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Chern-textured exciton insulators with valley spiral order in moiré materials

Ziwei Wang1,*, Yves H. Kwan2,*, Glenn Wagner3, Steven H. Simon1, Nick Bultinck4, and S. A. Parameswaran1

  • 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
  • 2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
  • 3Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
  • 4Department of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

  • *These authors contributed equally to this work.
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Phys. Rev. B 112, 035130 – Published 11 July, 2025

DOI: https://doi.org/10.1103/chsd-123t

Abstract

We explore the phase diagrams of moiré materials in search of a class of intervalley-coherent correlated insulating state: the Chern texture insulator (CTI). This phase of matter, proposed in a companion paper [Kwan et al., Phys. Rev. B 112, 035129 (2025).], breaks valley U(1) symmetry in a nontrivial fashion wherein the valley order parameter is forced to texture in momentum space as a consequence of band topology. Using detailed Hartree-Fock studies, we establish that the CTI emerges as an energetically competitive intermediate-coupling ground state in several moiré systems that lack a twofold rotation symmetry that forbids the single-particle topology essential to the formation of the CTI valley texture.

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Textured exciton insulators

Yves H. Kwan, Ziwei Wang, Glenn Wagner, Steven H. Simon, S. A. Parameswaran, and Nick Bultinck
Phys. Rev. B 112, 035129 (2025)

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References (109)

