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  • Letter
  • Open Access

Topological order without band topology in moiré graphene

Hui Liu1,*, Raul Perea-Causin1,†, Zhao Liu2,3,‡, and Emil J. Bergholtz1,§

  • 1Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden
  • 2Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310058, China
  • 3Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University, Hangzhou 310027, China

  • *Contact author: hui.liu@fysik.su.se
  • †Contact author: raul.perea.causin@fysik.su.se
  • ‡Contact author: zhaol@zju.edu.cn
  • §Contact author: emil.bergholtz@fysik.su.se
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Phys. Rev. Research 8, L032039 – Published 3 September, 2026

DOI: https://doi.org/10.1103/jx5w-35z9

Abstract

The discovery of zero-field fractional Chern insulators (FCIs) in moiré materials has attracted intense interest in the interplay between topology and correlations. Here, we demonstrate that fractionalized topological order can emerge under realistic conditions even within a topologically trivial moiré band. Employing exact diagonalization of long-range Coulomb interactions projected into a trivial band of twisted multilayer graphene, we identify a set of incompressible Laughlin-like ground states exhibiting fractional quantized Hall conductance. We trace the origin of this FCI phase to the strongly inhomogeneous distribution of quantum geometry, which reshapes interaction effects independently of the band topology. Extending this heuristic quantum-geometric mechanism, we demonstrate that similarly unexpected Laughlin-like FCIs can also be stabilized in higher-Chern-number moiré bands under experimentally accessible conditions. Our results establish realistic scenarios under which many-body topological order can emerge independently of single-particle band topology.

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References (99)

