Andrei, E. Y. et al. The marvels of moiré materials. Nat. Rev. Mater. 6, 201–206 (2021).
Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).
Angeli, M. & MacDonald, A. H. Γ valley transition metal dichalcogenide moiré bands. Proc. Natl Acad. Sci. 118, e2021826118 (2021).
Claassen, M., Xian, L., Kennes, D. M. & Rubio, A. Ultra-strong spin–orbit coupling and topological moiré engineering in twisted ZrS2 bilayers. Nat. Commun. 13, 4915 (2022).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. 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).
Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).
Zhang, X.-W. et al. Polarization-driven band topology evolution in twisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).
Carr, S. et al. Twistronics: manipulating the electronic properties of two-dimensional layered structures through their twist angle. Phys. Rev. B 95, 075420 (2017).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Song, Z.-D. & Bernevig, B. A. Magic-angle twisted bilayer graphene as a topological heavy fermion problem. Phys. Rev. Lett. 129, 047601 (2022).
Chou, Y.-Z. & Das Sarma, S. Kondo lattice model in magic-angle twisted bilayer graphene. Phys. Rev. Lett. 131, 026501 (2023).
Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).
Wang, P. et al. One-dimensional Luttinger liquids in a two-dimensional moiré lattice. Nature 605, 57–62 (2022).
Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).
Anderson, E. et al. Programming correlated magnetic states with gate-controlled moiré geometry. Science 381, 325–330 (2023).
Zhao, W. et al. Gate-tunable heavy fermions in a moiré Kondo lattice. Nature 616, 61–65 (2023).
Guo, Y. et al. Superconductivity in 5.0° twisted bilayer WSe2. Nature 637, 839–845 (2025).
Xia, Y. et al. Superconductivity in twisted bilayer WSe2. Nature 637, 833–838 (2025).
Sheng, D. N., Reddy, A. P., Abouelkomsan, A., Bergholtz, E. J. & Fu, L. Quantum anomalous Hall crystal at fractional filling of moiré superlattices. Phys. Rev. Lett. 133, 066601 (2024).
Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021).
Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2. Phys. Rev. X 13, 031037 (2023).
Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74–79 (2023).
Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).
Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023).
Wang, C. et al. Fractional Chern insulator in twisted bilayer MoTe2. Phys. Rev. Lett. 132, 036501 (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, J. et al. Fractional Chern insulators versus nonmagnetic states in twisted bilayer MoTe2. Phys. Rev. B 109, 045147 (2024).
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).
Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (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).
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).
Herzog-Arbeitman, J. et al. Moiré fractional Chern insulators. II. First-principles calculations and continuum models of rhombohedral graphene superlattices. Phys. Rev. B 109, 205122 (2024).
Jiang, Y. et al. 2D theoretically twistable material database. Preprint at https://arxiv.org/abs/2411.09741 (2024).
de la Flor, G., Souvignier, B., Madariaga, G. & Aroyo, M. I. Layer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Server. Acta Crystallogr. A 77, 559–571 (2021).
Chen, Z. Y., Yang, S. A. & Zhao, Y. X. Brillouin Klein bottle from artificial gauge fields. Nat. Commun. 13, 2215 (2022).
Zhang, C., Chen, Z. Y., Zhang, Z. & Zhao, Y. X. General theory of momentum-space nonsymmorphic symmetry. Phys. Rev. Lett. 130, 256601 (2023).
Xiao, Z., Zhao, J., Li, Y., Shindou, R. & Song, Z.-D. Spin space groups: full classification and applications. Phys. Rev. X 14, 031037 (2024).
Fonseca, A. G. et al. Weyl points on nonorientable manifolds. Phys. Rev. Lett. 132, 266601 (2024).
Kennes, D. M., Xian, L., Claassen, M. & Rubio, A. One-dimensional flat bands in twisted bilayer germanium selenide. Nat. Commun. 11, 1124 (2020).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
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).
Herzog-Arbeitman, J., Song, Z.-D., Elcoro, L. & Bernevig, B. A. Hofstadter topology with real space invariants and reentrant projective symmetries. Phys. Rev. Lett. 130, 236601 (2023).
Petralanda, U., Jiang, Y., Bernevig, B. A., Regnault, N. & Elcoro, L., Two-dimensional topological quantum chemistry and catalog of topological materials. Preprint at https://arxiv.org/abs/2411.08950 (2024).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).
