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Topological charge density waves at half-integer filling of a moiré superlattice

Nature Physics volume 18, pages 42–47 (2022)Cite this article

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Abstract

When a flat band is partially filled with electrons, strong Coulomb interactions between them may lead to the emergence of topological gapped states with quantized Hall conductivity. Such emergent topological states have been found in partially filled Landau levels1 and Hofstadter bands2,3; however, in both cases, a large magnetic field is required to produce the underlying flat band. The recent observation of quantum anomalous Hall effects in narrow-band moiré materials4,5,6,7 has led to the theoretical prediction that such phases could be realized at zero magnetic field8,9,10,11,12. Here we report the observation of insulators with Chern number C = 1 in the zero-magnetic-field limit at half-integer filling of the moiré superlattice unit cell in twisted monolayer–bilayer graphene7,13,14,15. Chern insulators in a half-filled band suggest the spontaneous doubling of the superlattice unit cell2,3,16, and our calculations find a ground state of the topological charge density wave at half-filling of the underlying band. The discovery of these topological phases at fractional superlattice filling enables the further pursuit of zero-magnetic-field phases that have fractional statistics that exist either as elementary excitations or bound to lattice dislocations.

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Fig. 1: Correlated states at half-integer filling of the moiré unit cell.
Fig. 2: SBCI state at ν = 3/2 in device D1 (θ = 1.25°).
Fig. 3: Evidence for non-trivial topology and ferromagnetism at ν = 7/2.
Fig. 4: Hartree–Fock calculation of ν = 7/2 SBCI.

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Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

    Article  ADS  Google Scholar 

  4. Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020).

    Article  ADS  Google Scholar 

  5. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

    Article  ADS  Google Scholar 

  6. Stepanov, P. et al. Competing zero-field Chern insulators in superconducting twisted bilayer graphene. Phys. Rev. Lett. 127, 197701 (2021).

    Article  ADS  Google Scholar 

  7. Polshyn, H. et al. Electrical switching of magnetic order in an orbital Chern insulator. Nature 588, 66–70 (2020).

    Article  ADS  Google Scholar 

  8. Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127 (2019).

    Article  ADS  Google Scholar 

  9. Ledwith, P. J., Tarnopolsky, G., Khalaf, E. & Vishwanath, A. Fractional Chern insulator states in twisted bilayer graphene: an analytical approach. Phys. Rev. Res. 2, 023237 (2020).

    Article  Google Scholar 

  10. Repellin, C. & Senthil, T. Chern bands of twisted bilayer graphene: fractional Chern insulators and spin phase transition. Phys. Rev. Res. 2, 023238 (2020).

    Article  Google Scholar 

  11. Abouelkomsan, A., Liu, Z. & Bergholtz, E. J. Particle-hole duality, emergent Fermi liquids, and fractional Chern insulators in moiré flatbands. Phys. Rev. Lett. 124, 106803 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  12. Wilhelm, P., Lang, T. C. & Läuchli, A. M. Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene. Phys. Rev. B 103, 125406 (2021).

    Article  ADS  Google Scholar 

  13. Chen, S. et al. Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nat. Phys. 17, 374–380 (2021).

    Article  Google Scholar 

  14. Xu, S. et al. Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene. Nat. Phys. 17, 619–626 (2021).

    Article  Google Scholar 

  15. He, M. et al. Competing correlated states and abundant orbital magnetism in twisted monolayer–bilayer graphene. Nat. Commun. 12, 4727 (2021).

    Article  ADS  Google Scholar 

  16. Kumar, A., Roy, R. & Sondhi, S. L. Generalizing quantum Hall ferromagnetism to fractional Chern bands. Phys. Rev. B 90, 245106 (2014).

    Article  ADS  Google Scholar 

  17. Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).

    Article  ADS  MATH  Google Scholar 

  18. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  Google Scholar 

  19. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  20. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895–900 (2020).

    Article  ADS  Google Scholar 

  21. Dean, C., Kim, P., Li, J. I. A. & Young, A. Fractional quantum Hall effects in graphene. in Fractional Quantum Hall Effects: New Developments 317–375 (World Scientific, 2020).

  22. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  23. Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    Article  ADS  Google Scholar 

  24. Chen, G. et al. Tunable ferromagnetism at non-integer filling of a moiré superlattice. Preprint at https://arxiv.org/abs/2012.10075 (2020).

  25. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

    Article  ADS  Google Scholar 

  26. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).

    Article  ADS  Google Scholar 

  27. Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Nat. Mater. 20, 940–944 (2021).

    Article  ADS  Google Scholar 

  28. Huang, X. et al. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 17, 715–719 (2021).

    Article  Google Scholar 

  29. Rademaker, L., Protopopov, I. V. & Abanin, D. A. Topological flat bands and correlated states in twisted monolayer-bilayer graphene. Phys. Rev. Res. 2, 033150 (2020).

    Article  Google Scholar 

  30. Ma, Z. et al. Topological flat bands in twisted trilayer graphene. Sci. Bull. 66, 18–22 (2021).

    Article  Google Scholar 

  31. Park, Y., Chittari, B. L. & Jung, J. Gate-tunable topological flat bands in twisted monolayer-bilayer graphene. Phys. Rev. B 102, 035411 (2020).

