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  • Review Article
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A microscopic perspective on moiré materials

Abstract

Contemporary quantum materials research is guided by themes of topology and electronic correlations. A confluence of these two themes is engineered in moiré materials, an emerging class of highly tunable, strongly correlated 2D materials designed by the rotational or lattice misalignment of atomically thin crystals. In moiré materials, dominant Coulomb interactions among electrons give rise to collective electronic phases, often with robust topological properties. Identifying the mechanisms responsible for these exotic phases is fundamental to our understanding of strongly interacting quantum systems and to our ability to engineer new material properties for potential future technological applications. In this Review, we highlight the contributions of local spectroscopic, thermodynamic and electromagnetic probes to the budding field of moiré materials research. These techniques have not only identified many of the underlying mechanisms of the correlated insulators, generalized Wigner crystals, unconventional superconductors, moiré ferroelectrics and topological orbital ferromagnets found in moiré materials, but have also uncovered fragile quantum phases that have evaded spatially averaged global probes. Furthermore, we highlight recently developed local probe techniques, including local charge sensing and quantum interference probes, that have uncovered new physical observables in moiré materials.

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Fig. 1: Phase diagrams of moiré materials.
Fig. 2: Flat electronic bands and cascades of electronic transitions in magic-angle twisted bilayer graphene.
Fig. 3: Mapping correlated insulators in moiré graphene.
Fig. 4: Imaging correlated insulators in moiré transition metal dichalcogenides.
Fig. 5: Local sensing of correlation-driven topological phases in moiré graphene.
Fig. 6: Imaging orbital ferromagnetism and moiré ferroelectricity.
Fig. 7: Spectroscopic probes of unconventional superconductivity in moiré graphene.

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References

  1. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    CAS  Google Scholar 

  2. Andrei, E. Y. et al. The marvels of moiré materials. Nat. Rev. Mater. 6, 201–206 (2021).

    CAS  Google Scholar 

  3. Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).

    CAS  PubMed  Google Scholar 

  4. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018). This article reports the observation of correlated electronic states (correlated insulators) in a moiré material (magic-angle twisted bilayer graphene).

    CAS  PubMed  Google Scholar 

  5. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018). This article reports the observation of a superconducting phase in a moiré material (magic-angle twisted bilayer graphene).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

  7. Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

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

    CAS  Google Scholar 

  10. Polshyn, H. et al. Topological charge density waves at half-integer filling of a moiré superlattice. Nat. Phys. 18, 42–47 (2022).

    CAS  Google Scholar 

  11. Liu, X. et al. Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature 583, 221–225 (2020).

    CAS  PubMed  Google Scholar 

  12. Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).

    CAS  PubMed  Google Scholar 

  13. Shen, C. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520–525 (2020).

    CAS  Google Scholar 

  14. Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020). This article reports the observation of a correlated electronic state (correlated insulator) in a transition metal dichalcogenide moiré material (twisted bilayer WSe2).

    CAS  PubMed  Google Scholar 

  15. Ghiotto, A. et al. Quantum criticality in twisted transition metal dichalcogenides. Nature 597, 345–349 (2021).

    CAS  PubMed  Google Scholar 

  16. Xu, Y. et al. A tunable bilayer Hubbard model in twisted WSe2. Nat. Nanotechnol. 17, 934–939 (2022).

    CAS  PubMed  Google Scholar 

  17. Li, H. et al. Mapping charge excitations in generalized Wigner crystals. Nat. Nanotechnol. https://doi.org/10.1038/s41565-023-01594-x (2024).

  18. Li, H. et al. Imaging moiré excited states with photocurrent tunnelling microscopy. Nat. Mater. 23, 633–638 (2024).

    CAS  PubMed  Google Scholar 

  19. Wang, P. et al. One-dimensional Luttinger liquids in a two-dimensional moiré lattice. Nature 605, 57–62 (2022).

    CAS  PubMed  Google Scholar 

  20. Yu, G. et al. Evidence for two dimensional anisotropic Luttinger liquids at millikelvin temperatures. Nat. Commun. 14, 7025 (2023).

    CAS  PubMed  PubMed Central  Google Scholar 

  21. Anderson, E. et al. Programming correlated magnetic states with gate-controlled moiré geometry. Science 381, 325–330 (2023).

    CAS  PubMed  Google Scholar 

  22. Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023). This article reports the observation of fractional Chern insulators at zero magnetic field (twisted bilayer MoTe2).

    CAS  PubMed  Google Scholar 

  23. Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).

    CAS  PubMed  Google Scholar 

  24. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021). This article reports the observation of highly tunable superconducting phases in magic-angle twisted trilayer graphene.

    CAS  PubMed  Google Scholar 

  25. Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021). This article reports the observation of highly tunable superconducting phases in magic-angle twisted trilayer graphene.

    CAS  PubMed  Google Scholar 

  26. Park, J. M. et al. Robust superconductivity in magic-angle multilayer graphene family. Nat. Mater. 21, 877–883 (2022).

