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Orbital magnetic states in moiré graphene systems

Nature Reviews Physics volume 3, pages 367–382 (2021)Cite this article

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Abstract

Moiré graphene systems have attracted considerable attention in the past 3 years because they exhibit exotic phenomena including correlated insulating states, unconventional superconductivity and the quantum anomalous Hall effect. All these phenomena are intimately related to the valley-spin-degenerate and topologically non-trivial flat bands in moiré graphene systems. When time-reversal symmetry is broken spontaneously, such flavour-degenerate topological flat bands exhibit unconventional orbital magnetism associated with real-space current-loop patterns on the moiré length scale. In this Perspective, we first survey key experimental progress on the correlated insulating states and the quantum anomalous Hall phenomena. Most of these phenomena are related to the moiré orbital magnetic states, which originate from the topological nature of the moiré flat bands. Finally, we discuss theoretical progress in the understanding of the correlated insulating and quantum anomalous Hall phenomena from the perspective of spontaneous symmetry breaking.

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Fig. 1: Moiré graphene systems.
Fig. 2: Correlated insulators in twisted graphene systems.
Fig. 3: Quantum anomalous Hall effects in moiré graphene systems.
Fig. 4: Band structures and topological properties of moiré graphene systems.
Fig. 5: Moiré orbital ferromagnetism.
Fig. 6: A pseudo-Landau-level picture for twisted bilayer graphene (TBG).

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References

  1. Varma, C. M. Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997).

    Article  ADS  Google Scholar 

  2. Sarma, S. D. & Pinczuk, A. Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-dimensional Semiconductor Structures (Wiley, 2008).

  3. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    Article  ADS  Google Scholar 

  4. 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).

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    Article  ADS  Google Scholar 

  7. Neupert, T., Santos, L., Chamon, C. & Mudry, C. Fractional quantum Hall states at zero magnetic field. Phys. Rev. Lett. 106, 236804 (2011).

    Article  ADS  Google Scholar 

  8. Tang, E., Mei, J.-W. & Wen, X.-G. High-temperature fractional quantum Hall states. Phys. Rev. Lett. 106, 236802 (2011).

    Article  ADS  Google Scholar 

  9. Sun, K., Gu, Z., Katsura, H. & Das Sarma, S. Nearly flatbands with nontrivial topology. Phys. Rev. Lett. 106, 236803 (2011).

    Article  ADS  Google Scholar 

  10. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  Google Scholar 

  11. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: electronic structure. Phys. Rev. Lett. 99, 256802 (2007).

    Article  ADS  Google Scholar 

  12. Mele, E. J. Commensuration and interlayer coherence in twisted bilayer graphene. Phys. Rev. B 81, 161405 (2010).

    Article  ADS  Google Scholar 

  13. Trambly de Laissardiere, G., Mayou, D. & Magaud, L. Localization of Dirac electrons in rotated graphene bilayers. Nano Lett. 10, 804–808 (2010).

    Article  ADS  Google Scholar 

  14. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Continuum model of the twisted graphene bilayer. Phys. Rev. B 86, 155449 (2012).

    Article  ADS  Google Scholar 

  15. San-Jose, P., González, J. & Guinea, F. Non-Abelian gauge potentials in graphene bilayers. Phys. Rev. Lett. 108, 216802 (2012).

    Article  ADS  Google Scholar 

  16. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  Google Scholar 

  17. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Song, Z. et al. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

    Article  ADS  Google Scholar 

  20. Ahn, J., Park, S. & Yang, B.-J. Failure of Nielsen–Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).

    Google Scholar 

  21. Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).

    Article  ADS  Google Scholar 

  22. Liu, J., Liu, J. & Dai, X. Pseudo Landau level representation of twisted bilayer graphene: band topology and implications on the correlated insulating phase. Phys. Rev. B 99, 155415 (2019).

    Article  ADS  Google Scholar 

  23. Tarnopolsky, G., Kruchkov, A. J. & Vishwanath, A. Origin of magic angles in twisted bilayer graphene. Phys. Rev. Lett. 122, 106405 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  27. 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 

  28. 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 

  29. Lee, J. Y. et al. Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene. Nat. Commun. 10, 5333 (2019).

    Article  ADS  Google Scholar 

  30. Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene. Phys. Rev. B 99, 235417 (2019).

