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Topological phase transition between Jain states and daughter states of the ν = 1/2 fractional quantum Hall state

Nature Physics volume 20, pages 1247–1252 (2024)Cite this article

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Abstract

The even-denominator fractional quantum Hall state at a filling factor of ν = 1/2 is an intriguing many-body phase in two-dimensional electron systems, as it appears in the ground state rather than an excited Landau level. It is observed in wide quantum wells where the electrons have a bilayer charge distribution with finite tunnelling between the layers. Whether this state takes an Abelian two-component form or a non-Abelian one-component form has been debated since its experimental discovery. Here we report the observation of the ν = 1/2 state that is flanked by numerous Jain-sequence composite fermion states at ν = p/(2p ± 1) up to ν = 8/17 and 9/17. As we raise the density, the ν = 1/2 state is strengthened, its energy gap increases up to 4 K and, at the same time, the 8/17 and 7/13 states abruptly become stronger than their neighbouring high-order fractions. The states at ν = 8/17 and 7/13 are the theoretically predicted, simplest daughter states of the one-component Pfaffian ν = 1/2 state. This means that our data suggest a topological phase transition of 8/17 and 7/13 states from the Jain-sequence states to the daughter states of the Pfaffian.

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Fig. 1: Evolution of two-dimensional electrons in a wide QW with varying densities.
Fig. 2: Energy gaps Δν of FQHSs in N = 0 LL.
Fig. 3: Topological phase transition in N = 0 LL.
Fig. 4: Effect of in-plane magnetic field.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

References

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  4. 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).

    Article  ADS  Google Scholar 

  5. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Hossain, M. S. et al. Unconventional anisotropic even-denominator fractional quantum Hall state in a system with mass anisotropy. Phys. Rev. Lett. 121, 256601 (2018).

    Article  ADS  Google Scholar 

  10. Shi, Q. et al. Odd- and even-denominator fractional quantum Hall states in monolayer WSe2. Nat. Nanotechnol. 15, 569–573 (2020).

    Article  ADS  Google Scholar 

  11. Dutta, B. et al. Distinguishing between non-Abelian topological orders in a quantum Hall system. Science 375, 193–197 (2022).

    Article  ADS  Google Scholar 

  12. Huang, K. et al. Valley isospin controlled fractional quantum Hall states in bilayer graphene. Phys. Rev. X 12, 031019 (2022); erratum 12, 049901 (2022).

    Google Scholar 

  13. Willett, R. L. et al. Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding. Phys. Rev. X 13, 011028 (2023).

    Google Scholar 

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

  15. Suen, Y. W., Engel, L. W., Santos, M. B., Shayegan, M. & Tsui, D. C. Observation of a ν = 1/2 fractional quantum Hall state in a double-layer electron system. Phys. Rev. Lett. 68, 1379–1382 (1992).

    Article  ADS  Google Scholar 

  16. Eisenstein, J. P., Boebinger, G. S., Pfeiffer, L. N., West, K. W. & He, S. New fractional quantum Hall state in double-layer two-dimensional electron systems. Phys. Rev. Lett. 68, 1383–1386 (1992).

    Article  ADS  Google Scholar 

  17. He, S., Das Sarma, S. & Xie, X. C. Quantized Hall effect and quantum phase transitions in coupled two-layer electron systems. Phys. Rev. B 47, 4394–4412 (1993).

    Article  ADS  Google Scholar 

  18. Halperin, B. I. Theory of the quantized Hall conductance. Helv. Phys. Acta 56, 75–102 (1983).

    Google Scholar 

  19. Greiter, M., Wen, X. G. & Wilczek, F. Paired Hall states in double-layer electron systems. Phys. Rev. B 46, 9586–9589 (1992).

    Article  ADS  Google Scholar 

  20. Greiter, M., Wen, X. G. & Wilczek, F. Paired Hall states. Nucl. Phys. B 374, 567–614 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  21. Suen, Y. W., Manoharan, H. C., Ying, X., Santos, M. B. & Shayegan, M. Origin of the ν = 1/2 fractional quantum Hall state in wide single quantum wells. Phys. Rev. Lett. 72, 3405–3408 (1994).

    Article  ADS  Google Scholar 

  22. Shabani, J. et al. Phase diagrams for the stability of the ν = 1/2 fractional quantum Hall effect in electron systems confined to symmetric, wide GaAs quantum wells. Phys. Rev. B 88, 245413 (2013).

