phase jumps are visible. Dashed lines are guides to theeye for these features. Least-squares fits of δG = δG0 cos (2πAB

Φ0+ θ0) are shown with highlighted stripes, and the extracted

change in phase ∆θ2π

are indicated for each discrete jump. Increasing magnetic field is expected to reduce the number of localizedquasiparticles; therefore the change in phase across each jump is predicted to be −θanyon.

created inside the interferometer to keep the total chargefixed, with one quasiparticle or quasihole created whenthe flux is changed by one flux quantum Φ0, resulting insignificant changes in interference behavior mediated bythe anyonic phase.

Motivated by these predictions, we operated the inter-ferometer in a wide range of magnetic field across theν = 1/3 fractional quantum Hall state. Bulk magneto-transport at ν = 1/3 with vanishing longitudinal resis-tance Rxx and a quantized plateau in the Hall resistanceRxy is shown in Fig. 3a, showing the range of magneticfield over which the ν = 1/3 state occurs in our sample.The conductance measured across the device across theν = 1/3 state is shown in Fig. 3b; this is the same mea-surement as shown in Fig. 2, but extended to higher andlower magnetic field. As discussed previously, near thecenter of the ν = 1/3 plateau the predominantly observedbehavior in the conductance is lines of constant phasewith negative slope consistent with Aharonov-Bohm in-terference [18–20, 58] with a small number of discrete

jumps attributed to quasiparticle transitions. The gatevoltage and magnetic field oscillation periods are ap-proximately three times larger than the integer periodsmeasured at ν = 1, consistent with interference of e/3fractionally charged quasiparticles, as is expected for theν = 1/3 state and consistent with previous experimen-tal observations of fractional charge [34, 58, 64–66]. Oneither side of this central region, however, the behav-ior changes significantly. The lines of constant phase losetheir negative slope; although there is still weak magneticfield dependence to the pattern, the magnetic field scaleover which the phase varies is much larger than in thecentral region, making the lines of constant phase nearlyflat; the oscillations depend primarily only on the sidegate voltage. It is noteworthy that, despite this conspic-uous change, the lines of constant phase are continuousacross the transition from the central Aharonov-Bohmregion to the upper and lower regions, which indicatesthat the oscillations are still due to interference of theedge state.

5

8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3

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dV

g (

mV

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-4-3-2-101234dG (x10-2 e2/h)

8.0 8.5 9.0 9.5

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5

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15

20

25

30

35

40

B (T)

Rxx (kW

)

60

70

80

90

Rxy (

kW

)

450mTa) b)

FIG. 3. Interference across the ν = 1/3 quantum Hall plateau. a) Bulk magnetransport showing longitudinal resistanceRxx and Hall resistance Rxy across the ν = 1/3 state. b) Conductance oscillations δG versus magnetic field B and side gatevoltage δVg (this side gate voltage variation is relative to -0.8V). The dashed lines indicate the approximate range over which thedevice appears to exhibit conventional Aharonov-Bohm interference with minimal influence of the anyonic phase contribution.The region over which this occurs is near the center of the plateau, and is highlighted in the bulk transport data in a).

Our experimental observation that negatively-slopedAharonov-Bohm interference occurs only in a finite rangeof magnetic field agrees with the predictions of [17]. Atfirst blush the behavior observed above and below thiscentral region seems to conflict with predictions: we ob-serve an interference pattern that becomes nearly inde-pendent of magnetic field, while [17] predicts that themagnetic field period will decrease from 3Φ0 in the cen-tral region to Φ0 in the upper and lower regions becausequasiparticles will be created with period Φ0. However,an additional key prediction in [17] is that the Φ0 oscil-lations will be extremely susceptible to thermal smear-ing, with the authors estimating a temperature scaleT0 ≈ 2mK (because our device is smaller than the oneconsidered in [17] this predicted temperature scale wouldbe T0 ≈ 4mK for our device, still much smaller thanour estimated electron temperature of T ≈ 22mK). Thisthermal smearing can be understood from the fact thatthe regime of constant density corresponds to the chemi-cal potential being at a position of high density of states,and thus small energy spacing between states, leading tothermal smearing. Therefore, the absence of Φ0 oscilla-tions at T ≈ 22mK is in fact in agreement with [17].

The fact that the lines of constant phase flatten outand become independent of magnetic field can be un-derstood based on the combined contribution of theAharonov-Bohm phase and anyonic phase (Eqn. 1).For the ν = 1/3 state, quasiparticles are predicted tocarry fractional charge e∗ = e/3 and fractional braid-ing statistics θanyon = 2π/3 [6]. Changing the mag-netic field to add one flux quantum to the device will

change the Aharonov-Bohm phase by 2π3 . Additionally,

in the lower field regimes one quasiparticle will be re-moved, and in the high field regime one quasihole willbe added, resulting in a phase shift of − 2π

3 and leavingthe total interference phase unchanged in both regimes.The Aharonov-Bohm phase varies continuously, while (inthe limit of zero temperature) the quasiparticle numberwill change discretely, leading to the predicted Φ0 oscil-lations [17, 18]; however, when the quasiparticle numberis thermally smeared, the average number of localizedquasiparticles will vary nearly continuously, leading to asmooth variation of the anyonic phase; in this case thesmoothly varying thermally-averaged anyonic phase can-cels the Aharonov-Bohm phase, leading to no change inθ as B is varied, consistent with our experimental obser-vations. Because each quasiparticle at the 1/3 state is avortex, this can also be understood based on the resultfrom [6] that the Berry phase of a vortex encircling aclosed path is equal to 2π〈qenc〉 where qenc is the chargeenclosed in the path, and the high and low field regionsthe electrostatics force density to remain fixed, and thus〈qenc〉 remains nearly constant.