    Article  ADS  Google Scholar 

  32. Liu, J., Ma, Z., Gao, J. & Dai, X. Quantum valley Hall effect, orbital magnetism, and anomalous Hall effect in twisted multilayer graphene systems. Phys. Rev. X 9, 031021 (2019).

    Google Scholar 

  33. Zhang, S. & Liu, J. Spin polarized nematic order, quantum valley Hall states, and field tunable topological transitions in twisted multilayer graphene systems. Preprint at https://arxiv.org/abs/2101.04711 (2021).

  34. Streda, P. Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C: Solid State Phys. 15, L1299–L1303 (1982).

    Article  ADS  Google Scholar 

  35. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Wannier, G. H. A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Stat. Sol. (B) 88, 757–765 (1978).

    Article  ADS  Google Scholar 

  37. Oshikawa, M. Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice. Phys. Rev. Lett. 84, 1535–1538 (2000).

    Article  ADS  Google Scholar 

  38. Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020).

    Article  ADS  Google Scholar 

  39. Tschirhart, C. L. et al. Imaging orbital ferromagnetism in a moiré Chern insulator. Science 372, 1323–1327 (2021).

    Article  ADS  Google Scholar 

  40. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021).

    Article  Google Scholar 

  41. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Article  ADS  Google Scholar 

  42. Pierce, A. T. et al. Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-021-01347-4 (2021).

  43. Kang, J. & Vafek, O. Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic-angle twisted bilayer graphene. Phys. Rev. B 102, 035161 (2020).

    Article  ADS  Google Scholar 

  44. Xie, M. & MacDonald, A. H. Nature of the correlated insulator states in twisted bilayer graphene. Phys. Rev. Lett. 124, 097601 (2020).

    Article  ADS  Google Scholar 

  45. Barkeshli, M. & Qi, X.-L. Topological nematic states and non-Abelian lattice dislocations. Phys. Rev. X 2, 031013 (2012).

    Google Scholar 

Download references

Acknowledgements

We are grateful to J. Zhu for fruitful discussions. A.F.Y. acknowledges support from the Office of Naval Research under award N00014-20-1-2609, and the Gordon and Betty Moore Foundation under award GBMF9471. M.P.Z. acknowledges support from the ARO under MURI W911NF-16-1-0361. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, via grant no. JPMXP0112101001; JSPS KAKENHI grant no. JP20H00354; and the CREST(JPMJCR15F3), JST. A.V. was supported by a Simons Investigator Award. P.L. was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program.

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Authors and Affiliations

  1. Department of Physics, University of California, Santa Barbara, CA, USA

    H. Polshyn, Y. Zhang, M. A. Kumar & A. F. Young

  2. Department of Physics, University of California, Berkeley, CA, USA

    T. Soejima & M. P. Zaletel

  3. Department of Physics, Harvard University, Cambridge, MA, USA

    P. Ledwith & A. Vishwanath

  4. Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan

    K. Watanabe

  5. International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    T. Taniguchi

  6. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    M. P. Zaletel

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Contributions

H.P., M.A.K. and Y.Z. fabricated the devices. H.P. performed the measurements, advised by A.F.Y. K.W. and T.T. grew the hBN crystals. T.S., P.L., M.P.Z. and A.V. contributed to the theoretical interpretation and performed the Hartree–Fock calculations. H.P., P.L. and A.F.Y. wrote the manuscript with inputs from all the other authors.

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Correspondence to A. F. Young.

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Extended data

Extended Data Fig. 1 tMBG devices.

Optical images of tMBG devices D1 (a) and D2 (b). Scale bars are 10 μm.

Extended Data Fig. 2 Spin- and valley- symmetry breaking.

a, Metallic phases with broken flavour (spin and valley) degeneracies can be highlighted using quantity n* = (neff − n)/n0, where neff is the effective carrier density which was extracted here from Ryx measured at B = 1 T in device D2. n* corresponds to the filling of the superlattice at which the band that is being filled becomes completely empty (full) for n-type (p-type) Fermi surface. Thus, integer values of n* that are intermediate between 0 and 4 indicate interaction-induced symmetry breaking of the flavour degeneracies. b, Diagram of flavour symmetry broken phases corresponding to (a). The boundaries were determined using n* as well as measured Rxx and Ryx. The incompressible states at commensurate fillings are labeled with the Chern numbers observed for B > 0. The blue region around the ν=2 insulating state corresponds to a two-fold degenerate state, which we identify as spin-polarized phase because of the spin polarization of ν = 2 (see (c)) and the absence of the anomalous Hall effect [7]. Smaller red regions around ν=1 and 3 states correspond to phases with fully lifted spin-and valley- degeneracies, in which a single C = 2 band is being filled. States at ν = 3/2 and 7/2 emerge within these regions. c, In-plane magnetic field dependence of the gap at ν=2 determined from the thermal activation of Rxx in device D1 at D=0.443 V/nm. Linear fit yields g-factor of 1.59 ± 0.05 which indicates large spin polarization. The determined value is lower than g = 2 which is expected for fully spin-polarized state. The exact origin of this discrepancy is not fully understood and could include orbital effects of the in-plane magnetic field.

