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).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
von Klitzing, K. Essay: quantum Hall effect and the new international system of units. Phys. Rev. Lett. 122, 200001 (2019).
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).
Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).
Wen, X.-G. Colloquium: zoo of quantum-topological phases of matter. Rev. Mod. Phys. 89, 041004 (2017).
Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).
Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, 1583–1586 (1984).
Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).
Jain, J. K. Composite fermion theory of exotic fractional quantum Hall effect. Annu. Rev. Condens. Matter Phys. 6, 39–62 (2015).
Lin, X., Du, R. & Xie, X. Recent experimental progress of fractional quantum Hall effect: 5/2 filling state and graphene. Natl Sci. Rev. 1, 564–579 (2014).
Moore, G. & Read, N. Nonabelions in the fractional quantum hall effect. Nucl. Phys. B 360, 362–396 (1991).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).
Halperin, B. I. in The Physics and Chemistry of Oxide Superconductors (eds Iye, Y. & Yasuoka, H.) 439–450 (Springer, 1992).
MacDonald, A. H. Introduction to the physics of the quantum Hall regime. Preprint at https://arxiv.org/abs/cond-mat/9410047 (1994).
Tong, D. Lectures on the quantum Hall effect. Preprint at https://arxiv.org/abs/1606.06687 (2016).
Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Bernevig, B. A. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).
Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).
Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).
Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).
Lau, C. N., Bockrath, M. W., Mak, K. F. & Zhang, F. Reproducibility in the fabrication and physics of moiré materials. Nature 602, 41–50 (2022).
Nuckolls, K. P. & Yazdani, A. A microscopic perspective on moiré materials. Nat. Rev. Mater. 9, 460–480 (2024).
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).
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).
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).
Chang, C.-Z., Liu, C.-X. & MacDonald, A. H. Colloquium: quantum anomalous Hall effect. Rev. Mod. Phys. 95, 011002 (2023).
Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
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).
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
Neupert, T., Santos, L., Chamon, C. & Mudry, C. Fractional quantum Hall states at zero magnetic field. Phys. Rev. Lett. 106, 236804 (2011).
Regnault, N. & Bernevig, B. A. Fractional Chern insulator. Phys. Rev. 1, 021014 (2011).
Sheng, D. N., Gu, Z.-C., Sun, K. & Sheng, L. Fractional quantum Hall effect in the absence of Landau levels. Nat. Commun. 2, 389 (2011).
Sun, K., Gu, Z., Katsura, H. & Das Sarma, S. Nearly flatbands with nontrivial topology. Phys. Rev. Lett. 106, 236803 (2011).
Tang, E., Mei, J.-W. & Wen, X.-G. High-temperature fractional quantum Hall states. Phys. Rev. Lett. 106, 236802 (2011).
Lin, Z., Yang, W., Lu, H., Zhai, D. & Yao, W. Fractional Chern insulator states in an isolated flat band of zero Chern number. Preprint at https://arxiv.org/abs/2505.09009 (2025).
Qi, X.-L. Generic wave-function description of fractional quantum anomalous Hall states and fractional topological insulators. Phys. Rev. Lett. 107, 126803 (2011).
Claassen, M., Lee, C. H., Thomale, R., Qi, X.-L. & Devereaux, T. P. Position-momentum duality and fractional quantum Hall effect in Chern insulators. Phys. Rev. Lett. 114, 236802 (2015).
Wang, J., Cano, J., Millis, A. J., Liu, Z. & Yang, B. Exact Landau level description of geometry and interaction in a flatband. Phys. Rev. Lett. 127, 246403 (2021).
Ledwith, P. J., Vishwanath, A. & Parker, D. E. Vortexability: a unifying criterion for ideal fractional Chern insulators. Phys. Rev. B 108, 205144 (2023).
Estienne, B., Regnault, N. & Crépel, V. Ideal Chern bands as Landau levels in curved space. Phys. Rev. Res. 5, L032048 (2023).
