Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Direct observation of anyonic braiding statistics

Matters Arising to this article was published on 27 November 2023

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.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

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.

References

  1. Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Nuovo Ciment. B 37, 1–23 (1977).

    ADS  Google Scholar 

  2. Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).

    ADS  MathSciNet  Google Scholar 

  3. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    ADS  Google Scholar 

  4. Laughlin, R. B. Anomolous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitation. Phys. Rev. Lett. 50, 1395–1398 (1983).

    ADS  Google Scholar 

  5. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  6. Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).

    ADS  Google Scholar 

  7. Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, 1583–1586 (1984).

    ADS  Google Scholar 

  8. Kjonsberg, H. & Leinaas, J. M. Charge and statistics of quantum Hall quasi-particles – numerical study of mean values and fluctuations. Nucl. Phys. B 559, 705–742 (1999).

    ADS  Google Scholar 

  9. Jeon, G. S., Graham, K. L. & Jain, J. K. Fractional statistics in the fractional quantum Hall effect. Phys. Rev. Lett. 91, 036801 (2003).

    ADS  Google Scholar 

  10. Jain, J. K. Composite Fermions (Cambridge Univ. Press, 2007).

  11. Das Sarma, S., Freedman, M. & Nayak, C. Topologically protected qubits from a possible non-Abelian fractional quantum Hall state. Phys. Rev. Lett. 94, 166802 (2005).

    ADS  Google Scholar 

  12. Stern, A. & Lindner, N. H. Topological quantum computation – from basic concepts to first experiments. Science 339, 1179–1184 (2013).

    ADS  Google Scholar 

  13. Bartolomei, H. et al. Fractional statistics in anyon collisions. Science 368, 173–177 (2020).

    ADS  MathSciNet  Google Scholar 

  14. Rosenow, B. & Stern, A. Flux superperiods and periodicity transitions in quantum Hall interferometers. Phys. Rev. Lett. 124, 106805 (2020).

    ADS  Google Scholar 

  15. Chamon, C. de C., Freed, D. E., Kivelson, S. A., Sondhi, S. L. & Wen, X. G. Two point-contact interferometer for quantum Hall systems. Phys. Rev. B 55, 2331–2342 (1997).

    ADS  Google Scholar 

  16. Halperin, B. I. & Rosenow, B. Influence of interactions on flux and back-gate period of quantum Hall interferometers. Phys. Rev. Lett. 98, 106801 (2007).

    ADS  Google Scholar 

  17. Halperin, B. I., Stern, A., Neder, I. & Rosenow, B. Theory of the Fabry–Perot quantum Hall interferometer. Phys. Rev. B 83, 155440 (2011).

    ADS  Google Scholar 

  18. Rosenow, B. & Simon, S. H. Telegraph noise and the Fabry–Perot quantum Hall interferometer. Phys. Rev. B 85, 201302 (2012).

    ADS  Google Scholar 

  19. Levkivskyi, I. P., Frohlich, J. & Sukhorukov, E. B. Theory of fractional quantum Hall interferometers. Phys. Rev. B 86, 245105 (2012).

    ADS  Google Scholar 

  20. von Keyserlingk, C. W., Simon, S. H. & Rosenow, B. Enhanced bulk–edge Coulomb coupling in fractional Fabry–Perot interferometers. Phys. Rev. Lett. 115, 126807 (2015).

    ADS  Google Scholar 

  21. Goldstein, M. & Gefen, Y. Suppression of interference in quantum Hall Mach–Zehnder geometry by upstream neutral modes. Phys. Rev. Lett. 117, 276804 (2016).

    ADS  Google Scholar 

  22. Litvin, L. V., Helzel, A., Tranitz, H. -P., Wegscheider, W. & Strunk, C. Edge–channel interference controlled by Landau level filling. Phys. Rev. B. 78, 075303 (2008).

    ADS  Google Scholar 

  23. Deviatov, E. V. & Lorke, A. Experimental realization of a Fabry–Perot type interferometer by copropogating edge states in the quantum Hall regime. Phys. Rev. B 77, 161302 (2008).

    ADS  Google Scholar 

  24. Zhang, Y. et al. Distinct signatures for Coulomb blockade and interference in electronic Fabry–Perot interferometers. Phys. Rev. B 79, 241304 (2009).

    ADS  Google Scholar 

  25. McClure, D. T. et al. Edge-state velocity and coherence in a quantum Hall Fabry–Perot interferometer. Phys. Rev. Lett. 103, 206806 (2009).

    ADS  Google Scholar 

  26. Lin, P. V., Camino, F. E. & Goldman, V. J. Electron interferometry in the quantum Hall regime: Aharonov–Bohm effect of interacting electrons. Phys. Rev. B 80, 125310 (2009).

    ADS  Google Scholar 

  27. An, S., et al., Braiding of Abelian and non-Abelian anyons in the fractional quantum Hall effect. Preprint at https://arxiv.org/abs/1112.3400 (2011).

  28. Ofek, N. et al. Role of interactions in an electron Fabry–Perot interferometer operating in the quantum Hall effect regime. Proc. Natl Acad. Sci. USA 107, 5276–5281 (2010).

    ADS  Google Scholar 

  29. Choi, H. K. et al. Robust electron pairing in the integer quantum Hall effect. Nat. Comm. 6, 7435 (2015).

    ADS  Google Scholar 

  30. Gurman, I., Sabo, R., Heiblum, M., Umansky, V. & Mahalu, D. Dephasing of an electronic two-path interferometer. Phys. Rev. B 93, 121412 (2016).

    ADS  Google Scholar 

  31. Sivan, I. et al. Observation of interaction-induced modulations of a quantum Hall liquid’s area. Nat. Comm. 7, 12184 (2016).

    ADS  Google Scholar 

  32. Sabo, R. et al. Edge reconstruction in fractional quantum Hall states. Nat. Phys. 13, 491 (2017).

    Google Scholar 

  33. Duprez, H. et al. Macroscopic electron quantum coherence in a solid-state circuit. Phys. Rev. X 9, 021030 (2019).

    Google Scholar 

  34. Bhattacharyya, R., Mitali, B., Heiblum, M., Mahalu, D. & Umansky, V. Melting of interference in the fractional quantum Hall effect: appearance of neutral modes. Phys. Rev. Lett. 122, 246801 (2019).

    ADS  Google Scholar 

  35. Roosli, M. P. et al. Observation of quantum Hall interferometer phase jumps due to a change in the number of localized bulk quasiparticles. Phys. Rev. B 101, 125302 (2020).

    ADS  Google Scholar 

  36. Kivelson, S. Semiclassical theory of localized many-anyon states. Phys. Rev. Lett. 65, 3369 (1990).

    ADS  Google Scholar 

  37. Kim, E. Aharonov–Bohm interference and fractional statistics in a quantum Hall interferometer. Phys. Rev. Lett. 97, 216404 (2006).

    ADS  Google Scholar 

  38. Stern, A. & Halperin, B. I. Proposed experiments to probe the non-Abelian ν = 5/2 quantum Hall state. Phys. Rev. Lett. 96, 016802 (2006).

    ADS  Google Scholar 

  39. Bonderson, P., Kitaev, A. & Shtengel, K. Detecting non-Abelian statistics in the ν = 5/2 fractional quantum Hall state. Phys. Rev. Lett. 96, 016803 (2006).

    ADS  Google Scholar 

  40. Bishara, W., Bonderson, P., Nayak, C., Shtengel, K. & Slingerland, J. K. Interferometric signature of non-Abelian anyons. Phys. Rev. B 80, 155303 (2009).

    ADS  Google Scholar 

  41. Willett, R. L., Pfeiffer, L. N. & West, K. W. Measurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillations. Proc. Natl Acad. Sci. USA 106, 8853–8858 (2009).

    ADS  Google Scholar 

  42. Stern, A., Rosenow, B., Ilan, R. & Halperin, B. I. Interference, Coulomb blockade, and the identification of non-Abelian quantum Hall states. Phys. Rev. B 82, 085321 (2010).

    ADS  Google Scholar 

  43. Willett, R. L., Nayak, C., Shtengel, K., Pfeiffer, L. N. & West, K. W. Magnetic-field-tuned Aharonov–Bohm oscillations and evidence for non-Abelian anyons at ν = 5/2. Phys. Rev. Lett. 111, 186401 (2013).

    ADS  Google Scholar 

  44. Willett, R.L., et al., Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding. Preprint at https://arxiv.org/abs/1905.10248v1 (2019).

  45. McClure, D. T., Chang, W., Marcus, C. M., Pfeiffer, L. N. & West, K. W. Fabry–Perot interferometry with fractional charges. Phys. Rev. Lett. 108, 256804 (2012).

    ADS  Google Scholar 

  46. Nakamura, J. et al. Aharonov–Bohm interference of fractional quantum Hall edge modes. Nat. Phys. 15, 563–569 (2019).

    Google Scholar 

  47. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B. 25, 2185–2190 (1982).

    ADS  Google Scholar 

  48. Rezayi, E. H. & Haldane, F. D. M. Incompressible states of the fractionally quantized Hall effect in the presence of impurities: a finite-size study. Phys. Rev. B. 32, 6924–6927 (1985).

    ADS  Google Scholar 

  49. MacDonald, A. H., Liu, K. L., Girvin, S. M. & Platzman, P. M. Disorder and the fractional quantum Hall effect: activation energies and the collapse of the gap. Phys. Rev. B. 33, 4014–4020 (1986).

    ADS  Google Scholar 

  50. Goldman, V. J. & Su, B. Resonant tunneling in the quantum Hall regime: measurement of fractional charge. Science 267, 1010–1012 (1995).

    ADS  Google Scholar 

  51. de-Picciotto, R. et al. Direct observation of a fractional charge. Nature 389, 162–164 (1997).

    ADS  Google Scholar 

  52. Saminadayar, L., Glattli, D. C., Lin, Y. & Etienne, B. Observation of the e/3 fractionally charged Laughlin quasiparticle. Phys. Rev. Lett. 79, 2526–2529 (1997).

    ADS  Google Scholar 

  53. Du, R. R., Stormer, H. L., Tsui, D. C., Pfeiffer, L. N. & West, K. W. Experimental evidence for new particles in the fractional quantum Hall effect. Phys. Rev. Lett. 70, 2944–2947 (1993).

    ADS  Google Scholar 

  54. Hu, Z., Rezayi, E. H., Wan, X. & Yang, K. Edge-mode velocities and thermal coherence of quantum Hall interferometers. Phys. Rev. B 80, 235330 (2009).

    ADS  Google Scholar 

  55. Park, J., Gefen, Y. & Sim, H. Topological dephasing in the ν = 2/3 fractional quantum Hall regime. Phys. Rev. B 92, 245437 (2015).

    ADS  Google Scholar 

  56. Bid, A. et al. Observation of neutral modes in the fractional quantum Hall regime. Nature 466, 585–590 (2010).

    ADS  Google Scholar 

  57. Inoue, H. et al. Proliferation of neutral modes in fractional quantum Hall states. Nat. Comm. 5, 4067 (2014).

    ADS  Google Scholar 

  58. Manfra, M. J. Molecular beam epitaxy of ultra-high-quality AlGaAs/GaAs heterostructures: enabling physics in low-dimensional electronic systems. Annu. Rev. Condens. Matter Phys. 5, 347–373 (2014).

    ADS  Google Scholar 

  59. Gardner, G. C., Fallahi, S., Watson, J. D. & Manfra, M. J. Modified MBE hardware and techniques and role of gallium purity for attainment of two dimensional electron gas mobility > 35 × 106 cm2/Vs in AlGaAs/GaAs quantum wells grown by MBE. J. Cryst. Growth 441, 71–77 (2016).

    ADS  Google Scholar 

  60. Sahasrabudhe, H. et al. Optimization of edge state velocity in the integer quantum Hall regime. Phys. Rev. B 97, 085302 (2018).

    ADS  Google Scholar 

  61. Jeon, G. S., Graham, K. L. & Jain, J. K. Berry phases for composite fermions: effective magnetic field and fractional statistics. Phys. Rev. B 70, 125316 (2004).

    ADS  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Contributions

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.

Corresponding author

Correspondence to M. J. Manfra.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Supplementary information

Supplementary Information

Supplementary Discussion 1–4.

Source data

Source Data Fig. 2

Numerical source data.

Source Data Fig. 3

Numerical source data.

Source Data Fig. 4

Numerical source data.

Source Data Extended Data Fig. 2

Numerical source data.

Source Data Extended Data Fig. 3

Numerical source data.

Source Data Extended Data Fig. 4

Numerical source data.

Source Data Extended Data Fig. 5

Numerical source data.

Source Data Extended Data Fig. 6

Numerical source data.

Source Data Extended Data Fig. 7

Numerical source data.

Source Data Extended Data Fig. 8

Numerical source data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-020-1019-1

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing