Frenkel, Y. I. & Kontorova, T. The model of dislocation in solid body. Zh. Eksp. Teor. Fiz. 8, 1340–1348 (1938).
McMillan, W. L. Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition. Phys. Rev. B 14, 1496–1502 (1976).
Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 18 (1980).
Bak, P. Commensurate phases, incommensurate phases and the devil’s staircase. Rep. Prog. Phys. 45, 587 (1982).
Alden, J. S. et al. Strain solitons and topological defects in bilayer graphene. Proc. Natl Acad. Sci. USA 110, 11256–11260 (2013).
Huang, F.-T. & Cheong, S.-W. Aperiodic topological order in the domain configurations of functional materials. Nat. Rev. Mater. 2, 17004 (2017).
Yoo, H. et al. Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene. Nat. Mater. 18, 448–453 (2019).
Engelke, R. et al. Topological nature of dislocation networks in two-dimensional moiré materials. Phys. Rev. B 107, 125413 (2023).
Zhu, Z., Carr, S., Massatt, D., Luskin, M. & Kaxiras, E. Twisted trilayer graphene: a precisely tunable platform for correlated electrons. Phys. Rev. Lett. 125, 116404 (2020).
Hao, Z. et al. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).
Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021).
Zhang, X. et al. Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene superlattices. Phys. Rev. Lett. 127, 166802 (2021).
Shin, J., Chittari, B. L. & Jung, J. Stacking and gate-tunable topological flat bands, gaps, and anisotropic strip patterns in twisted trilayer graphene. Phys. Rev. B 104, 045413 (2021).
Turkel, S. et al. Orderly disorder in magic-angle twisted trilayer graphene. Science 376, 193–199 (2022).
Park, J. M. et al. Robust superconductivity in magic-angle multilayer graphene family. Nat. Mater. 21, 877–883 (2022).
Li, Y. et al. Symmetry breaking and anomalous conductivity in a double-moiré superlattice. Nano Lett. 22, 6215–6222 (2022).
Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene. Nature 606, 494–500 (2022).
Lin, X., Li, C., Su, K. & Ni, J. Energetic stability and spatial inhomogeneity in the local electronic structure of relaxed twisted trilayer graphene. Phys. Rev. B 106, 075423 (2022).
Devakul, T. et al. Magic-angle helical trilayer graphene. Sci. Adv. 9, eadi6063 (2023).
Kim, H. et al. Imaging inter-valley coherent order in magic-angle twisted trilayer graphene. Nature 623, 942–948 (2023).
Nakatsuji, N., Kawakami, T. & Koshino, M. Multiscale lattice relaxation in general twisted trilayer graphenes. Phys. Rev. X 13, 041007 (2023).
Uri, A. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023).
Mao, Y., Guerci, D. & Mora, C. Supermoiré low-energy effective theory of twisted trilayer graphene. Phys. Rev. B 107, 125423 (2023).
Popov, F. K. & Tarnopolsky, G. Magic angles in equal-twist trilayer graphene. Phys. Rev. B 108, L081124 (2023).
Meng, H., Zhan, Z. & Yuan, S. Commensurate and incommensurate double moiré interference in twisted trilayer graphene. Phys. Rev. B 107, 035109 (2023).
Craig, I. M. et al. Local atomic stacking and symmetry in twisted graphene trilayers. Nat. Mater. 23, 323–330 (2024).
Park, C. Calculation of charge density wave phase diagram by interacting eigenmodes method. J. Phys. Condens. Matter 34, 315401 (2022).
Park, C. & Son, Y.-W. Condensation of preformed charge density waves in kagome metals. Nat. Commun. 14, 7309 (2023).
Kim, K. et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. USA 114, 3364–3369 (2017).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).
Zhang, S. et al. Domino-like stacking order switching in twisted monolayer–multilayer graphene. Nat. Mater. 21, 621–626 (2022).
Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).
Lipson, H. S. & Stokes, A. The structure of graphite. Proc. R. Soc. Lond. A. Math. Phys. Sci. 181, 101–105 (1942).