  1. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  2. M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019).
  3. X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019).
  4. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature (London) 590, 249 (2021).
  5. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras et al., Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  6. A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science 365, 605 (2019).
  7. M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020).
  8. H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu et al., Observation of fractionally quantized anomalous Hall effect, Nature (London) 622, 74 (2023).
  9. J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe et al., Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature (London) 622, 63 (2023).
  10. Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional Chern insulator in moiré MoTe2, Nature (London) 622, 69 (2023).
  11. F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2, Phys. Rev. X 13, 031037 (2023).
  12. Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous Hall effect in a graphene moire superlattice, Nature (London) 626, 759 (2024).
  13. Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and A. Yacoby, Fractional Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 600, 439 (2021).
  14. A. Jaoui, I. Das, G. Di Battista, J. Díez-Mérida, X. Lu, K. Watanabe, T. Taniguchi, H. Ishizuka, L. Levitov, and D. K. Efetov, Quantum critical behaviour in magic-angle twisted bilayer graphene, Nat. Phys. 18, 633 (2022).
  15. H. Polshyn, M. Yankowitz, S. Chen, Y. Zhang, K. Watanabe, T. Taniguchi, C. R. Dean, and A. F. Young, Large linear-in-temperature resistivity in twisted bilayer graphene, Nat. Phys. 15, 1011 (2019).
  16. E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta et al., Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices, Nature (London) 579, 359 (2020).
  17. J. M. Park, Y. Cao, L. Xia, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Magic-angle multilayer graphene: A robust family of moiré superconductors, arXiv:2112.10760.
  18. Y. Zhang, R. Polski, C. Lewandowski, A. Thomson, Y. Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi, J. Alicea et al., Promotion of superconductivity in magic-angle graphene multilayers, Science 377, 1538 (2022).
  19. K. F. Mak and J. Shan, Semiconductor moirématerials, Nat. Nanotechnol. 17, 686 (2022).
  20. S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47, 16419 (1993).
  21. Y. H. Kwan, Z. Wang, G. Wagner, S. H. Simon, S. A. Parameswaran, and N. Bultinck, preceding paper, Textured exciton insulators, Phys. Rev. B 112, 035129 (2025).
  22. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene, Phys. Rev. Lett. 124, 166601 (2020).
  23. A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019).
  24. J. Ahn, S. Park, and B.-J. Yang, 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).
  25. Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All magic angles in twisted bilayer graphene are topological, Phys. Rev. Lett. 123, 036401 (2019).
  26. Y. H. Kwan, G. Wagner, T. Soejima, M. P. Zaletel, S. H. Simon, S. A. Parameswaran, and N. Bultinck, Kekulé spiral order at all nonzero integer fillings in twisted bilayer graphene, Phys. Rev. X 11, 041063 (2021).
  27. K. P. Nuckolls, R. L. Lee, M. Oh, D. Wong, T. Soejima, J. P. Hong, D. Călugăru, J. Herzog-Arbeitman, B. A. Bernevig, K. Watanabe et al., Quantum textures of the many-body wavefunctions in magic-angle graphene, Nature (London) 620, 525 (2023).
  28. H. Kim, Y. Choi, É. Lantagne-Hurtubise, C. Lewandowski, A. Thomson, L. Kong, H. Zhou, E. Baum, Y. Zhang, L. Holleis et al., Imaging inter-valley coherent order in magic-angle twisted trilayer graphene, Nature (London) 623, 942 (2023).
  29. Z. Wang, Y. H. Kwan, G. Wagner, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Kekulé spirals and charge transfer cascades in twisted symmetric trilayer graphene, Phys. Rev. B 109, L201119 (2024).
  30. C. Zhang, T. Zhu, S. Kahn, S. Li, B. Yang, C. Herbig, X. Wu, H. Li, K. Watanabe, T. Taniguchi et al., Visualizing delocalized correlated electronic states in twisted double bilayer graphene, Nat. Commun. 12, 2516 (2021).
  31. C. Rubio-Verdú, S. Turkel, Y. Song, L. Klebl, R. Samajdar, M. S. Scheurer, J. W. F. Venderbos, K. Watanabe, T. Taniguchi, H. Ochoa et al., Moiré nematic phase in twisted double bilayer graphene, Nat. Phys. 18, 196 (2022).
