The interplay of hopping parameters that can give rise to flat bands in consequence of quantum interference in electronic, photonic, and other interesting materials has become an extensively studied topic. Most of the recognized structures having flat bands are lattices that can be understood by the mathematical theory of line graphs, such as the Lieb, kagome, and checkerboard lattices. Here, we demonstrate that the structures that can realize the same kinds of flat bands given by those well-known lattices hosting exotic quantum phases are more flexible. The flat bands belonging to the recognized structures can be ideally embedded into new structures that cannot be considered as the original ones in terms of a unitary transformation. The uncovered mechanism enriches the understanding of physics behind the localized quantum states and broadens the choice of materials that can be used for designing electronic and photonic devices from the zero band dispersion.