    Article  ADS  Google Scholar 

  23. Peterson, M. R. & Sarma, S. D. Quantum Hall phase diagram of half-filled bilayers in the lowest and the second orbital Landau levels: Abelian versus non-Abelian incompressible fractional quantum Hall states. Phys. Rev. B 81, 165304 (2010).

    Article  ADS  Google Scholar 

  24. Thiebaut, N., Regnault, N. & Goerbig, M. O. Fractional quantum Hall states versus Wigner crystals in wide quantum wells in the half-filled lowest and second Landau levels. Phys. Rev. B 92, 245401 (2015).

    Article  ADS  Google Scholar 

  25. Mueed, M. A. et al. Geometric resonance of composite fermions near the ν = 1/2 fractional quantum Hall state. Phys. Rev. Lett. 114, 236406 (2015).

    Article  ADS  Google Scholar 

  26. Mueed, M. A. et al. Geometric resonance of composite fermions near bilayer quantum Hall states. Phys. Rev. Lett. 117, 246801 (2016).

    Article  ADS  Google Scholar 

  27. Zhu, W., Liu, Z., Haldane, F. D. M. & Sheng, D. N. Fractional quantum Hall bilayers at half filling: tunneling-driven non-Abelian phase. Phys. Rev. B 94, 245147 (2016).

    Article  ADS  Google Scholar 

  28. Sharma, A., Balram, A. C. & Jain, J. K. Composite-fermion pairing at half-filled and quarter-filled lowest Landau level. Phys. Rev. B 109, 035306 (2024).

  29. Suen, Y. W. et al. Missing integral quantum Hall effect in a wide single quantum well. Phys. Rev. B 44, 5947–5950 (1991).

    Article  ADS  Google Scholar 

  30. Manoharan, H. C., Suen, Y. W., Santos, M. B. & Shayegan, M. Evidence for a bilayer quantum Wigner solid. Phys. Rev. Lett. 77, 1813–1816 (1996).

    Article  ADS  Google Scholar 

  31. Shayegan, M., Manoharan, H. C., Suen, Y. W., Lay, T. S. & Santos, M. B. Correlated bilayer electron states. Semicond. Sci. Technol. 11, 1539–1545 (1996).

    Article  Google Scholar 

  32. Hatke, A. T. et al. Microwave spectroscopic observation of a Wigner solid within the ν = 1/2 fractional quantum Hall effect. Phys. Rev. B 95, 045417 (2017).

    Article  ADS  Google Scholar 

  33. Halperin, B. I. Theories for ν = 1/2 in single- and double-layer systems. Surf. Sci. 305, 1–7 (1994).

    Article  ADS  Google Scholar 

  34. Chung, Y. J. et al. Ultra-high-quality two-dimensional electron systems. Nat. Mater. 20, 632–637 (2021).

    Article  ADS  Google Scholar 

  35. Levin, M. & Halperin, B. I. Collective states of non-Abelian quasiparticles in a magnetic field. Phys. Rev. B 79, 205301 (2009).

    Article  ADS  Google Scholar 

  36. Wen, X. G. Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995).

    Article  ADS  Google Scholar 

  37. Kumar, A., Csáthy, G. A., Manfra, M. J., Pfeiffer, L. N. & West, K. W. Nonconventional odd-denominator fractional quantum Hall states in the second Landau level. Phys. Rev. Lett. 105, 246808 (2010).

    Article  ADS  Google Scholar 

  38. Halperin, B. I., Lee, P. A. & Read, N. Theory of the half-filled Landau level. Phys. Rev. B 47, 7312–7343 (1993).

    Article  ADS  Google Scholar 

  39. 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).

    Article  ADS  Google Scholar 

  40. Manoharan, H. C., Shayegan, M. & Klepper, S. J. Signatures of a novel Fermi liquid in a two-dimensional composite particle metal. Phys. Rev. Lett. 73, 3270–3273 (1994).

    Article  ADS  Google Scholar 

  41. Villegas Rosales, K. A. et al. Fractional quantum Hall effect energy gaps: role of electron layer thickness. Phys. Rev. Lett. 127, 056801 (2021).

    Article  ADS  Google Scholar 

  42. Zhao, T., Kudo, K., Faugno, W. N., Balram, A. C. & Jain, J. K. Revisiting excitation gaps in the fractional quantum Hall effect. Phys. Rev. B 105, 205147 (2022).

    Article  ADS  Google Scholar 

  43. Lay, T. S., Jungwirth, T., Smrčka, L. & Shayegan, M. One-component to two-component transition of the ν = 2/3 fractional quantum Hall effect in a wide quantum well induced by an in-plane magnetic field. Phys. Rev. B 56, R7092–R7095 (1997).

