$Z_2$ spin liquid and chiral antiferromagnetic phase in Hubbard model on the honeycomb lattice: duality between Schwinger-fermion and Schwinger-boson representations

Understand

In our previous work, we identify the Sublattice-Pairing State (SPS) in Schwinger-fermion representation as the spin liquid phase discovered in recent numerical study on a honeycomb lattice.

• In this paper, we show that SPS is identical to the zero-flux $Z_2$ spin liquid in Schwinger-boson representation found by Wang\cite{Wang2010} by an explicit duality transformation.
• SPS is connected to an \emph{unusual} antiferromagnetic ordered phase, which we term as chiral-antiferromagnetic (CAF) phase, by an O(4) critical point.
• CAF phase breaks the SU(2) spin rotation symmetry completely and has three Goldstone modes.