Siddharth, A. P., Rahul, R. & Shivaji, L. S. Fractional quantum Hall physics in topological flat bands. C. R. Phys. 14, 816–839 (2013).
Wu, Y.-L., Bernevig, B. A. & Regnault, N. Zoology of fractional Chern insulators. Phys. Rev. B 85, 075116 (2012).
Behrmann, J., Liu, Z. & Bergholtz, E. J. Model fractional Chern insulators. Phys. Rev. Lett. 116, 216802 (2016).
Roy, R. Band geometry of fractional topological insulators. Phys. Rev. B 90, 165139 (2014).
Wang, Z. & Simon, S. H. A closed band-projected density algebra must be Girvin-MacDonald-Platzman. Phys. Rev. Lett. 134, 136502 (2025).
Simon, S. H., Harper, F. & Read, N. Fractional Chern insulators in bands with zero Berry curvature. Phys. Rev. B 92, 195104 (2015).
Bernevig, B. A. & Zhang, S.-C. Quantum spin Hall effect. Phys. Rev. Lett. 96, 106802 (2006).
Neupert, T., Chamon, C., Iadecola, T., Santos, L. H. & Mudry, C. Fractional (Chern and topological) insulators. Phys. Scr. 2015, 014005 (2015).
Stern, A. Fractional topological insulators: a pedagogical review. Annu. Rev. Condens. Matter Phys. 7, 349–368 (2016).
Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).
Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).
Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020).
Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).
Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).
Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021).
Polshyn, H. et al. Topological charge density waves at half-integer filling of a moiré superlattice. Nat. Phys. 18, 42–47 (2022).
Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).
Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).
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).
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).
Repellin, C. & Senthil, T. Chern bands of twisted bilayer graphene: fractional Chern insulators and spin phase transition. Phys. Rev. Res. 2, 023238 (2020).
Tarnopolsky, G., Kruchkov, A. J. & Vishwanath, A. Origin of magic angles in twisted bilayer graphene. Phys. Rev. Lett. 122, 106405 (2019).
Mera, B. & Ozawa, T. Kähler geometry and Chern insulators: relations between topology and the quantum metric. Phys. Rev. B 104, 045104 (2021).
Mera, B. & Ozawa, T. Engineering geometrically flat Chern bands with Fubini-Study Kähler structure. Phys. Rev. B 104, 115160 (2021).
Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).
Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).
Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. H. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).
Zhang, X.-W. et al. Polarization-driven band topology evolution in twisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).
Jia, Y. et al. Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe2. Phys. Rev. B 109, 205121 (2024).
Yu, H., Chen, M. & Yao, W. Giant magnetic field from moiré induced Berry phase in homobilayer semiconductors. Natl Sci. Rev. 7, 12–20 (2020).
Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).
Xu, C., Mao, N., Zeng, T. & Zhang, Y. Multiple Chern bands in twisted MoTe2 and possible non-Abelian states. Phys. Rev. Lett. 134, 066601 (2025).
Li, H., Kumar, U., Sun, K. & Lin, S.-Z. Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices. Phys. Rev. Res. 3, L032070 (2021).
Crépel, V. & Fu, L. Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides. Phys. Rev. B 107, L201109 (2023).
Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023).
Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).
Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74–79 (2023).
Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2. Phys. Rev. 13, 031037 (2023).
Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Foutty, B. A. et al. Mapping twist-tuned multiband topology in bilayer WSe2. Science 384, 343–347 (2024).
Knüppel, P. et al. Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayers. Nat. Commun. 16, 1959 (2025).
Anderson, E. et al. Programming correlated magnetic states with gate-controlled moiré geometry. Science 381, 325–330 (2023).
Redekop, E. et al. Direct magnetic imaging of fractional Chern insulators in twisted MoTe2. Nature 635, 584–589 (2024).
Ji, Z. et al. Local probe of bulk and edge states in a fractional Chern insulator. Nature 635, 578–583 (2024).
Kang, K. et al. Evidence of the fractional quantum spin Hall effect in moiré MoTe2. Nature 628, 522–526 (2024).
