When two graphene sheets are twisted to the magic angle near 1.1 degrees, their moiré superlattice hosts nearly flat bands in which interactions drive correlated insulating states, Chern insulators, and superconductivity. The 2019 study by Lu, Stepanov and co-workers on exceptionally twist-angle-uniform devices found superconducting domes adjacent to correlated insulators across a wide range of moiré filling factors, Hall slopes consistent with non-zero Chern numbers at odd fillings, and a hysteretic field-driven first-order transition at filling ν = −1, but could not determine the microscopic mechanism of superconductivity, could not establish why the ν = −3 and ν = −1 states lack activated (gapped) transport, and could not resolve quantized Hall transport. This problem asks for the microscopic characterization of these correlated states: identify the pairing mechanism and gap symmetry of the superconducting domes, distinguishing interaction-driven unconventional pairing from conventional electron–phonon pairing in metallic states that evolve adiabatically from the non-interacting flat bands; establish the criteria that determine whether a given integer filling is gapped or semi-metallic; and map the complete superconducting phase diagram and its relation to band topology. Later literature (2020–2026) closed parts of the original question — quantized anomalous Hall transport and Chern-domain imaging were achieved, and the odd-filling Chern states were identified as competing anomalous Hall ferromagnets — leaving the pairing mechanism, the metallicity criteria, and the phase-diagram completion as the open core.

Background

Twisted bilayer graphene (TBG) is formed by stacking two graphene sheets with a small relative rotation angle θ\theta. The mismatch creates a moiré superlattice — a long-wavelength periodic potential whose unit cell is much larger than graphene's — so that a modest carrier density fills one moiré band. At the so-called magic angle near 1.1°, first identified in Bistritzer and MacDonald's continuum model (2011), the low-energy moiré minibands become nearly flat (bandwidth of a few meV), so that Coulomb interactions dominate the kinetic energy. The two flat bands (one conduction, one valence) are approximately fourfold degenerate in spin and valley flavor; the filling factor ν counts electrons per moiré unit cell relative to charge neutrality, from ν = −4 to +4 across one flat band.

In this regime, interactions can lift the flavor degeneracy and open gaps at every integer ν, producing correlated insulators, much as in the fractional quantum Hall effect but at zero magnetic field. Because each valley-projected flat band carries non-trivial Berry curvature and (sublattice-resolved) Chern character, flavor polarization can also produce insulators with non-zero total Chern number C — Chern insulators — which exhibit quantized anomalous Hall conductance Ce2/hCe^2/h at zero field and orbital ferromagnetism (a ferromagnetic order in which the orbital, rather than spin, magnetic moment orders; Sharpe et al. reported anomalous Hall behavior near three-quarters filling, published 2019, preprint 2018). Theory describes these as anomalous Hall ferromagnets, states that break time-reversal symmetry by valley polarization (Bultinck, Chatterjee and Zaletel, 2020).

Cao et al. discovered in 2018 that magic-angle TBG also superconducts, with domes adjacent to correlated insulators at ν ≈ ±2, prompting comparison with doped Mott insulators. In 2019 Lu et al. (Nature 574, 653) fabricated devices of unprecedented twist-angle homogeneity (twist-angle variation Δθ<0.02\Delta\theta < 0.02° over ~10 µm) and observed: (i) resistive states at all integer ν including charge neutrality, with a superconducting dome near ν = −2 at critical temperatures up to 3 K and three additional low-temperature domes near ν = 0 and ν = ±1; (ii) Landau-fan slopes dn/dB\mathrm{d}n/\mathrm{d}B (the slopes of the lines in a plot of a transport feature's index versus magnetic field, which equal the Chern number) at ν = ±1 consistent with Chern numbers 1 and 2, though quantized Hall resistance and zero longitudinal resistance were not achieved because residual inhomogeneity remained; (iii) a sharp hysteretic resistance enhancement at ν = −1 above 3.6 T, signalling a field-driven first-order transition to an orbital magnetic state; and (iv) activated (gapped) transport at all integer fillings except ν = −3 and ν = −1, which remained semi-metallic. Their Discussion states that the consistent association of superconductivity with nearby correlated insulators suggests an exotic pairing mechanism, but that the data cannot rule out conventional electron–phonon coupling superconductivity in metallic states with quasiparticles evolving adiabatically from the non-interacting bands.

