Direct observation of anyonic braiding statistics

Abstract

Anyons are quasiparticles that, unlike fermions and bosons, show fractional statistics when two of them are exchanged. Here, we report the experimental observation of anyonic braiding statistics for the ν = 1/3 fractional quantum Hall state by using an electronic Fabry–Perot interferometer. Strong Aharonov–Bohm interference of the edge mode is punctuated by discrete phase slips that indicate an anyonic phase θanyon = 2π/3. Our results are consistent with a recent theory that describes an interferometer operated in a regime in which device charging energy is small compared to the energy of formation of charged quasiparticles, which indicates that we have observed anyonic braiding.

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Fig. 1: Quasiparticle braiding experiment.
Fig. 2: Conductance oscillations versus magnetic field and side gate voltage.
Fig. 3: Interference across the ν = 1/3 quantum Hall plateau.
Fig. 4: Dependence of oscillation amplitude on temperature.

Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

This work is supported by the US Department of Energy, Office of Science and Office of Basic Energy Sciences (award no. DE-SC0020138). G.C.G. acknowledges support from Microsoft Quantum. We thank B. Rosenow for valuable comments on an early version of this manuscript.

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J.N. and M.J.M. designed the heterostructures and experiments. S.L. and G.C.G. conducted the molecular beam epitaxy growth. J.N. fabricated the devices, performed the measurements and analysed the data with input from M.J.M. J.N. and M.J.M wrote the manuscript with input from all authors.

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Correspondence to M. J. Manfra.

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Extended data

Extended Data Fig. 1 Layer stack of the GaAs/AlGaAs heterostructure used for the experiments.

This structure utilizes three GaAs quantum wells: a primary 30nm well flanked by two 13nm screening wells to reduce the bulk-edge interaction in the interferometer. There are 25nm AlGaAs barriers between the main well and screening wells, and the total center-to-center setback of the screening wells from the main well is 48nm.

Extended Data Fig. 2 Repeatability of discrete phase jumps.

a, First scan measurement of conductance versus B and δVg. This is the same data in Fig. 2 of the main text. b, Second scan across the same range of magnetic field using the same QPC gate voltages. As can be seen from the data, the same pattern of discrete jumps appear in the second scan. The second scan was taken approximately one hour after the first scan. Values of Δθ/2π extracted from least squares fits are shown for both scans, and show similar values for each phase jump in both scans. Source data

Extended Data Fig. 3 Measurement of the energy gap for the ν= 1/3 fractional quantum Hall state.

The inset shows longitudinal resistance Rxx measured in a bulk region away from the interferometer at different temperatures. A linear fit of the data yields a gap of Δ = 5.5K. This is consistent with values measured in previous experiments at similar magnetic field. Source data

Extended Data Fig. 4 Conductance oscillations at different magnetic fields.

a, Conductance oscillations δG versus side gate voltage 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 at B = 9.3T (red). The side gate oscillation period δVg is significantly smaller in the low field and high field regions than in the central region, with δVg = 5.8mV at 8.4T, δVg = 8.5mV at 8.85T, and δVg = 5.4mV at 9.3T. The QPCs are tuned to approximately 90% transmission. b, Conductance G versus side gate voltage at zero magnetic field with the device operated in 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 of the gates to the edge. Source data

Extended Data Fig. 5 Simulations of interferometer behavior at ν = 1/3.

Conductance values are computed as a function of magnetic field B and side gate voltages δVg, taking into account both the Aharonov-Bohm phase and the contribution θanyon from 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 = e2/2C a) 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 this context a negative quasiparticle number indicates a population of quasiholes. In each case in the middle of the state there are no quasiparticles, resulting in conventional Aharonov-Bohm interference with 3Φ0 period, while at higher fields quasiholes form and at lower fields quasiparticles form, resulting in phase slips with Φ0 period. As temperature is elevated, the quasiparticle number is thermally smeared, making the Φ0 period phase slips unobservable and reducing the amplitude of the oscillations that occur as a function of δVg. e, Qualitative plot of the density of states versus energy. Source data

Extended Data Fig. 6 Measurements of interference at ν = 1.

a, Bulk quantum Hall transport showing the zero in Rxx and plateau in Rxy corresponding to the ν = 1 integer quantum Hall state. For this integer state, the bulk excitations and edge state current carrying particles are simply electrons, which obey fermionic statistics. b, Conductance oscillations versus magnetic field, showing an oscillation period ΔB =11mT. From this period the effective area AI of the interferometer can be extracted: AI = Φ0 ΔB. In 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 corresponding to 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 and low fields. This is consistent with the fact that electrons, which carry current and form localized states at ν = 1, are fermions who obey trivial braiding statistics, θfermion = 2π, making braiding unobservable and leading to no change in interference behavior. Source data

Extended Data 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 from a Fourier transform of the conductance versus side gate voltage data as a function of Vsd. The oscillation amplitude shows a node pattern as a function of Vsd from which the edge velocity may be extracted, yielding vedge = 8.3 × 103m/s. c, Differential conductance and d) oscillation amplitude versus Vsd at 8.85T giving vedge = 9.7 × 103m/s. e, Differential conductance and f) oscillation amplitude versus Vsd at 9.3T giving vedge = 9.3 × 103m/s. Evidently, the edge velocity does not change significantly across the ν = 1/3 quantum Hall plateau. Source data

Extended Data Fig. 8 Measurements of interference for a second device, taken from a different chip fabricated on the same 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, the device exhibits negatively sloped Aharonov-Bohm oscillations, which flatten out at higher and lower magnetic fields, consistent with 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 scan intended to improve signal to noise). Least-squares fits of the conductance on either side of the phase jump yields an extracted phase jump Δθ/2π = -0.32, yielding an anyonic phase θanyon = 2π × 0.32, consistent with theory. Source data

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Nakamura, J., Liang, S., Gardner, G.C. et al. Direct observation of anyonic braiding statistics. Nat. Phys. 16, 931–936 (2020). https://doi.org/10.1038/s41567-020-1019-1

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