The approximate range over which the negatively-sloped Aharonov-Bohm oscillations occur is marked withdashed lines in Fig. 3a, and has a span of approximately450mT. To make a quantitative comparison to theory,we compute the predicted width of the fixed ν region

from [17]: ∆Bconstant−ν =∆1/3Φ0CSW

νe2e∗ . In this expres-sion ∆1/3 is the energy gap of the ν = 1/3 state whichwe measure to be ≈ 5.5K (see Supp. Fig. 3), consis-tent with previous measurements of the ν = 1/3 gap [67].

6

CSW is the capacitance per unit area of the screening lay-ers to the quantum well which we calculate as CSW = 2ε

d ,with the factor of two accounting for the fact that thereare two screening layers and d = 48nm the setback of thescreening layers from the quantum well. Using the exper-imental values from the device gives a predicted value for∆Bfixed−ν ≈ 530mT, in good agreement with the exper-imentally observed range of Aharonov-Bohm interferenceof ≈ 450mT, which suggests that the experimentally ob-served transition in interference behavior can indeed beexplained by the model of [17].

Additionally, there is a moderate reduction in the sidegate voltage oscillation period in the high and low fieldregions compared to the central region. In Supp. Discus-sion 1 and Supp. Fig 4 we analyze this shift in period andextract the parameters αbulk relating the change in bulkcharge and αedge relating the change in charge at theedge to side gate voltage. Using these parameters, wefind that the shift in period is consistent with creationof quasiparticles and quasiholes with gate voltage, lead-ing to a change in period through the statistical phase.Also, we have performed numerical simulations of inter-ferometer behavior based on the models of [20] and [17]which show good qualitative agreement with experiment;see Supp. Discussion 2 and Supp. Fig. 5.

Taken together, our observations of discrete phasejumps near the center of ν = 1/3 along with the tran-sitions in behavior at high and low field in concordancewith the predictions of [17] give a consistent picture inwhich the statistical phase of anyonic quasiparticles con-tributes to the interference phase. Our data point to theimpact of braiding both in the regime where the chemi-cal potential is near the center of the energy gap, wherethe density of states is small and individual quasiparti-cle transitions can be resolved, and in the regimes aboveand below the gap where the density of states is highand a continuum of quasiparticle and quasihole statescontribute to the phase.

The arguments in [17] of a transition from a regime ofconstant ν to regimes of constant n when moving awayfrom the center of the state also apply to integer quan-tum Hall states. In Supp. Fig. 6 we show measurementsof interference as a function of B and Vg across the in-teger state ν = 1; in contrast to the fractional ν = 1/3case, the device exhibits no change in behavior and dis-plays negatively-sloped Aharonov-Bohm interference atthe high and low field extremes of the plateau. This isconsistent with the fact that the charge carriers and ex-cited states are electrons which obey fermionic statistics,making their braiding unobservable; θfermion = 2π.

TEMPERATURE DEPENDENCE

An additional observation is that the oscillation ampli-tudes decay with temperature much more sharply in the

high-field and low-field regions than in the central region.We measured the amplitude of the oscillations in each re-gion versus temperature; the oscillations decay approx-imately exponentially with T as temperature increases,and we can characterize each region by the temperaturedecay scale T0 assuming that the oscillation amplitude

varies as e−TT0 [18, 50, 51, 68]. We extract T0 through a

linear fit of the natural log of the oscillation amplitudeas a function of temperature; this data is shown in Fig.4. For the low-field region at 8.4T (blue) T0 = 31mK, forthe central region at 8.85T (black) T0 = 94mK, and forthe high-field region at 9.3T (red) T0 = 32mK. Differen-tial conductance measurements to extract the edge-statevelocity were performed [18, 31, 38], and indicate thatthe edge velocity does not vary significantly between thedifferent regions (see Supp. Discussion 3 and Supp. Fig.7). Evidently, the observed suppression of T0 by nearlya factor of 3 in the high and low field regions cannotbe explained by lower edge velocity. Based on the mea-sured edge velocities we calculate predicted temperaturedecay scales T0 of 76mK at 8.4T, 89mK at 8.85T, and85mK at 9.3T (see Supp. Discussion 2). In the centralregion the predicted value of T0 is close to the experi-mentally observed value, indicating that in this region ofconstant ν and a small number of quasiparticles, dephas-ing can primarily be attributed to thermal smearing ofthe edge state based on dwell time in the interferometer.In the high and low field regions, however, the experi-mental values are much smaller than the predicted val-ues, suggesting that there must be an additional source ofdephasing in these regions. The fact that this increaseddephasing occurs in the regions where a large number ofquasiparticles and quasiholes populate the interferome-ter, but not in the central region, suggests that it maybe explained by the topological dephasing proposed in[69], in which thermal fluctuations in localized quasipar-ticle number reduce interference visibility in Fabry-Perotinterferometers. This affirms the expectation that theregimes of constant density correspond to high quasipar-ticle DOS [17]. This dephasing is a remarkable exampleof the non-local influence of anyonic statistics: despitethe fact that the edge quasiparticles are well separatedby many magnetic lengths from quasiparticles inside thebulk of the interferometer such that there is minimal di-rect interaction, thermal fluctuations in Nqp neverthelesslead to rapid thermal dephasing of the interference signal.

The behavior of the device described here in the maintext was reproduced in a second device, including thechange in interference behavior from negatively-slopedAharonov-Bohm interference to flat lines of constantphase, the suppression of T0 outside the central region,and the observation of discrete phase jumps consistentwith the predicted anyonic phase at ν = 1/3; see Supp.Fig. 8. The possible influence of residual bulk-edge in-teraction is discussed in Supp. Discussion 4.

