  We classify the order parameters on the honeycomb lattice using the SO(4)
symmetry of the Hubbard model. We will focus on the topologically nontrivial
quantum spin Hall order and spin triplet superconductor, which together belong
to the (3, 3) representation of the SO(4). Depending on the microscopic
parameters, this (3, 3) order parameter has two types of ground states with
different symmetries: type A, the ground state manifold is [S^2xS^2]/Z_2; and
type B, with ground state manifold SO(3)xZ_2. We demonstrate that phase A is
adjacent to a Z_2xZ_2 topological phase with mutual semion statistics between
spin and charge excitations, while phase B is adjacent to a nonabelian phase
described by SU(2) Chern-Simons theory. Connections of our study to the recent
quantum Monte Carlo simulation on the Hubbard model on the honeycomb lattice
will also be discussed.
