  We classify all possible 36 gap-opening instabilities in graphene-like
structures in two dimensions, i.e., masses of Dirac Hamiltonian when the spin,
valley, and superconducting channels are included. These 36 order parameters
break up into 56 possible quintuplets of masses that add in quadrature, and
hence do not compete and thus can coexist. There is additionally a 6th
competing mass, the one added by Haldane to obtain the quantum Hall effect in
graphene without magnetic fields, that breaks time-reversal symmetry and
competes with all other masses in any of the quintuplets. Topological defects
in these 5-dimensional order parameters can generically bind excitations with
fractionalized quantum numbers. The problem simplifies greatly if we consider
spin-rotation invariant systems without superconductivity. In such simplified
systems, the possible masses are only 4 and correspond to the Kekul\'e
dimerization pattern, the staggered chemical potential, and the Haldane mass.
Vortices in the Kekul\'e pattern are topological defects that have Abelian
fractional statistics in the presence of the Haldane term. We calculate the
statistical angle by integrating out the massive fermions and constructing the
effective field theory for the system. Finally, we discuss how one can have
generically non-Landau-Ginzburg-type transitions, with direct transitions
between phases characterized by distinct order parameters.