    Article  ADS  Google Scholar 

  31. Cea, T., Walet, N. R. & Guinea, F. Twists and the electronic structure of graphitic materials. Nano Lett. 19, 8683–8689 (2019).

    Article  ADS  Google Scholar 

  32. Koshino, M. Band structure and topological properties of twisted double bilayer graphene. Phys. Rev. B 99, 235406 (2019).

    Article  ADS  Google Scholar 

  33. Ma, Z. et al. Topological flat bands in twisted trilayer graphene. Sci. Bull. 66, 18–22 (2020).

    Article  Google Scholar 

  34. Haddadi, F., Wu, Q., Kruchkov, A. J. & Yazyev, O. V. Moiré flat bands in twisted double bilayer graphene. Nano Lett. 20, 2410–2415 (2020).

    Article  ADS  Google Scholar 

  35. Zhang, Y.-H. & Senthil, T. Bridging Hubbard model physics and quantum Hall physics in trilayer graphene/h-BN moiré superlattice. Phys. Rev. B 99, 205150 (2019).

    Article  ADS  Google Scholar 

  36. Chittari, B. L., Chen, G., Zhang, Y., Wang, F. & Jung, J. Gate-tunable topological flat bands in trilayer graphene boron-nitride moiré superlattices. Phys. Rev. Lett. 122, 016401 (2019).

    Article  ADS  Google Scholar 

  37. Chen, G. et al. Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature 572, 215–219 (2019).

    Article  Google Scholar 

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

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  41. Liu, J. & Dai, X. Theories for the correlated insulating states and quantum anomalous Hall effect phenomena in twisted bilayer graphene. Phys. Rev. B 103, 035427 (2021).

    Article  ADS  Google Scholar 

  42. 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).

    Google Scholar 

  43. Angeli, M., Tosatti, E. & Fabrizio, M. Valley Jahn–Teller effect in twisted bilayer graphene. Phys. Rev. X 9, 041010 (2019).

    Google Scholar 

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

    Google Scholar 

  45. Bultinck, N., Chatterjee, S. & Zaletel, M. P. Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene. Phys. Rev. Lett. 124, 166601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  46. Zhang, Y.-H., Mao, D. & Senthil, T. Twisted bilayer graphene aligned with hexagonal boron nitride: anomalous Hall effect and a lattice model. Phys. Rev. Res. 1, 033126 (2019).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  48. 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).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

  53. Uchida, K., Furuya, S., Iwata, J.-I. & Oshiyama, A. Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes. Phys. Rev. B 90, 155451 (2014).

    Article  ADS  Google Scholar 

  54. Lee, J.-K. et al. The growth of AA graphite on (111) diamond. J. Chem. Phys. 129, 234709 (2008).

    Article  ADS  Google Scholar 

  55. Mora, C., Regnault, N. & Bernevig, B. A. Flatbands and perfect metal in trilayer moiré graphene. Phys. Rev. Lett. 123, 026402 (2019).

    Article  ADS  Google Scholar 

  56. Zhu, Z., Carr, S., Massatt, D., Luskin, M. & Kaxiras, E. Twisted trilayer graphene: a precisely tunable platform for correlated electrons. Phys. Rev. Lett. 125, 116404 (2020).

    Article  ADS  Google Scholar 

  57. Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).

    Article  ADS  Google Scholar 

  58. Lei, C., Linhart, L., Qin, W., Libisch, F. & MacDonald, A. H. Mirror symmetry breaking and stacking-shift dependence in twisted trilayer graphene. Preprint at https://arxiv.org/abs/2010.05787 (2020).

  59. Wu, F., Zhang, R.-X. & Das Sarma, S. Three-dimensional topological twistronics. Phys. Rev. Res. 2, 022010 (2020).

    Article  Google Scholar 

  60. Tsai, K.-T. et al. Correlated superconducting and insulating states in twisted trilayer graphene moire of moire superlattices. Preprint at https://arxiv.org/abs/1912.03375 (2019).

