General construction scheme for geometrically nontrivial flat band models

Abstract A singular flat band (SFB), a distinct class of the flat band, has been shown to exhibit various intriguing material properties characterized by the quantum distance. We present a general construction scheme for a tight-binding model hosting an SFB, where the quantum distance profile can be controlled. We first introduce how to build a compact localized state (CLS), endowing the flat band with a band-touching point and a specific value of the maximum quantum distance. Then, we develop a scheme designing a tight-binding Hamiltonian hosting an SFB starting from the obtained CLS, with the desired hopping range and symmetries. We propose several simple SFB models on the square and kagome lattices. Finally, we establish a bulk-boundary correspondence between the maximum quantum distance and the boundary modes for the open boundary condition, which can be used to detect the quantum distance via the electronic structure of the boundary states.