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.

# Non-Abelian states of matter

## Abstract

Quantum mechanics classifies all elementary particles as either fermions or bosons, and this classification is crucial to the understanding of a variety of physical systems, such as lasers, metals and superconductors. In certain two-dimensional systems, interactions between electrons or atoms lead to the formation of quasiparticles that break the fermion–boson dichotomy. A particularly interesting alternative is offered by 'non-Abelian' states of matter, in which the presence of quasiparticles makes the ground state degenerate, and interchanges of identical quasiparticles shift the system between different ground states. Present experimental studies attempt to identify non-Abelian states in systems that manifest the fractional quantum Hall effect. If such states can be identified, they may become useful for quantum computation.

## Relevant articles

• ### Precursors of Majorana modes and their length-dependent energy oscillations probed at both ends of atomic Shiba chains

Nature Nanotechnology Open Access 07 March 2022

• ### Interplay of filling fraction and coherence in symmetry broken graphene p-n junction

Communications Physics Open Access 01 October 2020

• ### Progress in Cooling Nanoelectronic Devices to Ultra-Low Temperatures

Journal of Low Temperature Physics Open Access 05 June 2020

## Access options

\$32.00

All prices are NET prices.

## References

1. Stern, A. Anyons and the quantum Hall effect — a pedagogical review. Ann. Phys. 323, 204–249 (2008). This paper is a pedagogical introduction to the concept of Abelian and non-Abelian anyons.

2. 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). This is a comprehensive review of non-Abelian states of matter and their relevance to quantum computation, and contains an extensive list of references.

3. Das Sarma, S., Freedman, M. & Nayak, C. Topological quantum computation. Phys. Today 59, 32–38 (2006).

4. Collins, G. P. Computing with quantum knots. Sci. Am. 4, 57 (2006).

5. Day, C. Devices based on the fractional quantum Hall effect may fulfill the promise of quantum computing. Phys. Today 58, 21–23 (2005).

6. Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003). This paper discusses the use of anyons for quantum computation.

7. Freedman, M. H. et al. Topological quantum computation. Bull. Am. Math. Soc. 40, 31–38 (2003).

8. 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). This paper proposes a qubit based on the ν = 5/2 state.

9. Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 1776–1779 (1987).

10. Xia, J. S. et al. Electron correlation in the second Landau level: a competition between many nearly degenerate quantum phases. Phys. Rev. Lett. 93, 176809 (2004).

11. Eisenstein, J. P., Cooper, K. B., Pfeiffer, L. N. & West, K. W. Insulating and fractional quantum Hall states in the first excited Landau level. Phys. Rev. Lett. 88, 076801 (2002).

12. Moore, G. & Read, N. Non-Abelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991). This paper introduced the concept of non-Abelian states of matter.

13. Read, N. Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p x + ip y paired superfluids. Phys. Rev. B 79, 045308 (2009).

14. Nayak, C. & Wilczek, F. 2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states. Nucl. Phys. B. 479, 529–553 (1996).

15. Ivanov, D. A. Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268–271 (2001).

16. Stern, A., von Oppen, F. & Mariani, E. Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics. Phys. Rev. B 70, 205338 (2004).

17. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000). This paper puts forward a composite fermion theory of the ν = 5/2 state.

18. Read, N. & Rezayi, E. Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level. Phys. Rev. B 59, 8084–8092 (1999). This paper describes a proposed series of non-Abelian quantum Hall states of matter.

19. Levin, M. A. & Wen, X. G. String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005).

20. Wen, X. G. Topological order and edge structure of ν = 1/2 quantum Hall state. Phys. Rev. Lett. 70, 355–358 (1993).

21. Greiter, M., Wen, X.-G. & Wilczek, F. Paired Hall state at half filling. Phys. Rev. Lett. 66, 3205–3208 (1991).

22. Morf, R. H. Transition from quantum Hall to compressible states in the second Landau level: new light on the ν = 5/2 enigma. Phys. Rev. Lett. 80, 1505–1508 (1998).

23. Storni, M., Morf, R. H. & Das Sarma, S. The fractional quantum Hall state at ν = 5/2 and the Moore–Read Pfaffian. Preprint at &lt;http://arxiv.org/abs/0812.2691&gt; (2008).

24. Rezayi, E. H. & Haldane, F. D. M. Incompressible paired Hall state, stripe order and the composite fermion liquid phase in half-filled Landau levels. Phys. Rev. Lett. 84, 4685–4688 (2000).

