Li, G. et al. Commentary of Van Hove singularities in twisted graphene layers. Nat. Phys. 6, 109–113 (2010).
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).
Chen, G. et al. Proof of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233 (2011).
Li, T. et al. Steady Mott transition in semiconductor moiré superlattices. Nature 597, 350–354 (2021).
Ghiotto, A. et al. Quantum criticality in twisted transition metallic dichalcogenides. Nature 597, 345–349 (2021).
Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).
Andrei, E. Y. et al. The marvels of moiré supplies. Nat. Rev. Mater. 6, 201–206 (2021).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
Balents, L., Dean, C. R., Efetov, D. Ok. & Younger, A. F. Superconductivity and powerful correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).
Zhang, Y., Yuan, N. F. Q. & Fu, L. Moiré quantum chemistry: cost switch in transition metallic dichalcogenide superlattices. Phys. Rev. B 102, 201115 (2020).
Novoselov, Ok. S. et al. Electrical subject impact in atomically skinny carbon movies. Science 306, 666–669 (2004).
Novoselov, Ok. S. et al. Two-dimensional fuel of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental statement of the quantum Corridor impact and Berry’s section in graphene. Nature 438, 201–204 (2005).
Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic, Z. Flat bands in barely twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010).
Mele, E. J. Commensuration and interlayer coherence in twisted bilayer graphene. Phys. Rev. B 81, 161405 (2010).
Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).
Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. H. Hubbard mannequin physics in transition metallic dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).
Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metallic dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).
Xian, L., Kennes, D. M., Tancogne-Dejean, N., Altarelli, M. & Rubio, A. Multiflat bands and powerful correlations in twisted bilayer boron nitride: doping-induced correlated insulator and superconductor. Nano Lett. 19, 4934–4940 (2019).
Zou, L., Po, H. C., Vishwanath, A. & Senthil, T. Band construction of twisted bilayer graphene: emergent symmetries, commensurate approximants, and Wannier obstructions. Phys. Rev. B 98, 085435 (2018).
Mak, Ok. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically skinny MoS2: a brand new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010).
Splendiani, A. et al. Rising photoluminescence in monolayer MoS2. Nano Lett. 10, 1271–1275 (2010).
Gustafsson, M. V. et al. Ambipolar Landau ranges and powerful band-selective service interactions in monolayer WSe2. Nat. Mater. 17, 411–415 (2018).
Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and different group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).
Mak, Ok. F., Xiao, D. & Shan, J. Gentle–valley interactions in 2D semiconductors. Nat. Photon. 12, 451–460 (2018).
Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metallic dichalcogenides. Nat. Phys. 10, 343–350 (2014).
Wang, L. et al. Correlated digital phases in twisted bilayer transition metallic dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Zhang, Z. et al. Flat bands in twisted bilayer transition metallic dichalcogenides. Nat. Phys. 16, 1093–1096 (2020).
Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).
Tang, Y. et al. Simulation of Hubbard mannequin physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).
Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).
Jin, C. et al. Commentary of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).
Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature 567, 81–86 (2019).
Tran, Ok. et al. Proof for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).
Seyler, Ok. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).
Kim, Ok. et al. van der Waals heterostructures with excessive accuracy rotational alignment. Nano Lett. 16, 1989–1995 (2016).
Cao, Y. et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016).
Chung, T.-F., Xu, Y. & Chen, Y. P. Transport measurements in twisted bilayer graphene: electron–phonon coupling and Landau stage crossing. Phys. Rev. B 98, 035425 (2018).
Rosenberger, M. R. et al. Twist angle-dependent atomic reconstruction and moiré patterns in transition metallic dichalcogenide heterostructures. ACS Nano 14, 4550–4558 (2020).
Li, H. et al. Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices. Nat. Mater. 20, 945–950 (2021).
Bai, Y. et al. Excitons in strain-induced one-dimensional moiré potentials at transition metallic dichalcogenide heterojunctions. Nat. Mater. 19, 1068–1073 (2020).
Zhang, Y., Devakul, T. & Fu, L. Spin-textured Chern bands in AB-stacked transition metallic dichalcogenide bilayers. Proc. Natl Acad. Sci. USA 118, e2112673118 (2021).
Pan, H., Wu, F. & Das Sarma, S. Band topology, Hubbard mannequin, Heisenberg mannequin, and Dzyaloshinskii–Moriya interplay in twisted bilayer WSe2. Phys. Rev. Res. 2, 033087 (2020).
Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metallic dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).
Pan, H., Wu, F. & Das Sarma, S. Quantum section diagram of a moiré-Hubbard mannequin. Phys. Rev. B 102, 201104 (2020).
