Fractionalization in graphene-like systems

I will discuss the possible interaction-driven instabilities that can arise in a system of massles Dirac fermions modeled by the extended Hubbard model on the $\pi$ flux square lattice and the honeycomb lattice, and their relevance to fractionalization in 2D graphene-like systems. Through numerical studies we have shown that these instabilities can result in a number of interesting phases. In addition to the charge density wave and various stripe phases these include the exotic ``quantum anomalous Hall'' (Haldane) phase and the dimerized ``Kekul\'e'' phase. A self consistent calculation inside the Kekul\'e phase on the $\pi$ flux lattice indicates that a discretized $U(1)$ vortex can be stabilized in this region leading to a zero-energy bound state with fractional charge $e/2$.