Phase diagram of the Hubbard model on a honeycomb lattice: A cluster slave-spin study

The cluster slave-spin method is implemented to research the ground state properties of the honeycomb lattice Hubbard model with doping $\delta$ and coupling $U$ being its parameters. At half-filling, a single direct and continuous phase transition between the semi-metal and antiferromagnetic (AFM) insulator is found at $U_{\text{AFM}}=2.43t$ that is in the Gross-Neveu-Yukawa universality class, where a relation between the staggered magnetization $M$ and the AFM energy gap $\Delta_{\text{AFM}}$ is established as $M \propto \Delta_{\text{AFM}}$, compared to $M \propto \Delta_{\text{AFM}} ( \ln{\Delta_{\text{AFM}}})^2$ in the square lattice case. A first-order semi-metal to the underlying paramagnetic (PM) insulator Mott transition is corroborated at $U_{\text{Mott}}=8.36t$, which is responsible for a broad crossover around $U_{c} = 5.4t$ between the weak- and strong-coupling regimes in the AFM state that increases with $\delta$, in contrast to the square lattice case. In the doped system, the compressibility $\kappa$ near the van Hove singularity at $\delta=1/4$ is suppressed substantially by the interaction before the semi-metal to AFM transition occurs, whereas $\kappa$ near the Dirac points is very close to the noninteracting one, indicating that the Dirac cone structure of the energy dispersion is rather robust. An overall phase diagram in the $U$-$\delta$ plane is presented, consisting of four regimes: the AFM insulator at $\delta=0$ for $U> U_{\text{AFM}}$, the AFM metal with compressibility $\kappa>0$ or $\kappa<0$, and the PM semi-metal, and the AFM metal with $\kappa<0$ only exists in an extremely small area near the phase boundary between the AFM and PM state.