Flat bands of a tight-binding electronic system with hexagonal structure

A constructive method to find flat bands is demonstrated for a model describing tight-binding electrons on the hexagonal lattice in which each atomic site has three orbitals. The system may represent sigma-electrons of an sp(2)-hybridized material but we assume transfer integrals between orbitals to be free parameters for general discussion. Finding highly degenerate eigenfunctions of each flat band is identified as finding a localized eigenfunction, which is determined as a basis function of an irreducible representation of, for example, a point group C-6nu. This approach yields not only flat bands on all over the k-space but also some partial flat bands only on a partial k-space. In the former case, if the flat band locates at the bottom (or the top) of a band structure, it is shown that Mielke's condition holds, allowing occurrence of the flat band ferromagnetism.