    CAS  PubMed  Google Scholar 

  27. Zhang, Y. et al. Promotion of superconductivity in magic-angle graphene multilayers. Science 377, 1538–1543 (2022).

    CAS  PubMed  Google Scholar 

  28. Uri, A. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023).

    CAS  PubMed  Google Scholar 

  29. Xia, L.-Q. et al. Helical trilayer graphene: a moiré platform for strongly-interacting topological bands. Preprint at https://arxiv.org/abs/2310.12204 (2023).

  30. Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    CAS  PubMed  Google Scholar 

  31. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020). This article reports the observation of correlated insulators (generalized Wigner crystals) at fractional fillings of a moiré flat band (AA-stacked WSe2/WS2).

    CAS  PubMed  Google Scholar 

  32. Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021). This article reports the observation of a Chern insulator in a transition metal dichalcogenide moiré material, showing a quantized anomalous Hall effect (AB-stacked MoTe2/WSe2).

    CAS  PubMed  Google Scholar 

  33. Zhao, W. et al. Realization of the Haldane Chern insulator in a moiré lattice. Nat. Phys. 20, 275–280 (2024).

    CAS  Google Scholar 

  34. Zhao, W. et al. Gate-tunable heavy fermions in a moiré Kondo lattice. Nature 616, 61–65 (2023).

    CAS  PubMed  Google Scholar 

  35. Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

    CAS  Google Scholar 

  36. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013). This article reports a high-electronic-quality, gate-tunable moiré material (bilayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

  37. Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013). This article reports a high-electronic-quality, gate-tunable moiré material (monolayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

  38. Ponomarenko, L. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013). This article reports a high-electronic-quality, gate-tunable moiré material (monolayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

  39. Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).

    CAS  Google Scholar 

  40. Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019). This article predicts a hierarchy of graphene-based moiré materials with flat moiré bands.

    CAS  Google Scholar 

  41. Ledwith, P. J., Vishwanath, A. & Khalaf, E. Family of ideal Chern flatbands with arbitrary Chern number in chiral twisted graphene multilayers. Phys. Rev. Lett. 128, 176404 (2022).

    CAS  PubMed  Google Scholar 

  42. Lian, B., Liu, Z., Zhang, Y. & Wang, J. Flat Chern band from twisted bilayer MnBi2Te4. Phys. Rev. Lett. 124, 126402 (2020).

    CAS  PubMed  Google Scholar 

  43. Zhang, Y.-H., Sheng, D. & Vishwanath, A. SU(4) chiral spin liquid, exciton supersolid, and electric detection in moiré bilayers. Phys. Rev. Lett. 127, 247701 (2021).

    CAS  PubMed  Google Scholar 

  44. Shi, J., Zhu, J. & MacDonald, A. Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride. Phys. Rev. B 103, 075122 (2021).

    CAS  Google Scholar 

  45. Can, O. et al. High-temperature topological superconductivity in twisted double-layer copper oxides. Nat. Phys. 17, 519–524 (2021).

    CAS  Google Scholar 

  46. Popov, F. K. & Tarnopolsky, G. Magic angle butterfly in twisted trilayer graphene. Phys. Rev. Res. 5, 043079 (2023).

    CAS  Google Scholar 

  47. Yang, C., May-Mann, J., Zhu, Z. & Devakul, T. Multi-moiré trilayer graphene: lattice relaxation, electronic structure, and magic angles. Preprint at https://arxiv.org/abs/2310.12961 (2023).

  48. Kim, K.-M. & Park, M. J. Controllable magnetic domains in twisted trilayer magnets. Phys. Rev. B 108, L100401 (2023).

    CAS  Google Scholar 

  49. Morell, E. S., Correa, J., Vargas, P., Pacheco, M. & Barticevic, Z. Flat bands in slightly twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010). This article predicts the electronic structure of twisted bilayer graphene, calculated at small commensurate twist angles.

    Google Scholar 

  50. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011). This article proposes the Bistritzer–MacDonald continuum model, which forms an essential theoretical starting point for understanding moiré materials.

    CAS  PubMed  PubMed Central  Google Scholar 

  51. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Google Scholar 

  52. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    CAS  Google Scholar 

  53. Bernevig, B. A. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  54. Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).

    PubMed  Google Scholar 

  55. Burkov, A. Topological semimetals. Nat. Mater. 15, 1145–1148 (2016).

    CAS  PubMed  Google Scholar 

  56. Yan, B. & Felser, C. Topological Materials: Weyl semimetals. Annu. Rev. Condens. Matter Phys. 8, 337–354 (2017).

    Google Scholar 

  57. Armitage, N., Mele, E. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    CAS  Google Scholar 

  58. Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 1, 126–143 (2019).