7

0 20 40 60 80 100 120

-3

-2

-1

0

B = 8.4T

B = 8.85T

B = 9.3T

ln(d

G)

(a.

u.)

T (mK)

T0 = 94mK

T0 = 31mK

T0 = 32mK

FIG. 4. Dependence of oscillation amplitude on tem-perature. The natural log of the oscillation amplitude δGat 8.4T, 8.85T, and 9.3T is plotted versus temperature. Datapoints are normalized to the amplitude at the lowest temper-ature and offset for clarity. The oscillation amplitudes showan approximately exponential decay with increasing tempera-ture. Dashed lines indicate linear fits from which the temper-ature decay scale T0 is extracted at each magnetic field. T0

is much larger in the central region than in the low and highfield regions, suggesting that there is an additional dephasingmechanism in these regions. This may be explained by topo-logical dephasing due to thermal smearing of the quasiparticlenumber. The QPCs are tuned to approximately 90% trans-mission at each temperature to maintain constant backscat-tering.

CONCLUSIONS

We have measured conductance oscillations in a Fabry-Perot interferometer across a wide range of magnetic fieldat the ν = 1/3 quantum Hall state. Near the center of thestate, we observe discrete jumps in the interference phaseconsistent with the anyonic braiding statistics of localizedquasiparticles, and we obtain θanyon = 2π×(0.31±0.04),which agrees with the theoretically predicted value ofθanyon = 2π

3 . When the magnetic field is moved awayfrom the center, we observe a change in interference be-havior from predominantly negatively sloped lines of con-stant phase to a phase that is nearly independent of B.This observation suggests that the 2DES transitions froma regime of constant filling factor at the center to regimesof constant density leading to a thermally smeared pop-ulation of quasiparticles (at low field) and quasiholes (athigh field), as predicted in a recent theoretical work [17].Extraction of the lever arms which parameterize the ef-

fects of gate voltage on charge in the device increasesour confidence in this analysis. In the low and high fieldregimes we observe a dramatic increase in thermal de-phasing evidenced by the suppression the temperaturedecay scale T0, which indicates that despite their largespatial separation from the interfering edge state, local-ized quasiparticles have a profound impact on interfer-ence behavior through their braiding statistics. Taken to-gether, our experimental observations are consistent withinterference of anyonic quasiparticles for which braidingstatistics contribute to the interference phase.

METHODS

The device used was fabricated using the followingsteps: (1) optical lithography and wet etching to de-fine the mesa; (2) deposition and annealing of Ni/Au/GeOhmic contacts; (3) electron beam lithography and elec-tron beam evaporation (5nm Ti/10nm Au) to define theinterferometer gates; (4) optical lithography and electronbeam evaporation (20nm Ti/150nm Au) to define bond-pads and surface gates around the Ohmic contacts; (5)mechanical polishing to thin the GaAs substrate; (6) op-tical lithography and electron beam evaporation (100nmTi/150nm Au) to define the backgates used to depletethe bottom screening well around the Ohmic contacts sothat only the primary quantum well is probed.

Standard low-frequency (f = 13Hz) 4-terminal and 2-terminal lock-in amplifier techniques were used to probethe diagonal resistance and conductance across the de-vice. Typically a 50pA excitation current was used formeasurements. A +600mV bias was applied to the QPCand side gates while the device was cooled from roomtemperature; this bias-cool technique results in an ap-proximately 600mV built-in bias on these gates, whichwas found to improve device stability.

ACKNOWLEDGEMENTS

This work is supported by the U.S. Department of En-ergy, Office of Science, Office of Basic Energy Sciences,under award number de-sc0020138. G. C. Gardner ac-knowledges support from Microsoft Quantum. The au-thors thank Bernd Rosenow for valuable comments on anearly version of this manuscript.

REFERENCES

∗ mmanfra@purdue.edu

8

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Supplementary Information for “Direct observation of anyonic braiding statistics atthe ν=1/3 fractional quantum Hall state”

arX

iv:2

006.

1411

5v1

[co

nd-m

at.m

es-h

all]

25

Jun

2020

2

10nm GaAs

65nm Al0.36Ga.64As

45nm Al0.36Ga.64As

30nm GaAs

25nm Al0.36Ga.64As

25nm Al0.36Ga.64As

Substrate

13nm GaAs

2nm AlAs

45nm Al0.36Ga.64As

2nm AlAs

13nm GaAs

2nm AlAs

2nm AlAs

Si 𝛿 -doping

Si 𝛿 -doping

Bottom Screening Well

Top Screening Well

Primary Well

SUPP. FIG. 1. Layer stack of the GaAs/AlGaAs heterostructure used for the experiments. This structure utilizesthree GaAs quantum wells: a primary 30nm well flanked by two 13nm screening wells to reduce the bulk-edge interaction inthe interferometer. There are 25nm AlGaAs barriers between the main well and screening wells, and the total center-to-centersetback of the screening wells from the main well is 48nm.

3

Scan 1

Scan 2

a)

8.75 8.80 8.85 8.90 8.95 9.00

-30

-25

-20

-15

-10

-5

dV

g (

V)

B (T)

-4

-3

-2

-1

0

1

2

3

4dG (x10-2e2/h)

Δ𝜃

2𝜋= −0.32

Δ𝜃

2𝜋= −0.35

Δ𝜃

2𝜋= −0.25

Δ𝜃

2𝜋= −0.29

8.75 8.80 8.85 8.90 8.95 9.00

-30

-25

-20

-15

-10

-5

dV

g (

mV

)

B (T)

-4

-3

-2

-1

0

1

2

3

4dG (x10-2e2/h)

Δ𝜃

2𝜋= −0.32

Δ𝜃

2𝜋= −0.38

Δ𝜃

2𝜋= −0.28

Δ𝜃

2𝜋= −0.29

b)

SUPP. FIG. 2. Repeatability of discrete phase jumps. a)First scan measurement of conductance versus B and δVg. Thisis the same data in Fig. 2 of the main text. b) Second scan across the same range of magnetic field using the same QPC gatevoltages. As can be seen from the data, the same pattern of discrete jumps appear in the second scan. The second scan wastaken approximately one hour after the first scan. Values of ∆θ

2πextracted from least squares fits are shown for both scans, and

show similar values for each phase jump in both scans.

