Phase diagram of graphene in the presence of spin-orbit coupling and electron-electron interaction

We theoretically investigate in this work the interplay between the intrinsic spin-orbit coupling (SOC) and electron-electron interaction on the magnetism in the graphene honeycomb lattice. Particularly, a phase diagram is here explored and studied by Kane-Mele-Hubbard (KMH) model combined with mean-field theory and self-consistent algorithm. Results show that the graphene undergoes a phase transition from the gapless semi-metal to topological band insulator (TBI) at a finite SOC and weak Coulomb energy U, and another transition from the TBI to antiferromagnetic ordered insulator at stronger Coulomb energy U observed as well. Moreover, our calculations also point out the prior developing orientation of magnetic moments in the in-plane direction driven by the SOC rather than that in the out-of-plane direction without the SOC.