- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Fractional Chern mosaic in supermoiré graphene

Yves H. Kwan1, Tixuan Tan2, and Trithep Devakul2,*

  • *Contact author: tdevakul@stanford.edu
PDF

Phys. Rev. Research 7, L032070 – Published 29 September, 2025

DOI: https://doi.org/10.1103/srgm-9ybc

Abstract

We propose the realization of a fractional Chern mosaic: a state characterized by a spatially varying topological order. This arises in trilayer graphene with small sequential twist angles, where a large supermoiré lattice emerges alongside conventional moiré lattices. We show that electron correlations drive local fractionalization on the moiré scale, while its pattern varies across the supermoiré scale.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (82)

  1. 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).
  2. K. F. Mak and J. Shan, Semiconductor moiré materials, Nat. Nanotechnol. 17, 686 (2022).
  3. D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. Basov, A. N. Pasupathy, and A. Rubio, Moiré heterostructures as a condensed-matter quantum simulator, Nat. Phys. 17, 155 (2021).
  4. N. P. Wilson, W. Yao, J. Shan, and X. Xu, Excitons and emergent quantum phenomena in stacked 2D semiconductors, Nature (London) 599, 383 (2021).
  5. 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).
  6. 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).
  7. X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019).
  8. G. Chen, A. L. Sharpe, P. Gallagher, I. T. Rosen, E. J. Fox, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi et al., Signatures of tunable superconductivity in a trilayer graphene moiré superlattice, Nature (London) 572, 215 (2019).
  9. H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu et al., Superconductivity in metallic twisted bilayer graphene stabilized by WSe2, Nature (London) 583, 379 (2020).
  10. 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).
  11. Z. Hao, A. 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).
  12. Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Unconventional superconductivity in twisted bilayer WSe2, Nature 637, 833 (2025).
  13. Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. Mandrus, K. Barmak, J. Hone, A. J. Millis, A. N. Pasupathy, and C. R. Dean, Superconductivity in 5.0° twisted bilayer WSe2, Nature (London) 637, 839 (2025).
  14. 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).
  15. 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).
  16. 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 bilayer, Nature (London) 622, 63 (2023).
  17. 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).
  18. 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).
  19. 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).
  20. 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).
  21. X. Lai, D. Guerci, G. Li, K. Watanabe, T. Taniguchi, J. Wilson, J. H. Pixley, and E. Y. Andrei, Imaging self-aligned moiré crystals and quasicrystals in magic-angle bilayer graphene on hBN heterostructures, arXiv:2311.07819.
  22. Z. Wang, Y. B. Wang, J. Yin, E. Tóvári, Y. Yang, L. Lin, M. Holwill, J. Birkbeck, D. J. Perello, S. Xu et al., Composite super-moiré lattices in double-aligned graphene heterostructures, Sci. Adv. 5, aay8897 (2019).
  23. L. Wang, S. Zihlmann, M.-H. Liu, P. Makk, K. Watanabe, T. Taniguchi, A. Baumgartner, and C. Schonenberger, New generation of moiré superlattices in doubly aligned hBN/graphene/hBN heterostructures, Nano Lett. 19, 2371 (2019).
  24. N. R. Finney, M. Yankowitz, L. Muraleetharan, K. Watanabe, T. Taniguchi, C. R. Dean, and J. Hone, Tunable crystal symmetry in graphene–boron nitride heterostructures with coexisting moiré superlattices, Nat. Nanotechnol. 14, 1029 (2019).
  25. M. M. Al Ezzi, J. Hu, A. Ariando, F. Guinea, and S. Adam, Topological flat bands in graphene super-moiré lattices, Phys. Rev. Lett. 132, 126401 (2024).
  26. D. Mao and T. Senthil, Quasiperiodicity, band topology, and moiré graphene, Phys. Rev. B 103, 115110 (2021).
  27. N. Leconte and J. Jung, Commensurate and incommensurate double moire interference in graphene encapsulated by hexagonal boron nitride, 2D Mater. 7, 031005 (2020).
  28. M. Andjelkovic, S. P. Milovanovic, L. Covaci, and F. M. Peeters, Double moiré with a twist: Supermoiré in encapsulated graphene, Nano Lett. 20, 979 (2020).