25. Tőke, C., Regnault, N. & Jain, J. K. Nature of excitations of the 5/2 fractional quantum Hall effect. Phys. Rev. Lett. 98, 036806 (2007).

26. Wójs, W. & Quinn, J. J. Landau level mixing in the ν = 5/2 fractional quantum Hall state. Phys. Rev. B 74, 235319 (2006).

27. Feiguin, A. E., Rezayi, E., Nayak, C. & Das Sarma, S. Density matrix renormalization group study of incompressible fractional quantum Hall states. Phys. Rev. Lett. 100, 166803 (2008).

28. Cooper, N. R., Wilkin, N. K. & Gunn, J. M. F. Quantum phases of vortices in rotating Bose–Einstein condensates. Phys. Rev. Lett. 87, 120405 (2001).

29. Gurarie, V. & Radzihovsky, L. Resonantly paired fermionic superfluids. Ann. Phys. 322, 2–119 (2007).

30. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008). This paper proposes a setting in which Majorana fermions are created in a hybrid system of a topological insulator and a superconductor.

31. Fu, L. & Kane, C. L. Probing neutral Majorana fermion edge modes with charge transport. Phys. Rev. Lett. 102, 216403 (2009).

32. Nilsson, J., Akhmerov, A. R. & Beenakker, C. W. Splitting of a Cooper pair by a pair of Majorana bound states. Phys. Rev. Lett. 101, 120403 (2008).

33. Sau, J. D., Lutchyn, R. M., Tewari, S. & Das Sarma, S. A generic new platform for topological quantum computation using semiconductor heterostructures. Preprint at &lt;http://arxiv.org/abs/0907.2239&gt; (2009).

34. Wilczek, F. Majorana returns. Nature Phys. 5, 614–618 (2009).

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

36. Heinonen, O. (ed.) Composite Fermions (World Scientific, 1998).

37. Boyarsky, A., Cheianov, V. & Fröhlich, J. Effective field theories for the ν = 5/2 edge. Phys. Rev. B 80, 233302 (2009).

38. Cooper, N. R. & Stern, A. Observable bulk signatures of non-Abelian quantum Hall states. Phys. Rev. Lett. 102, 176807 (2009).

39. Yang, K. & Halperin, B. I. Thermopower as a possible probe of non-Abelian quasiparticle statistics in fractional quantum Hall liquids. Phys. Rev. B 79, 115317 (2009).

40. Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Compressibility of the two-dimensional electron gas: measurements of the zero-field exchange energy and fractional quantum Hall gap. Phys. Rev. B 50, 1760–1778 (1994).

41. Wiegers, S. A. J. et al. Magnetization and energy gaps of a high-mobility 2D electron gas in the quantum limit. Phys. Rev. Lett. 79, 3238–3241 (1997).

42. 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, 2331–2343 (1997).

43. Fradkin, E. et al. A Chern–Simons effective field theory for the Pfaffian quantum Hall state. Nucl. Phys. B 516, 704–718 (1998).

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

45. 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).

46. Bonderson, P., Shtengel, K. & Slingerland, J. K. Probing non-Abelian statistics with quasiparticle interferometry. Phys. Rev. Lett. 97, 016401 (2006).

47. Ilan, R., Grosfeld, E., Schoutens, K. & Stern, A. Experimental signatures of non-Abelian statistics in clustered quantum Hall states. Phys. Rev. B 79, 245305 (2009).

48. Feldman, D. E. & Kitaev, A. Detecting non-Abelian statistics with an electronic Mach–Zehnder interferometer. Phys. Rev. Lett. 97, 186803 (2006).

49. Feldman, D. E., Gefen, Y., Kitaev, A., Law, K. T. & Stern, A. Shot noise in an anyonic Mach–Zehnder interferometer. Phys. Rev. B 76, 085333 (2007).

50. Neder, I. et al. Interference between two indistinguishable electrons from independent sources. Nature 448, 333–337 (2007).

51. Roulleau, P. et al. Direct measurement of the coherence length of edge states in the integer quantum Hall regime. Phys. Rev. Lett. 100, 126802 (2008).

52. Milovanović, M. & Read, N. Edge excitations of paired fractional quantum Hall states. Phys. Rev. B 53, 13559–13582 (1996).

53. Fendley, P., Fisher, M. P. A. & Nayak, C. Edge states and tunneling of non-Abelian quasiparticles in the ν = 5/2 quantum Hall state and p + ip superconductors. Phys. Rev. B. 75, 045317 (2007).

54. Fiete, G. A., Bishara, W. & Nayak, C. Multi-channel Kondo models in non-Abelian quantum Hall droplets. Phys. Rev. Lett. 101, 176801 (2008).

