Gapped $Z_2$ spin liquid and chiral antiferromagnetic phase in Hubbard model on the honeycomb lattice

In Schwinger-fermion representation we identify a $Z_2$ spin liquid called the sublattice-pairing state (SPS) as the gapped spin liquid phase discovered in recent Quantum Monte study of Hubbard model on a honeycomb lattice. We show that SPS is identical to the zero-flux $Z_2$ spin liquid state in Schwinger-boson representation by an explicit duality transformation. SPS is connected to an \emph{unusual} antiferromagnetic ordered phase, which we term as chiral-antiferromagnetic (CAF) phase, through an $O(4)$ critical point. CAF phase breaks $SU(2)$ spin rotation symmetry completely and has three Goldstone modes. Our results indicate that there is likely a hidden phase transition between CAF phase and the usual antiferromagnetic (Neel) phase at large $U/t$. We also propose numerical measurements to reveal the CAF phase and the hidden phase transition.