Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
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).
Castellanos-Gomez, A. et al. Van der waals heterostructures. Nat. Rev. Methods Prim. 2, 58 (2022).
Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).
Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).
Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).
Wu, F., MacDonald, A. H. & Martin, I. Theory of phonon-mediated superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 121, 257001 (2018).
Lian, B., Wang, Z. & Bernevig, B. A. Twisted bilayer graphene: a phonon-driven superconductor. Phys. Rev. Lett. 122, 257002 (2019).
Wu, F., Hwang, E. & Das Sarma, S. Phonon-induced giant linear-in-T resistivity in magic angle twisted bilayer graphene: ordinary strangeness and exotic superconductivity. Phys. Rev. B 99, 165112 (2019).
Shavit, G., Berg, E., Stern, A. & Oreg, Y. Theory of correlated insulators and superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 127, 247703 (2021).
Liu, C.-X., Chen, Y., Yazdani, A. & Bernevig, B. A. Electron–K-phonon interaction in twisted bilayer graphene. Phys. Rev. B 110, 045133 (2024).
Wang, Y.-J., Zhou, G.-D., Peng, S.-Y., Lian, B. & Song, Z.-D. Molecular pairing in twisted bilayer graphene superconductivity. Phys. Rev. Lett. 133, 146001 (2024).
Wang, Y.-J., Zhou, G.-D., Lian, B. & Song, Z.-D. Electron-phonon coupling in the topological heavy fermion model of twisted bilayer graphene. Phys. Rev. B 111, 035110 (2025).
Ochi, M., Koshino, M. & Kuroki, K. Possible correlated insulating states in magic-angle twisted bilayer graphene under strongly competing interactions. Phys. Rev. B 98, 081102 (2018).
Chou, Y.-Z., Wu, F., Sau, J. D. & Das Sarma, S. Acoustic-phonon-mediated superconductivity in bernal bilayer graphene. Phys. Rev. B 105, L100503 (2022).
Isobe, H., Yuan, N. F. Q. & Fu, L. Unconventional superconductivity and density waves in twisted bilayer graphene. Phys. Rev. X 8, 041041 (2018).
Angeli, M., Tosatti, E. & Fabrizio, M. Valley jahn-teller effect in twisted bilayer graphene. Phys. Rev. X 9, 041010 (2019).
Blason, A. & Fabrizio, M. Local kekulé distortion turns twisted bilayer graphene into topological mott insulators and superconductors. Phys. Rev. B 106, 235112 (2022).
Christos, M., Sachdev, S. & Scheurer, M. S. Nodal band-off-diagonal superconductivity in twisted graphene superlattices. Nat. Commun. 14, 7134 (2023).
Islam, S. F., Zyuzin, A. Y. & Zyuzin, A. A. Unconventional superconductivity with preformed pairs in twisted bilayer graphene. Phys. Rev. B 107, L060503 (2023).
Song, Z.-D. & Bernevig, B. A. Magic-angle twisted bilayer graphene as a topological heavy fermion problem. Phys. Rev. Lett. 129, 047601 (2022).
Dodaro, J. F., Kivelson, S. A., Schattner, Y., Sun, X.-Q. & Wang, C. Phases of a phenomenological model of twisted bilayer graphene. Phys. Rev. B 98, 075154 (2018).
You, Y.-Z. & Vishwanath, A. Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene. npj Quantum Mater. 4, 16 (2019).
Khalaf, E., Chatterjee, S., Bultinck, N., Zaletel, M. P. & Vishwanath, A. Charged skyrmions and topological origin of superconductivity in magic-angle graphene. Sci. Adv. 7, eabf5299 (2021).
Löthman, T., Schmidt, J., Parhizgar, F. & Black-Schaffer, A. M. Nematic superconductivity in magic-angle twisted bilayer graphene from atomistic modeling. Commun. Phys. 5, 92 (2022).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
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).
Hao, Z. et al. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).
Cao, Y., Park, J. M., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Pauli-limit violation and re-entrant superconductivity in moiré graphene. Nature 595, 526–531 (2021).
Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene. Nature 606, 494–500 (2022).