    Article  ADS  Google Scholar 

  44. Hasdemir, S. et al. ν = 1/2 fractional quantum Hall effect in tilted magnetic fields. Phys. Rev. B 91, 045113 (2015).

    Article  ADS  Google Scholar 

  45. Assouline, A. et al. Energy gap of the even-denominator fractional quantum Hall state in bilayer graphene. Phys. Rev. Lett. 132, 046603 (2024).

    Article  ADS  Google Scholar 

  46. Hu, Y. et al. High-resolution tunneling spectroscopy of fractional quantum Hall states. Preprint at https://arxiv.org/abs/2308.05789 (2023).

Download references

Acknowledgements

We acknowledge support by the US Department of Energy (DOE) Basic Energy Sciences via grant no. DEFG02-00-ER45841 for measurements, the National Science Foundation (NSF) grant nos. DMR 2104771 and ECCS 1906253 for sample characterization and the Eric and Wendy Schmidt Transformative Technology Fund for sample fabrication. The Princeton University portion of this research is funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative, grant GBMF9615.01, to L.N.P. Our measurements were partly performed at the National High Magnetic Field Laboratory (NHMFL), which is supported by the NSF Cooperative Agreement no. DMR 1644779, by the State of Florida, and by the DOE. This research is funded in part by QuantEmX grant from the Institute for Complex Adaptive Matter and the Gordon and Betty Moore Foundation through grant GBMF9616 to S.K.S., C.W., A.G., C.T.T. and C.S.C. We thank R. Nowell, G. Jones, A. Bangura and T. Murphy at NHMFL for technical assistance, and B. I. Halperin, J. K. Jain, P. Kumar, M. Levin and X.-G. Wen for illuminating discussions.

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ, USA

    S. K. Singh, C. Wang, C. T. Tai, C. S. Calhoun, K. A. Villegas Rosales, P. T. Madathil, A. Gupta, K. W. Baldwin, L. N. Pfeiffer & M. Shayegan

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  1. S. K. Singh
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Contributions

S.K.S. and M.S. conceived the work. S.K.S., C.W., C.T.T. and C.S.C. performed the low-temperature transport measurements. S.K.S. and K.W.B. fabricated the sample. L.N.P., K.W.B. and A.G. produced the molecular-beam epitaxy samples and characterized them. S.K.S. and M.S. analysed the data and wrote the manuscript with input from all co-authors.

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Correspondence to S. K. Singh or M. Shayegan.

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Extended data

Extended Data Fig. 1 Evolution of the ground state at ν = 1/2 in wide GaAs QWs.

a, Longitudinal resistance (Rxx) of 2D electrons confined to a 48-nm-wide GaAs QW shows the evolution from a compressible composite fermion Fermi sea to an incompressible phase, the 1/2 FQHS, as the 2D electron density is raised symmetrically. Except for the bottom trace, the traces are offset vertically for clarity. The 1/2 FQHS becomes stronger as the density is raised within an intermediate density range as evinced by the Rxx minimum at ν = 1/2 getting deeper and eventually touching zero. The top trace shows the Hall resistance (Rxy) for n = 3.42 × 1011 cm−2. Also noteworthy is that the numerous odd-denominator FQHSs at the lowest density n = 2.80 are also present together with the strong 1/2 FQHS at higher densities. The relative strengths of the odd-denominator FQHSs suggest that these are the usual Jain-sequence states at all densities. The inset shows the self-consistently calculated charge distribution for 2.80 (black) and 3.26 × 1011 cm−2 (green). b, Experimentally determined phase diagram depicting the phase boundaries between the composite fermion Fermi sea and the 1/2 FQHS, and the 1/2 FQHS and insulating phase, by dashed and dotted lines, respectively. The 1/2 FQHS is stable for an intermediate density range, and is sandwiched between the compressible composite fermion Fermi sea at low densities and the incompressible insulating phase at high densities. Within the intermediate density range, as the density is increased away from the compressible boundary, the 1/2 FQHS gets stronger, but eventually begins to weaken as the nearby insulating phase begins to engulf it; see Ref. 22 for details. Figure reproduced with permission from ref. 22, APS.