Extended Data Fig. 3 Ryx and Rxx measured in device D2 at selected magnetic fields.

Magnetic fields are indicated in panels. The Rxx (Ryx) data taken at finite fields is symmetrized (antisymmetrized) with respect to the magnetic field reversal.

Extended Data Fig. 4 SBCI state at ν =3/2.

a,b, Rxx (a) and Ryx (b) measured in the region around ν = 3/2 at B = 1.35 and 2 T in device D1. At B=1.35 T, weak peaks in Rxx and Ryx emerge at ν=3/2. At 2 T, the peak in Ryx is enhanced while Rxx shows a dip developing at ν=3/2. c, Ryx as a function of B measured near ν = 3/2 at D = 0.533 V/nm in device D2. Solid lines have slopes expected for C = ± 1 insulators.

Extended Data Fig. 5 Dependence of the anomalous Hall effect on n and D near ν = 7/2 in D2.

a, Ryx in the vicinity of a ν=7/2 state measured at B=0.15 T. b, D-field evolution of Ryx measured at n= 3.39 × 1012 cm−2. In this measurement, the fast sweep axis is D. The range of D at which the SBCI state is observed increases with B, indicating that the magnetic field stabilizes the state. c,d, The dependence of the magnetic hysteresis on n (c) and D (d). The values of (n, D) for each hysteresis loop are indicated by markers of the same color in (a). The curves are offset by 0.5h/e2 for clarity.

Extended Data Fig. 6 Details of the magnetic field evolution of Rxx (a) and Ryx (b,c) near ν = 7/2 in D2.

a,b,c, Ryx and Rxx measured at D = 0.466 V/nm. The corresponding contact configurations are shown in the diagrams above the plots. Rxx in (b) and (c) were measured on the opposite sides of the device. The tilted lines are guides for the eye and have the slope expected for C= ± 1 states. A number of fine features in (a-c), some of which persist down to zero field, align perfectly with these lines, providing further evidence for QAHE state at ν=7/2 which is deteriorated by the disorder. We speculate that the finite values of Rxx originate from the twist angle disorder which results in the presence of metallic regions in the device and hence interferes with the edge state transport. It is worth pointing out an interesting feature of Rxx: Rxx is low for B < 0, but becomes of the order of h/e2 for B > 0, as shown in (b). This trend reverses on the opposite side of the device, as shown in (c). A plausible scenario that can produce this behavior is illustrated in d and e. Let us consider a situation when the device region between contacts ‘b’ and ‘e’ is insulating in the bulk (shown in white) and has fully-developed edge states, while the adjacent regions remain metallic (shown in red and blue) due to the local twist angle variation. The color marks local electrical potential, increasing from blue to red. As a result of the edge state transport, there is a significant voltage drop ΔV ≈ Ih/e2 across the insulating region. In contrast to this, the electrical potential changes only gradually within the metallic regions since they have resistivity which is much smaller than h/e2. Because of the chiral nature of the edge states, Rxx ~ h/e2 when measured on the side of the device where the ‘cold’ edge state equilibrates with the metallic region, while Rxx ~ 0 on the other side of the device. We note that in this scenario, Ryx measured using contacts ‘b’ and ‘e’ is quantized.

Extended Data Fig. 7 Effects of reducing the area of D2 device.

a, Etching the device to reduce the active area of the device. Further Ryx and Rxx are measured between pairs of contacts (b,e) and (f,e) respectively. b,c,e,f, Comparison of Ryx (b,c) and Rxx (e,f) measured near ν= 7/2 before and after etching the device. Panels b and e show the same data that in Fig. E6, while the data shown in c and f was measured at D= 0.456 V/nm. The lines are guides for eye that correspond to C= ± 1 states. d Magnetic hysteresis measured near ν = 7/2 before and after etching show strong enhancement of Ryx jump at zero field. g,h,i Traces of Ryx (g) and Rxx (h,i) at selected fields which are indicated in panels. Plateau in Rxy and a dip in Rxx as a function of carrier density persist down to zero field.

Extended Data Table 1 Chern insulating states in a two dimensional lattice classified by whether quantum numbers C and s are integer (\({\mathbb{Z}}\)) or noninteger rational numbers (\({\mathbb{Q}}\backslash {\mathbb{Z}}\)). C is the net Chern number and s is the number of electrons bound to each lattice unit cell
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Supplementary Information

Supplementary discussion and Figs. 1 and 2.

Source data

Source data for Fig. 1

Data for Fig. 1b–f.

Source data for Fig. 2

Data for Fig. 2.

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Data for Fig. 4b–e.

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Polshyn, H., Zhang, Y., Kumar, M.A. et al. Topological charge density waves at half-integer filling of a moiré superlattice. Nat. Phys. 18, 42–47 (2022). https://doi.org/10.1038/s41567-021-01418-6

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