Flat bands in crystalline materials come in two kinds: atomically flat and topologically flat. Flat atomic bands are topologically trivial and commonly exist in layered materials and heavy fermion systems. Topological flat bands were recently discovered in twisted 2D materials, where the coexistence of nontrivial band topology and strong electronic correlation manifests kinds of exotic quantum phases, such as quantum anomalous Hall effect, magnetism, correlated insulating states, and superconductivity. In addition, a few 2D line-graph lattices with s orbitals were also proposed to have topological flat bands in the tight-binding approximation. Compared with the twisted superlattice, stoichiometric flat-band materials are much easier to synthesize and have a larger carrier density. Here I’ll introduce a general construction of flat bands in both 2D and 3D crystals. Using the magnetic topological quantum chemistry theory, we have a full classification of band topologies in paramagnetic and magnetic materials both with and without spin-orbit coupling. These advantages enable a complete understanding of flat-band features in most materials. By analyzing the geometry and symmetry properties, a high-throughput search and classification of topological flat-band materials were performed to build a material database.
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