Park, H. et al. Ferromagnetism and topology of the higher flat band in a fractional Chern insulator. Nat. Phys. https://doi.org/10.1038/s41567-025-02804-0 (2025).
Xu, F. et al. Interplay between topology and correlations in the second moiré band of twisted bilayer MoTe2. Nat. Phys. https://doi.org/10.1038/s41567-025-02803-1 (2025).
Kang, K. et al. Double quantum spin Hall phase in moiré WSe2. Nano Lett. 24, 14901–14907 (2024).
Abouelkomsan, A. & Fu, L. Non-Abelian spin Hall insulator. Phys. Rev. Res. 7, 023083 (2025).
Jian, C. M., Cheng, M. & Xu, C. Minimal fractional topological insulator in half-filled conjugate moiré Chern bands. Phys. Rev. X 15, 021063 (2025).
Sodemann Villadiego, I. Halperin states of particles and holes in ideal time reversal invariant pairs of Chern bands and the fractional quantum spin Hall effect in moiré MoTe2. Phys. Rev. B 110, 045114 (2024).
Zhang, Y.-H. Non-Abelian and Abelian descendants of a vortex spin liquid: fractional quantum spin Hall effect in twisted MoTe2. Phys. Rev. B 110, 155102 (2024).
Kwan, Y. H. et al. When could Abelian fractional topological insulators exist in twisted MoTe2 (and other systems). Preprint at https://arxiv.org/abs/2407.02560 (2024).
Min, H. & MacDonald, A. H. Chiral decomposition in the electronic structure of graphene multilayers. Phys. Rev. B 77, 155416 (2008).
Koshino, M. & McCann, E. Trigonal warping and Berry’s phase Nπ in ABC-stacked multilayer graphene. Phys. Rev. B 80, 165409 (2009).
Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429–433 (2021).
Han, T. et al. Orbital multiferroicity in pentalayer rhombohedral graphene. Nature 623, 41–47 (2023).
Zhang, F., Jung, J., Fiete, G. A., Niu, Q. & MacDonald, A. H. Spontaneous quantum Hall states in chirally stacked few-layer graphene systems. Phys. Rev. Lett. 106, 156801 (2011).
Han, T. et al. Large quantum anomalous Hall effect in spin-orbit proximitized rhombohedral graphene. Science 384, 647–651 (2024).
Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Superconductivity in rhombohedral trilayer graphene. Nature 598, 434–438 (2021).
Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).
Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (2024).
Lu, Z. et al. Extended quantum anomalous Hall states in graphene/hBN moiré superlattices. Nature 637, 1090–1095 (2025).
Xie, J. et al. Tunable fractional Chern insulators in rhombohedral graphene superlattices. Nat. Mater. 24, 1042–1048 (2025).
Choi, Y. et al. Electric field control of superconductivity and quantized anomalous Hall effects in rhombohedral tetralayer graphene. Nature 639, 342–347 (2025).
Aronson, S. H.et al. Displacement field-controlled fractional Chern insulators and charge density waves in a graphene/hBN moiré superlattice. Phys. Rev. X 15, 031026 (2024).
Dong, Z., Patri, A. S. & Senthil, T. Theory of quantum anomalous Hall phases in pentalayer rhombohedral graphene moiré structures. Phys. Rev. Lett. 133, 206502 (2024).
Zhou, B., Yang, H. & Zhang, Y. H. Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moir‚less limit and in Coulomb imprinted superlattice. Phys. Rev. Lett. 133, 206504 (2024).
Dong, J. et al. 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).
Halbertal, D. et al. Multilayered atomic relaxation in van der Waals heterostructures. Phys. Rev. 13, 011026 (2023).
Kwan, Y. H. et al. Moiré fractional Chern insulators III: Hartree-Fock phase diagram, magic angle regime for Chern insulator states, the role of the moir‚ potential and Goldstone gaps in rhombohedral graphene superlattices. Phys. Rev. B 112, 075109 (2025).
