Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).
Nuckolls, K. P. & Yazdani, A. A microscopic perspective on moiré materials. Nat. Rev. Mater. 9, 460–480 (2024).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023).
Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered ferroelectricity in bilayer boron nitride. Science 372, 1458–1462 (2021).
Vizner Stern, M. et al. Interfacial ferroelectricity by van der Waals sliding. Science 372, 1462–1466 (2021).
Song, T. et al. Direct visualization of magnetic domains and moiré magnetism in twisted 2D magnets. Science 374, 1140–1144 (2021).
Zhao, S. Y. F. et al. Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors. Science 382, 1422–1427 (2023).
Cummins, H. Z. Experimental studies of structurally incommensurate crystal phases. Phys. Rep. 185, 211–409 (1990).
Smaalen, S. V. Incommensurate Crystallography (Oxford Univ. Press, 2007).
Janssen, T., Chapuis, G. & de Boissieu, M. Aperiodic Crystals: From Modulated Phases to Quasicrystals: Structure and Properties 2nd edn (Oxford Univ. Press, 2018).
Onsager, L. Interpretation of the de Haas-van Alphen effect. Lond. Edinb. Dublin Philos. Mag. J. Sci. 43, 1006–1008 (1952).
Lifshitz, I. M. & Kosevich, A. M. On the theory of magnetic susceptibility of metals at low temperatures. Zh. Eksp. Teor. Fiz. 29, 730–742 (1955).
Leeb, V., Huber, N., Pfleiderer, C., Knolle, J. & Wilde, M. A. A field guide to non-Onsager quantum oscillations in metals. Adv. Phys. Res. 4, 2400134 (2025).
de Wolff, P. M. The pseudo-symmetry of modulated crystal structures. Acta Crystallogr. A 30, 777–785 (1974).
Janner, A. & Janssen, T. Symmetry of periodically distorted crystals. Phys. Rev. B 15, 643–658 (1977).
Hazzard, K. R. A. & Gadway, B. Synthetic dimensions. Phys. Today 76, 62–63 (2023).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).
Sugawa, S., Salces-Carcoba, F., Perry, A. R., Yue, Y. & Spielman, I. B. Second Chern number of a quantum-simulated non-Abelian Yang monopole. Science 360, 1429–1434 (2018).
Sakai, S., Takemori, N., Koga, A. & Arita, R. Superconductivity on a quasiperiodic lattice: extended-to-localized crossover of Cooper pairs. Phys. Rev. B 95, 024509 (2017).
Cao, Y. et al. Kohn-Luttinger mechanism driven exotic topological superconductivity on the Penrose lattice. Phys. Rev. Lett. 125, 017002 (2020).
Evans, H. T. & Allmann, R. The crystal structure and crystal chemistry of valleriite. Z. Kristallogr. Cryst. Mater. 127, 73–93 (1968).
Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).
Wiegers, G. A. Misfit layer compounds: structures and physical properties. Prog. Solid State Chem. 24, 1–139 (1996).
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).
Goldman, A. I. & Kelton, R. F. Quasicrystals and crystalline approximants. Rev. Mod. Phys. 65, 213–230 (1993).
Bancel, P. A., Heiney, P. A., Stephens, P. W., Goldman, A. I. & Horn, P. M. Structure of rapidly quenched Al-Mn. Phys. Rev. Lett. 54, 2422–2425 (1985).
Zhang, Y., Maharaj, A. V. & Kivelson, S. Disruption of quantum oscillations by an incommensurate charge density wave. Phys. Rev. B 91, 085105 (2015).
Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic, Z. Flat bands in slightly twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).
Devarakonda, A. et al. Clean 2D superconductivity in a bulk van der Waals superlattice. Science 370, 231–236 (2020).
Ma, K. et al. Two-dimensional superconductivity in a bulk superlattice van der Waals material Ba6Nb11Se28. Phys. Rev. Mater. 6, 044806 (2022).
Devarakonda, A. et al. Evidence of striped electronic phases in a structurally modulated superlattice. Nature 631, 526–530 (2024).
Yoo, H. et al. Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene. Nat. Mater. 18, 448–453 (2019).
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).
Khasanova, N. R. et al. A new structure type of the ternary sulfide Eu1.3Nb1.9S5. J. Solid State Chem. 164, 345–353 (2002).
Everson, M. P., Johnson, A., Lu, H.-A., Coleman, R. V. & Falicov, L. M. Magnetoquantum oscillations, magnetic breakdown, and Fermi-surface modifications in NbSe3. Phys. Rev. B. 36, 6953–6962 (1987).
Hill, S. et al. Quantum limit and anomalous field-induced insulating behavior in η-Mo4O11. Phys. Rev. B. 55, 2018–2031 (1997).
Huber, N. et al. Quantum oscillations of the quasiparticle lifetime in a metal. Nature 621, 276–281 (2023).
Gerhardts, R. R., Weiss, D. & Klitzing, K. V. Novel magnetoresistance oscillations in a periodically modulated two-dimensional electron gas. Phys. Rev. Lett. 62, 1173–1176 (1989).
Janssen, T. Aperiodic crystals: a contradictio in terminis? Phys. Rep. 168, 55–113 (1988).
Boada, O., Celi, A., Latorre, J. I. & Lewenstein, M. Quantum simulation of an extra dimension. Phys. Rev. Lett. 108, 133001 (2012).
Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).
Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).
Lu, J. P. & Birman, J. L. Electronic structure of a quasiperiodic system. Phys. Rev. B. 36, 4471–4474 (1987).
Batalla, E., Razavi, F. S. & Datars, W. R. Fermi surface of Hg3−δAsF6 and Hg3−δSbF6. Phys. Rev. B. 25, 2109–2118 (1982).
Kawamoto, T. et al. Fermi surface of the organic superconductor (MDT–ST)(I3)0.417 reconstructed by incommensurate potential. Phys. Rev. B 73, 024503 (2006).
Schnars, U. & Jüptner, W. Direct recording of holograms by a CCD target and numerical reconstruction. Appl. Opt. 33, 179–181 (1994).
Bousso, R. The holographic principle. Rev. Mod. Phys. 74, 825–874 (2002).
Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. J. High Energy Phys. 2015, 149 (2015).