    Google Scholar 

  59. Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019). This article reports the observation of the anomalous Hall effect and spontaneous ferromagnetic state in a moiré material (twisted bilayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

  61. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020). This article reports the observation of the quantized anomalous Hall effect in a moiré material (twisted bilayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

  62. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020). This article reports the observation of a sequence of correlated Chern insulators in magic-angle twisted bilayer graphene.

    CAS  PubMed  Google Scholar 

  63. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. 20, 488–494 (2021). This article reports the observation of a sequence of correlated Chern insulators in magic-angle twisted bilayer graphene.

    CAS  PubMed  Google Scholar 

  64. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021). This article reports the observation of symmetry-broken Chern insulators in magic-angle twisted bilayer graphene.

    CAS  Google Scholar 

  65. Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. 17, 710–714 (2021).

    CAS  Google Scholar 

  66. Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021).

    CAS  PubMed  Google Scholar 

  67. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021).

    CAS  PubMed  Google Scholar 

  68. Yu, J. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. 18, 825–831 (2022).

    CAS  Google Scholar 

  69. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021). This article reports the observation of a series of fractional Chern insulators in magic-angle twisted bilayer graphene.

    CAS  PubMed  PubMed Central  Google Scholar 

  70. Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74–79 (2023). This article reports the observation of the fractionally quantized anomalous Hall effect (twisted bilayer MoTe2).

    CAS  PubMed  Google Scholar 

  71. Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2Phys. Rev. X 13, 031037 (2023). This article reports the observation of the fractionally quantized anomalous Hall effect (twisted bilayer MoTe2).

    CAS  Google Scholar 

  72. Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).

    CAS  PubMed  Google Scholar 

  73. Goldman, H., Reddy, A. P., Paul, N. & Fu, L. Zero-field composite Fermi liquid in twisted semiconductor bilayers. Phys. Rev. Lett. 131, 136501 (2023).

    CAS  PubMed  Google Scholar 

  74. Dong, J., Wang, J., Ledwith, P. J., Vishwanath, A. & Parker, D. E. Composite Fermi liquid at zero magnetic field in twisted MoTe2. Phys. Rev. Lett. 131, 136502 (2023).

    CAS  PubMed  Google Scholar 

  75. Shabani, S. et al. Deep moiré potentials in twisted transition metal dichalcogenide bilayers. Nat. Phys. 17, 720–725 (2021).

    CAS  Google Scholar 

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

    CAS  PubMed  Google Scholar 

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

    CAS  Google Scholar 

  78. Lisi, S. et al. Observation of flat bands in twisted bilayer graphene. Nat. Phys. 17, 189–193 (2021).

    CAS  Google Scholar 

  79. Utama, M. I. B. et al. Visualization of the flat electronic band in twisted bilayer graphene near the magic angle twist. Nat. Phys. 17, 184–188 (2021).

    CAS  Google Scholar 

  80. Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020).

    CAS  PubMed  Google Scholar 

  81. Chen, C. et al. Strong inter-valley electron–phonon coupling in magic-angle twisted bilayer graphene. Preprint at https://arxiv.org/abs/2303.14903 (2023).

  82. Slot, M. et al. A quantum ruler for orbital magnetism in moiré quantum matter. Science 382, 81–87 (2023).

    CAS  PubMed  Google Scholar 

  83. Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene. Nature 606, 494–500 (2022).

    CAS  PubMed  Google Scholar 

  84. Nuckolls, K. P. et al. Quantum textures of the many-body wavefunctions in magic-angle graphene. Nature 620, 525–532 (2023).

    CAS  PubMed  Google Scholar 

  85. Kim, H. et al. Imaging inter-valley coherent order in magic-angle twisted trilayer graphene. Nature 623, 942–948 (2023).

    CAS  PubMed  Google Scholar 

  86. Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

    CAS  PubMed  Google Scholar 

  87. Pierce, A. T. et al. Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene. Nat. Phys. 17, 1210–1215 (2021).

    CAS  Google Scholar 

  88. Rozen, A. et al. Entropic evidence for a Pomeranchuk effect in magic-angle graphene. Nature 592, 214–219 (2021).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

  90. Grover, S. et al. Chern mosaic and Berry-curvature magnetism in magic-angle graphene. Nat. Phys. 18, 885–892 (2022).

    CAS  Google Scholar 

  91. Song, T. et al. Direct visualization of magnetic domains and moiré magnetism in twisted 2D magnets. Science 374, 1140–1144 (2021).

    CAS  PubMed  Google Scholar 

  92. Huang, M. et al. Revealing intrinsic domains and fluctuations of moiré magnetism by a wide-field quantum microscope. Nat. Commun. 14, 5259 (2023).

    CAS  PubMed  PubMed Central  Google Scholar 

  93. Woods, C. et al. Charge-polarized interfacial superlattices in marginally twisted hexagonal boron nitride. Nat. Commun. 12, 347 (2021). This article reports the observation of a moiré ferroelectric (twisted hexagonal boron nitride).

