Quantum spin Hall, triplet superconductor, and topological liquids on the honeycomb lattice

We classify the order parameters on the honeycomb lattice using the SO(4) symmetry of the Hubbard model. We 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) symmetry. Depending on the microscopic parameters, this $(3,3)$ order parameter has two types of ground states with different symmetries: type A, with ground-state manifold [S²S²]/Z₂, and type B, with ground-state manifold SO(3)Z₂. We demonstrate that phase A is adjacent to a Z₂Z₂ 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 are also discussed.