  Recent advances in ultracold atoms in optical lattices and developments in
surface science have allowed for the creation of artificial lattices as well as
the control of many-body interactions. Such systems provide new settings to
investigate interaction-driven instabilities and non-trivial topology. In this
paper, we explore the interplay between molecular electric dipoles on a
two-dimensional triangular lattice with fermions hopping on the dual decorated
honeycomb lattice which hosts Dirac and flat band states. We show that
short-range dipole-dipole interaction can lead to ordering into various stripe
and vortex crystal ground states. We study these ordered states and their
thermal transitions as a function of the interaction range using simulated
annealing and Monte Carlo methods. For the special case of zero wave vector
ferrodipolar order, incorporating dipole-electron interactions and integrating
out the electrons leads to a six-state clock model for the dipole ordering.
Finally, we discuss the impact of the various dipole orders on the electronic
band structure and the local tunneling density of states. Our work may be
relevant to studies of "molecular graphene" -- CO molecules arranged on the
Cu(111) surface -- which have been explored using scanning tunneling
spectroscopy, as well as ultracold molecule-fermion mixtures in optical
lattices.