    CAS  PubMed  PubMed Central  Google Scholar 

  94. Vizner Stern, M. et al. Interfacial ferroelectricity by van der Waals sliding. Science 372, 1462 (2021). This article reports the observation of a moiré ferroelectric (twisted hexagonal boron nitride).

    CAS  Google Scholar 

  95. Weston, A. et al. Interfacial ferroelectricity in marginally twisted 2D semiconductors. Nat. Nanotechnol. 17, 390–395 (2022).

    CAS  PubMed  PubMed Central  Google Scholar 

  96. Deb, S. et al. Cumulative polarization in conductive interfacial ferroelectrics. Nature 612, 465–469 (2022).

    CAS  PubMed  Google Scholar 

  97. Moore, S. et al. Nanoscale lattice dynamics in hexagonal boron nitride moiré superlattices. Nat. Commun. 12, 5741 (2021).

    CAS  PubMed  PubMed Central  Google Scholar 

  98. Lau, C. N., Bockrath, M. W., Mak, K. F. & Zhang, F. Reproducibility in the fabrication and physics of moiré materials. Nature 602, 41–50 (2022).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

  100. Xi, X. et al. Strongly enhanced charge-density-wave order in monolayer NbSe2. Nat. Nanotechnol. 10, 765–769 (2015).

    CAS  PubMed  Google Scholar 

  101. Ugeda, M. M. et al. Characterization of collective ground states in single-layer NbSe2. Nat. Phys. 12, 92–97 (2016).

    CAS  Google Scholar 

  102. Chen, Y. et al. Strong correlations and orbital texture in single-layer 1T-TaSe2. Nat. Phys. 16, 218–224 (2020).

    CAS  Google Scholar 

  103. Călugăru, D. et al. Spectroscopy of twisted bilayer graphene correlated insulators. Phys. Rev. Lett. 129, 117602 (2022).

    PubMed  Google Scholar 

  104. Hong, J. P. et al. Detecting symmetry breaking in magic angle graphene using scanning tunneling microscopy. Phys. Rev. Lett. 129, 147001 (2022).

    CAS  PubMed  Google Scholar 

  105. Li, H. et al. Imaging two-dimensional generalized Wigner crystals. Nature 597, 650–654 (2021).  This article reports a new atomic-scale charge-sensing technique using scanning tunnelling microscopy.

    CAS  PubMed  Google Scholar 

  106. Kuwabara, M., Clarke, D. R. & Smith, D. Anomalous superperiodicity in scanning tunneling microscope images of graphite. Appl. Phys. Lett. 56, 2396–2398 (1990).

    CAS  Google Scholar 

  107. Rong, Z. Y. & Kuiper, P. Electronic effects in scanning tunneling microscopy: moiré pattern on a graphite surface. Phys. Rev. B 48, 17427–17431 (1993).

    CAS  Google Scholar 

  108. Xhie, J., Sattler, K., Ge, M. & Venkateswaran, N. Giant and supergiant lattices on graphite. Phys. Rev. B 47, 15835–15841 (1993).

    CAS  Google Scholar 

  109. Li, G. et al. Observation of van Hove singularities in twisted graphene layers. Nat. Phys. 6, 109–113 (2010). This article reports the spectroscopic characterization of twisted graphene.

    Google Scholar 

  110. Wong, D. et al. Local spectroscopy of moiré-induced electronic structure in gate-tunable twisted bilayer graphene. Phys. Rev. B 92, 155409 (2015). This article reports the experimental verification of the BistritzerMacDonald continuum model (twisted bilayer graphene).

    Google Scholar 

  111. Li, Y. et al. Observation of coexisting Dirac bands and moiré flat bands in magic‐angle twisted trilayer graphene. Adv. Mater. 34, 2205996 (2022).

    CAS  Google Scholar 

  112. Stansbury, C. H. et al. Visualizing electron localization of WS2/WSe2 moiré superlattices in momentum space. Sci. Adv. 7, eabf4387 (2021).

    CAS  PubMed  PubMed Central  Google Scholar 

  113. Xie, Y. et al. Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene. Nature 572, 101–105 (2019).

    CAS  PubMed  Google Scholar 

  114. Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

    CAS  PubMed  Google Scholar 

  115. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019).

    CAS  Google Scholar 

  116. Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

    CAS  PubMed  Google Scholar 

  117. Kang, J., Bernevig, B. A. & Vafek, O. Cascades between light and heavy fermions in the normal state of magic-angle twisted bilayer graphene. Phys. Rev. Lett. 127, 266402 (2021).

    CAS  PubMed  Google Scholar 

  118. Song, Z.-D. & Bernevig, B. A. Magic-angle twisted bilayer graphene as a topological heavy fermion problem. Phys. Rev. Lett. 129, 047601 (2022). This article proposes the topological heavy-fermion model of magic-angle twisted bilayer graphene.

    CAS  PubMed  Google Scholar 

  119. Datta, A., Calderon, M. J., Camjayi, A. & Bascones, E. Heavy quasiparticles and cascades without symmetry breaking in twisted bilayer graphene. Nat. Commun. 14, 5036 (2023).

