Superconductivity, superfluidity and quantum geometry in twisted multilayer systems

Superconductivity has been observed in moiré systems such as twisted bilayer graphene, which host flat, dispersionless electronic bands. In parallel, theory work has discovered that superconductivity and superfluidity of flat-band systems can be made possible by the quantum geometry and topology of the band structure. These recent key developments are merging into a flourishing research topic: understanding the possible connection and ramifications of quantum geometry on the induced superconductivity and superfluidity in moiré multilayer and other flat-band systems. This article presents an introduction to how quantum geometry governs superconductivity and superfluidity in platforms including, and beyond, graphene. Ultracold gases are introduced as a complementary platform for quantum geometric effects and a comparison is made to moiré materials. An outlook sketches the prospects of twisted multilayer systems in providing the route to room-temperature superconductivity. Flat bands enhance the effect of electronic interactions and have emerged as a promising platform for superconductivity. This Review explains the quantum geometric origin of flat-band superconductivity and superfluidity, and discusses its relevance in graphene and ultracold gas moiré systems. Bands of (quasi-)flat dispersion dramatically enhance Cooper pairing as their (nearly) vanishing kinetic energy allows interaction effects to dominate. Superfluidity and stable supercurrents are possible in a flat band if the band has non-trivial quantum geometry: the related overlap of the Wannier functions facilitates movement of interacting particles even when non-interacting particles would be localized. Twisted bilayer graphene exhibits nearly flat bands at its Fermi energy for small twist angles. Theory work suggests that quantum geometry is essential for the experimentally observed twisted bilayer graphene superconductivity and that a topological invariant called Euler class provides a lower bound for its superfluid weight. Ultracold gases offer another promising platform for highly controllable studies of superfluidity in moiré geometries. The first experiments are on the way. A flat-band dispersion together with a quantum geometry that guarantees superfluidity are powerful guidelines for the search of superconductivity at elevated temperatures. Bands of (quasi-)flat dispersion dramatically enhance Cooper pairing as their (nearly) vanishing kinetic energy allows interaction effects to dominate. Superfluidity and stable supercurrents are possible in a flat band if the band has non-trivial quantum geometry: the related overlap of the Wannier functions facilitates movement of interacting particles even when non-interacting particles would be localized. Twisted bilayer graphene exhibits nearly flat bands at its Fermi energy for small twist angles. Theory work suggests that quantum geometry is essential for the experimentally observed twisted bilayer graphene superconductivity and that a topological invariant called Euler class provides a lower bound for its superfluid weight. Ultracold gases offer another promising platform for highly controllable studies of superfluidity in moiré geometries. The first experiments are on the way. A flat-band dispersion together with a quantum geometry that guarantees superfluidity are powerful guidelines for the search of superconductivity at elevated temperatures.