Zhou, H. et al. Isospin magnetism and spin-polarized superconductivity in bernal bilayer graphene. Science 375, 774–778 (2022).
Zhang, Y. et al. Enhanced superconductivity in spin–orbit proximitized bilayer graphene. Nature 613, 268–273 (2023).
Arora, H. S. et al. Superconductivity in metallic twisted bilayer graphene stabilized by WSe2. Nature 583, 379–384 (2020).
Holleis, L. et al. Nematicity and orbital depairing in superconducting bernal bilayer graphene. Nature Physics 21, 444–450 (2025).
Li, C. et al. Tunable superconductivity in electron-and hole-doped bernal bilayer graphene. Nature 631, 300–306 (2024).
Su, R., Kuiri, M., Watanabe, K., Taniguchi, T. & Folk, J. Superconductivity in twisted double bilayer graphene stabilized by WSe2. Nat. Mater. 22, 1332–1337 (2023).
Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Superconductivity in rhombohedral trilayer graphene. Nature 598, 434–438 (2021).
Han, T. et al. Signatures of chiral superconductivity in rhombohedral graphene. Nature 643, 654–661 (2025).
Choi, Y. et al. Superconductivity and quantized anomalous Hall effect in rhombohedral graphene. Nature 639, 342–347 (2025).
Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. H. Hubbard model physics in transition metal 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 metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).
Tang, Y. et al. Simulation of hubbard model 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).
Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).
Li, T. et al. Continuous mott transition in semiconductor moiré superlattices. Nature 597, 350–354 (2021).
Ghiotto, A. et al. Quantum criticality in twisted transition metal dichalcogenides. Nature 597, 345–349 (2021).
Xu, Y. et al. A tunable bilayer Hubbard model in twisted WSe2. Nat. Nanotechnol. 17, 934–939 (2022).
Li, T. et al. Quantum anomalous hall effect from intertwined moiré bands. Nature 600, 641–646 (2021).
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. X 13, 031037 (2023).
Kang, K. et al. Evidence of the fractional quantum spin hall effect in moiré MoTe2. Nature 628, 522–526 (2024).
Wietek, A. et al. Tunable stripe order and weak superconductivity in the moiré hubbard model. Phys. Rev. Res. 4, 043048 (2022).
Klebl, L., Fischer, A., Classen, L., Scherer, M. M. & Kennes, D. M. Competition of density waves and superconductivity in twisted tungsten diselenide. Phys. Rev. Res. 5, L012034 (2023).
Zhou, B. & Zhang, Y.-H. Chiral and nodal superconductors in the t−J model with valley contrasting flux on a triangular moiré lattice. Phys. Rev. B 108, 155111 (2023).
Bélanger, M., Fournier, J. & Sénéchal, D. Superconductivity in the twisted bilayer transition metal dichalcogenide WSe2: a quantum cluster study. Phys. Rev. B 106, 235135 (2022).
Chen, F. & Sheng, D. N. Singlet, triplet, and pair density wave superconductivity in the doped triangular-lattice moiré system. Phys. Rev. B 108, L201110 (2023).
Zegrodnik, M. & Biborski, A. Mixed singlet-triplet superconducting state within the moiré t − J − U model applied to twisted bilayer WSe2. Phys. Rev. B 108, 064506 (2023).
Akbar, W., Biborski, A., Rademaker, L. & Zegrodnik, M. Topological superconductivity with mixed singlet-triplet pairing in moiré transition-metal-dichalcogenide bilayers. Phys. Rev. B 110, 064516 (2024).
Wu, Y.-M., Wu, Z. & Yao, H. Pair-density-wave and chiral superconductivity in twisted bilayer transition metal dichalcogenides. Phys. Rev. Lett. 130, 126001 (2023).
Xie, Y.-M. & Law, K. T. Orbital fulde-ferrell pairing state in moiré ising superconductors. Phys. Rev. Lett. 131, 016001 (2023).
Schrade, C. & Fu, L. Nematic, chiral, and topological superconductivity in twisted transition metal dichalcogenides. Phys. Rev. B 110, 035143 (2024).
Crépel, V., Guerci, D., Cano, J., Pixley, J. & Millis, A. Topological superconductivity in doped magnetic moiré semiconductors. Phys. Rev. Lett. 131, 056001 (2023).