  32. S.-y. Li, Z. Wang, Y. Xue, Y. Wang, S. Zhang, J. Liu, Z. Zhu, K. Watanabe, T. Taniguchi, H.-j. Gao et al., Imaging topological and correlated insulating states in twisted monolayer-bilayer graphene, Nat. Commun. 13, 4225 (2022).
  33. This should be contrasted against the ETI setting, where single-valley Ĉ2zT̂ symmetry is essential [21].
  34. Y.-H. Zhang, D. Mao, and T. Senthil, Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous Hall effect and a lattice model, Phys. Rev. Res. 1, 033126 (2019).
  35. Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Nearly flat Chern bands in moiré superlattices, Phys. Rev. B 99, 075127 (2019).
  36. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  37. G. W. Burg, J. Zhu, T. Taniguchi, K. Watanabe, A. H. MacDonald, and E. Tutuc, Correlated insulating states in twisted double bilayer graphene, Phys. Rev. Lett. 123, 197702 (2019).
  38. Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene, Nature (London) 583, 215 (2020).
  39. R. Du, J. Xiao, D. Zhang, X. Cai, S. Jiang, F. Lian, K. Watanabe, T. Taniguchi, L. Wang, and G. Yu, Ferroelectricity in twisted double bilayer graphene, 2D Mater. 11, 025015 (2024).
  40. M. He, Y. Li, J. Cai, Y. Liu, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Symmetry breaking in twisted double bilayer graphene, Nat. Phys. 17, 26 (2021).
  41. L. Liu, S. Zhang, Y. Chu, C. Shen, Y. Huang, Y. Yuan, J. Tian, J. Tang, Y. Ji, R. Yang et al., Isospin competitions and valley polarized correlated insulators in twisted double bilayer graphene, Nat. Commun. 13, 3292 (2022).
  42. X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo, D. Haei Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Tunable spin-polarized correlated states in twisted double bilayer graphene, Nature (London) 583, 221 (2020).
  43. P. Rickhaus, F. K. de Vries, J. Zhu, E. Portoles, G. Zheng, M. Masseroni, A. Kurzmann, T. Taniguchi, K. Watanabe, A. H. MacDonald et al., Correlated electron-hole state in twisted double-bilayer graphene, Science 373, 1257 (2021).
  44. C. Shen, Y. Chu, Q. Wu, N. Li, S. Wang, Y. Zhao, J. Tang, J. Liu, J. Tian, K. Watanabe et al., Correlated states in twisted double bilayer graphene, Nat. Phys. 16, 520 (2020).
  45. Y. Zhu, Y. Chen, Q. Li, Y. Chen, Y. Huang, W. Zhu, D. An, J. Song, Q. Gan, K. Wang et al., Tunable multi-bands in twisted double bilayer graphene, 2D Mater. 9, 034001 (2022).
  46. M. He, J. Cai, Y.-H. Zhang, Y. Liu, Y. Li, T. Taniguchi, K. Watanabe, D. H. Cobden, M. Yankowitz, and X. Xu, Symmetry-broken Chern insulators in twisted double bilayer graphene, Nano Lett. 23, 11066 (2023).
  47. M. Koshino, Band structure and topological properties of twisted double bilayer graphene, Phys. Rev. B 99, 235406 (2019).
  48. J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum valley Hall effect, orbital magnetism, and anomalous Hall effect in twisted multilayer graphene systems, Phys. Rev. X 9, 031021 (2019).
  49. N. R. Chebrolu, B. L. Chittari, and J. Jung, Flat bands in twisted double bilayer graphene, Phys. Rev. B 99, 235417 (2019).
  50. J. Y. Lee, E. Khalaf, S. Liu, X. Liu, Z. Hao, P. Kim, and A. Vishwanath, Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene, Nat. Commun. 10, 5333 (2019).
  51. F. Haddadi, Q. Wu, A. J. Kruchkov, and O. V. Yazyev, Moiré flat bands in twisted double bilayer graphene, Nano Lett. 20, 2410 (2020).
  52. For all materials considered, we model the displacement field by a linearly varying potential that differs by ΔV between the outermost layers.
  53. J. Jung and A. H. MacDonald, Accurate tight-binding models for the π bands of bilayer graphene, Phys. Rev. B 89, 035405 (2014).
  54. N. N. T. Nam and M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Phys. Rev. B 96, 075311 (2017).
  55. S. Carr, S. Fang, Z. Zhu, and E. Kaxiras, Exact continuum model for low-energy electronic states of twisted bilayer graphene, Phys. Rev. Res. 1, 013001 (2019).
  56. S. Chen, M. He, Y.-H. Zhang, V. Hsieh, Z. Fei, K. Watanabe, T. Taniguchi, D. H. Cobden, X. Xu, C. R. Dean, and M. Yankowitz, Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene, Nat. Phys. 17, 374 (2021).
  57. M. He, Y.-H. Zhang, Y. Li, Z. Fei, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Competing correlated states and abundant orbital magnetism in twisted monolayer-bilayer graphene, Nat. Commun. 12, 4727 (2021).