  1. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  2. E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nat. Rev. Mater. 6, 201 (2021).
  3. K. F. Mak and J. Shan, Semiconductor moiré materials, Nat. Nanotechnol. 17, 686 (2022).
  4. 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).
  5. H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous Hall effect, Nature (London) 622, 74 (2023).
  6. J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature (London) 622, 63 (2023).
  7. 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).
  8. 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 multilayer graphene, Nature (London) 626, 759 (2024).
  9. 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).
  10. E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Observation of fractional Chern insulators in a van der Waals heterostructure, Science 360, 62 (2018).
  11. H. Park, W. Li, C. Hu, C. Beach, M. Gonçalves, J. F. Mendez-Valderrama, J. Herzog-Arbeitman, T. Taniguchi, K. Watanabe, D. Cobden, L. Fu, B. A. Bernevig, N. Regnault, J.-H. Chu, D. Xiao, and X. Xu, Observation of dissipationless fractional Chern insulator, Nat. Phys. 22, 389 (2026).
  12. N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  13. Z. Liu and E. J. Bergholtz, Recent developments in fractional Chern insulators, in Encyclopedia of Condensed Matter Physics, edited by T. Chakraborty, 2nd ed. (Academic Press, Oxford, 2024), pp. 515–538.
  14. E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, Int. J. Mod. Phys. B 27, 1330017 (2013).
  15. S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, C. R. Phys. 14, 816 (2013).
  16. A. Kol and N. Read, Fractional quantum Hall effect in a periodic potential, Phys. Rev. B 48, 8890 (1993).
  17. E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  18. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  19. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  20. D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum Hall effect in the absence of Landau levels, Nat. Commun. 2, 389 (2011).
  21. G. Möller and N. R. Cooper, Composite fermion theory for bosonic quantum Hall states on lattices, Phys. Rev. Lett. 103, 105303 (2009).
  22. E. Kapit and E. Mueller, Exact parent Hamiltonian for the quantum Hall states in a lattice, Phys. Rev. Lett. 105, 215303 (2010).
  23. A. Abouelkomsan, Z. Liu, and E. J. Bergholtz, Particle-hole duality, emergent Fermi liquids, and fractional Chern insulators in moiré flatbands, Phys. Rev. Lett. 124, 106803 (2020).
  24. C. Repellin and T. Senthil, Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition, Phys. Rev. Res. 2, 023238 (2020).
  25. P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional Chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Res. 2, 023237 (2020).
  26. Z. Liu, A. Abouelkomsan, and E. J. Bergholtz, Gate-tunable fractional Chern insulators in twisted double bilayer graphene, Phys. Rev. Lett. 126, 026801 (2021).
  27. H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices, Phys. Rev. Res. 3, L032070 (2021).
  28. P. Wilhelm, T. C. Lang, and A. M. Läuchli, Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene, Phys. Rev. B 103, 125406 (2021).
  29. A. P. Reddy, N. Paul, A. Abouelkomsan, and L. Fu, Non-Abelian fractionalization in topological minibands, Phys. Rev. Lett. 133, 166503 (2024).
  30. C. Wang, X.-W. Zhang, X. Liu, J. Wang, T. Cao, and D. Xiao, Higher Landau-level analogs and signatures of non-Abelian states in twisted bilayer MoTe2, Phys. Rev. Lett. 134, 076503 (2025).
  31. C. Xu, N. Mao, T. Zeng, and Y. Zhang, Multiple Chern bands in twisted MoTe2 and possible non-Abelian states, Phys. Rev. Lett. 134, 066601 (2025).
  32. C.-E. Ahn, W. Lee, K. Yananose, Y. Kim, and G. Y. Cho, Non-Abelian fractional quantum anomalous Hall states and first Landau level physics of the second moiré band of twisted bilayer MoTe2, Phys. Rev. B 110, L161109 (2024).
  33. H. Liu, Z. Liu, and E. J. Bergholtz, Non-Abelian fractional Chern insulators and competing states in flat moiré bands, Phys. Rev. Lett. 135, 106604 (2025).
  34. F. Chen, W.-W. Luo, W. Zhu, and D. N. Sheng, Robust non-Abelian even-denominator fractional Chern insulator in twisted bilayer MoTe2, Nat. Commun. 16, 2115 (2025).
  35. Z. Liu, B. Mera, M. Fujimoto, T. Ozawa, and J. Wang, Theory of generalized Landau levels and its implications for non-Abelian states, Phys. Rev. X 15, 031019 (2025).
  36. S. Niu, Y. Peng, and D. N. Sheng, Quantum phase diagram and non-Abelian Moore-Read state in double twisted bilayer graphene, Phys. Rev. B 112, 035415 (2025).
  37. H. Liu, R. Perea-Causin, and E. J. Bergholtz, Parafermions in moiré minibands, Nat. Commun. 16, 1770 (2025).
  38. K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin Hall effect in moiré MoTe2, Nature (London) 628, 522 (2024).