4

1.0 1.5 2.0

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

Data

Linear fit

log(R

xx)

T -1(K-1)

8.0 8.5 9.00

1

2

3

4 757mK

705mK

655mK

543mK

475mK

334mK

Rxx (

kW

)

B (T)

Δ = 5.5K

SUPP. FIG. 3. Measurement of the energy gap for the ν = 1/3 fractional quantum Hall state. The inset showslongitudinal resistance Rxx measured in a bulk region away from the interferometer at different temperatures. A linear fit of

the data to the form Rxx = R0e−∆2kT yields a gap of ∆ =5.5K. This is consistent with values measured in previous experiments

at similar magnetic field [1].

5

SUPPLEMENTARY DISCUSSION 1: ANALYSISOF PERIODS AND LEVER ARMS

We discuss the change in interference behavior fromthe central region, where the device exhibits primarilyAharonov-Bohm interference behavior with a few phasejumps, to the high and low field regions, where the lines ofconstant phase flatten out. Interestingly, despite the linesof constant phase remaining continuous, the side gateoscillation period becomes smaller in the high and lowfield regions relative to the central region, with periodsof 5.8mV at 8.4T, 8.5mV at 8.85T, and 5.4mV at 9.3T.In Supp. Fig. 4a line cuts of conductance versus gatevoltage are shown at these magnetic fields illustratingthe change in period. On the other hand, the model of[2] suggests that the side gate oscillation period will bethe same in the central region as in the upper and lowerfield regions, because the authors assume that the sidegate couples only to the edge of the interferometer; underthis assumption the side gate voltage will only affect theAharonov-Bohm phase, making the variation of θ withVg the same in each region. However, in a real device thegates do not affect just area; they also have some effect onthe charge in the bulk of the interferometer. In the highand low field regions this will lead to an additional changein phase with gate voltage due to changes in localizedquasiparticle number. To analyze the effect this will haveon the side gate oscillation period, in Supp. Eqn. 1 wetake the derivative of θ (from Eqn. 1 in the main text)with respect to side gate voltage:

∂θ

∂Vg= 2π

e∗

e

B

Φ0

∂AI∂Vg

+ θanyon∂〈NL〉∂Vg

(1)

Here with 〈NL〉 we take the thermally averaged num-ber of quasiparticles to account for the fact that in thehigh and low field regimes significant thermal smearingis expected, and 〈NL〉 will not necesarily be an integer[2].

In order to determine whether change in localizedquasiparticle number with gate voltage can explain theobserved change in period, we determine the parameterαbulk ≡ ∂qbulk

∂Vgwhich paramaterizes how the bulk charge

inside the interferometer qbulk changes with Vg. To deter-mine αbulk we have operated the device at zero magneticfield in the Coulomb blockade regime [3] with the QPCstuned to weak tunneling; in this regime there is one con-ductance peak each time the number of electrons in thedevice changes by one. Inverting the Coulomb blockadeoscillation period gives the total lever arm coupling theside gates to the interferometer, αtotal = 1

∆VCB= ∂qtotal

∂Vg.

Zero-field Coulomb blockade oscillations are plotted in4b; the 5.4mV period yields αtotal =0.19mV−1. How-ever, qtotal is a combination of charge at the edge andcharge in the bulk, qtotal = qedge + qbulk, so to determine

αbulk we must also determine αedge ≡ ∂qedge∂Vg

. To extract

αedge we operate the device as an Aharonov-Bohm in-terferometer at the integer quantum Hall state ν = 1; inthis regime the interference phase and thus the oscillationperiod depends only on change in interference area andnot on changes in charges localized in the bulk [4]. Sincefor an integer state each oscillation period correspondsto changing the enclosed flux by one, αedge = n∂AI∂Vg

=nΦ0

B1

∆Vν=1with n the electron density (we assume that

the electrostatics which determine the coupling of thegate to the edge and to the bulk do not change signifi-cantly with magnetic field). AB interference oscillationsfor the integer state ν = 1 at B = 3.1T are show in Supp.Fig. 4c; the period of 8.0 mV gives ∂A

∂Vg= 0.167µm2V−1

and αedge = 0.12mV−1. Finally, we calculate αbulk =αtotal − αedge =0.19mV−1−0.12mV−1 =0.07mV−1. Thefact that αedge is significantly larger than αbulk indicatesthat, as expected, the primary action of the side gates isto change the area of the interferometer, and the changein bulk charge is comparatively small.