Extended Data Fig. 2 Evolution of an electron system in a 70-nm-wide GaAs QW with varying density.

a,b, Longitudinal (Rxx) and Hall resistance (Rxy) as a function of 1/ν in a 70-nm-wide GaAs QW. Traces are offset vertically for clarity. Rxx for the lowest density n = 1.09 shows a compressible ν = 1/2 composite fermion Fermi sea, with the characteristic Jain-sequence states up to ν = 10/19 and 9/19 on its flanks. Note that the Rxx minima at ν = 8/17 and 7/13 are strong, but their strength relative to other, nearby FQHSs is consistent with what is expected for Jain-sequence states. As the density is increased, the compressible composite fermion Fermi sea makes a transition into an incompressible FQHS at ν = 1/2, which gets stronger as the density is further raised. The odd-denominator FQHSs flanking the 1/2 FQHS undergo a similar evolution as displayed in Fig. 1e, where the Jain-sequence states get weaker because of the increased electron layer thickness, but the 8/17 and 7/13 FQHSs suddenly transition into anomalously strong FQHSs, distinct from Jain-sequence states. Quantitatively, owing to the smaller well width, the transitions of the ν = 8/17 and 7/13 FQHSs are both shifted to higher densities compared to the 72.5-nm-wide sample of Fig. 1. In panel b, self-consistently calculated charge distributions for a 2DES confined to a 70-nm-wide GaAs QW at n = 1.24 and 1.60 × 1011 cm−2 are shown as bottom and top insets, respectively.

Source data

Extended Data Fig. 3 Evolution of an electron system in a 70-nm-wide GaAs QW with in-plane magnetic field.

a,b, Longitudinal (Rxx) and Hall (Rxy) resistances are shown as a function of 1/ν at various tilt angles for n = 1.24 × 1011 cm−2. Qualitatively, once again the sample shows a similar evolution of FQHSs at and around ν = 1/2 as the 72.5-nm-wide sample, albeit at higher tilt angles.

Source data

Extended Data Fig. 4 Longitudinal (Rxx) and Hall resistance (Rxy) as a function of perpendicular magnetic field for ultrahigh quality 2D electrons in a 72.5-nm-wide GaAs QW.

The strong FQHS observed at ν = 1/2 and flanked by numerous high-order Jain-sequence FQHSs up to ν = 9/19 and 10/19 (as shown in the top right inset) on its sides attest to the ultra high quality of our sample. Top left inset shows the self-consistently calculated charge distribution for n = 1.16 × 1011 cm−2 electrons confined to a 72.5-nm-wide GaAs QW.

Source data

Extended Data Fig. 5 Arrhenius plots to extract the energy gaps of FQHSs in the N = 0 LL for n = 1.17.

a-c, Temperature dependence of Rxx minima for the FQHSs on the low-field side of ν = 1/2, at ν = 1/2, and on the high-field side of ν = 1/2, respectively. The red lines through the data points are fits to the data points in the activated regimes for different fillings, and their slopes yield the energy gaps Δν, determined from \({R}_{xx}\propto {e}^{-{\Delta }_{\nu }/2kT}\). Note in panel a that the ν = 7/13 FQHS is weaker than the 6/11 FQHS and has a smaller energy gap. The activated behavior of Rxx at the different FQHSs allows us to extract an energy gap, but one must be careful with the energy gaps of the weakest FQHSs which mostly serve as a measure of their strength. Because of the limited available temperature range and the effect of disorder, the small energy gaps of the higher-order FQHSs have a large error bar.

Source data

Extended Data Fig. 6 Arrhenius plots to extract the energy gaps of FQHSs in the N = 0 LL for n = 1.35.

Data are for the 72.5-nm-wide GaAs QW with n = 1.35 × 1011 cm−2. a-c, Temperature dependence of Rxx minima for the FQHSs on the low-field side of ν = 1/2, at ν = 1/2, and on the high-field side of ν = 1/2, respectively. The red lines through the data points are fits to the data points in the activated regimes for different fillings, and their slopes yield the energy gaps Δν, determined from \({R}_{xx}\propto {e}^{-{\Delta }_{\nu }/2kT}\). Note in panel a that, at this density, the 7/13 FQHS has a larger gap than the 6/11 FQHS. On the high-field side of ν = 1/2 (panel c), only the ν = 8/17 FQHS yields an activation, and the other FQHSs at higher fields are consumed by the ensuing insulating phase; see Fig. 1e of main text.

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Singh, S.K., Wang, C., Tai, C.T. et al. Topological phase transition between Jain states and daughter states of the ν = 1/2 fractional quantum Hall state. Nat. Phys. 20, 1247–1252 (2024). https://doi.org/10.1038/s41567-024-02517-w

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