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

We employ the cluster slave-spin method to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\delta$ and coupling strength $U$ being its parameters. In addition to a crossover reflected in the behavior of the antiferromagnetic gap $\Delta_{\text{AFM}}$, this property can also be observed in the energetics of the cluster slave-spin Hamiltonian -- the antiferromagnetism at small $U$ is due to the potential energy gain while that in the strong coupling limit is driven by the kinetic energy gain, which is consistent with the results from the cluster dynamical mean-field theory calculation and the quantum Monte Carlo simulation. We find the interaction $U_{c}$ for the crossover in the AFM state, separating the weak- and strong- coupling regimes, almost remains unchanged upon doping, and it is smaller than the critical coupling strength $U_{\text{Mott}}$ for the first-order metal-insulator Mott transition in the half-filled paramagnetic state. At half-filling, a relationship between the staggered magnetization $M$ and $\Delta_{\text{AFM}}$ is established in the small $U$ limit to nullify the Hartree-Fock theory, and a first-order Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at $U_{\text{Mott}}=10t$. Finally, an overall phase diagram in the $U$-$\delta$ plane is presented, which is composed of four regimes: the antiferromagnetic insulator, the antiferromagnetic metal with the compressibility $\kappa>0$ or $\kappa<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the antiferromagnetic metal to the paramagnetic metal, (ii) between the antiferromagnetic metal phases with positive and negative $\kappa$, and (iii) separating the antiferromagnetic insulating phase from the antiferromagnetic metal phase.