  61. Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moiré superlattice bands in layered two-dimensional materials. Phys. Rev. B 89, 205414 (2014).

    Article  ADS  Google Scholar 

  62. Moon, P. & Koshino, M. Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys. Rev. B 90, 155406 (2014).

    Article  ADS  Google Scholar 

  63. Moon, P. & Koshino, M. Optical absorption in twisted bilayer graphene. Phys. Rev. B 87, 205404 (2013).

    Article  ADS  Google Scholar 

  64. Ohta, T. et al. Evidence for interlayer coupling and moiré periodic potentials in twisted bilayer graphene. Phys. Rev. Lett. 109, 186807 (2012).

    Article  ADS  Google Scholar 

  65. Luican, A. et al. Single-layer behavior and its breakdown in twisted graphene layers. Phys. Rev. Lett. 106, 126802 (2011).

    Article  ADS  Google Scholar 

  66. Li, G. et al. Observation of Van Hove singularities in twisted graphene layers. Nat. Phys. 6, 109–113 (2010).

    Article  Google Scholar 

  67. Yan, W. et al. Angle-dependent Van Hove singularities in a slightly twisted graphene bilayer. Phys. Rev. Lett. 109, 126801 (2012).

    Article  ADS  Google Scholar 

  68. Angeli, M. et al. Emergent D6 symmetry in fully relaxed magic-angle twisted bilayer graphene. Phys. Rev. B 98, 235137 (2018).

    Article  ADS  Google Scholar 

  69. 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. https://doi.org/10.1038/s41563-020-00911-2(2021).

  70. Das, I. et al. Symmetry broken Chern insulators and magic series of Rashba-like Landau level crossings in magic angle bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-021-01186-3 (2021).

  71. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  75. Wu, F. & Das Sarma, S. Collective excitations of quantum anomalous Hall ferromagnets in twisted bilayer graphene. Phys. Rev. Lett. 124, 046403 (2020).

    Article  ADS  Google Scholar 

  76. Zhang, Y., Jiang, K., Wang, Z. & Zhang, F. Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: a Hartree–Fock study. Phys. Rev. B 102, 035136 (2020).

    Article  ADS  Google Scholar 

  77. Hejazi, K., Chen, X. & Balents, L. Hybrid Wannier Chern bands in magic angle twisted bilayer graphene and the quantized anomalous Hall effect. Preprint at https://arxiv.org/abs/2007.00134 (2020).

  78. Chen, S. et al. Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-020-01062-6 (2020).

  79. Shi, Y. et al. Tunable van Hove singularities and correlated states in twisted trilayer graphene. Preprint at https://arxiv.org/abs/2004.12414 (2020).

  80. Koshino, M. et al. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

    Google Scholar 

  81. Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moiré superlattice bands in layered two-dimensional materials. Phys. Rev. B 89, 205414 (2014).

    Article  ADS  Google Scholar 

  82. Moon, P. & Koshino, M. Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys. Rev. B 90, 155406 (2014).

    Article  ADS  Google Scholar 

  83. Hejazi, K., Liu, C., Shapourian, H., Chen, X. & Balents, L. Multiple topological transitions in twisted bilayer graphene near the first magic angle. Phys. Rev. B 99, 035111 (2019).

    Article  ADS  Google Scholar 

  84. Nam, N. N. T. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayer graphene. Phys. Rev. B 96, 075311 (2017).

    Article  ADS  Google Scholar 

  85. Fang, S. & Kaxiras, E. Electronic structure theory of weakly interacting bilayers. Phys. Rev. B 93, 235153 (2016).

    Article  ADS  Google Scholar 

  86. Lucignano, P., Alfè, D., Cataudella, V., Ninno, D. & Cantele, G. Crucial role of atomic corrugation on the flat bands and energy gaps of twisted bilayer graphene at the magic angle θ ~ 1.08°. Phys. Rev. B 99, 195419 (2019).

    Article  ADS  Google Scholar 

  87. Yu, R., Qi, X. L., Bernevig, A., Fang, Z. & Dai, X. Equivalent expression of \({{\mathbb{z}}}_{2}\) topological invariant for band insulators using the non-Abelian Berry connection. Phys. Rev. B 84, 075119 (2011).