Slagle, Ok. & Fu, L. Cost switch excitations, pair density waves, and superconductivity in moiré supplies. Phys. Rev. B 102, 235423 (2020).
Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).
Zhang, Y.-H., Sheng, D. N. & Vishwanath, A. SU(4) chiral spin liquid, exciton supersolid, and electrical detection in moiré bilayers. Phys. Rev. Lett. 127, 247701 (2021).
Xu, Y. et al. Tunable bilayer Hubbard mannequin physics in twisted WSe2. Preprint at https://arxiv.org/abs/2202.02055 (2022).
Mak, Ok. F. & Shan, J. Photonics and optoelectronics of 2D semiconductor transition metallic dichalcogenides. Nat. Photon. 10, 216–226 (2016).
Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).
Li, T. et al. Cost-order-enhanced capacitance in semiconductor moiré superlattices. Nat. Nanotechnol. 16, 1068–1072 (2021).
Stepanov, P. et al. Untying the insulating and superconducting orders in magic-angle graphene. Nature 583, 375–378 (2020).
Saito, Y., Ge, J., Watanabe, Ok., Taniguchi, T. & Younger, A. F. Impartial superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. 16, 926–930 (2020).
Li, T. et al. Quantum anomalous Corridor impact from intertwined moiré bands. Nature 600, 641–646 (2021).
Mott, N. F. Steel–insulator transition. Rev. Mod. Phys. 40, 677–683 (1968).
Imada, M., Fujimori, A. & Tokura, Y. Steel–insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
Senthil, T. Concept of a steady Mott transition in two dimensions. Phys. Rev. B 78, 045109 (2008).
Mishmash, R. V., González, I., Melko, R. G., Motrunich, O. I. & Fisher, M. P. A. Steady Mott transition between a metallic and a quantum spin liquid. Phys. Rev. B 91, 235140 (2015).
Lee, P. A. Moiré bands in transitional metallic dichalcogenides: steady Mott transition, quantum anomalous Corridor and extra. Journal Membership for Condensed Matter Physics https://doi.org/10.36471/JCCM_September_2021_03 (2021).
Balents, L. Spin liquids in annoyed magnets. Nature 464, 199–208 (2010).
Szasz, A., Motruk, J., Zaletel, M. P. & Moore, J. E. Chiral spin liquid section of the triangular lattice Hubbard mannequin: a density matrix renormalization group examine. Phys. Rev. 10, 021042 (2020).
Yiqing Zhou, D. N., Sheng & Kim, E.-A. Quantum phases of transition metallic dichalcogenide moiré programs. Phys. Rev. Lett. 128, 157602 (2021).
Xu, Y. et al. Steel–insulator transition with cost fractionalization. Preprint at https://arxiv.org/abs/2106.14910 (2021).
Morales-Durán, N., MacDonald, A. H. & Potasz, P. Steel–insulator transition in transition metallic dichalcogenide heterobilayer moiré superlattices. Phys. Rev. B 103, L241110 (2021).
Pan, H. & Das Sarma, S. Interplay-driven filling-induced metal-insulator transitions in 2D moiré lattices. Phys. Rev. Lett. 127, 096802 (2021).
Wietek, A. et al. Mott insulating states with competing orders within the triangular lattice Hubbard mannequin. Phys. Rev. 11, 041013 (2021).
Zang, J., Wang, J., Cano, J., Georges, A. & Millis, A. J. Dynamical imply subject idea of moiré bilayer transition metallic dichalcogenides: section diagram, resistivity, and quantum criticality. Phys. Rev. X 12, 021064 (2022).
Ahn, Seongjin & Sarma, S. D. Dysfunction induced two-dimensional metallic–insulator transition in moiré transition metallic dichalcogenide multilayers. Phys. Rev. B 105, 115114 (2021).
Huang, Y., Skinner, B. & Shklovskii, B. I. Conductivity of two-dimensional small hole semiconductors and topological insulators in robust Coulomb dysfunction. Preprint at https://arxiv.org/abs/2201.11652 (2022).
Zhang, Y., Liu, T. & Fu, L. Digital buildings, cost switch, and cost order in twisted transition metallic dichalcogenide bilayers. Phys. Rev. B 103, 155142 (2021).
Padhi, B., Chitra, R. & Phillips, P. W. Generalized Wigner crystallization in moiré supplies. Phys. Rev. B 103, 125146 (2021).
Huang, X. et al. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 17, 715–719 (2021).
Li, H. et al. Imaging two-dimensional generalized Wigner crystals. Nature 597, 650–654 (2021).
Liu, E. et al. Excitonic and valley-polarization signatures of fractional correlated digital phases in a WSe2/WS2 moiré superlattice. Phys. Rev. Lett. 127, 037402 (2021).