Subsequent experiments sharpened the landscape: screening experiments (Stepanov et al., 2020) quenched the correlated insulators with a nearby metallic layer while superconductivity persisted with comparable TcT_{\mathrm c}; scanning tunnelling microscopy found superconducting spectra inconsistent with conventional s-wave pairing (Oh et al., 2021) and nematicity (rotational-symmetry breaking) intertwined with superconductivity (Cao et al., 2021); scanning superconducting-quantum-interference-device (SQUID) microscopy imaged the orbital magnetism and a Chern-domain mosaic (Grover et al., 2022); and devices aligned to hexagonal boron nitride (hBN) realized a full cascade of zero-field Chern insulators at all odd fillings (Serlin et al., 2020; Stepanov et al., 2021; Zhang et al., 2026). What remains unsettled is the microscopic origin of superconductivity itself, the rules governing gapped versus semi-metallic integer fillings, and the completion of the superconducting phase diagram across the flat bands.

Problem Statement

Determine the microscopic origin of the correlated-state phenomenology of magic-angle twisted bilayer graphene as reported by Lu et al. (2019), namely: (a) identify the pairing mechanism and gap symmetry of its superconducting domes — in particular, distinguish an exotic, interaction-linked unconventional pairing associated with the nearby symmetry-broken correlated insulators from conventional electron–phonon superconductivity in metallic states whose quasiparticles evolve adiabatically from the non-interacting flat bands; (b) establish the criteria — interaction strength relative to bandwidth, twist-angle inhomogeneity, disorder, and subleading symmetry breaking — that determine whether a given integer moiré filling factor hosts a gapped insulating state or a semi-metallic correlated state, explaining why ν = −3 and ν = −1 lack activated transport in randomly aligned devices while other integer fillings are gapped; and (c) map the complete superconducting phase diagram across the flat bands, including the domes that were only developing in the 2019 devices, and establish its relation to the Chern-number band topology and interaction parameters of the flat bands. The common objective is a microscopic, predictive account of which correlated or superconducting state appears at each filling and why. Two components of the original 2019 question are already resolved by later work — quantized anomalous Hall transport of the odd-filling Chern states and imaging of their domain structure were achieved in 2020–2025, and the hysteretic ν = −1 field-driven transition is understood as competition among nearly degenerate anomalous-Hall-ferromagnetic Chern states — so the completion boundary of the present problem lies in items (a)–(c).