    Article  ADS  Google Scholar 

  88. Soluyanov, A. A. & Vanderbilt, D. Smooth gauge for topological insulators. Phys. Rev. B 85, 115415 (2012).

    Article  ADS  Google Scholar 

  89. Taherinejad, M., Garrity, K. F. & Vanderbilt, D. Wannier center sheets in topological insulators. Phys. Rev. B 89, 115102 (2014).

    Article  ADS  Google Scholar 

  90. 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 

  91. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    Article  ADS  Google Scholar 

  92. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    Article  ADS  Google Scholar 

  93. Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  94. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

    Article  ADS  Google Scholar 

  95. Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and Wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).

    Article  ADS  Google Scholar 

  96. Song, Z.-D., Elcoro, L., Xu, Y.-F., Regnault, N. & Bernevig, B. A. Fragile phases as affine monoids: classification and material examples. Phys. Rev. X 10, 031001 (2020).

    Google Scholar 

  97. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    Article  ADS  Google Scholar 

  98. Uchida, K., Furuya, S., Iwata, J.-I. & Oshiyama, A. Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes. Phys. Rev. B 90, 155451 (2014).

    Article  ADS  Google Scholar 

  99. Rademaker, L. & Mellado, P. Charge-transfer insulation in twisted bilayer graphene. Phys. Rev. B 98, 235158 (2018).

    Article  ADS  Google Scholar 

  100. Haldane, F. D. M. & Rezayi, E. H. Periodic Laughlin–Jastrow wave functions for the fractional quantized Hall effect. Phys. Rev. B 31, 2529–2531 (1985).

    Article  ADS  Google Scholar 

  101. 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 

  102. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    Article  ADS  Google Scholar 

  103. Lian, B., Xie, F. & Bernevig, B. A. Landau level of fragile topology. Phys. Rev. B 102, 041402 (2020).

    Article  ADS  Google Scholar 

  104. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

    Article  ADS  Google Scholar 

  105. Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 123, 237002 (2019).

    Article  ADS  Google Scholar 

  106. Julku, A., Peltonen, T. J., Liang, L., Heikkilä, T. T. & Törmä, P. Superfluid weight and Berezinskii–Kosterlitz–Thouless transition temperature of twisted bilayer graphene. Phys. Rev. B 101, 060505 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  109. Zhu, J., Su, J.-J. & MacDonald, A. H. Voltage-controlled magnetic reversal in orbital Chern insulators. Phys. Rev. Lett. 125, 227702 (2020).

    Article  ADS  Google Scholar 

  110. Tschirhart, C. et al. Imaging orbital ferromagnetism in a moiré Chern insulator. Preprint at https://arxiv.org/abs/2006.08053 (2020).

  111. Li, S.-Y. et al. Experimental evidence for orbital magnetic moments generated by moiré-scale current loops in twisted bilayer graphene. Phys. Rev. B 102, 121406 (2020).

    Article  ADS  Google Scholar 

  112. Ceresoli, D., Thonhauser, T., Vanderbilt, D. & Resta, R. Orbital magnetization in crystalline solids: multi-band insulators, Chern insulators, and metals. Phys. Rev. B 74, 024408 (2006).

    Article  ADS  Google Scholar 

  113. Macdonald, A. H. Introduction to the physics of the quantum Hall regime. Preprint at https://arxiv.org/abs/cond-mat/9410047 (1994).

  114. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. & Ong, N. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539 (2010).

    Article  ADS  Google Scholar 

  115. Liu, J. & Dai, X. Anomalous Hall effect, magneto-optical properties, and nonlinear optical properties of twisted graphene systems. npj Comput. Mater. 6, 57 (2020).

    Article  ADS  Google Scholar 

  116. He, W.-Y., Goldhaber-Gordon, D. & Law, K. T. Giant orbital magnetoelectric effect and current-induced magnetization switching in twisted bilayer graphene. Nat. Commun. 11, 1650 (2020).

    Article  ADS  Google Scholar 

  117. Su, Y. & Lin, S.-Z. Current-induced reversal of anomalous Hall conductance in twisted bilayer graphene. Phys. Rev. Lett. 125, 226401 (2020).

