Interaction driven instabilities of a Dirac semi-metal

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.