Emery, V. J., Kivelson, S. A. & Tranquada, J. M. Stripe phases in high-temperature superconductors. Proc. Natl Acad. Sci. USA 96, 8814 (1999).
Koulakov, A. A., Fogler, M. M. & Shklovskii, B. I. Cost density wave in two-dimensional electron liquid in weak magnetic subject. Phys. Rev. Lett. 76, 499–502 (1996).
Kivelson, S. A., Fradkin, E. & Emery, V. J. Digital liquid-crystal phases of a doped Mott insulator. Nature 393, 550–553 (1998).
Matty, M. & Kim, E.-A. Melting of generalized Wigner crystals in transition metallic dichalcogenide heterobilayer Moiré programs. Preprint at https://arxiv.org/abs/2112.08624 (2021).
Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Nat. Mater. 20, 940–944 (2021).
Camjayi, A., Haule, Ok., Dobrosavljević, V. & Kotliar, G. Coulomb correlations and the Wigner–Mott transition. Nat. Phys. 4, 932–935 (2008).
Musser, S., Senthil, T & Chowdhury, D. Concept of a steady bandwidth-tuned Wigner–Mott transition. Preprint at https://arxiv.org/abs/2111.09894 (2021).
Wigner, E. On the interplay of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).
Tang, Y. et al. Dielectric disaster on the Mott and Wigner transitions in a moiré superlattice. Preprint at https://arxiv.org/abs/2201.12510 (2022).
Kane, C. L. & Mele, E. J. Quantum spin corridor impact in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Hohenadler, M. & Assaad, F. F. Correlation results in two-dimensional topological insulators. J. Phys. Condens. Matter 25, 143201 (2013).
Witczak-Krempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated quantum phenomena within the robust spin-orbit regime. Annu. Rev. Condens. Matter Phys. 5, 57–82 (2014).
Pan, H., Xie, M., Wu, F. & Sarma, S. D. Topological phases in AB-stacked MoTe2/WSe2: ℤ2 topological insulators, Chern insulators, and topological cost density waves. Preprint at https://arxiv.org/abs/2111.01152 (2021).
Regnault, N. & Bernevig, B. A. Fractional chern insulator. Phys. Rev. 1, 021014 (2011).
Serlin, M. et al. Intrinsic quantized anomalous Corridor impact in a moiré heterostructure. Science 367, 900–903 (2020).
Sharpe Aaron, L. et al. Emergent ferromagnetism close to three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Ezawa, M., Tanaka, Y. & Nagaosa, N. Topological section transition with out hole closing. Sci. Rep. 3, 2790 (2013).
Xie, Y.-M., Zhang, C.-P., Hu, J.-X., Mak, Ok. F. & Regulation, Ok. Valley polarized quantum anomalous Corridor state in moiré MoTe2/WSe2 heterobilayers. Phys. Rev. Lett. 128, 026402 (2021).
Chang, Y.-W. & Chang, Y.-C. Concept of quantum anomalous Corridor impact and electric-field-induced section transition in AB-stacked MoTe2/WSe2 moire heterobilayers. Preprint at https://arxiv.org/abs/2203.10088 (2022).
Rademaker, L. Spin–orbit coupling in transition metallic dichalcogenide heterobilayer flat bands. Phys. Rev. B 105, 195428 (2022).
Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in moiré heterojunctions. Phys. Rev. Lett. 118, 147401 (2017).
Yu, H., Liu, G.-B., Tang, J., Xu, X. & Yao, W. Moiré excitons: from programmable quantum emitter arrays to spin–orbit-coupled synthetic lattices. Sci. Adv. 3, e1701696 (2017).
Ruiz-Tijerina, D. A. & Fal’ko, V. I. Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides. Phys. Rev. B 99, 125424 (2019).
Shimazaki, Y. et al. Optical signatures of periodic cost distribution in a Mott-like correlated insulator state. Phys. Rev. 11, 021027 (2021).
Wilson, N. P., Yao, W., Shan, J. & Xu, X. Excitons and emergent quantum phenomena in stacked 2D semiconductors. Nature 599, 383–392 (2021).
Huang, D., Choi, J., Shih, C.-Ok. & Li, X. Excitons in semiconductor moiré superlattices. Nat. Nanotechnol. 17, 227–238 (2022).
Tang, Y. et al. Tuning layer-hybridized moiré excitons by the quantum-confined Stark impact. Nat. Nanotechnol. 16, 52–57 (2021).
Zhang, L. et al. Van der Waals heterostructure polaritons with moiré-induced nonlinearity. Nature 591, 61–65 (2021).
Liu, E. et al. Signatures of moiré trions in WSe2/MoSe2 heterobilayers. Nature 594, 46–50 (2021).
