  Two-dimensional Graphene is fascinating because of its unique electronic
properties. From a fundamental perspective, one among them is the geometric
phase structure near the Dirac points in the Brillouin zone, owing to the SU(2)
nature of the Dirac cone wave functions. We ask if there are geometric phase
structures in two dimensions which go beyond that of a Dirac cone. Here we
write down a family of three-band continuum models of non-interacting fermions
which have more intricate geometric phase structures. This is connected to the
SU(3) nature of the wavefunctions near three-fold degeneracies. We also give a
tight-binding free fermion model on a two-dimensional Graphene-like lattice
where the three-fold degeneracies are realized at fine-tuned points. Away from
them, we obtain new "three-band" Dirac cone structures with associated
non-standard Landau level quantization, whose organization is strongly affected
by the non-SU(2) or beyond-Dirac geometric phase structure of the fine-tuned
points.
