  As a minimal fermionic model with kinetic frustration, we study a system of
spinless fermions in the lowest band of a triangular lattice with long-range
repulsion. We find that the combination of interactions and kinetic frustration
leads to spontaneous symmetry breaking in various ways. Time-reversal symmetry
can be broken by two types of loop current patterns, a chiral one and one that
breaks the translational lattice symmetry. Moreover, the translational symmetry
can also be broken by a density wave forming a kagome pattern or by a
Peierls-type trimerization characterized by enhanced correlations among the
sites of certain triangular plaquettes (giving a plaquette-centered density
wave). We map out the phase diagram as it results from leading order
Ginzburg-Landau mean-field theory. Several experimental realizations of the
type of system under study are possible with ultracold atoms in optical
lattices.