Boehm, H. P. & Coughlin, R. W. Enthalpy difference of hexagonal and rhombohedral graphite. Carbon 2, 1–6 (1964).
Li, H. et al. Global control of stacking-order phase transition by doping and electric field in few-layer graphene. Nano Lett. 20, 3106–3112 (2020).
Nery, J. P., Calandra, M. & Mauri, F. Ab-initio energetics of graphite and multilayer graphene: stability of Bernal versus rhombohedral stacking. 2D Mater. 8, 035006 (2021).
Kolmogorov, A. N. & Crespi, V. H. Registry-dependent interlayer potential for graphitic systems. Phys. Rev. B 71, 235415 (2005).
Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Independent superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. 16, 926–930 (2020).
Kaliteevski, M. A., Enaldiev, V. & Fal’ko, V. I. Twirling and spontaneous symmetry breaking of domain wall networks in lattice-reconstructed heterostructures of two-dimensional materials. Nano Lett. 23, 8875–8880 (2023).
Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020).
Lu, C., Chang, C.-P., Huang, Y.-C., Chen, R.-B. & Lin, M. Influence of an electric field on the optical properties of few-layer graphene with AB stacking. Phys. Rev. B 73, 144427 (2006).
Aoki, M. & Amawashi, H. Dependence of band structures on stacking and field in layered graphene. Solid State Commun. 142, 123–127 (2007).
Martin, I., Blanter, Y. M. & Morpurgo, A. Topological confinement in bilayer graphene. Phys. Rev. Lett. 100, 036804 (2008).
Zhang, F., MacDonald, A. H. & Mele, E. J. Valley Chern numbers and boundary modes in gapped bilayer graphene. Proc. Natl Acad. Sci. USA 110, 10546–10551 (2013).
San-Jose, P. & Prada, E. Helical networks in twisted bilayer graphene under interlayer bias. Phys. Rev. B 88, 121408 (2013).
Huang, S. et al. Topologically protected helical states in minimally twisted bilayer graphene. Phys. Rev. Lett. 121, 037702 (2018).
Tsim, B., Nam, N. N. T. & Koshino, M. Perfect one-dimensional chiral states in biased twisted bilayer graphene. Phys. Rev. B 101, 125409 (2020).
Zou, K., Zhang, F., Clapp, C., MacDonald, A. & Zhu, J. Transport studies of dual-gated ABC and ABA trilayer graphene: band gap opening and band structure tuning in very large perpendicular electric fields. Nano Lett. 13, 369–373 (2013).
Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Superconductivity in rhombohedral trilayer graphene. Nature 598, 434–438 (2021).
Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429–433 (2021).
Sau, J. D., Lutchyn, R. M., Tewari, S. & Das Sarma, S. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).
Woods, C. R. et al. Commensurate–incommensurate transition in graphene on hexagonal boron nitride. Nat. Phys. 10, 451–456 (2014).
Meyer, J. C. et al. The structure of suspended graphene sheets. Nature 446, 60–63 (2007).
Brown, L. et al. Twinning and twisting of tri- and bilayer graphene. Nano Lett. 12, 1609–1615 (2012).
Sung, S. H., Schnitzer, N., Brown, L., Park, J. & Hovden, R. Stacking, strain, and twist in 2D materials quantified by 3D electron diffraction. Phys. Rev. Mater. 3, 064003 (2019).
Lin, J. et al. AC/AB stacking boundaries in bilayer graphene. Nano Lett. 13, 3262–3268 (2013).
Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515 (2001).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Trambly de Laissardière, G., Mayou, D. & Magaud, L. Localization of Dirac electrons in rotated graphene bilayers. Nano Lett. 10, 804–808 (2010).
OpenCollab. ARPACK-ng: high-performance eigenvalue solver for large sparse matrices. GitHub https://github.com/opencollab/arpack-ng (2023).
Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. in Modern Software Tools for Scientific Computing (eds Arge, E., Bruaset, A. M. & Langtangen, H. P.) 163–202 (Springer, 1997).