Wang, X. et al. Moiré trions in MoSe2/WSe2 heterobilayers. Nat. Nanotechnol. 16, 1208–1213 (2021).
Brotons-Gisbert, M. et al. Moiré-trapped interlayer trions in a charge-tunable WSe2/MoSe2 heterobilayer. Phys. Rev. 11, 031033 (2021).
Marcellina, E. et al. Proof for moiré trions in twisted MoSe2 homobilayers. Nano Lett. 21, 4461–4468 (2021).
Gu, J. et al. Dipolar excitonic insulator in a moiré lattice. Nat. Phys. 18, 395–400 (2022).
Zuocheng, Z. et al. Correlated interlayer exciton insulator in double layers of monolayer WSe2 and moiré WS2/WSe2. Preprint at https://arxiv.org/abs/2108.07131 (2021).
Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).
Dutta, O. et al. Non-standard Hubbard fashions in optical lattices: a overview. Rep. Prog. Phys. 78, 066001 (2015).
Xie, M. & MacDonald, A. H. Electrical reservoirs for bilayer excitons. Phys. Rev. Lett. 121, 067702 (2018).
Ma, L. et al. Strongly correlated excitonic insulator in atomic double layers. Nature 598, 585–589 (2021).
Zeng, Y. & MacDonald, A. H. Electrically managed two-dimensional electron-hole fluids. Phys. Rev. B 102, 085154 (2020).
Eisenstein, J. P. & MacDonald, A. H. Bose–Einstein condensation of excitons in bilayer electron programs. Nature 432, 691–694 (2004).
Zhang, Y.-H. Doping a Mott insulator with excitons in moiré bilayer: fractional superfluid, impartial Fermi floor and Mott transition. Preprint at https://arxiv.org/abs/2204.10937 (2022).
Angeli, M. & MacDonald Allan, H. Γ valley transition metallic dichalcogenide moiré bands. Proc. Natl Acad. Sci. USA 118, e2021826118 (2021).
Kumar, A., Hu, N. C., MacDonald, A. H. & Potter, A. C. Gate-tunable heavy fermion quantum criticality in a moiré Kondo lattice. Preprint at https://arxiv.org/abs/2110.11962 (2021).
Dalal, A. & Ruhman, J. Orbitally selective Mott section in electron-doped twisted transition metal-dichalcogenides: a attainable realization of the Kondo lattice mannequin. Phys. Rev. Res. 3, 043173 (2021).
Zhang, Y.-H. & Vishwanath, A. Electrical detection of spin liquids in double moiré layers. Preprint at https://arxiv.org/abs/2005.12925 (2020).
Xia, F., Wang, H., Hwang, J. C. M., Neto, A. H. C. & Yang, L. Black phosphorus and its isoelectronic supplies. Nat. Rev. Phys. 1, 306–317 (2019).
Chaves, A. et al. Bandgap engineering of two-dimensional semiconductor supplies. npj 2D Mater. Appl. 4, 29 (2020).
McGuire, M. A. Crystal and magnetic buildings in layered, transition metallic dihalides and trihalides. Crystals 7, 121 (2017).
Mak, Ok. F., Shan, J. & Ralph, D. C. Probing and controlling magnetic states in 2D layered magnetic supplies. Nat. Rev. Phys. 1, 646–661 (2019).
Hejazi, Ok., Luo, Z.-X. & Balents, L. Noncollinear phases in moiré magnets. Proc. Natl Acad. Sci. USA 117, 10721 (2020).
Tong, Q., Liu, F., Xiao, J. & Yao, W. Skyrmions within the moiré of van der Waals 2D magnets. Nano Lett. 18, 7194–7199 (2018).
Akram, M. et al. Moiré skyrmions and chiral magnetic phases in twisted CrX3 (X = I, Br, and Cl) bilayers. Nano Lett. 21, 6633–6639 (2021).
Wang, C., Gao, Y., Lv, H., Xu, X. & Xiao, D. Stacking area wall magnons in twisted van der Waals magnets. Phys. Rev. Lett. 125, 247201 (2020).
Xu, Y. et al. Coexisting ferromagnetic–antiferromagnetic state in twisted bilayer CrI3. Nat. Nanotechnol. 17, 143–147 (2022).
Xie, H. et al. Twist engineering of the two-dimensional magnetism in double bilayer chromium triiodide homostructures. Nat. Phys. 18, 30–36 (2022).
Track, T. et al. Direct visualization of magnetic domains and moiré magnetism in twisted 2D magnets. Science 374, 1140–1144 (2021).
Edelberg, D. et al. Approaching the intrinsic restrict in transition metallic diselenides through level defect management. Nano Lett. 19, 4371–4379 (2019).
