Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2D Fermi Systems with Quadratic Band Crossings

In the light of recent progress in the search for topologically nontrival states of matter, discovering and/or understanding new mechanisms which could stabilize these exotic states has become increasingly important. We have investigated two-dimensional semi-metallic fermionic systems with a quadratic band-crossing point in the single-particle energy spectrum. At the noninteracting level, this quadratic band-crossing point is found to be topologically stable for a Berry flux $2\pi$ if the point symmetry group has either fourfold or sixfold rotational symmetry. However, this putative topologically stable free-fermion quadratic band-crossing point is marginally unstable in the presence of arbitrarily weak short-range repulsive interactions. For spinless fermions in the weak-coupling limit, an insulating quantum anomalous Hall phase is stabilized with a nontrivial Chern number. For relatively stronger coupling, a semi-metallic nematic phase with spontaneous rotational symmetry breaking occurs. For spin-$1/2$ fermions, two additional phases, the Z$_2$ quantum-spin-Hall phase and the nematic-spin-nematic phase, are found.