    Article  ADS  Google Scholar 

  118. Huang, C., Wei, N. & MacDonald, A. Current driven magnetization reversal in orbital Chern insulators. Phys. Rev. Lett. 126, 056801 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  119. Ying, X., Ye, M. & Balents, L. Current switching of valley polarization in twisted bilayer graphene. Preprint at https://arxiv.org/abs/2101.01790 (2021).

  120. Kraut, W. & von Baltz, R. Anomalous bulk photovoltaic effect in ferroelectrics: a quadratic response theory. Phys. Rev. B 19, 1548–1554 (1979).

    Article  ADS  Google Scholar 

  121. Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors. Phys. Rev. B 61, 5337–5352 (2000).

    Article  ADS  Google Scholar 

  122. Gao, Y., Zhang, Y. & Xiao, D. Tunable layer circular photogalvanic effect in twisted bilayers. Phys. Rev. Lett. 124, 077401 (2020).

    Article  ADS  Google Scholar 

  123. Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

    Google Scholar 

  124. Yuan, N. F. Q. & Fu, L. Model for the metal–insulator transition in graphene superlattices and beyond. Phys. Rev. B 98, 045103 (2018).

    Article  ADS  Google Scholar 

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

    Google Scholar 

  126. Xu, X. Y., Law, K. T. & Lee, P. A. Kekulé valence bond order in an extended Hubbard model on the honeycomb lattice with possible applications to twisted bilayer graphene. Phys. Rev. B 98, 121406 (2018).

    Article  ADS  Google Scholar 

  127. Huang, T., Zhang, L. & Ma, T. Antiferromagnetically ordered Mott insulator and d + id superconductivity in twisted bilayer graphene: a quantum Monte Carlo study. Sci. Bull. 64, 310–314 (2019).

    Article  Google Scholar 

  128. 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).

    Article  ADS  Google Scholar 

  129. Venderbos, J. W. F. & Fernandes, R. M. Correlations and electronic order in a two-orbital honeycomb lattice model for twisted bilayer graphene. Phys. Rev. B 98, 245103 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  131. 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 

  132. Jian, C.-M. & Xu, C. Moire Mott insulators viewed as the surface of three dimensional symmetry protected topological phases. Preprint at https://arxiv.org/abs/1810.03610 (2018).

  133. Liu, S., Khalaf, E., Lee, J. Y. & Vishwanath, A. Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality. Phys. Rev. Res. 3, 013033 (2021).

    Article  Google Scholar 

  134. Chatterjee, S., Bultinck, N. & Zaletel, M. P. Symmetry breaking and skyrmionic transport in twisted bilayer graphene. Phys. Rev. B 101, 165141 (2020).

    Article  ADS  Google Scholar 

  135. Alavirad, Y. & Sau, J. Ferromagnetism and its stability from the one-magnon spectrum in twisted bilayer graphene. Phys. Rev. B 102, 235123 (2020).

    Article  ADS  Google Scholar 

  136. Repellin, C., Dong, Z., Zhang, Y.-H. & Senthil, T. Ferromagnetism in narrow bands of moiré superlattices. Phys. Rev. Lett. 124, 187601 (2020).

    Article  ADS  Google Scholar 

  137. Angeli, M., Tosatti, E. & Fabrizio, M. Valley Jahn–Teller effect in twisted bilayer graphene. Phys. Rev. X 9, 041010 (2019).

    Google Scholar 

  138. Lu, C. et al. Chiral so (4) spin-charge density wave and degenerate topological superconductivity in magic-angle-twisted bilayer-graphene. Preprint at https://arxiv.org/abs/2003.09513 (2020).

  139. Da Liao, Y. et al. Correlation-induced insulating topological phases at charge neutrality in twisted bilayer graphene. Phys. Rev. X 11, 011014 (2021).

    Google Scholar 

  140. 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 

  141. Seo, K., Kotov, V. N. & Uchoa, B. Ferromagnetic Mott state in twisted graphene bilayers at the magic angle. Phys. Rev. Lett. 122, 246402 (2019).