  39. Y. Wang, J. Choe, E. Anderson, W. Li, J. Ingham, E. A. Arsenault, Y. Li, X. Hu, T. Taniguchi, K. Watanabe, X. Roy, D. Basov, D. Xiao, R. Queiroz, J. C. Hone, X. Xu, and X. Y. Zhu, Hidden states and dynamics of fractional fillings in twisted MoTe2 bilayers, Nature (London) 641, 1149 (2025).
  40. Z. Tešanović, F. Axel, and B. I. Halperin, “Hall crystal” versus Wigner crystal, Phys. Rev. B 39, 8525 (1989).
  41. X.-Y. Song, C.-M. Jian, L. Fu, and C. Xu, Intertwined fractional quantum anomalous Hall states and charge density waves, Phys. Rev. B 109, 115116 (2024).
  42. D. N. Sheng, A. P. Reddy, A. Abouelkomsan, E. J. Bergholtz, and L. Fu, Quantum anomalous Hall crystal at fractional filling of moiré superlattices, Phys. Rev. Lett. 133, 066601 (2024).
  43. Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov, N. Regnault, and B. A. Bernevig, Moiré fractional Chern insulators. III. Hartree-Fock phase diagram, magic angle regime for Chern insulator states, role of moiré potential, and Goldstone gaps in rhombohedral graphene superlattices, Phys. Rev. B 112, 075109 (2025).
  44. Z. Dong, A. S. Patri, and T. Senthil, Theory of quantum anomalous Hall phases in pentalayer rhombohedral graphene moiré structures, Phys. Rev. Lett. 133, 206502 (2024).
  45. J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous Hall crystals in rhombohedral multilayer graphene. I. Interaction-driven Chern bands and fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 133, 206503 (2024).
  46. Z. Dong, A. S. Patri, and T. Senthil, Stability of anomalous Hall crystals in multilayer rhombohedral graphene, Phys. Rev. B 110, 205130 (2024).
  47. B. Zhou, H. Yang, and Y.-H. Zhang, Fractional quantum anomalous Hall effect in rhombohedral multilayer graphene in the moiréless limit, Phys. Rev. Lett. 133, 206504 (2024).
  48. T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous Hall crystals in rhombohedral multilayer graphene. II. General mechanism and a minimal model, Phys. Rev. B 110, 205124 (2024).
  49. T. Tan and T. Devakul, Parent Berry curvature and the ideal anomalous Hall crystal, Phys. Rev. X 14, 041040 (2024).
  50. N. Paul, G. Shavit, and L. Fu, Designing Hall crystals with variable Chern numbers, Nat. Commun. 16, 8103 (2025).
  51. B. Zhou and Y.-H. Zhang, New classes of quantum anomalous Hall crystals in multilayer graphene, Phys. Rev. Lett. 135, 036501 (2025).
  52. Y. Zeng, D. Guerci, V. Crépel, A. J. Millis, and J. Cano, Sublattice structure and topology in spontaneously crystallized electronic states, Phys. Rev. Lett. 132, 236601 (2024).
  53. Z. Lu, T. Han, Y. Yao, J. Yang, J. Seo, L. Shi, S. Ye, K. Watanabe, T. Taniguchi, and L. Ju, Extended quantum anomalous Hall states in graphene/hBN moiré superlattices, Nature (London) 637, 1090 (2025).
  54. 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).
  55. 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).
  56. D. Waters, A. Okounkova, R. Su, B. Zhou, J. Yao, K. Watanabe, T. Taniguchi, X. Xu, Y.-H. Zhang, J. Folk, and M. Yankowitz, Chern insulators at integer and fractional filling in moiré pentalayer graphene, Phys. Rev. X 15, 011045 (2025).
  57. Z. Liu, E. J. Bergholtz, H. Fan, and A. M. Läuchli, Fractional Chern insulators in topological flat bands with higher Chern number, Phys. Rev. Lett. 109, 186805 (2012).
  58. A. Sterdyniak, C. Repellin, B. A. Bernevig, and N. Regnault, Series of Abelian and non-Abelian states in C>1 fractional Chern insulators, Phys. Rev. B 87, 205137 (2013).
  59. J. Behrmann, Z. Liu, and E. J. Bergholtz, Model fractional Chern insulators, Phys. Rev. Lett. 116, 216802 (2016).
  60. R. Perea-Causin, H. Liu, and E. J. Bergholtz, Quantum anomalous Hall crystals in moiré bands with higher Chern number, Nat. Commun. 16, 6875 (2025).
  61. S. Niu, J. Alicea, D. N. Sheng, and Y. Peng, Quantum anomalous Hall effects and emergent SU(2) Hall ferromagnets at fractional filling of helical trilayer graphene, Phys. Rev. Lett. 135, 146505 (2025).
  62. J. D. Choi, N. Morales-Durán, Y. H. Kwan, A. J. Millis, N. Regnault, and D. Guerci, Higher Chern bands in helical homotrilayer transition metal dichalcogenides, Phys. Rev. B 112, 205122 (2025).
  63. J. Dong, L. Liu, J. Zhu, Z. Pan, Y. Hong, F. Wang, Z. Ren, Z. Jia, K. Watanabe, T. Taniguchi, L. Du, D. Shi, W. Yang, and G. Zhang, Observation of integer and fractional Chern insulators in high Chern number flatbands, arXiv:2507.09908.
  64. W. Wang, Y. Wang, Z. Zhang, Z. Huo, G. Zhou, S. Zhang, K. Watanabe, T. Taniguchi, X. Yang, Q. Dai, X. C. Xie, K. Liu, Z. Song, and X. Lu, Programmable quantum anomalous hall insulator in twisted crystalline flatbands, Phys. Rev. X 16, 011015 (2026).
  65. N. Liu, Z. Chen, J. Ding, W. Zhou, H. Xiang, X. Fang, L. Wu, X. Zhan, L. Zhang, Q. Chen, K. Watanabe, T. Taniguchi, N. Xin, and S. Xu, Diverse high-Chern-number quantum anomalous Hall insulators in twisted rhombohedral graphene, arXiv:2507.11347.
  66. H. Liu, C. Fulga, E. J. Bergholtz, and J. Asboth, Topological fine structure of an energy band, SciPost Phys. 20, 172 (2026).