Han, Z., Herzog-Arbeitman, J., Bernevig, B. A. & Kivelson, S. A. “Quantum geometric nesting” and solvable model flat-band systems. Phys. Rev. X 14, 041004 (2024).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, 13115 (1993).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558 (1993).
Kresse, G. & Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys. Rev. B 49, 14251 (1994).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B 67, 155108 (2003).
Ozaki, T. & Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B 69, 195113 (2004).
Ozaki, T. & Kino, H. Efficient projector expansion for the ab initio LCAO method. Phys. Rev. B 72, 045121 (2005).
Lejaeghere, K. et al. Reproducibility in density functional theory calculations of solids. Science 351, aad3000 (2016).
Batzner, S. et al. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).
Liu, J., Fang, Z., Weng, H. & Wu, Q., DPmoire: a tool for constructing accurate machine learning force fields in moiré systems. Preprint at https://arxiv.org/abs/2412.19333 (2025).
Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Busch, C., Fröhlich, C. & Hulliger, F. Struktur, elektrische und thermoelektrische Eigenschaften von SnSe2. Helv. Phys. Acta 34, 359–368 (1961).
Su, Y., Ebrish, M. A., Olson, E. J. & Koester, S. J. SnSe2 field-effect transistors with high drive current. Appl. Phys. Lett. 103, 263104 (2013).
Zeng, J. et al. Gate-induced interfacial superconductivity in 1T-SnSe2. Nano Lett. 18, 1410–1415 (2018).
Wu, H. et al. Spacing dependent and cation doping independent superconductivity in intercalated 1T 2D SnSe2. 2D Mater. 6, 045048 (2019).
Huang, Y. et al. Universal mechanical exfoliation of large-area 2D crystals. Nat. Commun. 11, 2453 (2020).
Lv, H. Y., Lu, W. J., Shao, D. F., Lu, H. Y. & Sun, Y. P. Strain-induced enhancement in the thermoelectric performance of a ZrS2 monolayer. J. Mater. Chem. C 4, 4538–4545 (2016).
Mañas-Valero, S., García-López, V., Cantarero, A. & Galbiati, M. Raman Spectra of ZrS2 and ZrSe2 from bulk to atomically thin layers. Appl. Sci. 6, 264 (2016).
Alsulami, A. et al. Lattice transformation from 2D to quasi 1D and phonon properties of exfoliated ZrS2 and ZrSe2. Small 19, 2205763 (2023).
Xia, C. et al. The characteristics of n- and p-type dopants in SnS2 monolayer nanosheets. Phys. Chem. Chem. Phys. 16, 19674–19680 (2014).
Huang, F. et al. Mechanically exfoliated few-layer SnS2 and integrated van der Waals electrodes for ultrahigh responsivity phototransistors. ACS Appl. Electron. Mater. 4, 5333–5339 (2022).
Kang, M. et al. Electrical characterization of multilayer HfSe2 field-effect transistors on SiO2 substrate. Appl. Phys. Lett. 106, 143108 (2015).
Zhao, X., Xia, C., Wang, T., Dai, X. & Yang, L. Characteristics of n- and p-type dopants in 1T-HfS2 monolayer. J. Alloys Compd. 689, 302–306 (2016).
Wu, N. et al. Strain effect on the electronic properties of 1T-HfS2 monolayer. Physica E 93, 1–5 (2017).
Singh, D. & Ahuja, R. Enhanced optoelectronic and thermoelectric properties by intrinsic structural defects in monolayer HfS2. ACS Appl. Energy Mater. 2, 6891–6903 (2019).
Zhao, Q. et al. Thickness-induced structural phase transformation of layered gallium telluride. Phys. Chem. Chem. Phys. 18, 18719–18726 (2016).
Kariyado, T. & Vishwanath, A. Flat band in twisted bilayer Bravais lattices. Phys. Rev. Res. 1, 033076 (2019).
Kariyado, T. Twisted bilayer BC3: valley interlocked anisotropic flat bands. Phys. Rev. B 107, 085127 (2023).
Fujimoto, M., Kawakami, T. & Koshino, M. Perfect one-dimensional interface states in a twisted stack of three-dimensional topological insulators. Phys. Rev. Res. 4, 043209 (2022).
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).
