Even denominator fractional quantum Hall states in higher Landau levels of graphene


An important development in the field of the fractional quantum Hall effect was the proposal that the 5/2 state observed in the Landau level with orbital index n = 1 of two-dimensional electrons in a GaAs quantum well1 originates from a chiral p-wave paired state of composite fermions that are topological bound states of electrons and quantized vortices. The excitations of this state, which is theoretically described by a ‘Pfaffian’ wavefunction2 or its hole partner called the anti-Pfaffian3,4, are neither fermions nor bosons but Majorana quasiparticles obeying non-Abelian braid statistics5. This has inspired ideas for fault-tolerant topological quantum computation6 and has also instigated a search for other states with exotic quasiparticles. Here we report experiments on monolayer graphene that show clear evidence for unexpected even denominator fractional quantum Hall physics in the n = 3 Landau level. We numerically investigate the known candidate states for the even denominator fractional quantum Hall effect, including the Pfaffian, the particle–hole symmetric Pfaffian and the 221-parton states, and conclude that, among these, the 221-parton appears a potentially suitable candidate to describe the experimentally observed state. Like the Pfaffian, this state is believed to harbour quasi-particles with non-Abelian braid statistics7.

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Fig. 1: Longitudinal (σxx) and Hall (σxy) conductivity of device D1.
Fig. 2: Hall plateaus at half filling of the n = 3 Landau levels.
Fig. 3: Overlaps and transport gaps at half filling of the n = 3 Landau level.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


  1. 1.

    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  Article  Google Scholar 

  2. 2.

    Moore, G. & Read, N. Nonabelions in the fractional quantum hall effect. Nucl. Phys. B 360, 362–396 (1991).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Levin, M., Halperin, B. I. & Rosenow, B. Particle–hole symmetry and the pfaffian state. Phys. Rev. Lett. 99, 236806 (2007).

    ADS  Article  Google Scholar 

  4. 4.

    Lee, S.-S., Ryu, S., Nayak, C. & Fisher, M. P. A. Particle–hole symmetry and the ν = 5/2 quantum Hall state. Phys. Rev. Lett. 99, 236807 (2007).

    ADS  Article  Google Scholar 

  5. 5.

    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).

    ADS  Article  Google Scholar 

  6. 6.

    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  Article  Google Scholar 

  7. 7.

    Wen, X. G. Non-Abelian statistics in the fractional quantum Hall states. Phys. Rev. Lett. 66, 802–805 (1991).

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Feldman, B. E., Krauss, B., Smet, J. H. & Yacoby, A. Unconventional sequence of fractional quantum Hall states in suspended graphene. Science 337, 1196–1199 (2012).

    ADS  Article  Google Scholar 

  9. 9.

    Feldman, B. E. et al. Fractional quantum Hall phase transitions and four-flux states in graphene. Phys. Rev. Lett. 111, 076802 (2013).

    ADS  Article  Google Scholar 

  10. 10.

    Jain, J. K. Composite-fermion approach for the fractional quantum hall effect. Phys. Rev. Lett. 63, 199–202 (1989).

    ADS  Article  Google Scholar 

  11. 11.

    Ki, D.-K., Fal’ko, V. I., Abanin, D. A. & Morpurgo, A. F. Observation of even denominator fractional quantum Hall effect in suspended bilayer graphene. Nano Lett. 14, 2135–2139 (2014).

    ADS  Article  Google Scholar 

  12. 12.

    Kim, Y. et al. Fractional quantum Hall states in bilayer graphene probed by transconductance fluctuations. Nano Lett. 15, 7445–7451 (2015).

    ADS  Article  Google Scholar 

  13. 13.

    Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature 549, 360–364 (2017).

    ADS  Article  Google Scholar 

  14. 14.

    Li, J. I. A. et al. Even denominator fractional quantum Hall states in bilayer graphene. Science 358, 648–652 (2017).

    ADS  Article  Google Scholar 

  15. 15.

    Falson, J. et al. Even-denominator fractional quantum Hall physics in ZnO. Nat. Phys. 11, 347–351 (2015).

    Article  Google Scholar 

  16. 16.

    Falson, J. et al. A cascade of phase transitions in an orbitally mixed half-filled Landau level. Sci. Adv. 4, eaat8742 (2018).

    ADS  Article  Google Scholar 

  17. 17.

    Zibrov, A. A. et al. Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene. Nat. Phys. 14, 930–935 (2018).

    Article  Google Scholar 

  18. 18.

    Banerjee, M. et al. Observation of half-integer thermal Hall conductance. Nature 559, 205–210 (2018).

    ADS  Article  Google Scholar 

  19. 19.

    Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene. Nat. Phys. 7, 693–696 (2011).

    Article  Google Scholar 

  20. 20.

    Amet, F. et al. Composite fermions and broken symmetries in graphene. Nat. Commun. 6, 5838 (2015).

    ADS  Article  Google Scholar 

  21. 21.

    Lilly, M. P., Cooper, K. B., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field. Phys. Rev. Lett. 83, 824–827 (1999).

    ADS  Article  Google Scholar 

  22. 22.

    Pan, W. et al. Exact quantization of the even-denominator fractional quantum Hall state at ν = 5/2 Landau level filling factor. Phys. Rev. Lett. 83, 3530–3533 (1999).

    ADS  Article  Google Scholar 

  23. 23.

    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  Article  Google Scholar 

  24. 24.

    Knoester, M. E., Papić, Z. & Morais Smith, C. Electron–solid and electron–liquid phases in graphene. Phys. Rev. B 93, 155141 (2016).

    ADS  Article  Google Scholar 

  25. 25.

    Balram, A. C., Töke, C., Wójs, A. & Jain, J. K. Spontaneous polarization of composite fermions in the n = 1 Landau level of graphene. Phys. Rev. B 92, 205120 (2015).

    ADS  Article  Google Scholar 

  26. 26.

    Haldane, F. D. M. Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. 51, 605–608 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Peterson, M. R. & Nayak, C. More realistic hamiltonians for the fractional quantum Hall regime in GaAs and graphene. Phys. Rev. B 87, 245129 (2013).

    ADS  Article  Google Scholar 

  28. 28.

    Jain, J. K. Incompressible quantum Hall states. Phys. Rev. B 40, 8079–8082 (1989).

    ADS  Article  Google Scholar 

  29. 29.

    Wu, Y., Shi, T. & Jain, J. K. Non-Abelian parton fractional quantum Hall effect in multilayer graphene. Nano Lett. 17, 4643–4647 (2017).

    ADS  Article  Google Scholar 

  30. 30.

    Balram, A. C. & Jain, J. K. Nature of composite fermions and the role of particle–hole symmetry: a microscopic account. Phys. Rev. B 93, 235152 (2016).

    ADS  Article  Google Scholar 

  31. 31.

    Morf, R. H., d’Ambrumenil, N. & Das Sarma, S. Excitation gaps in fractional quantum Hall states: an exact diagonalization study. Phys. Rev. B 66, 075408 (2002).

    ADS  Article  Google Scholar 

  32. 32.

    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 

  33. 33.

    Wen, X. G. & Zee, A. Shift and spin vector: new topological quantum numbers for the Hall fluids. Phys. Rev. Lett. 69, 953–956 (1992).

    ADS  Article  Google Scholar 

  34. 34.

    Balram, A. C., Barkeshli, M. & Rudner, M. S. Parton construction of a wave function in the anti-pfaffian phase. Phys. Rev. B 98, 035127 (2018).

    ADS  Article  Google Scholar 

  35. 35.

    Castellanos-Gomez, A. et al. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping. 2D Mater. 1, 011002 (2014).

    Article  Google Scholar 

  36. 36.

    Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Kim, Y. et al. Charge inversion and topological phase transition at a twist angle induced van Hove singularity of bilayer graphene. Nano Lett. 16, 5053–5059 (2016).

    ADS  Article  Google Scholar 

  38. 38.

    Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

    ADS  Article  Google Scholar 

  39. 39.

    Masubuchi, S. et al. Autonomous robotic searching and assembly of two-dimensional crystals to build van der Waals superlattices. Nat. Commun. 9, 1413 (2018).

    ADS  Article  Google Scholar 

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The authors acknowledge useful discussions with K. von Klitzing, I. Sodemann, Y.-H. Wu and J. Zhu, and assistance for sample preparation from S. Göres and M. Hagel. The authors thank S. Masubuchi and T. Machida for input on the ELVACITE stamp method for fabrication of the van der Waals heterostructure. J.H.S. is grateful for financial support from the graphene flagship. Y.K. thanks the Humboldt Foundation and A.C.B. the Villum Foundation for support. The Center for Quantum Devices is funded by the Danish National Research Foundation. This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 1104931001 (TOPDYN)). The work at Penn State was supported by the US Department of Energy under grant no. DE-SC0005042. Some portions of this research were conducted with Advanced CyberInfrastructure computational resources provided by The Institute for CyberScience at The Pennsylvania State University. The authors thank the authors of the DiagHam package, which was used for some of the numerical calculations. The growth of hBN crystals was supported by the Elemental Strategy Initiative conducted by MEXT (Japan) and CREST (JPMJCR15F3, JST).

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The experiments were designed by Y.K. and J.H.S., and were carried out in the laboratory by Y.K. The theory was performed by A.C.B. and J.K.J. The calculations were run by A.C.B. The hBN bulk crystal was synthesized by T.T. and K.W. The manuscript was written with contributions from Y.K., A.C.B., J.K.J. and J.H.S.

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Correspondence to Jurgen H. Smet.

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Supplementary information

Supplementary Figs. 1–10, supplementary references 1–45

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Kim, Y., Balram, A.C., Taniguchi, T. et al. Even denominator fractional quantum Hall states in higher Landau levels of graphene. Nature Phys 15, 154–158 (2019). https://doi.org/10.1038/s41567-018-0355-x

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