  Multilayer graphene systems with a rhombohedral stacking order harbor nearly
flat bands in their single-particle spectrum. We propose ansatz states to
describe the surface-localized states of flat band electrons. The absence of
kinetic dispersion near the fermi level leaves the interaction as a dominate
mechanism to govern the low energy physics of a low density electron system. We
build up an effective lattice model in two interacting low-energy bands, where
the full terms of the Coulomb interaction, including those long-range and
off-diagonal parts, have been considered. The interaction matrix coefficients
in the many-body Hamiltonian model are directly calculated for a trilayer
system using orthonormal Wannier basis. We then present a flat-band projection
to yield an interaction-only lattice model for flat band electrons. We find
that this limited model might energetically favor a ferromagnetic quantum
crystal under certain conditions.
