Flat band construction scheme characterized by quantum distance

Flat bands, usually considered topologically trivial, can be nontrivial in the perspective of another geometric quantity called the quantum distance. We present a general construction scheme for a tight-binding model hosting a flat band with a quadratic band crossing characterized by the maximum quantum distance. First, we introduce how to build a compact localized state(CLS) leading to the quadratic band touching and the specific value of the maximum quantum distance, where the CLS is the characteristic eigenstate of a flat band. Second, we develop a systematic strategy to obtain a tight-binding Hamiltonian with the desired hopping range and symmetry starting from the given CLS. The scheme can be applied to any dimensions and lattice structure. Using the scheme, we propose several simple flat band models on the square and kagome lattices, where the maximum quantum distance is controllable by tuning the next nearest neighbor hopping parameters. Finally, we discuss how to realize our flat band models in artificial systems.