Interaction-driven instabilities of a Dirac semimetal

We explore the possible particle-hole instabilities that can arise in a system of massless Dirac fermions on both the honeycomb and π-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 ``Kekul\'e'' phase over a significant portion of the phase diagram and discuss the possibility of its spinful counterpart, the ``spin Kekul\'e'' phase. On the π-flux square lattice we also find various stripe phases, which do not occur on the honeycomb lattice. The Kekul\'e phase is described by a Z₃ order parameter whose singly quantized vortices carry fractional charge ±e/2. On the π-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.