  We investigate the stability of a quadratic band-crossing point (QBCP) in 2D
fermionic systems. At the non-interacting level, we show that a QBCP exists and
is topologically stable for a Berry flux $\pm 2\pi$, if the point symmetry
group has either fourfold or sixfold rotational symmetries. This putative
topologically stable free-fermion QBCP is marginally unstable to {\em
arbitrarily weak} short-range repulsive interactions. We consider both spinless
and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous
Hall phase, a quantum spin Hall phase, a nematic phase, and a
nematic-spin-nematic phase.
