  We explore the possible particle-hole instabilities that can arise in a
system of massless Dirac fermions on both the honeycomb and pi-flux square
lattices with short range interactions. Through analytical and numerical
studies we show that these instabilities can result in a number of interesting
phases. In addition to the previously identified charge and spin density wave
phases and the exotic `quantum anomalous Hall' (Haldane) phase, we establish
the existence of the dimerized Kekule phase over a significant portion of the
phase diagram and discuss the possibility of its spinful counterpart, the `spin
Kekule' phase. On the pi-flux square lattice we also find various stripe
phases, which do not occur on the honeycomb lattice. The Kekule phase is
described by a Z3 order parameter whose singly quantized vortices carry
fractional charge +/- e/2. On the pi-flux lattice the analogous dimerized phase
is described by a Z4 order parameter. We perform a fully self-consistent
calculation of the vortex structure inside the dimerized phase and find that
close to the core the vortex resembles a familiar superconducting U(1) vortex,
but at longer length scales a clear Z4 structure emerges with domain walls
along the lattice diagonals.