    Article  ADS  Google Scholar 

  142. Yuan, N. F., Isobe, H. & Fu, L. Magic of high-order Van Hove singularity. Nat. Commun. 10, 5769 (2019).

    Article  ADS  Google Scholar 

  143. Chichinadze, D. V., Classen, L. & Chubukov, A. V. Valley magnetism, nematicity, and density wave orders in twisted bilayer graphene. Phys. Rev. B 102, 125120 (2020).

    Article  ADS  Google Scholar 

  144. Soejima, T., Parker, D. E., Bultinck, N., Hauschild, J. & Zaletel, M. P. Efficient simulation of moiré materials using the density matrix renormalization group. Phys. Rev. B 102, 205111 (2020).

    Article  ADS  Google Scholar 

  145. Xie, F. et al. TBG VI: an exact diagonalization study of twisted bilayer graphene at non-zero integer fillings. Preprint at https://arxiv.org/abs/2010.00588 (2020).

  146. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-020-01129-4 (2021).

  147. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour hund’s coupling, correlated Chern gaps, and diffusivity in moiré flat bands. Preprint at https://arxiv.org/abs/2008.12296 (2020).

  148. Chen, B.-B. et al. Realization of topological Mott insulator in a twisted bilayer graphene lattice model. Preprint at https://arxiv.org/abs/2011.07602 (2020).

  149. Lian, B. et al. TBG IV: exact insulator ground states and phase diagram of twisted bilayer graphene. Preprint at https://arxiv.org/abs/2009.13530 (2020).

  150. Bernevig, B. A., Song, Z., Regnault, N. & Lian, B. TBG III: interacting Hamiltonian and exact symmetries of twisted bilayer graphene. Preprint at https://arxiv.org/abs/2009.12376 (2020).

  151. Bernevig, B. A. et al. TBG V: exact analytic many-body excitations in twisted bilayer graphene Coulomb Hamiltonians: charge gap, goldstone modes and absence of Cooper pairing. Preprint at https://arxiv.org/abs/2009.14200 (2020).

  152. Parker, D. E., Soejima, T., Hauschild, J., Zaletel, M. P. & Bultinck, N. Strain-induced quantum phase transitions in magic angle graphene. Preprint at https://arxiv.org/abs/2012.09885 (2020).

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

    Article  ADS  Google Scholar 

  154. 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 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  158. Wu, F. Topological chiral superconductivity with spontaneous vortices and supercurrent in twisted bilayer graphene. Phys. Rev. B 99, 195114 (2019).

    Article  ADS  Google Scholar 

  159. Hsu, Y.-T., Wu, F. & Das Sarma, S. Topological superconductivity, ferromagnetism, and valley-polarized phases in moiré systems: renormalization group analysis for twisted double bilayer graphene. Phys. Rev. B 102, 085103 (2020).

    Article  ADS  Google Scholar 

  160. Khalaf, E., Chatterjee, S., Bultinck, N., Zaletel, M. P. & Vishwanath, A. Charged skyrmions and topological origin of superconductivity in magic angle graphene. Preprint at https://arxiv.org/abs/2004.00638 (2020).

  161. Herzog-Arbeitman, J., Song, Z.-D., Regnault, N. & Bernevig, B. A. Hofstadter topology: non-crystalline topological materials in the moiré era. Phys. Rev. Lett. 125, 236804 (2020).

    Article  ADS  Google Scholar 

  162. Wu, Q., Liu, J., Guan, Y. & Yazyev, O. V. Landau levels as a probe for band topology in graphene moiré superlattices. Phys. Rev. Lett. 126, 056401 (2021).

    Article  ADS  Google Scholar 

  163. Khalaf, E., Bultinck, N., Vishwanath, A. & Zaletel, M. P. Soft modes in magic angle twisted bilayer graphene. Preprint at https://arxiv.org/abs/2009.14827 (2020).

  164. Vafek, O. & Kang, J. Renormalization group study of hidden symmetry in twisted bilayer graphene with Coulomb interactions. Phys. Rev. Lett. 125, 257602 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  165. 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).

  166. Hao, Z. et al. Electric field tunable superconductivity in alternating twist magic-angle trilayer graphene. Science https://doi.org/10.1126/science.abg0399 (2021).