Guo, Z. & Liu, J. Correlation stabilized anomalous Hall crystal in bilayer graphene. Preprint at https://arxiv.org/abs/2409.14658 (2024).
Yu, J., Herzog-Arbeitman, J., Kwan, Y. H., Regnault, N. & Bernevig, B. A. Moiré fractional Chern insulators IV: fluctuation-driven collapse of FCIs in multi-band exact diagonalization calculations on rhombohedral graphene. Phys. Rev. B 112, 075110 (2025).
Han, T. et al. Signatures of chiral superconductivity in rhombohedral graphene. Nature 643, 654–661 (2025).
Das Sarma, S. & Xie, M. Thermal crossover from a Chern insulator to a fractional Chern insulator in pentalayer graphene. Phys. Rev. B 110, 155148 (2024).
Patri, A. S., Dong, Z. & Senthil, T. Extended quantum anomalous Hall effect in moiré structures: phase transitions and transport. Phys. Rev. B 110, 245115 (2024).
Chen, F., Luo, W.-W., Zhu, W. & Sheng, D. N. Robust non-Abelian even-denominator fractional Chern insulator in twisted bilayer MoTe2. Nat. Commun. 16, 2115 (2025).
Reddy, A. P., Paul, N., Abouelkomsan, A. & Fu, L. Non-Abelian fractionalization in topological minibands. Phys. Rev. Lett. 133, 166503 (2024).
Wang, C. et al. Higher Landau-level analogs and signatures of non-Abelian states in twisted bilayer MoTe2. Phys. Rev. Lett. 134, 076503 (2025).
Inbar, A. et al. The quantum twisting microscope. Nature 614, 682–687 (2023).
Jiang, Y. et al. 2D theoretically twistable material database. Preprint at https://arxiv.org/abs/2411.09741 (2024).
Sterdyniak, A., Repellin, C., Bernevig, B. A. & Regnault, N. Series of Abelian and non-Abelian states in C>1 fractional Chern insulators. Phys. Rev. B 87, 205137 (2013).
Song, X.-Y., Zhang, Y.-H. & Senthil, T. Phase transitions out of quantum Hall states in moiré materials. Phys. Rev. B 109, 085143 (2024).
Feldman, D. E. & Halperin, B. I. Fractional charge and fractional statistics in the quantum Hall effects. Rep. Prog. Phys. 84, 076501 (2021).
Clarke, D. J., Alicea, J. & Shtengel, K. Exotic non-Abelian anyons from conventional fractional quantum Hall states. Nat. Commun. 4, 1348 (2013).
Choi, Y. et al. Superconductivity and quantized anomalous Hall effect in rhombohedral graphene. Nature 639, 342–347 (2025).
Xu, F. et al. Signatures of unconventional superconductivity near reentrant and fractional quantum anomalous Hall insulators. Preprint at https://arxiv.org/abs/2504.06972 (2025).
Morales-Durán, N., Shi, J. & MacDonald, A. H. Fractionalized electrons in moiré materials. Nat. Rev. Phys. 6, 349–351 (2024).
Morales-Durán, N., Wei, N., Shi, J. & MacDonald, A. H. Magic angles and fractional Chern insulators in twisted homobilayer transition metal dichalcogenides. Phys. Rev. Lett. 132, 096602 (2024).
Reddy, A. P., Alsallom, F., Zhang, Y., Devakul, T. & Fu, L. Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2. Phys. Rev. B 108, 085117 (2023).
Wang, C. et al. Fractional Chern insulator in twisted bilayer MoTe2. Phys. Rev. Lett. 132, 036501 (2024).
Yu, J. et al. Fractional Chern insulators versus nonmagnetic states in twisted bilayer MoTe2. Phys. Rev. B 109, 045147 (2024).
Călugăru, D. et al. A new moiré platform based on M-point twisting. Nature 643, 376–381 (2025).
Lei, C., Mahon, P. T. & MacDonald, A. H. Moiré band theory for M-valley twisted transition metal dichalcogenides. Preprint at https://arxiv.org/abs/2411.18828 (2024).