  29. J. Shi, J. Zhu, and A. MacDonald, Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride, Phys. Rev. B 103, 075122 (2021).
  30. S. Grover, M. Bocarsly, A. Uri, P. Stepanov, G. Di Battista, I. Roy, J. Xiao, A. Y. Meltzer, Y. Myasoedov, K. Pareek et al., Chern mosaic and Berry-curvature magnetism in magic-angle graphene, Nat. Phys. 18, 885 (2022).
  31. A. Rothstein, C. Schattauer, R. J. Dolleman, S. Trellenkamp, F. Lentz, K. Watanabe, T. Taniguchi, D. M. Kennes, B. Beschoten, C. Stampfer, and F. Libisch, Band gap formation in commensurate twisted bilayer graphene/hBN moiré lattices, Phys. Rev. B 109, 155139 (2024).
  32. C. Mora, N. Regnault, and B. A. Bernevig, Flatbands and perfect metal in trilayer moiré graphene, Phys. Rev. Lett. 123, 026402 (2019).
  33. Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxiras, Twisted trilayer graphene: A precisely tunable platform for correlated electrons, Phys. Rev. Lett. 125, 116404 (2020).
  34. Z. Zhu, P. Cazeaux, M. Luskin, and E. Kaxiras, Modeling mechanical relaxation in incommensurate trilayer van der Waals heterostructures, Phys. Rev. B 101, 224107 (2020).
  35. X. Zhang, K.-T. Tsai, Z. Zhu, W. Ren, Y. Luo, S. Carr, M. Luskin, E. Kaxiras, and K. Wang, Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene superlattices, Phys. Rev. Lett. 127, 166802 (2021).
  36. 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).
  37. N. Nakatsuji, T. Kawakami, and M. Koshino, Multiscale lattice relaxation in general twisted trilayer graphenes, Phys. Rev. X 13, 041007 (2023).
  38. D. Guerci, Y. Mao, and C. Mora, Chern mosaic and ideal flat bands in equal-twist trilayer graphene, Phys. Rev. Res. 6, L022025 (2024).
  39. Y. Mao, D. Guerci, and C. Mora, Supermoiré low-energy effective theory of twisted trilayer graphene, Phys. Rev. B 107, 125423 (2023).
  40. F. K. Popov and G. Tarnopolsky, Magic angles in equal-twist trilayer graphene, Phys. Rev. B 108, L081124 (2023).
  41. F. K. Popov and G. Tarnopolsky, Magic angle butterfly in twisted trilayer graphene, Phys. Rev. Res. 5, 043079 (2023).
  42. D. Foo, Z. Zhan, M. M. Al Ezzi, L. Peng, S. Adam, and F. Guinea, Extended magic phase in twisted graphene multilayers, Phys. Rev. Res. 6, 013165 (2024).
  43. D. Guerci, Y. Mao, and C. Mora, Nature of even and odd magic angles in helical twisted trilayer graphene, Phys. Rev. B 109, 205411 (2024).
  44. M. Long, A. Jimeno-Pozo, H. Sainz-Cruz, P. A. Pantaleon, and F. Guinea, Evolution of superconductivity in twisted graphene multilayers, Proc. Natl. Acad. Sci. USA 121, e2405259121 (2024).
  45. D. Park, C. Park, E. Ko, K. Yananose, R. Engelke, X. Zhang, K. Davydov, M. Green, S. H. Park, J. H. Lee et al., Tunable incommensurability and spontaneous symmetry breaking in the reconstructed moiré-of-moiré lattices, Nature 641, 896 (2025).
  46. W. Ren, K. Davydov, Z. Zhu, J. Ma, K. Watanabe, T. Taniguchi, E. Kaxiras, M. Luskin, and K. Wang, Tunable inter-moiré physics in consecutively-twisted trilayer graphene, Phys. Rev. B 110, 115404 (2024).
  47. I. M. Craig, M. Van Winkle, C. Groschner, K. Zhang, N. Dowlatshahi, Z. Zhu, T. Taniguchi, K. Watanabe, S. M. Griffin, and D. K. Bediako, Local atomic stacking and symmetry in twisted graphene trilayers, Nat. Mater. 23, 323 (2024).
  48. C.-Y. Hao, Z. Zhan, P. A. Pantaleón, J.-Q. He, Y.-X. Zhao, K. Watanabe, T. Taniguchi, F. Guinea, and L. He, Robust flat bands in twisted trilayer graphene quasicrystals, Nat. Commun. 15, 8437 (2024).
  49. A. Uri, S. C. de la Barrera, M. T. Randeria, D. Rodan-Legrain, T. Devakul, P. J. Crowley, N. Paul, K. Watanabe, T. Taniguchi, R. Lifshitz et al., Superconductivity and strong interactions in a tunable moiré quasicrystal, Nature (London) 620, 762 (2023).
  50. C. Yang, J. May-Mann, Z. Zhu, and T. Devakul, Multi-moiré trilayer graphene: Lattice relaxation, electronic structure, and magic angles, Phys. Rev. B 110, 115434 (2024).
  51. 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).
  52. 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).
  53. A. Dunbrack and J. Cano, Intrinsically multilayer moiré heterostructures, Phys. Rev. B 107, 235425 (2023).
  54. J. Shin, Y. Park, B. L. Chittari, J.-H. Sun, and J. Jung, Electron-hole asymmetry and band gaps of commensurate double moire patterns in twisted bilayer graphene on hexagonal boron nitride, Phys. Rev. B 103, 075423 (2021).