We calculate the expected periods in each regime atν = 1/3 by ∆Vg = 2π( ∂θ

∂Vg)−1. In the central region

quasiparticles are unlikely to be created because of thelarge energy gap, so only the first term on the right-handside of Supp. Eqn. 1 contributes, whereas in the low-field/high-field regions quasiparticles/quasiholes will becreated and contribute to the phase, so both terms willcontribute. For the central region then the predicted pe-riod is ∆Vg = Φ0

Bee∗ (∂AI∂Vg

)−1 ≈ 8.4mV, in good agreement

with the measured value of 8.5mV. This agreement withthe model suggests that interference in the central regioncan indeed be understood as the Aharononv-Bohm ef-fect at constant ν of e/3 quasiparticles, consistent withtheoretical predictions of fractional charge [5] as well asprevious experiments in interferometry [6, 7] and otherexperimental observations of fractional charge [8–10]. At8.4T and 9.3T, taking into account the creation of quasi-particles, we calculate ∆Vg = 1

BΦ0

e∗e

∂AI∂Vg

+θanyon

2π

∂NL∂Vg

=

1B

3Φ0

∂AI∂Vg

+αbulk; here we have used e∗

e = 13 , θanyon = 2π

3 ,

and ∂NL∂Vg

= ee∗αbulk. This equation yields a predicted

δVg of ≈ 5.5mV at 8.4T and ≈ 5.1mV at 9.3T, in goodagreement with the experimental values of 5.8mV and5.4mV. This agreement between predicted and observedoscillation periods in each region is strong support for thepicture in [2] of a region of constant n and a quasipar-ticle population at low field, a region of constant ν nearthe center of the state, and a region of constant n and aquasihole population at high field.

Additionally, these extracted lever arms can be usedanalyze quantitatively the slope of the quasiparticle tran-sition lines in Fig. 2 of the main text. These transitionswill occur when it becomes energetically favorable for aquasiparticle to be created at a certain place in the de-

6

-40 -30 -20 -10 0

-0.10

-0.05

0.00

0.05

0.10

0.15

dG

(e

2/h

)

dVg (mV)

B = 8.4T DVg = 5.8mV

B = 8.85T DVg = 8.5mV

B = 9.3T DVg = 5.4mV

-40 -30 -20 -10 0-0.10

-0.05

0.00

0.05

0.10

0.15

dG

(e

2/h

)

dVg (mV)

0.00

0.05

0.10

0.15

0.20

0.25

G (

2e

2/h

)

a) b)

c)

B = 0 ΔVg = 5.3mV

B = 3.1T ΔVg = 8.0mV

SUPP. FIG. 4. Conductance oscillations at different magnetic fields. a) Conductance oscillations δG versus side gatevoltage δVg in the low-field region at B = 8.4T (blue), in the central region at B = 8.85T (black), and in the high-field region atB = 9.3T (red). The side gate oscillation period ∆Vsidegates is significantly smaller in the low field and high field regions thanin the central region, with ∆Vg = 5.8mV at 8.4T, ∆Vg = 8.5mV at 8.85T, and ∆Vg = 5.4mV at 9.3T. The QPCs are tuned toapproximately 90% transmission. b) Conductance G versus side gate voltage at zero magnetic field with the device operatedin the Coulomb blockade regime. Unlike other data presented in this work, the oscillations shown here are due to resonant

tunneling of electrons rather than interference, and the QPCs are tuned weak tunneling, G << e2

h. The Coulomb blockade

oscillations have a period of 5.3mV, which is used to obtain the total lever arm αtotal of the gates to the interferometer. c)Aharonov-Bohm interference oscillations at ν = 1. The oscillations period of 8.0mV is used to obtain the lever arm αedge ofthe gates to the edge.

vice, therefore the transition lines will correspond to linesof constant electrostatic energy associated with charge onthe device. This electrostatic energy comes from accu-mulating charge on the device due to changes in the con-densate charge density with magnetic field, which may becompensated for by the creation of quasiparticles. Thecharge on the 2DES is thus a combination of the conden-sate charge density and the charge associated with eachlocalized quasiparticle:

q2DES =eνAIB

Φ0+ e∗Nqp (2)

Furthermore, we must consider the net charge, qnet,the difference between the charge in the 2DES (fromSupp. Eqn. 2) and the background charge:

qnet = q2DES−qback =eνAB

Φ0+e∗Nqp−qdonor−eαbulkδVg

(3)Here the background charge is a combination of the

charge from the donors, qdonor, and the effect of the gatevoltage. We follow [4] in treating the gates as creatingsome effective additional background charge eαbulkδVg.Since the changes in localized quasiparticle number oc-cur when the electrostatic energy cost exceeds the en-

ergy cost to create a quasiparticle, Supp. Eqn. 3 im-plies that the localized quasiparticle transitions will occuracross lines with a slope

dVgdB = νA

Φ0αbulk. Using ν = 1/3,

αbulk = 0.07mV−1 (discussed above), and area extractedfrom the AB oscillations at ν = 1 A = Φ0

∆Bν=1≈ 0.38µm2

(Supp. Fig. 6b), we obtaindVgdB ≈ 0.44mV/mT. Exper-

imentally, the observed phase jumps occur with a slopeof approximately 0.5 mV/mT, in good agreement withthe predicted value. This is strong evidence that the dis-crete phase jumps do indeed correspond to changes in thenumber of localized quasiparticles inside the interferom-eter.

7

SUPPLEMENTARY DISCUSSION 2:SIMULATIONS

To further validate application of the model of [2] toour experimental results, we have performed simulationsof interferometer behavior to model the conductance ver-sus gate voltage Vg and magnetic field B. The startingpoint for this is the equation for the interference phasedifference which determines conductance oscillations:

θ = 2πe∗

e

AIB

Φ0+Nqpθanyon (4)

The conductance for an interferometer varies as δG =a0cos(θ), with a0 an amplitude that depends on thebackscattering of the QPCs. In our system thermal fluc-tuations inNqp may be important, so we need to calculatea thermal average (because we use low-frequency mea-surement techniques, this thermal average should cor-respond to the experimentally measured conductance).Following [4], we compute the thermal expectation value〈δG〉:

e〈δG〉 =1

Z

+∞∑

Nqp=−∞e

−E(Nqp)

kBT cos (2πe∗

e

AIB

Φ0+Nqpθanyon)

(5)

Z =+∞∑

Nqp=−∞e

−E(Nqp)

kBT (6)

