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.

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.

Your institute does not have access to this article

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Characteristics of non-Abelian systems.
Figure 2: Interferometric measurement setups in quantum Hall systems.
Figure 3: Experimental results from interferometers.

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.

    ADS  MathSciNet  CAS  Article  Google Scholar 

  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.

    ADS  MathSciNet  CAS  Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    ADS  MathSciNet  CAS  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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.

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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.

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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.

    ADS  CAS  Article  Google Scholar 

  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.

    ADS  CAS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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 <http://arxiv.org/abs/0812.2691> (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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  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.

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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 <http://arxiv.org/abs/0907.2239> (2009).

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

    ADS  CAS  Article  Google Scholar 

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

    Book  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

  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 <http://arxiv.org/abs/0911.0345> (2009).

    Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

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

    Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  CAS  Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Ethics declarations

Competing interests

The author declares no competing financial interests.

Additional information

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

About this article

Cite this article

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

Download citation

  • Published:

  • Issue Date:

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

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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