  58. H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L. Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. H. MacDonald, and A. F. Young, Electrical switching of magnetic order in an orbital Chern insulator, Nature (London) 588, 66 (2020).
  59. H. Polshyn, Y. Zhang, M. A. Kumar, T. Soejima, P. Ledwith, K. Watanabe, T. Taniguchi, A. Vishwanath, M. P. Zaletel, and A. F. Young, Topological charge density waves at half-integer filling of a moiré superlattice, Nat. Phys. 18, 42 (2022).
  60. S. Xu, M. M. Al Ezzi, N. Balakrishnan, A. Garcia-Ruiz, B. Tsim, C. Mullan, J. Barrier, N. Xin, B. A. Piot, T. Taniguchi et al., Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene, Nat. Phys. 17, 619 (2021).
  61. C. Zhang, T. Zhu, T. Soejima, S. Kahn, K. Watanabe, T. Taniguchi, A. Zettl, F. Wang, M. P. Zaletel, and M. F. Crommie, Local spectroscopy of a gate-switchable moiré quantum anomalous Hall insulator, Nat. Commun. 14, 3595 (2023).
  62. P. J. Ledwith, A. Vishwanath, and E. Khalaf, Family of ideal Chern flatbands with arbitrary Chern number in chiral twisted graphene multilayers, Phys. Rev. Lett. 128, 176404 (2022).
  63. R. Su, D. Waters, B. Zhou, K. Watanabe, T. Taniguchi, Y.-H. Zhang, M. Yankowitz, and J. Folk, Moiré-driven topological electronic crystals in twisted graphene, Nature (London) 637, 1084 (2025).
  64. L.-Q. Xia, S. C. de la Barrera, A. Uri, A. Sharpe, Y. H. Kwan, Z. Zhu, K. Watanabe, T. Taniguchi, D. Goldhaber-Gordon, L. Fu, T. Devakul, and P. Jarillo-Herrero, Topological bands and correlated states in helical trilayer graphene, Nat. Phys. 21, 239 (2025).
  65. T. Devakul, P. J. Ledwith, L.-Q. Xia, A. Uri, S. C. de la Barrera, P. Jarillo-Herrero, and L. Fu, Magic-angle helical trilayer graphene, Sci. Adv. 9, eadi6063 (2023).
  66. N. Nakatsuji, T. Kawakami, and M. Koshino, Multiscale lattice relaxation in general twisted trilayer graphenes, Phys. Rev. X 13, 041007 (2023).
  67. Y. H. Kwan, P. J. Ledwith, C. F. B. Lo, and T. Devakul, Strong-coupling topological states and phase transitions in helical trilayer graphene, Phys. Rev. B 109, 125141 (2024).
  68. The interlayer tunneling here does not include effects of the momentum dependence of the tunneling [64]. See Ref.  [69] for a discussion of this correction.
  69. See Supplemental Material at http://link.aps.org/supplemental/10.1103/chsd-123t for numerical results with the “graphene” subtraction scheme, gauge fixing, and other additional results.
  70. J. Yu, J. Herzog-Arbeitman, M. Wang, O. Vafek, B. A. Bernevig, and N. Regnault, Fractional Chern insulators versus nonmagnetic states in twisted bilayer MoTe2, Phys. Rev. B 109, 045147 (2024).
  71. Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe2, Phys. Rev. B 109, 205121 (2024).
  72. F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Phys. Rev. Lett. 122, 086402 (2019).
  73. C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Fractional Chern insulator in twisted bilayer MoTe2, Phys. Rev. Lett. 132, 036501 (2024).
  74. A. P. Reddy, F. Alsallom, Y. Zhang, T. Devakul, and L. Fu, Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2, Phys. Rev. B 108, 085117 (2023).
  75. Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant superconductivity in moiré graphene, Nature (London) 595, 526 (2021).
  76. Z. Hao, A. M. Zimmerman, P. Ledwith, E. Khalaf, D. H. Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene, Science 371, 1133 (2021).
  77. H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Evidence for unconventional superconductivity in twisted trilayer graphene, Nature (London) 606, 494 (2022).
  78. Y. Li, S. Zhang, F. Chen, L. Wei, Z. Zhang, H. Xiao, H. Gao, M. Chen, S. Liang, D. Pei, L. Xu et al., Observation of coexisting Dirac bands and moiré flat bands in magic-angle twisted trilayer graphene, Adv. Mater. 34, 2205996 (2022).
  79. J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. I. A. Li, Zero-field superconducting diode effect in small-twist-angle trilayer graphene, Nat. Phys. 18, 1221 (2022).
  80. X. Liu, N. J. Zhang, K. Watanabe, T. Taniguchi, and J. I. A. Li, Isospin order in superconducting magic-angle twisted trilayer graphene, Nat. Phys. 18, 522 (2022).
  81. C. Shen, P. J. Ledwith, K. Watanabe, T. Taniguchi, E. Khalaf, A. Vishwanath, and D. K. Efetov, Dirac spectroscopy of strongly correlated phases in twisted trilayer graphene, Nat. Mater. 22, 316 (2023).
  82. S. Turkel, J. Swann, Z. Zhu, M. Christos, K. Watanabe, T. Taniguchi, S. Sachdev, M. S. Scheurer, E. Kaxiras, C. R. Dean, and A. N. Pasupathy, Orderly disorder in magic-angle twisted trilayer graphene, Science 376, 193 (2022).