  55. Y. H. Kwan, G. Wagner, N. Chakraborty, S. H. Simon, and S. Parameswaran, Domain wall competition in the Chern insulating regime of twisted bilayer graphene, Phys. Rev. B 104, 115404 (2021).
  56. G. Shavit and Y. Oreg, Domain formation driven by the entropy of topological edge modes, Phys. Rev. Lett. 128, 156801 (2022).
  57. Q. Tong, H. Yu, Q. Zhu, Y. Wang, X. Xu, and W. Yao, Topological mosaics in moiré superlattices of van der Waals heterobilayers, Nat. Phys. 13, 356 (2017).
  58. S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, C. R. Phys. 14, 816 (2013).
  59. E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, Int. J. Mod. Phys. B 27, 1330017 (2013).
  60. Z. Liu and E. J. Bergholtz, Recent developments in fractional Chern insulators, Encyclo. Condens. Matter Phys. 1, 515 (2024).
  61. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  62. 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).
  63. N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  64. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  65. J. C. Hoke, Y. Li, Y. Hu, J. May-Mann, K. Watanabe, T. Taniguchi, T. Devakul, and B. E. Feldman, Imaging supermoire relaxation and conductive domain walls in helical trilayer graphene, arXiv:2410.16269.
  66. M. Koshino, Interlayer interaction in general incommensurate atomic layers, New J. Phys. 17, 015014 (2015).
  67. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  68. See Supplemental Material at http://link.aps.org/supplemental/10.1103/srgm-9ybc for details of the momentum dependent tunneling term, and additional Hartree-Fock and exact diagonalization results.
  69. 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).
  70. 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).
  71. P. J. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogical introduction, Ann. Phys. 435, 168646 (2021).
  72. Z. Wang, Y. H. Kwan, G. Wagner, S. H. Simon, N. Bultinck, and S. A. Parameswaran, Chern-textured exciton insulators with valley spiral order in moiré materials, arXiv:2406.15342.
  73. J. Yu, B. A. Foutty, Z. Han, M. E. Barber, Y. Schattner, K. Watanabe, T. Taniguchi, P. Phillips, Z.-X. Shen, S. A. Kivelson et al., Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene, Nat. Phys. 18, 825 (2022).
  74. J. Yu, B. A. Foutty, Y. H. Kwan, M. E. Barber, K. Watanabe, T. Taniguchi, Z.-X. Shen, S. A. Parameswaran, and B. E. Feldman, Spin skyrmion gaps as signatures of strong-coupling insulators in magic-angle twisted bilayer graphene, Nat. Commun. 14, 6679 (2023).
  75. D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Cascade of electronic transitions in magic-angle twisted bilayer graphene, Nature (London) 582, 198 (2020).
  76. Y. Choi, H. Kim, Y. Peng, A. Thomson, C. Lewandowski, R. Polski, Y. Zhang, H. S. Arora, K. Watanabe, T. Taniguchi et al., Correlation-driven topological phases in magic-angle twisted bilayer graphene, Nature (London) 589, 536 (2021).
  77. Y. Choi, H. Kim, C. Lewandowski, Y. Peng, A. Thomson, R. Polski, Y. Zhang, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Interaction-driven band flattening and correlated phases in twisted bilayer graphene, Nat. Phys. 17, 1375 (2021).
  78. 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).
  79. H. Kim, Y. Choi, Étienne Lantagne-Hurtubise, C. Lewandowski, A. Thomson, L. Kong, H. Zhou, E. Baum, Y. Zhang, L. Holleis, K. Watanabe, T. Taniguchi, A. F. Young, J. Alicea, and S. Nadj-Perge, Imaging inter-valley coherent order in magic-angle twisted trilayer graphene, Nature (London) 623, 942 (2023)
  80. S. Turkel, J. Swann, Z. Zhu, M. Christos, K. Watanabe, T. Taniguchi, S. Sachdev, M. S. Scheurer, E. Kaxiras, C. R. Dean et al., Orderly disorder in magic-angle twisted trilayer graphene, Science 376, 193 (2022).
  81. K. P. Nuckolls, M. Oh, D. Wong, B. Lian, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Strongly correlated Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 588, 610 (2020).
  82. 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).

Outline

Information

Phys. Rev. Research 7, L032070– Published 29 September, 2025
  • Received 8 February 2025
  • Accepted 8 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation