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.
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum hall effect. Phys. Rev. Lett. 59, 1776–1779 (1987).
Moore, G. & Read, N. Nonabelions in the fractional quantum hall effect. Nucl. Phys. B 360, 362–396 (1991).
Levin, M., Halperin, B. I. & Rosenow, B. Particle–hole symmetry and the pfaffian state. Phys. Rev. Lett. 99, 236806 (2007).
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).
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).
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).
Wen, X. G. Non-Abelian statistics in the fractional quantum Hall states. Phys. Rev. Lett. 66, 802–805 (1991).
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).
Feldman, B. E. et al. Fractional quantum Hall phase transitions and four-flux states in graphene. Phys. Rev. Lett. 111, 076802 (2013).
Jain, J. K. Composite-fermion approach for the fractional quantum hall effect. Phys. Rev. Lett. 63, 199–202 (1989).
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).
Kim, Y. et al. Fractional quantum Hall states in bilayer graphene probed by transconductance fluctuations. Nano Lett. 15, 7445–7451 (2015).
Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature 549, 360–364 (2017).
Li, J. I. A. et al. Even denominator fractional quantum Hall states in bilayer graphene. Science 358, 648–652 (2017).
Falson, J. et al. Even-denominator fractional quantum Hall physics in ZnO. Nat. Phys. 11, 347–351 (2015).
Falson, J. et al. A cascade of phase transitions in an orbitally mixed half-filled Landau level. Sci. Adv. 4, eaat8742 (2018).
Zibrov, A. A. et al. Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene. Nat. Phys. 14, 930–935 (2018).
Banerjee, M. et al. Observation of half-integer thermal Hall conductance. Nature 559, 205–210 (2018).
Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene. Nat. Phys. 7, 693–696 (2011).
Amet, F. et al. Composite fermions and broken symmetries in graphene. Nat. Commun. 6, 5838 (2015).
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).
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).
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).
Knoester, M. E., Papić, Z. & Morais Smith, C. Electron–solid and electron–liquid phases in graphene. Phys. Rev. B 93, 155141 (2016).
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).
Haldane, F. D. M. Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. 51, 605–608 (1983).
Peterson, M. R. & Nayak, C. More realistic hamiltonians for the fractional quantum Hall regime in GaAs and graphene. Phys. Rev. B 87, 245129 (2013).
Jain, J. K. Incompressible quantum Hall states. Phys. Rev. B 40, 8079–8082 (1989).
Wu, Y., Shi, T. & Jain, J. K. Non-Abelian parton fractional quantum Hall effect in multilayer graphene. Nano Lett. 17, 4643–4647 (2017).
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).
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).
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).
Wen, X. G. & Zee, A. Shift and spin vector: new topological quantum numbers for the Hall fluids. Phys. Rev. Lett. 69, 953–956 (1992).
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).
Castellanos-Gomez, A. et al. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping. 2D Mater. 1, 011002 (2014).
Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).
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).
Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).
Masubuchi, S. et al. Autonomous robotic searching and assembly of two-dimensional crystals to build van der Waals superlattices. Nat. Commun. 9, 1413 (2018).
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).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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