  83. S.-J. Yang, J.-H. Jung, E. Lee, E. Han, M.-Y. Choi, D. Jung, S. Choi, J.-H. Park, D. Oh, S. Noh et al., Wafer-scale programmed assembly of one-atom-thick crystals, Nano Lett. 22, 1518 (2022).
  84. N. J. Zhang, J.-X. Lin, D. V. Chichinadze, Y. Wang, K. Watanabe, T. Taniguchi, L. Fu, and J. I. A. Li, Angle-resolved transport non-reciprocity and spontaneous symmetry breaking in twisted trilayer graphene, Nat. Mater. 23, 356 (2024).
  85. X. Li, F. Wu, and A. H. MacDonald, Electronic structure of single-twist trilayer graphene, arXiv:1907.12338.
  86. E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vishwanath, Magic angle hierarchy in twisted graphene multilayers, Phys. Rev. B 100, 085109 (2019).
  87. D. Calugaru, F. Xie, Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field, Phys. Rev. B 103, 195411 (2021).
  88. M. Xie and A. H. MacDonald, Nature of the correlated insulator states in twisted bilayer graphene, Phys. Rev. Lett. 124, 097601 (2020).
  89. J. Kang and O. Vafek, Strong coupling phases of partially filled twisted bilayer graphene narrow bands, Phys. Rev. Lett. 122, 246401 (2019).
  90. N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath, and M. P. Zaletel, Ground state and hidden symmetry of magic-angle graphene at even integer filling, Phys. Rev. X 10, 031034 (2020).
  91. B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yazdani, and B. A. Bernevig, Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram, Phys. Rev. B 103, 205414 (2021).
  92. T. Wang, D. E. Parker, T. Soejima, J. Hauschild, S. Anand, N. Bultinck, and M. P. Zaletel, 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).
  93. Reference [69] details how we implement a smooth but nonperiodic gauge fixing.
  94. We briefly discuss the situation where multiple HF bands are proximate in energy in Sec.  3g.
  95. The hole order parameter would differ by a minus sign.
  96. Within spin-collinear HF calculations, states with different IVC angles in the two spin sectors are degenerate with SU(2)s-symmetric IVC states.
  97. C. Fang, M. J. Gilbert, and B. A. Bernevig, Bulk topological invariants in noninteracting point group symmetric insulators, Phys. Rev. B 86, 115112 (2012).
  98. Y. H. Kwan, Z. Wang, G. Wagner, S. H. Simon, S. A. Parameswaran, and N. Bultinck (unpublished).
  99. D. Guerci, Y. Mao, and C. Mora, Chern mosaic and ideal flat bands in equal-twist trilayer graphene, Phys. Rev. Res. 6, L022025 (2024).
  100. E. Anderson, F.-R. Fan, J. Cai, W. Holtzmann, T. Taniguchi, K. Watanabe, D. Xiao, W. Yao, and X. Xu, Programming correlated magnetic states with gate-controlled moiré geometry, Science 381, 325 (2023).
  101. K. Kang, B. Shen, Y. Qiu, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Observation of the fractional quantum spin Hall effect in moiré MoTe2, Nature (London) 628, 522 (2024).
  102. Our definition of T̂ for this subsection interchanges (K↑)↔(K′↓) and satisfies T̂2=1.
  103. T. Wang, T. Devakul, M. P. Zaletel, and L. Fu, Topological magnets and magnons in twisted bilayer MoTe2 and WSe2, arXiv:2306.02501.
  104. X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted MoTe2 and WSe2, Nat. Commun. 15, 4223 (2024).
  105. T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. Devakul, K. Watanabe, T. Taniguchi, L. Fu et al., Quantum anomalous Hall effect from intertwined moiré bands, Nature (London) 600, 641 (2021).
  106. G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Global phase diagram of the normal state of twisted bilayer graphene, Phys. Rev. Lett. 128, 156401 (2022).
  107. D. Calugaru, N. Regnault, M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, A. Yazdani, O. Vafek, and B. A. Bernevig, Spectroscopy of twisted bilayer graphene correlated insulators, Phys. Rev. Lett. 129, 117602 (2022).
  108. J. P. Hong, T. Soejima, and M. P. Zaletel, Detecting symmetry breaking in magic angle graphene using scanning tunneling microscopy, Phys. Rev. Lett. 129, 147001 (2022).
  109. If the underlying Chern bands instead arise from opposite spin projections, then spin-resolved ARPES would be able to detect the spin polarization pattern.

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Phys. Rev. B 112, 035130– Published 11 July, 2025
  • Received 31 October 2024
  • Revised 2 March 2025
  • Accepted 11 March 2025
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