We have found that superconductivity and superfluidity are connected to quantum geometry [1,2], and shown that superconductivity is possible also in a flat band where individual electrons would not move. These results may be relevant for explaining the observation of superconductivity in twisted bilayer graphene [3]; we discuss our new results on this [4] and other quantum materials [5]. The quantum transport in flat band shows unique behavior [6]: while supercurrent can flow, quasiparticle transport is highly suppressed even in non-equilibrium conditions. This may have important consequences for superconducting devices. We have predicted that flat band systems as part of Josephson junctions can lead to behavior distinct from the dispersive case and discuss a related experiment [7]. Furthermore, we present results on superconductivity in surface flat bands of rhombohedral graphite [8]. We also discuss new phenomena unique to flat bands [9].

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[2] P Törmä, Phys. Rev. Lett. 131, 240001 (2023); J. Yu, B.A. Bernevig, R. Queiroz, E. Rossi, P. Törmä, B.-J. Yang, njp Quantum Materials 10, 101 (2025).

[3] A. Julku, T.J. Peltonen, L. Liang, T.T. Heikkilä, P. Törmä, Phys. Rev. B 101, 060505(R) (2020); X. Hu, T. Hyart, D.I. Pikulin, E. Rossi, Phys. Rev. Lett. 123, 237002 (2019); F. Xie, Z. Song, B. Lian, B.A. Bernevig, Phys. Rev. Lett. 124, 167002 (2020); P. Törmä, S. Peotta, B.A. Bernevig, Nat. Rev. Phys. 4, 528 (2022).

[4] A. Datta et al., in preparation (2026).

[5] M. Gutierrez-Amigo et al., in preparation (2026).

[6] V.A.J. Pyykkönen, S. Peotta, P. Törmä, Phys. Rev. Lett. 130, 216003 (2023).

[7] P. Virtanen, R.P.S. Penttilä, P. Törmä, A. Díez-Carlón, D.K. Efetov, T.T. Heikkilä, Phys. Rev. B 112, L100502 (2025); A. Diez-Carlon, J. Diez-Merida, P. Rout, D. Sedov, P. Virtanen, S. Banerjee, R.P.S. Penttilä, P. Altpeter, K. Watanabe, T. Taniguchi, S.-Y. Yang, K.T. Law, T.T. Heikkilä, P. Törmä, M.S. Scheurer, D.K. Efetov, Phys. Rev. X 15, 041033 (2025).

[8] G. Jiang, T.T. Heikkilä, P. Törmä, Phys. Rev. B 113, L041111 (2026).

[9] G. Jiang, A. Dunbrack, T.T. Heikkilä, P. Törmä, in preparation (2026).