55. Feldman, D. E. & Li, F. Charge-statistics separation and probing non-Abelian states. Phys. Rev. B 78, 161304 (2008).

56. Dolev, M. et al. Observation of a quarter of an electron charge at the ν = 5/2 quantum Hall state. Nature 452, 829–834 (2008).

57. Radu, I. P. et al. Quasi-particle properties from tunneling in the ν = 5/2 fractional quantum Hall state. Science 320, 899–902 (2008).

58. Camino, F. E., Zhou, W. & Goldman, V. J. Quantum transport in electron Fabry–Perot interferometers. Phys. Rev. B 76, 155305 (2007).

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

60. Goldman, V. J. Superperiods and quantum statistics of Laughlin quasiparticles. Phys. Rev. B 75, 045334 (2007).

61. Ofek, N. et al. The role of interactions in an electronic Fabry–Perot interferometer operating in the quantum Hall effect regime. Preprint at &lt;http://arxiv.org/abs/0911.0794&gt; (2009).

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

63. 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).

64. Willett, R. L., Pfeiffer, L. N. & West, K. W. Alternating e/4 and e/2 period interference oscillations consistent with filling factor 5/2 non-Abelian quasiparticles. Preprint at &lt;http://arxiv.org/abs/0911.0345&gt; (2009).

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

66. Rosenow, B., Halperin, B., Simon, S. & Stern, A. Bulk-edge coupling in the non-Abelian ν = 5/2 quantum Hall interferometer. Phys. Rev. Lett. 100, 226803 (2008).

67. Overbosch, B. J. & Wen, X. G. Dynamical and scaling properties of ν = 5/2 interferometer. Preprint at &lt;http://arxiv.org/abs/0706.4339&gt; (2007).

68. Rosenow, B., Halperin, B., Simon, S. & Stern, A. Exact solution for bulk-edge coupling in the non-Abelian ν=5/2 quantum Hall interferometer. Phys. Rev. B 80, 155305 (2009).

69. Bishara, W. & Nayak, C. Odd–even crossover in a non-Abelian ν = 5/2 interferometer. Phys. Rev. B 80, 155304 (2009).

70. Saarikoski, H., Tölö, E., Harju, A. & Räsänen, E. Pfaffian and fragmented states at ν = 5/2 in quantum Hall droplets. Phys. Rev. B 78, 195321 (2008).

71. Overbosch, B. J. & Bais, F. A. Inequivalent classes of interference experiments with non-Abelian anyons. Phys. Rev. A 64, 062107 (2001).

72. Feiguin, A. E., Rezayi, E., Yang, K., Nayak, C. & Das Sarma, S. Spin polarization of the ν = 5/2 quantum Hall state. Phys. Rev. B 79, 115322 (2009).

73. Dean, C. R. et al. Contrasting behavior of the ν = 5/2 and 7/3 fractional quantum Hall effect in a tilted field. Phys. Rev. Lett. 101, 186806 (2008).

## Acknowledgements

This work was supported by the US–Israel Binational Science Foundation, the Minerva foundation and Microsoft's Station Q.

Authors

## Ethics declarations

### Competing interests

The author declares no competing financial interests.

Reprints and permissions information is available at http://www.nature.com/reprints. Correspondence should be addressed to the author (adiel.stern@weizmann.ac.il).

## Rights and permissions

Reprints and Permissions

Stern, A. Non-Abelian states of matter. Nature 464, 187–193 (2010). https://doi.org/10.1038/nature08915

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/nature08915

• ### Precursors of Majorana modes and their length-dependent energy oscillations probed at both ends of atomic Shiba chains

• Lucas Schneider
• Philip Beck
• Roland Wiesendanger

Nature Nanotechnology (2022)

• ### Classical non-Abelian braiding of acoustic modes

• Ze-Guo Chen
• Ruo-Yang Zhang
• Guancong Ma

Nature Physics (2022)

• ### Zero-bias conductance modification in the quantum-dot system with side-coupled Majorana bound states

• Cui Jiang
• Guang-Yi Meng
• Lian-Lian Zhang

Applied Physics A (2022)

• ### Zero-bias peaks at zero magnetic field in ferromagnetic hybrid nanowires

• S. Vaitiekėnas
• Y. Liu
• C. M. Marcus

Nature Physics (2021)

• ### How Should We Choose the Boundary Conditions in a Simulation Which Could Detect Anyons in One and Two Dimensions?

• Riccardo Fantoni

Journal of Low Temperature Physics (2021)