Main

For more than four decades, quasiparticles carrying fractional charge and obeying fractional statistics have captivated the condensed matter physics community1. Their most prevalent types are the Abelian anyons, which exhibit quantized exchange phases lying between those of bosons and fermions. Even more notable are non-Abelian anyons that can fundamentally transform the many-body wavefunction through particle exchanges, processing quantum information in a topologically protected manner2. The FQH systems have emerged as a leading platform for realizing and manipulating these exotic quasiparticles, owing to high electron mobility, long coherence times and exceptional controllability3. Fractional charge was first observed by means of shot-noise measurements at odd-denominator filling factors expected to host Abelian states4,5 and later at the even-denominator filling (\nu =\frac{5}{2}) in GaAs (refs. 6,7), a leading candidate for non-Abelian topological order8.

Direct measurements of anyonic exchange statistics require phase-sensitive techniques such as quantum Hall interferometry in the Aharonov–Bohm regime, in which Coulomb interactions are sufficiently weak for the interferometer area to remain constant as B is varied9. Seminal works by Nakamura et al. demonstrated Aharonov–Bohm interference of fractionally charged quasiparticles using a GaAs FPI at filling (\nu =\frac{1}{3}) (ref. 10), and braiding (double-exchange) phases in a subsequent study11. These findings were generalized to different filling factors12 and also observed in alternative platforms, first in integer13,14,15 and then in fractional fillings16,17,18, and interferometer architectures19,20. In parallel, time-domain braiding experiments21,22,23,24,25,26 also support anyonic quasiparticle statistics in Abelian FQH states27. At even-denominator fillings, Fabry–Pérot interferometry studies at (\nu =\frac{5}{2}) in GaAs have reported signatures consistent with non-Abelian statistics28. However, the interpretation of those experiments remains challenging, primarily because of the absence of robust Aharonov–Bohm interference.

Even-denominator states have been observed in several FQH platforms, including GaAs (ref. 29), ZnO (ref. 30), graphene31, bilayer graphene32,33,34,35 and WSe2 (ref. 36). In GaAs narrow quantum wells, thermal-transport measurements37,38 consistently support a non-Abelian topological order known as PH-Pfaffian39. Distinct non-Abelian orders known as Moore–Read Pfaffian40 and anti-Pfaffian41 are indicated by daughter states42 in bilayer graphene35,43 and GaAs wide quantum wells44. Specifically, bilayer graphene realizes quantized plateaus at seven half-integer filling factors in the zeroth Landau level. Moreover, the presumed topological orders alternate between Pfaffian and anti-Pfaffian, offering a rich playground for interference studies of non-Abelian anyons.

In this work, we report the observation of robust Aharonov–Bohm oscillations at two even-denominator FQH plateaus in bilayer graphene. Using a gate-defined FPI in a high-mobility bilayer-graphene-based vdW heterostructure, we perform a detailed study of the interference patterns as a function of magnetic field, area and density. At both fillings, we observe the unexpected Aharonov–Bohm periodicity ΔΦ = 2Φ0 when the magnetic field and the density are varied together to maintain contact filling. The most conservative interpretation of these measurements is the interference of quasiparticles with charge ({e}^{{* }=\frac{1}{2}e), twice the charge expected theoretically40 and observed in earlier shot-noise and single-electron transistor measurements in GaAs (refs. 6,7). However, this frequency could also originate from ({e}}{* }=\frac{1}{4}e) quasiparticles performing an even number of loops.

This finding prompted us to study the nearby odd-denominator states at Landau-level fillings of (\nu =\frac{1}{3}) and (\frac{2}{3}), for which we found Aharonov–Bohm periodicities corresponding to interference of quasiparticles with charges ({e}^{* }=\frac{1}{3}e) and (\frac{2}{3}e), respectively. Across the three fillings, the interfering charge follows e* = ν**e instead of the minimal charges of bulk quasiparticles, which are (\frac{1}{3}e), (\frac{1}{4}e) and (\frac{1}{3}e) for these states. All even-denominator and odd-denominator states for this study are indicated in the phase diagram of bilayer graphene (see Supplementary Information Section 1). We note that, in GaAs, shot-noise measurements at hole-conjugate states also find a partitioned charge of ν**e (ref. 45) but interference at (\nu =\frac{2}{3}) shows e* = e (ref. 10). Finally, by tuning the electron density independently of the magnetic field, we deviate from the fixed-filling constraint, thereby introducing localized bulk quasiparticles46. Unlike the integer case, we observe a statistical contribution to the interference phase of fractional fillings, supporting their anyonic character.

Design and measurement phase-space of an FPI

The FPI device is constructed on a high-mobility vdW heterostructure, with bilayer graphene as the active 2D layer, which is encapsulated between hexagonal boron nitride (hBN) dielectric layers, whereas conductive graphite layers on the top and bottom serve as gates. The heterostructure design and nanofabrication techniques follow those detailed in our previous study16, with measurements conducted under a perpendicular magnetic field up to B = 12 T and at a base temperature of T = 10 mK.

A false-colour scanning electron microscopy image of the FPI is shown in Fig. 1a. The top graphite layer is divided into eight distinct regions by 40-nm-wide etched trenches, with each region contacted by means of air bridges. Together with a global graphite back gate, these eight top gates enable capacitive tuning of the potential and displacement fields across various regions of the bilayer graphene. The filling factor inside the interferometer is controlled by the centre gate, whereas the left and right gates set the outer fillings. The two quantum point contacts (QPCs) are formed by the left and right split gates, which set the filling underneath to zero, thereby guiding the counter-propagating edge modes on opposite sides into close proximity and introducing tunnelling between them. An extra plunger gate (denoted PG) allows fine control over the area enclosed by the interfering quantum Hall edge mode. Figure 1b provides a zoomed-in view of the interfering region, lithographically defined to be 1 μm2 (see Supplementary Information Section 2). Two air bridges, denoted LBG and RBG, positioned 200 nm above the QPC regions, act as gates, fine-tuning the transmission tR, tL of each QPC independently. In the measurements presented in this study, tR,L are set to 0.5–0.7 for the interference at the integer states and 0.6–0.9 for the interference at the fractional states (see Supplementary Information Section 3).

Fig. 1: FPI based on the bilayer graphene.

a, Measurement configurations depicted as a false-colour scanning electron microscopy image of the FPI in a bilayer graphene heterostructure (see inset). The top graphite layer (purple) is divided into eight separate regions by means of etched trenches, better seen in b. Each region acts as a gate, electrostatically tuned by means of air bridges (blue) to define the interferometer. The device dimensions are indicated by the white scale bar of length 3 μm. In the quantum Hall regime, current ISD applied through an ohmic contact (yellow) propagates by means of edge modes and is partitioned by two QPCs formed by the left and right split gates (LSG and RSG), resulting in oscillating diagonal resistance ({R}_{{\rm{D}}}=({V}_{{\rm{D}}}^{{+}-{V}_{{\rm{D}}}}{-})/{I}_{{\rm{SD}}}). b, Magnification of the interfering region near the centre gate (CG). Left and right air bridges (LBG and RBG, shown in green) are suspended 200 nm above each QPC region, fine-tuning the saddle-point potential. The lithographic interference area, determined by CG area, is 1 μm2. Scale bar, 0.3 µm. c, Measurement phase space defined by B, VPG and VCG. RD is measured along planes defined by (\alpha =\frac{\partial B}{\partial {V}_{{\rm{CG}}}}) and shown as 2D B∣α–VPG pajamas.

We inject a bias current ISD, which propagates along the FQH edge modes with an anticlockwise (clockwise) chirality for electron (hole) carriers, impinging on the FPI as illustrated in Fig. 1a. Current is collected on the other side of the interferometer by a single ground while measuring the diagonal resistance ({R}_{{\rm{D}}}=\frac{{V}_{{\rm{D}}}^{{+}-{V}_{{\rm{D}}}}{-}}{{I}_{{\rm{SD}}}}) to reveal interference. In the low backscattering regime, RD includes an oscillatory contribution ΔRD ∝ cosθ, in which the interference phase θ is composed of both Aharonov–Bohm (denoted ‘AB’) and statistical phases, that is9,47,48,

$$\theta ={\theta }_{{\rm{AB}}}+{\theta }_{{\rm{stat}}}=2{\rm{\pi }}\frac{{e}^{* }}{e}\frac{AB}{{\varPhi }_{0}}+{N}_{{\rm{qp}}}{\theta }_{{\rm{anyon}}},$$

(1)

in which A is the interfering area, Nqp is the integer number of localized quasiparticles within the interference loop and θanyon is the braiding phase. For non-Abelian quasiparticles, RD is predicted to follow a more intricate pattern that differs for even and odd Nqp (refs. 49,50).

We perform measurements of RD in the 3D parameter space spanned by the magnetic field B, the plunger gate voltage VPG and the centre gate voltage VCG; see Fig. 1c. To disentangle the two terms in θ of equation (1), we follow lines of different slopes (\alpha =\frac{\partial B}{\partial {V}_{{\rm{CG}}}}) in the B–VCG plane. The Aharonov–Bohm contribution is isolated at the critical trajectories αc, for which charges are continuously added to the interference loop to maintain constant fillings. Along α ≠ αc trajectories, the filling factor deviates from the rational value defining the quantum Hall plateau through the Hall conductance. All such trajectories are measured within the incompressible region, as shown in Supplementary Information Section 4.

Consequently, RD follows the well-known ‘pajama pattern’ in the B–VPG plane with a flux periodicity set by the interfering quasiparticle charge e*. Deviations from this trajectory introduce bulk quasiparticles, Nqp, which are expected to manifest individually through phase slips and which alter the average flux periodicity. Other notable trajectories include constant density, α = ∞, and constant magnetic field, α = 0, illustrated in Fig. 1c.

Even-denominator Aharonov–Bohm interference

We begin the Fabry–Pérot interferometry study of even-denominator states at the filling factor (\nu =-,\frac{1}{2}) owing to its simple edge structure, which consists only of fractional modes. Figure 2a shows the longitudinal resistance R**xx and Hall resistance R**xy measured at 11 T on the right side of the FPI. The data clearly reveal fully developed integer and fractional quantum Hall states at ν = −1, (-\frac{2}{3}), (-\frac{1}{2}) and (-\frac{1}{3}). Figure 2b presents an R**xx fan diagram, which we use to extract the constant-filling-factor trajectories. We define ({\alpha }_{{\rm{c}}}=\frac{{\varPhi }_{0}}{\nu e}C), with (C=\frac{1}{A}\frac{{\rm{d}}Q}{{\rm{d}}{V}_{{\rm{RG}}}}) the capacitance per unit area between the right gate and the bilayer graphene underneath, extracted from the Streda formula for each fractional state as the centre of the incompressible region, whose boundaries are indicated by dashed red lines (see Supplementary Information Section 5).

Fig. 2: Even-denominator Aharonov–Bohm interference.

a, Longitudinal resistance R**xx and Hall resistance R**xy measured at 11 T on the right side of the FPI, clearly showing fully developed even-denominator and odd-denominator quantum Hall states at (\nu =-,\frac{2}{3}), (-\frac{1}{2}) and (-\frac{1}{3}). b, R**xx fan diagram performed on the right side of the FPI between 10.5 and 11.5 T. Dashed red lines indicate the boundaries for each quantum Hall state. c, ΔRD at (\nu =-,\frac{1}{2}) shown as a (B{| }_{{\alpha }_{{\rm{c}}}}-{V}_{{\rm{PG}}}) pajama plot, showing clear Aharonov–Bohm oscillations. Inset, 2D-FFT analysis used to extract the magnetic-field periodicity (\frac{{\varPhi }_{0}}{\Delta B}) shown on the lower-right side of the pajama. d,e, Same as a,b for the electron-doped filling factors (\nu =\frac{4}{3}), (\frac{3}{2}) and (\frac{5}{3}). f, Same as c for (\nu =\frac{3}{2}) with partitioning of the fractional inner mode. a.u., arbitrary units.

