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In this work, we develop a systematic method of constructing flat-band modelswith and without band crossings. Our construction scheme utilizes the symmetryand spatial shape of a compact localized state (CLS) and also the singularityof the flat-band wave function obtained by a Fourier transform of the CLS(FT-CLS). In order to construct a flat-band model systematically using theseingredients, we first choose a CLS with a specific symmetry representation in agiven lattice. Then, the singularity of FT-CLS indicates whether the resultingflat band exhibits a band crossing point or not. A tight-binding Hamiltonianwith the flat band corresponding to the FT-CLS is obtained by introducing a setof basis molecular orbitals, which are orthogonal to the FT-CLS. Ourconstruction scheme can be systematically applied to any lattice so that itprovides a powerful theoretical framework to study exotic properties of bothgapped and gapless flat bands arising from their wave function singularities.

General construction of flat bands with and without band crossings based on wave function singularity