Here E is the energy of the device as a function of Nqp.As in the main text, a negative quasiparticle number herecorresponds to a population of quasiholes. We define anenergy similar to that in [2], but for the interferometerrather than the bulk energy per unit area:

E = E0 +e2

2Cδq2net + ∆qp|Nqp| (7)

E0 is an offset accounting for the energy of the conden-sate which does not depend on the number of quasiparti-cles, so it is left out of the simulation. The absolute valueof Nqp is multiplied by ∆qp to account for the fact thatquasiholes (corresponding to negative Nqp) also cost en-ergy ∆qp, and for simplicity we set the energy associatedwith creating a quasiparticle ∆qp to be half of the fullgap, ∆qp = ∆

2 . This is a simplification which assumesthat the energy cost for creating quasiparticles is thesame as for quasiholes, although numerical results haveindicated that quasiparticles have a higher energy costthan quasiholes [11]. Nevertheless, the width of the con-stant ν region over which no quasiparticles or quasiholesare created is determined by the full gap which is a sum

of the quasiparticle and quasihole gaps, ∆ = ∆qp + ∆qh,and this full gap is the value measured in transport. So,asymmetry in the energies of quasiparticles and quasi-holes do not affect quantitative comparison between thetheory of [2] and our experimental results.

As discussed in the Supp. Discussion 1, qnet is thedifference between the charge in the 2DES and the com-pensating background charge (including that created bythe side gate), and is written in Supp. Eqn. 3. The terme2

2C δq2net in Supp. Eqn. 7 gives the energy cost associated

with building up excess charge on the device. In the pres-ence of screening layers separated from the main well bydistance d, we estimate the characteristic capacitance tobe C = 2εAI

d .

Simulations of conductance versus magnetic field andside gate voltage are performed by numerically evaluat-ing Supp. Eqn. 5 and Supp. Eqn. 6 at each value of Band δVg. The area AI is computed as AI = A0 + ∂A

∂VgδVg.

Rather than performing an infinite sum, we sum over Nqpfrom -20 to +20 to make the computation possible; thisis justified because states with large numbers of quasi-particles are exponentially suppressed. Simulations areperformed for the ν = 1/3 state, so based on theoreti-cal expectations we set θanyon = 2π

3 and e∗ = e/3. Thevalue of 5.5K for ∆ extracted from the bulk transportgap measurement in Supp. Fig. 3 was used, giving avalue of 2.75K for ∆qp. A 2DES of 0.7 × 1011cm−2 isassumed, which sets the background charge qdonor.

In Supp. Fig. 5a, b, and c we show the results ofthe simulations at different temperatures. The energyscale which primarily determines the thermal smearing

effect of temperature is e2

2C , so we set the ratio of kT toe2

2C at 0.002, 0.02, and 0.1 in a, b, and c. In order tomake the behavior in the simulations easier to see in theplots, these simulations are performed with a device withsmaller area than the real device; we set A0=0.1µm2,whereas for the real device A0 ≈ 0.38µm2 based on theAharonov-Bohm periods. We first focus on the low-temperature simulation in a). Qualitative features ofthe simulation match the experiment: negatively-slopedAharonov-Bohm oscillations of period 3Φ0 occur nearthe center, and this behavior is confined to a 530mT re-gion (consistent with the value calculated from the modelin [2] and close to the experimentally observed value of≈ 450mT). Above and below this region, the simulationin a) shows sharply defined discrete jumps in phase oc-curring with period Φ0, consistent with the findings of[2]. As temperature is increased in b) and c), the tran-sitions in these phase jumps become thermally smearedtogether, such that at the highest value of temperaturesimulated the transitions in phase are nearly completelysmeared out, and the simulation shows nearly flat lines ofconstant conductance. This is in good qualitative agree-ment with the experimental results.

An additional subtle feature in the simulations is that

8

De

nsity o

f S

tate

s

Energy

High field

Center

Low field

8.2 8.4 8.6 8.8 9.0-6

-4

-2

0

2

4

6

<N

QP>

B (T)

T/Ec = 0.002

T/Ec = 0.02

T/Ec = 0.1

a)

b)

c)

d)

e)𝜇

Quasiparticle region

Φ0 period

Pure AB region

3Φ0 period

Quasihole region

Φ0 period

Hole-like

statesParticle-like

states

SUPP. FIG. 5. Simulations of interferometer behavior at ν = 1/3. Conductance values are computed as a function ofmagnetic field B and side gate voltages Vg, taking into account both the Aharonov-Bohm phase and the contribution θanyonfrom braiding around localized quasiparticles inside the bulk of the interferometer. Simulations are performed at different ratios

of the temperature kBT to the interferometer charging energy Ec = e2

2Ca) 0.002 b) 0.02 and c) 0.1. d) Plot of the thermal

expectation value of the number of localized quasiparticles inside the interferometer for different ratios of kBT/Ec; in thiscontext a negative quasiparticle number indicates a population of quasiholes. In each case in the middle of the state there areno quasiparticles, resulting in conventional Aharonov-Bohm interference with 3Φ0 period, while at higher fields quasiholes formand at lower fields quasiparticles form, resulting in phase slips with Φ0 period. As temperature is elevated, the quasiparticlenumber is thermally smeared, making the Φ0 period phase slips unobservable and reducing the amplitude of the oscillationsthat occur as a function of Vg. e) Qualitative plot of the density of states versus energy.

the transition from the central Aharonov-Bohm region toquasiparticle and quasihole regions occurs across a linewith positive slope in the Vg − B plane, due to the cou-pling of the side gate to the bulk included in the sim-ulation, αbulk. This behavior is in fact observed in theexperimental data in the transition to the high-field re-gion. In the low-field transition this behavior is less clearbecause the transition appears to occur more smoothlyrather than abruptly, but a positive slope is still observ-able.