Focusing on the (-\frac{1}{2}) state, Fig. 2c shows the interference pattern as a function of VPG and (B{| }_{{\alpha }_{{\rm{c}}}}), in which the αc constraint indicates that VCG is adjusted to maintain constant filling. Specifically, we present the data as ΔRD = RD − ⟨RD⟩, subtracting the average value at each magnetic field. The positive slope of the pajama indicates Aharonov–Bohm-dominated interference, because increasing VPG decreases the interference area for hole-doped states. The measured visibility, defined as Visibility = (Gmax − Gmin)/(Gmax + Gmin − 2Gouter), in which Gmax and Gmin are the maximum and minimum diagonal conductance values, respectively, and Gouter represents the conductance of any fully transmitted outer edge modes, is around 1.9%, comparable with that at integers and odd-denominator states (see Supplementary Information Section 6), and the corresponding edge mode velocity of vedge = 7.95 × 103 m s−1, approximately an order of magnitude smaller than that observed at integer filling, is extracted from source-drain bias VSD-dependent RD (see Supplementary Information Section 7). To extract the flux periodicity, we perform a 2D fast Fourier transform (2D-FFT), shown in the inset of Fig. 2c as a function of (\frac{{\varPhi }_{0}}{\Delta B}) and (\frac{1}{\Delta {V}_{{\rm{PG}}}}). From the magnetic-field periodicity, we extract (A\frac{{\varPhi }_{0}}{\Delta \varPhi }\approx -,0.53,{{\rm{\mu }}{\rm{m}}}^{2}). The lithographic area A ≈ 1 μm2 agrees with that extracted from interference at ν = −1 to within 2% (see Supplementary Information Section 8). Using the same area at (\nu =-,\frac{1}{2}) yields the unexpected flux periodicity ΔΦ = (1.89 ± 0.26)Φ0 ≈ 2Φ0. We found that this unexpected 2Φ0 flux periodicity was robust against changes in the compressibility of the bulk, magnetic field and in plunger-gate spectroscopy13 (see Supplementary Information Sections 10, 11 and 12). Furthermore, our transmission study (Supplementary Information Section 9) shows qualitatively similar interference patterns over the experimentally available tR,L range. At very low t, at which we expect electron-dominated tunnelling, the visibility is lost.

As shown in Supplementary Fig. 3, nearly all QPCs show resonances, probably because of tunnelling by means of localized states near the saddle-point potential. Such resonances can affect the leading tunnelling channel and the total inference phase51. Nevertheless, irrespective of the precise tunnelling mechanism across the QPC, the magnetic-field periodicity of the Aharonov–Bohm oscillations at a constant filling provides a direct and robust measure of the fractional charges that dominate tunnelling and interference processes. As long as the QPC environment is approximately constant, owing to low cross-capacitance with the plunger gate and small magnetic-field variations, the total variations in the interference phase owing to Aharonov–Bohm effect and bulk quasiparticles remain unaffected.

Following the first term in equation (1), this periodicity suggests an interfering quasiparticle charge of ({e}^{* }=\frac{1}{2}e), which tunnels across the QPCs to form an interference loop. Quasiparticles with this charge exist as bulk excitations at half-filling, arising from the fusion of two fundamental quasiparticles carrying charge (\frac{1}{4}e), in all Abelian or non-Abelian FQH candidate states. Alternatively, this periodicity could also arise in a scenario in which non-Abelian (\frac{1}{4}e) quasiparticles interfere. In that case, when a non-zero number of non-Abelian quasiparticles are localized in the bulk, there are several degenerate ground states. Fluctuations between these ground states on the timescale of the measurement could suppress the 4Φ0 periodicity that arises from a single winding of (\frac{1}{4}e) quasiparticles while not affecting the 2Φ0 periodicity arising from double windings or (\frac{1}{2}e) quasiparticles49,50.

