Chiral d-wave RVB state on honeycomb lattice as a generalized staggered flux phase

We show the chiral d-wave RVB state on honeycomb lattice stands as a natural generalization of the staggered flux phase on square lattice. Although the state is generated from a time reversal symmetry broken mean field ansatz, it actually represents a fully symmetric spin liquid state with a positive definite wave function in the sense of Marshall sign rule for unfrustrated antiferromagnets. The evolution of the state with the parameter $\mathrm{\Delta }\mathrm{/}\chi$ follows exactly the same manner as that of the staggered flux phase on square lattice. The critical pairing strength corresponding to the $\pi$-flux phase is found to be $\mathrm{\Delta }\mathrm{/}\chi =\sqrt{2}$. As a result of the geometric frustration between neighboring plaquette on honeycomb lattice, a direct generalization of the U(1) staggered flux pattern on square lattice to honeycomb lattice is impossible. Replacing it is the chiral d-wave state with $Z_2$ gauge structure. However, this $Z_2$ gauge structure is found to be ineffective after Gutzwiller projection and the system does not support topological degeneracy. The chiral d-wave RVB state is also found to be a rather good variational state for the Heisenberg model on honeycomb lattice. The spin correlation of the chiral d-wave state is found to be greatly enhanced as compared to the mean field prediction.