    CAS  PubMed  PubMed Central  Google Scholar 

  120. Hu, H. et al. Symmetric Kondo lattice states in doped strained twisted bilayer graphene. Phys. Rev. Lett. 131, 166501 (2023).

    CAS  PubMed  Google Scholar 

  121. Zhou, G.-D. & Song, Z.-D. Kondo phase in twisted bilayer graphene. Phys. Rev. B 109, 045419 (2024).

    CAS  Google Scholar 

  122. Huang, C. et al. Evolution from quantum anomalous Hall insulator to heavy-fermion semimetal in magic-angle twisted bilayer graphene. Phys. Rev. B 109, 125404 (2024).

    CAS  Google Scholar 

  123. Chou, Y.-Z. & Sarma, S. D. Kondo lattice model in magic-angle twisted bilayer graphene. Phys. Rev. Lett. 131, 026501 (2023).

    CAS  PubMed  Google Scholar 

  124. Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Independent superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. 16, 926–930 (2020).

    CAS  Google Scholar 

  125. Stepanov, P. et al. Untying the insulating and superconducting orders in magic-angle graphene. Nature 583, 375–378 (2020).

    CAS  PubMed  Google Scholar 

  126. Tarnopolsky, G., Kruchkov, A. J. & Vishwanath, A. Origin of magic angles in twisted bilayer graphene. Phys. Rev. Lett. 122, 106405 (2019). This article proposes the chiral model of magic-angle twisted bilayer graphene.

    CAS  PubMed  Google Scholar 

  127. Bultinck, N. et al. Ground state and hidden symmetry of magic-angle graphene at even integer filling. Phys. Rev. X 10, 031034 (2020).

    CAS  Google Scholar 

  128. Shirane, G. et al. Two-dimensional antiferromagnetic quantum spin-fluid state in La2CuO4. Phys. Rev. Lett. 59, 1613–1616 (1987).

    CAS  PubMed  Google Scholar 

  129. Chakravarty, S., Halperin, B. I. & Nelson, D. R. Low-temperature behavior of two-dimensional quantum antiferromagnets. Phys. Rev. Lett. 60, 1057–1060 (1988).

    CAS  PubMed  Google Scholar 

  130. Manousakis, E. The spin-½ Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Mod. Phys. 63, 1–62 (1991).

    CAS  Google Scholar 

  131. Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands. Phys. Rev. Lett. 122, 246401 (2019).

    CAS  PubMed  Google Scholar 

  132. Liu, X. et al. Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet. Science 375, 321–326 (2022).

    CAS  PubMed  Google Scholar 

  133. Kwan, Y. H. et al. Kekulé spiral order at all nonzero integer fillings in twisted bilayer graphene. Phys. Rev. X 11, 041063 (2021).

    CAS  Google Scholar 

  134. Wagner, G., Kwan, Y. H., Bultinck, N., Simon, S. H. & Parameswaran, S. Global phase diagram of the normal state of twisted bilayer graphene. Phys. Rev. Lett. 128, 156401 (2022).

    CAS  PubMed  Google Scholar 

  135. Wang, T. et al. Ground-state order in magic-angle graphene at filling ν = −3: a full-scale density matrix renormalization group study. Phys. Rev. B 108, 235128 (2023).

    CAS  Google Scholar 

  136. Kwan, Y. H. et al. Electron-phonon coupling and competing Kekulé orders in twisted bilayer graphene. Preprint at https://arxiv.org/abs/2303.13602 (2023).

  137. Christos, M., Sachdev, S. & Scheurer, M. S. Nodal band-off-diagonal superconductivity in twisted graphene superlattices. Nat. Commun. 14, 7134 (2023).

    CAS  PubMed  Google Scholar 

  138. Wigner, E. On the interaction of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).

    CAS  Google Scholar 

  139. Monarkha, Y. P. & Syvokon, V. A two-dimensional Wigner crystal. Low Temp. Phys. 38, 1067–1095 (2012).

  140. Grimes, C. & Adams, G. Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons. Phys. Rev. Lett. 42, 795–798 (1979).

    CAS  Google Scholar 

  141. Andrei, E. et al. Observation of a magnetically induced Wigner solid. Phys. Rev. Lett. 60, 2765–2768 (1988).

    CAS  PubMed  Google Scholar 

  142. Tsui, Y.-C. et al. Direct observation of a magnetic-field-induced Wigner crystal. Nature 628, 287–292 (2024).

    CAS  PubMed  Google Scholar 

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

    CAS  PubMed  Google Scholar 

  144. Cui, Y.-T., Ma, E. Y. & Shen, Z.-X. Quartz tuning fork based microwave impedance microscopy. Rev. Sci. Instrum. 87, 063711 (2016).