In Supp. Fig. 5d, line cuts of the thermally aver-aged quasiparticle number 〈Nqp〉 are plotted versus Bfrom the simulations at each temperature. At low tem-peratures this forms a staircase-like function with verysharp transitions when it becomes energetically favorableto change the number of localized quasiparticles, whereasat high temperatures these transitions become thermallysmeared such that the evolution becomes quite smooth.While our simplified model and simulations likely donot capture all of the physics of the device, we believethat this picture of the average quasiparticle number be-coming thermally smeared at high temperature smeared

should hold. Additionally, it is possible that other mech-anisms, such as charge noise, may result in smearing of〈Nqp〉 on the measurement time scale.

9

3.42 3.44 3.46 3.48 3.50 3.52

B (T)

-10

-8

-6

-4

-2

0

2

4

6

8

10

dG (x10-2e2/h)

3.12 3.14 3.16 3.18 3.20 3.22 3.24

B (T)

2.70 2.72 2.74 2.76 2.78 2.80

-30

-25

-20

-15

-10

-5

dV

g (

mV

)

B (T)

2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

0

1

2

3

4

B (T)

Rxx (kW

)

15

20

25

30

Rxy (

kW

)

a)

c) d) e)

3.18 3.20 3.22 3.24

-0.02

-0.01

0.00

0.01

0.02

0.03

dG

(e

2/h

)B (T)

DB = 11mTb)

SUPP. FIG. 6. Measurements of interference at ν = 1. a) Bulk quantum Hall transport showing the zero in Rxxand plateau in Rxy corresponding to the ν = 1 integer quantum Hall state. For this integer state, the bulk excitations andedge state current carrying particles are simply electrons, which obey fermionic statistics. b) Conductance oscillations versusmagnetic field, showing an oscillation period ∆B =11mT. From this period the effective area AI of the interferometer can beextracted: AI = Φ0

∆BIn c), d), and e) we show conductance versus B and δVg across the interferometer in the low field region

of the plateau, near the center of the plateau, and on the high-field side of the plateau; the region on the plateau correspondingto each pajama plot is shown in a). In each of these regions the device exhibits negatively sloped Aharonov-Bohm oscillations.This contrasts with the data shown in the main text for the ν = 1/3 state where lines of constant phase flatten out at high andlow fields. This is consistent with the fact that electrons, which carry current and form localized states at ν = 1, are fermionswho obey trivial braiding statistics, θfermion = 2π, making braiding unobservable and leading to no change in interferencebehavior.

10

SUPPLEMENTARY DISCUSSION 3: VELOCITYMEASUREMENTS

In the main text we have discussed the observationthat the temperature decay scale is much smaller in thehigh and low field regions than in the central region atν = 1/3, which suggests that topological dephasing dueto thermal smearing of the localized quasiparticle num-ber inside the interferometer may contribute in the highand low field regions. However, another possible explana-tion for the change in T0 is that the edge velocity mightbe much larger at the center, and decrease in the highand low field regions; if this were the case, T0 would de-crease simply due to the additional thermal smearing ofthe edge state (that said, such a large and non-monotonicchange in the velocity over such a relatively small rangeof B would be rather surprising). In order to determineif the change in T0 can be explained by changes in theedge velocity vedge, we have performed differential con-ductance measurements in each region from which vedgecan be extracted [12]; this has been performed previ-ously for integer quantum Hall states in Aharonov-Bohminterferometers [6, 13, 14]. We consider first the case ofan integer charge edge state. When a finite source drainvoltage bias is applied, the energy of the injected edgeelectrons changes, and this leads to a shift in the phasefrom the edge dispersion, δθ = δε∂k∂εL = δεL

~vedge . This

leads to an additional interference pattern that occurs asa function of source-drain bias Vsd which can be observedin the differential conductance measurement, resulting ina checkerboard pattern in the measured differential con-ductance as a function of Vsd and δVg. Thus nodes inthe conductance oscillations which occur as a functionof δVg occur at certain values of Vsd, and the spacingbetween nodes can be used to extract the edge velocity[6, 12, 14, 15]:

vedge =e∗L∆Vsd

2π~(8)

Here L is the perimeter of the interferometer, esti-mated based on the area extracted from Aharonov-Bohminterference measurents L = 4

√AI ≈ 2.5µm. Differen-

tial conductance measurements are shown at ν = 1/3 inSupp. Fig. 7. Also plotted is the oscillation amplitude(extracted from a Fourier transform of the data) versusVsd, which enables convenient extraction of ∆Vsd [14].This is shown for the low-field region in a) and b), for thecentral region in c) and d), and for the high-field regionin e) and f). Using Supp. Eqn. 8 yields edge veloci-ties of 8.3× 103m/s in the low-field region at B = 8.4T,9.7 × 103m/s in the central region at B = 8.85T, and9.3×103m/s in the low-field region at B = 9.3T. The factthat the velocity does not change significantly with mag-netic field indicates that a change in edge velocity cannotaccount for the large change in temperature decay scale

T0 between regions. The expected T0 can be calculatedas T0 = h

2πkBτ1g [12, 16] where g is the scaling exponent

of the edge state, g = 13 for ν = 1/3 [12], and τ = L

vedge

is the time for the edge state to traverse the interferom-eter. This yields predicted T0 based on thermal smear-ing of the edge state of 76mK at B = 8.4T, 89mK at8.85T, and 85mK at 9.3T. The predicted value of 89mKin the central region at 8.85T is close to the experimen-tally observed T0 of 94mK, indicating that the decay ofamplitude in the region where the device is nearly free ofquasiparticles can be attributed to thermal smearing ofthe edge. We have observed similar agreement betweenpredicted and observed T0 at the integer quantum Hallstate ν = 1 [6]. However, the experimentally observedT0 of 31mK in the low-field region and 32mK in the highfield region at ν = 1/3 are much smaller than the val-ues predicted for thermal smearing of the edge, indicat-ing that another dephasing mechanism must be at play.This provides further support for the theory that topo-logical dephasing due to thermally smearing of localizedquasiparticles contributes to dephasing in these regions.

