Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Orbital magnetic states in moiré graphene systems

## 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.

This is a preview of subscription content, access via your institution

## Relevant articles

• ### Alternating twisted multilayer graphene: generic partition rules, double flat bands, and orbital magnetoelectric effect

npj Computational Materials Open Access 13 May 2022

## Access options

\$32.00

All prices are NET prices.

## 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).

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).

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).

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

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

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

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

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

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

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).

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

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

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).

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

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

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

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

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

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).

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).

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).

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

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

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

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

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).

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).

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).

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

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

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

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

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).

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).

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).

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

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).

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

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).

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).

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

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

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

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).

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

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).

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

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

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

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).

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

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

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).

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

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).

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).

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

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

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

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

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

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

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

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).

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

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

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

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

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).

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).

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).

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

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).

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

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

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).

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).

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

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

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).

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

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

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).

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

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

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).

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).

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).

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

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).

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).

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).

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

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

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).

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).

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

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

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

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).

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).

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).

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).

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).

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

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

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).

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

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

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).

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

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

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).

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).

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).

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).

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

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

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).

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

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

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

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

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).

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).

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).

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

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).

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).

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).

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).

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

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

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

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

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).

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).

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).

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).

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).

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

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

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

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

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

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

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

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).

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).

## 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).

## Author information

Authors

### Contributions

The authors contributed equally to all aspects of the article.

### Corresponding authors

Correspondence to Jianpeng Liu or Xi Dai.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Peer review information

Nature Reviews Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

### Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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.

## Rights and permissions

Reprints and Permissions

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

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s42254-021-00297-3

• ### Alternating twisted multilayer graphene: generic partition rules, double flat bands, and orbital magnetoelectric effect

• Bo Xie
• Ran Peng
• Jianpeng Liu

npj Computational Materials (2022)

• ### Reproducibility in the fabrication and physics of moiré materials

• Chun Ning Lau
• Marc W. Bockrath
• Fan Zhang

Nature (2022)