References

  1. Lu, X., Stepanov, P., Yang, W., Xie, M., Aamir, M. A., Das, I., Urgell, C., Watanabe, K., Taniguchi, T., Zhang, G., Bachtold, A., MacDonald, A. H. & Efetov, D. K. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019). DOI 10.1038/s41586-019-1695-0, https://doi.org/10.1038/s41586-019-1695-0 (preprint: arXiv:1903.06513, https://arxiv.org/abs/1903.06513).
  2. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proceedings of the National Academy of Sciences 108, 12233–12237 (2011). DOI 10.1073/pnas.1108174108, https://doi.org/10.1073/pnas.1108174108.
  3. Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E. & Jarillo-Herrero, P. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018). DOI 10.1038/nature26160, https://doi.org/10.1038/nature26160.
  4. Sharpe, A. L., Fox, E. J., Barnard, A. W., Finney, J., Watanabe, K., Taniguchi, T., Kastner, M. A. & Goldhaber-Gordon, D. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019). DOI 10.1126/science.aaw3780, https://doi.org/10.1126/science.aaw3780.
  5. Serlin, M., Tschirhart, C. L., Polshyn, H., Zhang, Y., Zhu, J., Watanabe, K., Taniguchi, T., Balents, L. & Young, A. F. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020). DOI 10.1126/science.aay5533, https://doi.org/10.1126/science.aay5533.
  6. Nuckolls, K. P., Oh, M., Wong, D., Lian, B., Watanabe, K., Taniguchi, T., Bernevig, B. A. & Yazdani, A. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020). DOI 10.1038/s41586-020-3028-8, https://doi.org/10.1038/s41586-020-3028-8.
  7. Bultinck, N., Chatterjee, S. & Zaletel, M. P. Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene. Physical Review Letters 124, 166601 (2020). DOI 10.1103/PhysRevLett.124.166601, https://doi.org/10.1103/PhysRevLett.124.166601.
  8. Stepanov, P., Das, I., Lu, X., Fahimniya, A., Watanabe, K., Taniguchi, T., Koppens, F. H. L., Lischner, J., Levitov, L. & Efetov, D. K. Untying the insulating and superconducting orders in magic-angle graphene. Nature 583, 375–378 (2020). DOI 10.1038/s41586-020-2459-6, https://doi.org/10.1038/s41586-020-2459-6 (preprint: arXiv:1911.09198, https://arxiv.org/abs/1911.09198).
  9. Stepanov, P., Xie, M., Taniguchi, T., Watanabe, K., Lu, X., MacDonald, A. H., Bernevig, B. A. & Efetov, D. K. Competing zero-field Chern insulators in superconducting twisted bilayer graphene. Physical Review Letters 127, 197701 (2021). DOI 10.1103/PhysRevLett.127.197701, https://doi.org/10.1103/PhysRevLett.127.197701.
  10. Cao, Y., Rodan-Legrain, D., Park, J. M., Yuan, N. F. Q., Watanabe, K., Taniguchi, T., Fernandes, R. M., Fu, L. & Jarillo-Herrero, P. Nematicity and competing orders in superconducting magic-angle graphene. Science 372, 264–271 (2021). DOI 10.1126/science.abc2836, https://doi.org/10.1126/science.abc2836.
  11. Oh, M., Nuckolls, K. P., Wong, D., Lee, R. L., Liu, X., Watanabe, K., Taniguchi, T. & Yazdani, A. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021). DOI 10.1038/s41586-021-04121-x, https://doi.org/10.1038/s41586-021-04121-x.
  12. Grover, S., Bocarsly, M., Uri, A., Stepanov, P., Di Battista, G., Roy, I., Xiao, J., Meltzer, A. Y., Myasoedov, Y., Pareek, K., Watanabe, K., Taniguchi, T., Yan, B., Stern, A., Berg, E., Efetov, D. K. & Zeldov, E. Chern mosaic and Berry-curvature magnetism in magic-angle graphene. Nature Physics 18, 885–892 (2022). DOI 10.1038/s41567-022-01635-7, https://doi.org/10.1038/s41567-022-01635-7.
  13. Datta, A., Calderón, M. J., Camjayi, A. & Bascones, E. Heavy quasiparticles and cascades without symmetry breaking in twisted bilayer graphene. Nature Communications 14, 5036 (2023). DOI 10.1038/s41467-023-40754-4, https://doi.org/10.1038/s41467-023-40754-4.
  14. Ledwith, P. J., Dong, J., Vishwanath, A. & Khalaf, E. Nonlocal moments and Mott semimetal in the Chern bands of twisted bilayer graphene. Physical Review X 15, 021087 (2025). DOI 10.1103/PhysRevX.15.021087, https://doi.org/10.1103/PhysRevX.15.021087.
  15. Zhang, Z., Yang, J., Xie, B., Feng, Z., Zhang, S., Watanabe, K., Taniguchi, T., Yang, X., Dai, Q., Liu, D., Liu, K., Song, Z., Liu, T., Liu, J. & Lu, X. Cascade of zero-field Chern insulators in magic-angle bilayer graphene. National Science Review 13, nwaf265 (2026; published online 3 July 2025). DOI 10.1093/nsr/nwaf265, https://doi.org/10.1093/nsr/nwaf265.
  16. Park, J. M., Sun, S., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Experimental evidence for nodal superconducting gap in moiré graphene. Science Advances 12, eadv8376 (2026). DOI 10.1126/science.adv8376, https://doi.org/10.1126/science.adv8376.