Lee, P. A. & Wen, X.-G. Unusual superconducting state of underdoped cuprates. Phys. Rev. Lett. 78, 4111 (1997).
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Millis, A. J., Girvin, S. M., Ioffe, L. B. & Larkin, A. I. Anomalous charge dynamics in the superconducting state of underdoped cuprates. J. Phys. Chem. Solids 59, 1742–1744 (1998).
Yip, S. K. & Sauls, J. A. Nonlinear Meissner effect in CuO superconductors. Phys. Rev. Lett. 69, 2264 (1992).
Xu, D., Yip, S. K. & Sauls, J. A. Nonlinear Meissner effect in unconventional superconductors. Phys. Rev. B 51, 16233 (1995).
Hardy, W. N., Bonn, D. A., Morgan, D. C., Liang, R. & Zhang, K. Precision measurements of the temperature dependence of λ in YBCO: strong evidence for nodes in the gap function. Phys. Rev. Lett. 70, 3999 (1993).
Uemura, Y. J. et al. Universal correlations between Tc and ns/m* (carrier density over effective mass) in high-Tc cuprate superconductors. Phys. Rev. Lett. 62, 2317 (1989).
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
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).
Park, J. M. et al. Robust superconductivity in magic-angle multilayer graphene family. Nat. Mater. 21, 877–883 (2022).
Zhang, Y. et al. Promotion of superconductivity in magic-angle graphene multilayers. Science 377, 1538–1543 (2022).
Cao, Y. et al. Nematicity and competing orders in superconducting magic-angle graphene. Science 372, 264–271 (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).
Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).
Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).
Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene. Nature 606, 494–500 (2022).
Schoelkopf, R. J., Wahlgren, P., Kozhevnikov, A. A., Delsing, P. & Prober, D. E. The radio-frequency single-electron transistor (RF-SET): a fast and ultrasensitive electrometer. Science 280, 1238–1242 (1998).
Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Fast single-charge sensing with a RF quantum point contact. Appl. Phys. Lett. 91, 162101 (2007).
Vigneau, F. et al. Probing quantum devices with radio-frequency reflectometry. Appl. Phys. Rev. 10, 021305 (2023).
Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene. Science 351, 1058–1061 (2016).
Annunziata, A. J. et al. Tunable superconducting nanoinductors. Nanotechnology 21, 445202 (2010).
Singh, G. et al. Competition between electron pairing and phase coherence in superconducting interfaces. Nat. Commun. 9, 407 (2018).
Weitzel, A. et al. Sharpness of the Berezinskii–Kosterlitz–Thouless transition in disordered NbN films. Phys. Rev. Lett. 131, 186002 (2023).
Haller, R. et al. Phase-dependent microwave response of a graphene Josephson junction. Phys. Rev. Res. 4, 013198 (2022).
Phan, D. et al. Detecting induced p ± ip pairing at the Al–InAs interface with a quantum microwave circuit. Phys. Rev. Lett. 128, 107701 (2022).
Bøttcher, C. G. L. et al. Circuit quantum electrodynamics detection of induced two-fold anisotropic pairing in a hybrid superconductor-ferromagnet bilayer. Nat. Phys. 20, 1609–1615 (2024).
Tanaka, M. et al. Superfluid stiffness of magic-angle twisted bilayer graphene. Nature https://doi.org/10.1038/s41586-024-08494-7 (2025).
Kreidel, M. et al. Measuring kinetic inductance and superfluid stiffness of two-dimensional superconductors using high-quality transmission-line resonators. Preprint at https://arxiv.org/abs/2407.09916 (2024).
Tinkham, M. Introduction to Superconductivity 2nd edn (Dover Publications, 2004).
Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).
Probst, S., Song, F. B., Bushev, P. A., Ustinov, A. V. & Weides, M. Efficient and robust analysis of complex scattering data under noise in microwave resonators. Rev. Sci. Instrum. 86, 024706 (2015).
Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of two-dimensional superfluids. Phys. Rev. Lett. 39, 1201 (1977).
Turkel, S. et al. Orderly disorder in magic-angle twisted trilayer graphene. Science 376, 193–199 (2022).
Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020).
Homes, C. C. et al. A universal scaling relation in high-temperature superconductors. Nature 430, 539–541 (2004).
Dordevic, S. V., Basov, D. N. & Homes, C. C. Do organic and other exotic superconductors fail universal scaling relations? Sci. Rep. 3, 1713 (2013).
Božović, I., He, X., Wu, J. & Bollinger, A. T. Dependence of the critical temperature in overdoped copper oxides on superfluid density. Nature 536, 309–311 (2016).
Bidinosti, C. P., Hardy, W. N., Bonn, D. A. & Liang, R. Magnetic field dependence of λ in YBa2Cu3O6.95: results as a function of temperature and field orientation. Phys. Rev. Lett. 83, 3277–3280 (1999).
Oates, D. E., Park, S.-H. & Koren, G. Observation of the nonlinear Meissner effect in YBCO thin films: evidence for a d-wave order parameter in the bulk of the cuprate superconductors. Phys. Rev. Lett. 93, 197001 (2004).
Wilcox, J. A. et al. Observation of the non-linear Meissner effect. Nat. Commun. 13, 1201 (2022).
Dahm, T. & Scalapino, D. J. Theory of microwave intermodulation in a high-Tc superconducting microstrip resonator. Appl. Phys. Lett. 69, 4248–4250 (1996).
Bae, S. et al. Dielectric resonator method for determining gap symmetry of superconductors through anisotropic nonlinear Meissner effect. Rev. Sci. Instrum. 90, 043901 (2019).
de Visser, P. J. et al. Evidence of a nonequilibrium distribution of quasiparticles in the microwave response of a superconducting aluminum resonator. Phys. Rev. Lett. 112, 047004 (2014).
Turneaure, S. J., Lemberger, T. R. & Graybeal, J. M. Effect of thermal phase fluctuations on the superfluid density of two-dimensional superconducting films. Phys. Rev. Lett. 84, 987–990 (2000).
Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).
Peotta, S., Huhtinen, K.-E. & Törmä, P. Quantum geometry in superfluidity and superconductivity. Preprint at https://arxiv.org/abs/2308.08248 (2023).
Verma, N., Guerci, D. & Queiroz, R. Geometric stiffness in interlayer exciton condensates. Phys. Rev. Lett. 132, 236001 (2023).
Mendez-Valderrama, J. F., Mao, D. & Chowdhury, D. Low-energy optical sum rule in moiré graphene. Phys. Rev. Lett. 133, 196501 (2023).
Mao, D. & Chowdhury, D. Diamagnetic response and phase stiffness for interacting isolated narrow bands. Proc. Natl Acad. Sci. USA 120, e2217816120 (2023).
Mao, D. & Chowdhury, D. Upper bounds on superconducting and excitonic phase stiffness for interacting isolated narrow bands. Phys. Rev. B 109, 024507 (2024).
Ioffe, L. B. & Millis, A. J. d-wave superconductivity in doped Mott insulators. J. Phys. Chem. Solids 63, 2259–2268 (2002).
Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Benfatto, L., Castellani, C. & Giamarchi, T. Kosterlitz–Thouless behavior in layered superconductors: the role of the vortex core energy. Phys. Rev. Lett. 98, 117008 (2007).
Hetel, I., Lemberger, T. R. & Randeria, M. Quantum critical behaviour in the superfluid density of strongly underdoped ultrathin copper oxide films. Nat. Phys. 3, 700–702 (2007).
Hazra, T., Verma, N. & Randeria, M. Upper bounds on the superfluid stiffness and superconducting Tc: applications to twisted-bilayer graphene and ultra-cold Fermi gases. Phys. Rev. X 9, 031049 (2019).
Mahmood, F., He, X., Bozovic, I. & Armitage, N. P. Locating the missing superconducting electrons in the overdoped cuprates La2−xSrxCuO4. Phys. Rev. Lett. 122, 027003 (2019).
Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 123, 237002 (2019).
Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).
Ganiev, O. K. & Yavidov, B. Superfluid density and critical current density in superconducting cuprates with an extended d-wave pairing symmetry. Eur. Phys. J. B 94, 116 (2021).