To test the generality of these findings, we investigated the (\nu =\frac{3}{2}) plateau (on the electron side), which exhibits a gap comparable with (\nu =-,\frac{1}{2}) (refs. 33,34). Similar to the previous case, Fig. 2d shows R**xx and R**xy measured at 11 T, revealing well-developed FQH states at (\nu =\frac{4}{3}), (\frac{3}{2}) and (\frac{5}{3}). Figure 2e presents an R**xx fan diagram, which we use to extract αc as before. Figure 2f shows the interference pattern as a function of VPG and (B{| }_{{\alpha }_{{\rm{c}}}}) that arises when the QPCs are tuned to partition the fractional inner edge mode (see Supplementary Information Section 13 for the interference of the integer outer edge). The slope of the pajama pattern with 5.6% visibility is opposite to (\nu =-,\frac{1}{2}), indicating Aharonov–Bohm-dominated interference for electron-doped states (see Supplementary Information Section 14). The 2D-FFT, shown in the inset of Fig. 2f, yields the magnetic-field periodicity (A\frac{{\varPhi }_{0}}{\Delta \varPhi }\approx 0.42,{{\rm{\mu m}}}^{2}). Estimating the interfering area based on the integers ν = 1 and 2, we find A = 0.99 ± 0.10 μm2, consistent with the lithographic area (see Supplementary Information Section 15). Using this area at (\nu =\frac{3}{2}), we conclude ΔΦ = (2.35 ± 0.78)Φ0 ≈ 2Φ0. Temperature-dependence measurements showed a reduction in visibility with increasing temperature, whereas the magnetic-field periodicity remained constant (see Supplementary Information Sections 16 and 17).

These two measurements consistently show periodicities close to 2Φ0 and not the expected 4Φ0. The observations reflect the interference of (\frac{1}{2}e) quasiparticles at (\nu =-,\frac{1}{2}) and (\frac{3}{2}). We note that the topological orders of electrons at both fillings are believed to be Pfaffians35,43 but their edge structures at the boundary to ν = 0 are qualitatively different. In particular, a Pfaffian order of electrons at (\nu =-,\frac{1}{2}) is equivalent to an anti-Pfaffian of holes. Consequently, it exhibits an edge state with three upstream Majorana fermions at a ν = 0 boundary. Insofar as the identification of these states is accurate, our experiment effectively examines two distinct non-Abelian topological orders.

Comparing fractional quasiparticle interference

At both even-denominator filling factors, the observed Aharonov–Bohm periodicity is consistent with an interfering quasiparticle charge that matches the Landau-level filling factor ({\nu }_{{\rm{LL}}}=\frac{1}{2}). In this study, νLL is the filling factor of the partially filled Landau level νLL = ν − ⌊ν⌋. The interfering charge at (\nu =\frac{1}{3}) also follows the filling factor10,17,18. We extended the study to states at (\nu =-,\frac{2}{3}) and (\frac{5}{3}) to determine whether their interfering charge is also set by the filling or by the minimal bulk excitation. We refer to these states as ‘hole-conjugate’ based on their presumed topological orders and edge structures, which are related to the ‘particle-like’ states at (\nu =-,\frac{1}{3}) and (\frac{4}{3}) by a hole conjugation in the partially occupied Landau level. As there is no particle–hole symmetry, the microscopic wavefunctions at these fillings are not related by hole conjugation.

Figure 3a,b shows the extracted flux periodicities for all six fractional fillings in constant-filling measurements. The values for the odd denominators are extracted from the pajama patterns in Fig. 3c–f through the 2D-FFTs shown in Fig. 3g–j, assuming the same interference areas for hole-doped and electron-doped states as before. The results confirm the interference of e* = νLLe quasiparticles in all states included in our study. A recent experiment on the hole-conjugate states (\nu =\frac{2}{3}), (\frac{3}{5}) and (\frac{4}{7}) in GaAs using a Mach–Zehnder interferometer also observed interference of e* = νLLe quasiparticles20. It is not understood why non-fundamental quasiparticles should dominate the interference, as our measurements at half-filled and hole-conjugate states indicate. We point out that previous interference experiments at the hole-conjugate (\nu =\frac{2}{3}) state in GaAs reported the periodicity ΔΦ = Φ0 corresponding to the interference of electrons10. Moreover, Mach–Zehnder interference of the higher particle-like Jain states (\nu =\frac{2}{5}) and (\frac{3}{7}) observed ΔΦ = 5Φ0 and 7Φ0 (ref. 19), corresponding to the fundamental quasiparticle charge instead of νLLe.

