Symmetry-protected flatband condition for Hamiltonians with local symmetry

We derive symmetry-based conditions for tight-binding Hamiltonians with flatbands to have compact localized eigenstates occupying a single unit cell. The conditions are based on unitary operators commuting with the Hamiltonian and associated with local symmetries that guarantee compact localized states and a flatband. We illustrate the conditions for compact localized states and flatbands with simple Hamiltonians with given symmetries. We also apply these results to general cases such as the Hamiltonian with long-range hoppings and higher-dimensional Hamiltonian.