![]() Symmetry-based indicators of band topology in the 230 space groups. Topological classification of crystalline insulators through band structure combinatorics. Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. Catalogue of flat band stoichiometric materials. Orbital design of flat bands in non-line-graph lattices via line-graph wavefunctions. Artificial flat band systems: from lattice models to experiments. Strongly correlated flat-band systems: the route from Heisenberg spins to Hubbard electrons. From dia- to paramagnetic orbital susceptibility of massless fermions. Raoux, A., Morigi, M., Fuchs, J.-N., Piéchon, F. Z Q topological invariants for polyacetylene, kagome and pyrochlore lattices. Isolated flat bands and spin-1 conical bands in two-dimensional lattices. Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Electronic properties of an amorphous solid. Spin-orbit-induced topological flat bands in line and split graphs of bipartite lattices. Fragile topology in line-graph lattices with two, three, or four gapped flat bands. Band touching from real-space topology in frustrated hopping models. Hyperbolic lattices in circuit quantum electrodynamics. Bosonic condensation and disorder-induced localization in a flat band. Observation of flat bands due to band hybridization in the 3d-electron heavy-fermion compound CaCu 3Ru 4O 12. Designer flat bands in quasi-one-dimensional atomic lattices. ![]() Exact results for the U = infinity Hubbard model. ![]() Exact ground states for the Hubbard model on the Kagome lattice. Ferromagnetism in the Hubbard model on line graphs and further considerations. Anderson localization in tight-binding models with flat bands. ![]() Inverse Anderson transition caused by flatbands. Flat bands and Wigner crystallization in the honeycomb optical lattice. Ferromagnetic ground states for the Hubbard model on line graphs. Topology-bounded superfluid weight in twisted bilayer graphene. All magic angles in twisted bilayer graphene are topological. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Origin of mott insulating behavior and superconductivity in twisted bilayer graphene. Unconventional superconductivity in magic-angle graphene superlattices. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Moiré bands in twisted double-layer graphene. Finally, we show that the set of all perfectly flat bands is finitely generated and construct the corresponding bases for all 1,651 Shubnikov space groups.īistritzer, R. We also derive criteria for the existence of symmetry-protected band touching points between the flat and dispersive bands, and identify the gapped flat bands as prime candidates for fragile topological phases. Using topological quantum chemistry, we build a complete topological classification in terms of symmetry eigenvalues of all the gapped and gapless flat bands. Our prescription encapsulates and generalizes the various flat-band models in the literature and is applicable to systems with any orbital content, with or without spin–orbit coupling. Here we present a generic theoretical technique for constructing perfectly flat bands from bipartite crystalline lattices. However, existing theoretical models for obtaining these ‘flat bands’ in crystals are often too restrictive for experimental realizations. Materials that have non-dispersing bands in their electronic band structure, such as twisted bilayer graphene, are prime candidates for strongly interacting physics. Exotic phases of matter can emerge from the interplay between strong electron interactions and non-trivial topology.
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