Phase diagram of the Hubbard model on honeycomb lattice

In this paper we generalized the slave-particle technique to study the phase diagram of the Hubbard model on honeycomb lattice which may contain charge fluctuations. For large $U$, we have antiferromagnetic order phase. As we decrease $U$ below $U_{c2}\simeq 3t$, the system undergoes a first order phase transition into a gapped spin rotation invariant phase. Under a semiclassical approximation of the slave-particle approach, we find that such phase breaks the translation symmetry, the parity and the lattice rotation symmetry. However, beyond the semiclassical approximation, a $Z_2$ spin liquid that does not break any lattice symmetry is also possible.