11

-50 -25 0 25 50

0.4

0.6

0.8

1.0

1.2

1.4

oscill

ation a

mplit

ude (

a. u.)

VSD (mV)

-50 -25 0 25 50

0.2

0.3

0.4

0.5

oscill

ation a

mplit

ude (

a. u.)

VSD (mV)

-30 -25 -20 -15 -10 -5

-100

-75

-50

-25

0

25

50

75

100

VS

D (m

V)

dVg (mV)

-3.2

-1.9

-0.6

0.7

2.0

3.3

dG (x10-2 e2/h)

-25 -20 -15 -10 -5

-100

-75

-50

-25

0

25

50

75

100V

SD (m

V)

dVg (mV)

-0.5

-0.3

-0.1

0.1

0.3

0.5

dG (x10-2 e2/h)

-25 -20 -15 -10 -5

-100

-75

-50

-25

0

25

50

75

100

VS

D (m

V)

dVg (mV)

-1.8

-1.3

-0.8

-0.3

0.2

0.7

1.2

1.7

dG (x10-2 e2/h)

-50 -25 0 25 50

0.2

0.3

0.4

0.5

0.6

0.7

oscill

ation a

mplit

ude (

a. u.)

VSD (mV)

ΔVSD = 41μV

ΔVSD = 48μV

ΔVSD = 46μV

a) b)

c) d)

e) f)

B = 8.4T

B = 8.85T

B = 9.3T

Low-field

region

Central

region

High-field

region

SUPP. FIG. 7. Differential conductance measurements at ν = 1/3. a) Differential conductance ∂I∂Vsd

as a function of

side gate voltage δVg and source-drain bias Vsd at B = 8.4T in the low-field region. b) Conductance oscillation amplitude froma FFT of the conductance versus side gate voltage data as a function of Vsd. The oscillation amplitude shows a node patternas a function of Vsd from which the edge velocity may be extracted, yielding vedge = 8.3 × 103m/s. c) Differential conductanceand d) oscillation amplitude versus Vsd at 8.85T giving vedge = 9.7 × 103m/s. e) Differential conductance and f) oscillationamplitude versus Vsd at 9.3T giving vedge = 9.3× 103m/s. Evidently, the edge velocity does not change significantly across theν = 1/3 quantum Hall plateau.

12

8.550 8.575 8.600 8.625 8.650 8.675

-30

-25

-20

-15

-10

-5

dV

g (

mV

)

B (T)

Δ𝜃

2𝜋= −0.32

7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6

0

1

2

3

B (T)

Rxx (kW

)

7.0

7.5

8.0

Rxy (

kW

)

430mTa)

b) c)

SUPP. FIG. 8. Measurements of interference for a second device, taken from a different chip fabricated on thesame wafer. a) Conductance across the interferometer versus magnetic field B and side gate voltage δVg; δVg is relative to-1.0V. Behavior is similar to that observed in the device described in the main text: in a finite region with width ≈ 430mT, thedevice exhibits negatively sloped Aharonov-Bohm oscillations, which flatten out at higher and lower magnetic fields, consistentwith the creation of quasipaticles and quasiholes. b) Bulk magnetotransport showing Rxx (red) and Rxy (blue) for device B.The region near the center of the ν = 1/3 state where the negatively sloped Aharonov-Bohm oscillations occur is highlighted.c) zoomed-in view of a clear phase jump in the data (this jump is also visible in b), but the data in c) is a different scanintended to improve signal to noise). Least-squares fits of the conductance on either side of the phase jump yields an extractedphase jump ∆θ

2π= −0.32, yielding an anyonic phase θanyon = 2π × 0.32, consistent with theory.

13

SUPPLEMENTARY DISCUSSION 4: POSSIBLEBULK-EDGE INTERACTION EFFECTS

We discuss another possible explanation for some ofthe experimental observations. It is worth mentioningthat there is another mechanism which can cause discretechanges in phase in quantum Hall interferometers, evenfor integer quantum Hall states. In devices which are in-termediate between the Aharonov-Bohm and Coulomb-dominated regimes, creation of a localized charge insidethe interferometer causes the area of the interferometerto change due to finite bulk-edge coupling, resulting in areduction in the Aharonov-Bohm phase and visible dis-crete changes in the interference phase. This mechanismdoes not depend on exotic braiding statistics [4]. How-ever, increasing the magnetic field should tend to removeparticle-like quasiparticles or create hole-like quasipar-ticles; in either case each excitation will lead to an anincrease in phase when magnetic field is increased, be-cause the decrease in Nqp would be accompanied by com-pensatory increase in AI . However, this is inconsistentwith our observation of negative changes in phase acrosseach discrete phase jump, and also inconsistent with thefact that these discrete phase jumps occur in the regionwhere the Aharonov-Bohm phase shows clear negativeslope, indicating minimal bulk-edge interaction. Never-theless it is possible that some residual bulk-edge inter-action may have a small effect on the observed phasejumps. In [4] it was found that the observed jumpsin phase when changing quasiparticle number should be∆θ = θanyon × (1 − KIL

KI), where KIL

KIis the ratio of the

bulk-edge interaction strength KI to the characteristicenergy cost for charging the edge KI . Thus, residualbulk-edge interaction would result in a slightly smallerobserved change in phase. This might account for thefact that the majority of the observed phase jumps areslightly smaller than 2π

3 .

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