    PubMed  Google Scholar 

  145. Halperin, B. I. & Jain, J. K. Fractional Quantum Hall Effects: New Developments (World Scientific, 2020).

  146. Tong, D. Lectures on the quantum Hall effect. Preprint at https://arxiv.org/abs/1606.06687 (2016).

  147. Rachel, S. Interacting topological insulators: a review. Rep. Prog. Phys. 81, 116501 (2018).

    PubMed  Google Scholar 

  148. Kometter, C. R. et al. Hofstadter states and re-entrant charge order in a semiconductor moiré lattice. Nat. Phys. 19, 1861–1867 (2023).

    CAS  Google Scholar 

  149. Foutty, B. A. et al. Mapping twist-tuned multiband topology in bilayer WSe2. Science 384, 343–347 (2024).

    CAS  PubMed  Google Scholar 

  150. Streda, P. Theory of quantised Hall conductivity in two dimensions. J. Phys. C Solid State Phys. 15, L717–L721 (1982).

    CAS  Google Scholar 

  151. Zhang, C. et al. Local spectroscopy of a gate-switchable moiré quantum anomalous Hall insulator. Nat. Commun. 14, 3595 (2023).

    CAS  PubMed  Google Scholar 

  152. Zhang, C. et al. Manipulation of chiral interface states in a moiré quantum anomalous Hall insulator. Nat. Phys. https://doi.org/10.1038/s41567-024-02444-w (2024).

  153. Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (2024). This article reports the observation of fractional Chern insulators showing fractional quantum anomalous Hall effects in moiré graphene (rhombohedral pentalayer graphene aligned to hexagonal boron nitride).

    CAS  PubMed  Google Scholar 

  154. Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429–433 (2021).

    CAS  PubMed  Google Scholar 

  155. Tschirhart, C. et al. Intrinsic spin Hall torque in a moiré Chern magnet. Nat. Phys. 19, 807–813 (2023).

    CAS  Google Scholar 

  156. Zhang, D., Schoenherr, P., Sharma, P. & Seidel, J. Ferroelectric order in van der Waals layered materials. Nat. Rev. Mater. 8, 25–40 (2023).

    Google Scholar 

  157. Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered ferroelectricity in bilayer boron nitride. Science 372, 1458–1462 (2021). This article reports the observation of a moiré ferroelectric (twisted hexagonal boron nitride).

    CAS  Google Scholar 

  158. Kim, D. S. et al. Electrostatic moiré potential from twisted hexagonal boron nitride layers. Nat. Mater. 23, 65–70 (2024).

    CAS  PubMed  Google Scholar 

  159. Wang, X. et al. Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides. Nat. Nanotechnol. 17, 367–371 (2022).

    CAS  PubMed  Google Scholar 

  160. Ko, K. et al. Operando electron microscopy investigation of polar domain dynamics in twisted van der Waals homobilayers. Nat. Mater. 22, 992–998 (2023).

    CAS  PubMed  Google Scholar 

  161. Cao, Y. et al. Nematicity and competing orders in superconducting magic-angle graphene. Science 372, 264–271 (2021).

    CAS  PubMed  Google Scholar 

  162. Cao, Y., Park, J. M., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Pauli-limit violation and re-entrant superconductivity in moiré graphene. Nature 595, 526–531 (2021).

    CAS  PubMed  Google Scholar 

  163. Su, R., Kuiri, M., Watanabe, K., Taniguchi, T. & Folk, J. Superconductivity in twisted double bilayer graphene stabilized by WSe2. Nat. Mater. 22, 1332–1337 (2023).

    CAS  PubMed  Google Scholar 

  164. Rodan-Legrain, D. et al. Highly tunable junctions and non-local Josephson effect in magic-angle graphene tunnelling devices. Nat. Nanotechnol. 16, 769–775 (2021).

    CAS  PubMed  Google Scholar 

  165. de Vries, F. K. et al. Gate-defined Josephson junctions in magic-angle twisted bilayer graphene. Nat. Nanotechnol. 16, 760–763 (2021).

    PubMed  Google Scholar 

  166. Díez-Mérida, J. et al. Symmetry-broken Josephson junctions and superconducting diodes in magic-angle twisted bilayer graphene. Nat. Commun. 14, 2396 (2023).

    PubMed  Google Scholar 

  167. Portolés, E. et al. A tunable monolithic SQUID in twisted bilayer graphene. Nat. Nanotechnol. 17, 1159–1164 (2022).

    PubMed  Google Scholar 

  168. Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).

    CAS  Google Scholar 

  169. Tsuei, C. & Kirtley, J. Pairing symmetry in cuprate superconductors. Rev. Mod. Phys. 72, 969–1016 (2000).

    CAS  Google Scholar 

  170. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    CAS  Google Scholar 

  171. Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).

    CAS  Google Scholar 

  172. Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

    CAS  PubMed  Google Scholar 

  173. Johnston, D. C. The puzzle of high temperature superconductivity in layered iron pnictides and chalcogenides. Adv. Phys. 59, 803–1061 (2010).