Fig. 3: Interference of e* = νLLe quasiparticles in various FQH states.

a, Magnetic-field periodicities (\frac{{\varPhi }_{0}}{\Delta B}) at constant filling extracted from 2D-FFT analyses at (\nu =-,\frac{2}{3}), (-\frac{1}{2}) and (-\frac{1}{3}). Error bars correspond to the variance of the Gaussian fit obtained by fitting the 2D-FFT with a Gaussian function (see Supplementary Information Section 8). b, Same for (\nu =\frac{4}{3}), (\frac{3}{2}) and (\frac{5}{3}) with partitioning of the fractional inner mode. Error bars correspond to the variance of the Gaussian fit obtained by fitting the 2D-FFT with a Gaussian function (see Supplementary Information Section 15). c–f, ΔRD shown as ({V}_{{\rm{PG}}}-B{| }_{{\alpha }_{{\rm{c}}}}) pajamas for (\nu =-,\frac{2}{3}), (-\frac{1}{3}), (\frac{4}{3}) and (\frac{5}{3}), respectively. At (\nu =\frac{5}{3}), α deviated from αc by 3%. g–j, Corresponding 2D-FFTs, along with the extracted magnetic-field periodicities. We show all 2D pajama plots measured with partitioning of all of the fractional modes, before subtracting the average value at each magnetic field to discuss how changes in B and VCG affect the QPC transmission (see Supplementary Information Section 18). a.u., arbitrary units.

Theoretically, the question of which type of quasiparticle tunnel is addressed on the basis of the renormalization of bare tunnelling amplitudes by the interactions intrinsic to fractional edge modes. The bare tunnelling amplitudes for different quasiparticles are non-universal and hard to calculate reliably. Their renormalization, encoded by means of a scaling dimension of tunnelling operators, is the same for fundamental and ({e}^{{* }=\frac{2}{3}e) quasiparticles at the (\nu =\frac{2}{3}) edge52. It is possible that interactions across the QPC tip the balance in favour of ({e}}{* }=\frac{2}{3}e) tunnelling. Alternatively, when both tunnelling processes occur with comparable probabilities, the 3Φ0 periodicity expected for (\frac{1}{3}e) quasiparticles could be thermally suppressed because it requires exciting a neutral mode, which propagates with a much smaller velocity than the charge mode. At half-filling, the scaling dimensions of ({e}^{{* }=\frac{1}{4}e) quasiparticles depend on which topological state is realized but their numerical values are generally close to those of ({e}}{* }=\frac{1}{2}e) tunnelling. Different interactions across the QPC could favour either and a similar thermal suppression may affect the (\frac{1}{4}e) quasiparticle, which also excites a neutral mode. We also warn that the scaling dimensions extracted from experiments often deviate substantially from theoretical expectations.

Statistical phase from bulk anyons in FPI

Interference of fractional quasiparticles fundamentally differs from that of electrons by quantum statistical effects, that is, the second term in equation (1). Interfering quasiparticles acquire a quantized phase change for each localized anyon in the interferometer bulk. For electron interference, this extra phase is an unobservable multiple of 2π independent of the bulk anyon type. To observe such contributions, we operate the FPI at α ≠ αc, such that tuning the magnetic field or VCG causes the filling factor to deviate slightly from the rational value ((\nu =\frac{p}{q})) reflected by the bulk Hall conductivity. These deviations introduce excess charge carriers in the form of quasiparticles inside the interference loop. Each well-isolated quasiparticle in the bulk is expected to result in a sharp phase jump in the interference pattern. Similar to previous studies11,16,18,19, we observe such discrete phase slips at (\nu =-,\frac{1}{3}), which are analysed in Supplementary Information Sections 19 and 20. Introducing quasiparticles at a constant rate along a fixed α trajectory alters the overall slope of the constant-phase lines in the pajama pattern. This change in slope is more robust against fluctuations in the number of bulk quasiparticles around its mean than individual phase slips. Such fluctuations can broaden and even entirely wash out phase slips (see Supplementary Information Section 21). We attribute the absence of phase slips at both half-fillings and hole-conjugate states to such fluctuations and proceed with a careful analysis of the slope.