  67. B. Michen and J. C. Budich, Quantum Hall effect without Chern bands, Phys. Rev. Lett. 135, 186603 (2025).
  68. S. H. Simon, F. Harper, and N. Read, Fractional Chern insulators in bands with zero Berry curvature, Phys. Rev. B 92, 195104 (2015).
  69. Z. Lin, H. Lu, W. Yang, D. Zhai, and W. Yao, Fractional Chern insulator states in an isolated flat band of zero Chern number, Newton 2, 100339 (2025).
  70. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jx5w-35z9 for additional results for the trivial band, the realistic and ideal C = 2 band systems, and information for tilted samples.
  71. C. Repellin, B. A. Bernevig, and N. Regnault, Z2 fractional topological insulators in two dimensions, Phys. Rev. B 90, 245401 (2014).
  72. 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).
  73. J. Wang and Z. Liu, Hierarchy of ideal flatbands in chiral twisted multilayer graphene models, Phys. Rev. Lett. 128, 176403 (2022).
  74. N. R. Chebrolu, B. L. Chittari, and J. Jung, Flat bands in twisted double bilayer graphene, Phys. Rev. B 99, 235417 (2019).
  75. M. G. Scheer and B. Lian, Twistronics of Kekulé graphene: Honeycomb and Kagome flat bands, Phys. Rev. Lett. 131, 266501 (2023).
  76. B.-T. Chen, M. G. Scheer, and B. Lian, Intervalley-coupled twisted bilayer graphene from substrate commensuration, Phys. Rev. Lett. 136, 066401 (2026).
  77. E. J. Bergholtz and A. Karlhede, Quantum Hall system in Tao-Thouless limit, Phys. Rev. B 77, 155308 (2008).
  78. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  79. K. Kudo, H. Watanabe, T. Kariyado, and Y. Hatsugai, Many-body Chern number without integration, Phys. Rev. Lett. 122, 146601 (2019).
  80. A. Abouelkomsan, K. Yang, and E. J. Bergholtz, Quantum metric induced phases in moiré materials, Phys. Rev. Res. 5, L012015 (2023).
  81. H. Liu, K. Yang, A. Abouelkomsan, Z. Liu, and E. J. Bergholtz, Broken symmetry in ideal Chern bands, Phys. Rev. B 111, L201105 (2025).
  82. J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact Landau level description of geometry and interaction in a flatband, Phys. Rev. Lett. 127, 246403 (2021).
  83. G. Ji and B. Yang, Quantum metric induced hole dispersion and emergent particle-hole symmetry in topological flat bands, arXiv:2409.08324.
  84. A. M. Läuchli, Z. Liu, E. J. Bergholtz, and R. Moessner, Hierarchy of fractional Chern insulators and competing compressible states, Phys. Rev. Lett. 111, 126802 (2013).
  85. A. Abouelkomsan, A. P. Reddy, L. Fu, and E. J. Bergholtz, Band mixing in the quantum anomalous Hall regime of twisted semiconductor bilayers, Phys. Rev. B 109, L121107 (2024).
  86. C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, Maximally localized Wannier functions, interaction models, and fractional quantum anomalous Hall effect in twisted bilayer MoTe2, Proc. Natl. Acad. Sci. USA 121, e2316749121 (2024).
  87. 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).
  88. J. Jung and A. H. MacDonald, Accurate tight-binding models for the π bands of bilayer graphene, Phys. Rev. B 89, 035405 (2014).
  89. 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).
  90. G. Möller and N. R. Cooper, Fractional Chern insulators in Harper-Hofstadter bands with higher Chern number, Phys. Rev. Lett. 115, 126401 (2015).
  91. M. Song and K. Chang, Fractional Chern insulators transition in non-ideal flat bands of twisted mono-bilayer graphene, arXiv:2511.12231.
  92. I. A. McDonald and F. D. M. Haldane, Topological phase transition in the ν = 2/3 quantum Hall effect, Phys. Rev. B 53, 15845 (1996).
  93. Y. Liu and Z. Zhu, Engineering fractional Chern insulators through periodic strain in monolayer graphene and transition metal dichalcogenides, Phys. Rev. B 112, L041123 (2025).
  94. B. I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett. 52, 1583 (1984).
  95. S. Kourtis and M. Daghofer, Combined topological and Landau order from strong correlations in Chern bands, Phys. Rev. Lett. 113, 216404 (2014).
  96. H. Lu, H.-Q. Wu, B.-B. Chen, W. Yao, and Z. Y. Meng, Generic (fractional) quantum anomalous Hall crystals from interaction-driven band folding, Phys. Rev. B 113, 245105 (2026).
  97. A. P. Reddy, D. N. Sheng, A. Abouelkomsan, E. J. Bergholtz, and L. Fu, Anti-topological crystal and non-Abelian liquid in twisted semiconductor bilayers, Nat. Commun. 17, 3814 (2026).
  98. R. Sohal, L. H. Santos, and E. Fradkin, Chern-Simons composite fermion theory of fractional Chern insulators, Phys. Rev. B 97, 125131 (2018).
  99. T. Lu and L. H. Santos, Fractional Chern insulators in twisted bilayer MoTe2: A composite fermion perspective, Phys. Rev. Lett. 133, 186602 (2024).

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Phys. Rev. Research 8, L032039– Published 3 September, 2026
  • Received 17 October 2025
  • Accepted 3 June 2026
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