    CAS  Google Scholar 

  174. Wen, H.-H. & Li, S. Materials and novel superconductivity in iron pnictide superconductors. Annu. Rev. Condens. Matter Phys. 2, 121–140 (2011).

    CAS  Google Scholar 

  175. Stewart, G. Superconductivity in iron compounds. Rev. Mod. Phys. 83, 1589–1652 (2011).

    CAS  Google Scholar 

  176. Si, Q., Yu, R. & Abrahams, E. High-temperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. 1, 1 (2016).

    Google Scholar 

  177. Fernandes, R. M. et al. Iron pnictides and chalcogenides: a new paradigm for superconductivity. Nature 601, 35–44 (2022).

    CAS  PubMed  Google Scholar 

  178. Stewart, S. G. Heavy-fermion systems. Rev. Mod. Phys. 56, 755–787 (1984).

    CAS  Google Scholar 

  179. Petrovic, C. et al. Heavy-fermion superconductivity in CeCoIn5 at 2.3 K. J. Phys. Condens. Matter 13, L337 (2001).

    CAS  Google Scholar 

  180. White, B. et al. Unconventional superconductivity in heavy-fermion compounds. Physica C 514, 246–278 (2015).

    CAS  Google Scholar 

  181. Jérome, D. The physics of organic superconductors. Science 252, 1509–1514 (1991).

    PubMed  Google Scholar 

  182. Singleton, J. & Mielke, C. Quasi-two-dimensional organic superconductors: a review. Contemp. Phys. 43, 63–96 (2002).

    CAS  Google Scholar 

  183. Arora, H. S. et al. Superconductivity in metallic twisted bilayer graphene stabilized by WSe2. Nature 583, 379–384 (2020).

    CAS  PubMed  Google Scholar 

  184. Liu, X. et al. Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening. Science 371, 1261–1265 (2021).

    CAS  PubMed  Google Scholar 

  185. Anderson, P. W. Theory of dirty superconductors. J. Phys. Chem. Solids 11, 26–30 (1959).

    CAS  Google Scholar 

  186. Zeljkovic, I. & Hoffman, J. E. Interplay of chemical disorder and electronic inhomogeneity in unconventional superconductors. Phys. Chem. Chem. Phys. 15, 13462–13478 (2013).

    CAS  PubMed  Google Scholar 

  187. Xu, C. & Balents, L. Topological superconductivity in twisted multilayer graphene. Phys. Rev. Lett. 121, 087001 (2018).

    CAS  PubMed  Google Scholar 

  188. Wu, F., MacDonald, A. H. & Martin, I. Theory of phonon-mediated superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 121, 257001 (2018).

    CAS  PubMed  Google Scholar 

  189. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, 031089 (2018).

    CAS  Google Scholar 

  190. Isobe, H., Yuan, N. F. & Fu, L. Unconventional superconductivity and density waves in twisted bilayer graphene. Phys. Rev. X 8, 041041 (2018).

    CAS  Google Scholar 

  191. Liu, C.-C., Zhang, L.-D., Chen, W.-Q. & Yang, F. Chiral spin density wave and d+id superconductivity in the magic-angle-twisted bilayer graphene. Phys. Rev. Lett. 121, 217001 (2018).

    CAS  PubMed  Google Scholar 

  192. Gonzalez, J. & Stauber, T. Kohn–Luttinger superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 122, 026801 (2019).

    CAS  PubMed  Google Scholar 

  193. Guinea, F. & Walet, N. R. Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, 13174–13179 (2018).

    CAS  PubMed  PubMed Central  Google Scholar 

  194. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

    CAS  PubMed  Google Scholar 

  195. Kennes, D. M., Lischner, J. & Karrasch, C. Strong correlations and d+id superconductivity in twisted bilayer graphene. Phys. Rev. B 98, 241407 (2018).

    CAS  Google Scholar 

  196. Peltonen, T. J., Ojajärvi, R. & Heikkilä, T. T. Mean-field theory for superconductivity in twisted bilayer graphene. Phys. Rev. B 98, 220504 (2018).

    CAS  Google Scholar 

  197. Lian, B., Wang, Z. & Bernevig, B. A. Twisted bilayer graphene: a phonon-driven superconductor. Phys. Rev. Lett. 122, 257002 (2019).

    CAS  PubMed  Google Scholar 

  198. Khalaf, E., Chatterjee, S., Bultinck, N., Zaletel, M. P. & Vishwanath, A. Charged skyrmions and topological origin of superconductivity in magic-angle graphene. Sci. Adv. 7, eabf5299 (2021).

    CAS  PubMed  PubMed Central  Google Scholar 

  199. Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

    CAS  Google Scholar 

  200. Fischer, O., Kugler, M., Maggio-Aprile, I., Berthod, C. & Renner, C. Scanning tunneling spectroscopy of high-temperature superconductors. Rev. Mod. Phys. 79, 353–419 (2007).