The change of the slope provides crucial insights into which quasiparticles enter the interference loop as the filling factor varies. In Supplementary Information Section 21, we show theoretically that the slope can even distinguish between different bulk quasiparticle types being introduced as ν changes. For the case in which interfering quasiparticles entering equation (1) carry charge e* = νLLe and fundamental quasiparticles are introduced into the bulk, we find

$$\begin{array}{ll}\text{Integer edge modes}: & \frac{{\varPhi }_{0}}{\Delta B}=A=\text{constant},\ \text{Fractional edge modes}: & \frac{{\varPhi }_{0}}{\Delta B}=({\nu }_{{\rm{LL}}}-\nu )A+\nu \frac{{\alpha }_{{\rm{c}}}}{\alpha }A.\end{array}$$

(2)

Because the mutual statistics with the ({e}^{* }=\frac{1}{2}e) quasiparticles with all other quasiparticles is Abelian, equation (2) holds for all paired states. The second line matches the phase θ = 2π⟨N⟩, with N the number of electrons in the loop, generalizing the result in ref. 46. If filling-factor deviations introduce quasiparticles other than the fundamental ones into the bulk, the slope on the right-hand side (ν**αcA) changes. We warn that strong bulk–edge coupling would also affect the slope and lead to an α dependence of the Aharonov–Bohm pattern even at integer fillings. We analyse this scenario in detail in Supplementary Information Sections 22 and 23 and show that it does not apply to our observations in the fractional case (see Supplementary Information Section 24).

We extract (\frac{{\varPhi }_{0}}{\Delta B}) from the 2D-FFT for the fillings (\nu =\frac{4}{3}), (\frac{3}{2}) and (\frac{5}{3}), for partitioning of both the fractional inner modes and the integer outer modes. Figure 4 shows our results for each α, with (\frac{{\varPhi }_{0}}{\Delta B}) obtained from the 2D-FFT of the corresponding pajama patterns. In Supplementary Information Section 25, we show all 2D pajama images and corresponding 2D-FFT analyses for all fractional filling factors discussed in this study. For all of the integer outer modes, ΔB is independent of the α, as expected. By contrast, for the fractional modes, all measurements collapse into a single linear dependence on (\frac{1}{\alpha }) as in the second line of equation (2). Their slope deviates by 15% from the numerical value expected on the basis of the bulk capacitance C, obtained through the Streda formula for the region to the right of the FPI. This discrepancy can arise from boundary effects of the comparatively small centre gate, small changes in the interference area with VCG and bulk–edge couplings (see Supplementary Information Section 26). At (\nu =\frac{3}{2}) with an interfering inner mode, the observed slope matches the theoretically expected statistical contribution of bulk ({e}^{{* }=\frac{1}{4}e) quasiparticles to the interference of Abelian ({e}}{* }=\frac{1}{2}e) quasiparticles.

Fig. 4: Statistical contribution to the interference of fractional quasiparticles.

Magnetic-field periodicities (\frac{{\varPhi }_{0}}{\Delta B}) obtained along different trajectories α from 2D-FFTs for the fractional inner and integer outer modes of (\nu =\frac{4}{3}), (\frac{3}{2}) and (\frac{5}{3}). Error bars correspond to the variance of the Gaussian fit obtained by fitting the 2D-FFT with a Gaussian function (see Supplementary Information Sections 15, 25).

Conclusions