Xia, Y. et al. Superconductivity in twisted bilayer WSe2. Nature 637, 833–838 (2025).
Guo, Y. et al. Superconductivity in 5.0° twisted bilayer WSe2. Nature 637, 839–845 (2025).
Pan, H., Wu, F. & Das Sarma, S. Band topology, hubbard model, heisenberg model, and dzyaloshinskii-moriya interaction in twisted bilayer WSe2. Phys. Rev. Res. 2, 033087 (2020).
Zang, J., Wang, J., Cano, J. & Millis, A. J. Hartree-fock study of the moiré hubbard model for twisted bilayer transition metal dichalcogenides. Phys. Rev. B 104, 075150 (2021).
Kiese, D., He, Y., Hickey, C., Rubio, A. & Kennes, D. M. TMDs as a platform for spin liquid physics: A strong coupling study of twisted bilayer WSe2. APL Mater. 10, 031113 (2022).
Pan, H., Wu, F. & Das Sarma, S. Quantum phase diagram of a moiré-hubbard model. Phys. Rev. B 102, 201104 (2020).
Kundu, S., Naik, M. H., Krishnamurthy, H. R. & Jain, M. Moiré induced topology and flat bands in twisted bilayer WSe2: a first-principles study. Phys. Rev. B 105, L081108 (2022).
Zhang, X.-W. et al. Polarization-driven band topology evolution in twisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).
Foutty, B. A. et al. Mapping twist-tuned multiband topology in bilayer WSe2. Science 384, 343–347 (2024).
Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).
Qiu, W.-X., Li, B., Luo, X.-J. & Wu, F. Interaction-driven topological phase diagram of twisted bilayer MoTe2. Phys. Rev. X 13, 041026 (2023).
Xu, C., Li, J., Xu, Y., Bi, Z. & Zhang, Y. Maximally localized wannier functions, interaction models, and fractional quantum anomalous hall effect in twisted bilayer MoTe2. Proc. Natl Acad. Sci. 121, e2316749121 (2024).
Crépel, V. & Millis, A. Bridging the small and large in twisted transition metal dichalcogenide homobilayers: a tight binding model capturing orbital interference and topology across a wide range of twist angles. Phys. Rev. Res. 6, 033127 (2024).
Koshino, M. et al. Maximally localized wannier orbitals and the extended hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).
Kang, J. & Vafek, O. Symmetry, maximally localized wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).
Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175 (1957).
McMillan, W. Transition temperature of strong-coupled superconductors. Phys. Rev. 167, 331 (1968).
Cea, T. & Guinea, F. Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene. Proc. Natl Acad. Sci. 118, e2107874118 (2021).
Ghazaryan, A., Holder, T., Serbyn, M. & Berg, E. Unconventional superconductivity in systems with annular fermi surfaces: Application to rhombohedral trilayer graphene. Phys. Rev. Lett. 127, 247001 (2021).
Guo, H., Zhu, X., Feng, S. & Scalettar, R. T. Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice. Phys. Rev. B 97, 235453 (2018).
Ray, S., Jung, J. & Das, T. Wannier pairs in superconducting twisted bilayer graphene and related systems. Phys. Rev. B 99, 134515 (2019).
Chatterjee, S., Wang, T., Berg, E. & Zaletel, M. P. Inter-valley coherent order and isospin fluctuation mediated superconductivity in rhombohedral trilayer graphene. Nat. Commun. 13, 6013 (2022).
Kohn, W. & Luttinger, J. New mechanism for superconductivity. Phys. Rev. Lett. 15, 524 (1965).
Monthoux, P., Balatsky, A. & Pines, D. Toward a theory of high-temperature superconductivity in the antiferromagnetically correlated cuprate oxides. Phys. Rev. Lett. 67, 3448 (1991).
Scalapino, D. J. The case for dx2- y2 pairing in the cuprate superconductors. Phys. Rep. 250, 329–365 (1995).
Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Chiu, C.-K., Teo, J. C., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Chen, K. S. et al. Unconventional superconductivity on the triangular lattice hubbard model. Phys. Rev. B 88, 041103 (2013).