    CAS  Google Scholar 

  201. Di Battista, G. et al. Revealing the thermal properties of superconducting magic-angle twisted bilayer graphene. Nano Lett. 22, 6465–6470 (2022).

    PubMed  Google Scholar 

  202. Deutscher, G. Andreev–Saint–James reflections: a probe of cuprate superconductors. Rev. Mod. Phys. 77, 109–135 (2005).

    CAS  Google Scholar 

  203. Zhou, H. et al. Imaging quantum oscillations and millitesla pseudomagnetic fields in graphene. Nature 624, 275–281 (2023).

    CAS  PubMed  PubMed Central  Google Scholar 

  204. Bocarsly, M. et al. De Haas–van Alphen spectroscopy and magnetic breakdown in moiré graphene. Science 383, 42–48 (2024).

    CAS  PubMed  Google Scholar 

  205. Inbar, A. et al. The quantum twisting microscope. Nature 614, 682–687 (2023). This article reports the quantum twisting microscope.

    CAS  PubMed  Google Scholar 

  206. Zhao, S. F. et al. Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors. Science 382, 1422–1427 (2023).

    CAS  PubMed  Google Scholar 

  207. Wang, X. et al. Light-induced ferromagnetism in moiré superlattices. Nature 604, 468–473 (2022).

    CAS  PubMed  Google Scholar 

  208. Xiong, R. et al. Correlated insulator of excitons in WSe2/WS2 moiré superlattices. Science 380, 860–864 (2023).

    CAS  PubMed  Google Scholar 

  209. Crépel, V., Regnault, N. & Queiroz, R. Chiral limit and origin of topological flat bands in twisted transition metal dichalcogenide homobilayers. Commun. Phys. 7, 146 (2024).

    Google Scholar 

  210. Dong, J. et al. Anomalous Hall crystals in rhombohedral multilayer graphene I: interaction-driven Chern bands and fractional quantum Hall states at zero magnetic field. Preprint at https://arxiv.org/abs/2311.05568 (2023).

  211. Kwan, Y. H. et al. Moiré fractional Chern insulators III: Hartree–Fock phase diagram, magic angle regime for Chern insulator states, the role of the moiré potential and Goldstone gaps in rhombohedral graphene superlattices. Preprint at https://arxiv.org/abs/2312.11617 (2023).

  212. Reddy, A. P., Alsallom, F., Zhang, Y., Devakul, T. & Fu, L. Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2. Phys. Rev. B 108, 085117 (2023).

    CAS  Google Scholar 

  213. Zhou, B., Yang, H. & Zhang, Y.-H. Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moiréless limit and in Coulomb imprinted superlattice. Preprint at https://arxiv.org/abs/2311.04217 (2023).

  214. Morales-Durán, N., Wei, N. & MacDonald, A. H. Magic angles and fractional Chern insulators in twisted homobilayer transition metal dichalcogenides. Phys. Rev. Lett. 132, 096602 (2024).

    PubMed  Google Scholar 

  215. Kang, K. et al. Evidence of the fractional quantum spin Hall effect in moiré MoTe2. Nature 628, 522–526 (2024). This article reports the observation of the fractional quantum spin Hall effect (twisted bilayer MoTe2).

    CAS  PubMed  Google Scholar 

  216. Reddy, A. P., Paul, N., Abouelkomsan, A. & Fu, L. Non-Abelian fractionalization in topological minibands. Preprint at https://arxiv.org/abs/2403.00059 (2024).

  217. Fujimoto, M. et al. Higher vortexability: zero field realization of higher Landau levels. Preprint at https://arxiv.org/abs/2403.00856 (2024).

  218. May-Mann, J., Stern, A. & Devakul, T. Theory of half-integer fractional quantum spin Hall insulator edges. Preprint at https://arxiv.org/abs/2403.03964 (2024).

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Acknowledgements

The authors are grateful to all collaborators on this subject, particularly to D. Wong, M. Oh, R. L. Lee, T. Soejima, J. P. Hong, D. Călugăru, J. Herzog-Arbeitman, C. Chen, Y. Xie, B. Lian, Y. Chen, O. Vafek, N. Regnault, M. Zaletel and B. A. Bernevig. In addition, the authors thank J. G. Checkelsky for fruitful discussions during the writing of this Review. K.P.N. acknowledges support from the MIT Pappalardo Fellowship in Physics. A.Y. acknowledges funding from Gordon and Betty Moore Foundation’s EPiQS initiative grant GBMF9469, DOE-BES grant DE-FG02-07ER46419, ONR grant N00014-21-1-2592, NSF-MRSEC through the Princeton Center for Complex Materials grant NSFDMR-2011750, the US Army Research Office MURI project under grant number W911NF-21-2-0147 and NSF grant DMR-2312311.

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Nuckolls, K.P., Yazdani, A. A microscopic perspective on moiré materials. Nat Rev Mater 9, 460–480 (2024). https://doi.org/10.1038/s41578-024-00682-1

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