  Recently two interesting candidate quantum phases --- the chiral spin density
wave state featuring anomalous quantum Hall effect and the d+id superconductor
--- were proposed for the Hubbard model on the honeycomb lattice at 1/4 doping.
Using a combination of exact diagonalization, density matrix renormalization
group, the variational Monte Carlo method and quantum field theories, we study
the quantum phase diagrams of both the Hubbard model and t-J model on the
honeycomb lattice at 1/4-doping. The main advantage of our approach is the use
of symmetry quantum numbers of ground state wavefunctions on finite size
systems (up to 32 sites) to sharply distinguish different quantum phases. Our
results show that for $1\lesssim U/t< 40$ in the Hubbard model and for $0.1<
J/t<0.80(2)$ in the t-J model, the quantum ground state is either a chiral spin
density wave state or a spin-charge-Chern liquid, but not a d+id
superconductor. However, in the t-J model, upon increasing $J$ the system goes
through a first-order phase transition at $J/t=0.80(2)$ into the d+id
superconductor. Here the spin-charge-Chern liquid state is a new type of
topologically ordered quantum phase with Abelian anyons and fractionalized
excitations. Experimental signatures of these quantum phases, such as tunneling
conductance, are calculated. These results are discussed in the context of
1/4-doped graphene systems and other correlated electronic materials on the
honeycomb lattice.