Venderley, J. & Kim, E.-A. Density matrix renormalization group study of superconductivity in the triangular lattice hubbard model. Phys. Rev. B 100, 060506 (2019).
Fallahazad, B. et al. Shubnikov–de haas oscillations of high-mobility holes in monolayer and bilayer WSe2: Landau level degeneracy, effective mass, and negative compressibility. Phys. Rev. Lett. 116, 086601 (2016).
Mounet, N. et al. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246–252 (2018).
Cheng, M., Sun, K., Galitski, V. & Das Sarma, S. Stable topological superconductivity in a family of two-dimensional fermion models. Phys. Rev. B 81, 024504 (2010).
Deutscher, G. Percolation and Superconductivity, 95–113 (Springer, 1984).
Alexander, S. Superconductivity of networks. A percolation approach to the effects of disorder. Phys. Rev. B 27, 1541–1557 (1983).
Fischer, A. et al. Theory of intervalley-coherent afm order and topological superconductivity in tWSe2. Preprint at https://arxiv.org/abs/2412.14296 (2024).
Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).
Julku, A., Peotta, S., Vanhala, T. I., Kim, D.-H. & Törmä, P. Geometric origin of superfluidity in the Lieb-lattice flat band. Phys. Rev. Lett. 117, 045303 (2016).
Liang, L. et al. Band geometry, berry curvature, and superfluid weight. Phys. Rev. B 95, 024515 (2017).
Herzog-Arbeitman, J., Peri, V., Schindler, F., Huber, S. D. & Bernevig, B. A. Superfluid weight bounds from symmetry and quantum geometry in flat bands. Phys. Rev. Lett. 128, 087002 (2022).
Huhtinen, K.-E., Herzog-Arbeitman, J., Chew, A., Bernevig, B. A. & Törmä, P. Revisiting flat band superconductivity: dependence on minimal quantum metric and band touchings. Phys. Rev. B 106, 014518 (2022).
Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).
Kasahara, Y. et al. Majorana quantization and half-integer thermal quantum hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).
Banerjee, M. et al. Observed quantization of anyonic heat flow. Nature 545, 75–79 (2017).
Banerjee, M. et al. Observation of half-integer thermal hall conductance. Nature 559, 205–210 (2018).
Wollman, D. A., Van Harlingen, D. J., Lee, W. C., Ginsberg, D. M. & Leggett, A. J. Experimental determination of the superconducting pairing state in YBCO from the phase coherence of YBCO-Pb dc SQUIDs. Phys. Rev. Lett. 71, 2134–2137 (1993).
Van Harlingen, D. J. Phase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors-evidence for \({{{{\rm{d}}}}}_{{x}^{2}-{y}^{2}}\) symmetry. Rev. Mod. Phys. 67, 515 (1995).
Tsuei, C. & Kirtley, J. Pairing symmetry in cuprate superconductors. Rev. Mod. Phys. 72, 969 (2000).
Kim, S., Mendez-Valderrama, J. F., Wang, X. & Chowdhury, D. Theory of correlated insulators and superconductor at ν = 1 in twisted WSe2. Nat. Commun. 16, 1701 (2025)
Myerson-Jain, N. & Xu, C. Superconductor-insulator transition in the TMD moiré systems and the deconfined quantum critical point. Preprint at https://arxiv.org/abs/2406.12971 (2024).
Zhu, J., Chou, Y.-Z., Xie, M. & Sarma, S. D. Superconductivity in twisted transition metal dichalcogenide homobilayers. Phys. Rev. B 111, L060501 (2025).
Christos, M., Bonetti, P. M. & Scheurer, M. S. Approximate symmetries, insulators, and superconductivity in the continuum-model description of twisted WSe2. Phys. Rev. Lett. 135, 046503 (2025).
Xie, F. et al. Superconductivity in twisted WSe2 from topology-induced quantum fluctuations. Phys. Rev. Lett. 134, 136503 (2025).
Guerci, D., Kaplan, D., Ingham, J., Pixley, J. & Millis, A. J. Topological superconductivity from repulsive interactions in twisted WSe2. Preprint at https://arxiv.org/abs/2408.16075 (2024).