  167. Li, X., Wu, F. & MacDonald, A. H. Electronic structure of single-twist trilayer graphene. Preprint at https://arxiv.org/abs/1907.12338 (2019).

  168. Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).

    Article  ADS  Google Scholar 

  169. Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).

    Article  ADS  Google Scholar 

  170. Jin, C. et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).

    Article  ADS  Google Scholar 

  171. Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature 567, 81–86 (2019).

    Article  ADS  Google Scholar 

  172. Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    Article  Google Scholar 

  173. 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 

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

    Article  ADS  Google Scholar 

  175. An, L. et al. Interaction effects and superconductivity signatures in twisted double-bilayer WSe2. Nanoscale Horiz. 5, 1309–1316 (2020).

    Article  ADS  Google Scholar 

  176. Li, H. et al. Imaging moiré flat bands in 3D reconstructed WSe2/WS2 superlattices. Preprint at https://arxiv.org/abs/2007.06113 (2020).

  177. Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Preprint at https://arxiv.org/abs/2007.12068 (2020).

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

    Article  ADS  Google Scholar 

  179. Rickhaus, P. et al. Density-wave states in twisted double-bilayer graphene. Preprint at https://arxiv.org/abs/2005.05373 (2020).

  180. Huang, M. et al. Giant nonlinear Hall effect in twisted WSe2. Preprint at https://arxiv.org/abs/2006.05615 (2020).

  181. Hu, J.-X., Zhang, C.-P., Xie, Y.-M. & Law, K. T. Nonlinear Hall effects in strained twisted bilayer WSe2. Preprint at https://arxiv.org/abs/2004.14140 (2020).

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Acknowledgements

J.L. acknowledges a start-up grant from ShanghaiTech University and the National Key R&D programme of China (Grant No. 2020YFA0309601). X.D. acknowledges financial support from the Hong Kong Research Grants Council (Project No. GRF16300918 and No. 16309020).

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Authors and Affiliations

  1. School of Physical Science and Technology, ShanghaiTech University, Shanghai, China

    Jianpeng Liu

  2. ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai, China

    Jianpeng Liu

  3. Department of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

    Xi Dai

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Glossary

Berry phases

The Berry phase of a Bloch eigenstate is a gauge-invariant phase angle accumulated after an adiabatic and cyclic evolution of the Bloch state in a vector parameter space, which can be the Brillouin zone in a crystalline solid.

Chern numbers

In topological band theory, the Chern number C of an energy band in a 2D crystalline solid is defined as the integration of the Berry curvature over the Brillouin zone.

Density matrix renormalization group

A numerical variational technique devised to solve for the Hamiltonians of low-dimensional quantum many-body systems based on efficient truncations of the many-body Hilbert space.

Filling

The filling factor p is defined as the number of electrons or holes per moiré supercell divided by the fourfold spin-valley degeneracy. p is positive/negative for electron/hole filling with respect to charge neutrality.

Hubbard model

A simple model of interacting particles in a lattice, with only two terms in the Hamiltonian: a kinetic term allowing for hopping of particles between sites of the lattice, and a potential term consisting of an on-site interaction.

Landau levels

(LL). The quantized energy levels of a 2D electron gas subject to strong perpendicular magnetic fields.

Superconducting domes

A dome-shaped superconducting region that appears in the phase diagrams of many unconventional superconductors such as cuprate superconductors and iron-based superconductors.

Superfluid weight

Non-zero superfluid weight (Ds) is a defining property of superconductors and leads to the Meissner effect and dissipationless transport. It can be more rigorously defined as the change in the free energy density \({\mathcal{F}}\) due to the motion of Cooper pairs with uniform momentum ps: \({\mathcal{F}}={D}_{{\rm{s}}}{p}_{{\rm{s}}}^{2}\,/\,8\).

Van Hove singularities

A singularity (non-smooth point) in the density of states of a crystalline solid.

Weyl nodes

In band theory, a twofold band degeneracy point at an arbitrary point in the Brillouin zone of a 3D crystalline solid.

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Liu, J., Dai, X. Orbital magnetic states in moiré graphene systems. Nat Rev Phys 3, 367–382 (2021). https://doi.org/10.1038/s42254-021-00297-3

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