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

The cluster slave-spin method is employed 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. At half-filling, a relation between the staggered magnetization $M$ and the antiferromagnetic (AFM) gap ${\mathrm{\Delta }}_{\text{AFM}}$ is established in the small $U$ limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at $U_{\text{Mott}}=10t$. The interaction $U_c$ for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than $U_{\text{Mott}}$ at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U-\delta$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\kappa >0$ or $\kappa <0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\kappa$, and (iii) separating the AFM insulating phase at $\delta =0$ from the AFM metal phase for $\delta >0$.

Figure 1

Schematic illustration of the (a) two- and (b) four-site cluster.

Figure 2

(a) The quasiparticle weight $Z$ and the generalized Gutzwiller factor $g_t$, (b) the AFM energy gap ${\mathrm{\Delta }}_{\text{AFM}}/t$, (c) the staggered magnetization $M$, (d) the holon-doublon correlator between the nearest neighbors $C_{12}$ and the next nearest neighbors $C_{14}$, (e) the expectation value of the cluster slave-spin Hamiltonian $\langle H_{\text{2-site}}^S\rangle$ and $\langle H_{\text{4-site}}^S\rangle /2$, and (f) the double occupancy $\langle D\rangle$ as functions of $U$ at $\delta =0.02$ in the AFM state obtained by the two-site (blue) and the four-site (red) cluster. For comparison, the staggered magnetization within the HF theory at $\delta =0.02$ is plotted in (c) as a black line.

Figure 3

(a) The quasiparticle weight $Z$, (b) the AFM energy gap ${\mathrm{\Delta }}_{\text{AFM}}/t$, (c) the staggered magnetization $M$, (d) the holon-doublon correlator between the nearest neighbors $C_{12}$, (e) the double occupancy per site, and (f) the expectation value of the cluster slave-spin Hamiltonian as functions of $U$ at a set of doping concentrations $\delta =0.02$ (red), 0.05 (green), 0.1 (blue), 0.2 (violet) in the AFM state and $\delta =0.02$ (black) in the PM state obtained by the four-site cluster.

Figure 4

The difference of the kinetic energy, interaction potential, and the total energy of the cluster slave-spin Hamiltonian between the AFM and PM state as functions of $U$ at $\delta =0.02$ obtained by the four-site cluster.

Figure 5

(a) and (b) The electron momentum distribution $n_d\left(\mathit{k}\right)$ versus $U$ at $\delta =0.02$ obtained by the two- and four-site cluster, respectively.

Figure 6

(a) The hopping probability between the nearest neighbors $|\langle c_{i\sigma }^{†}{c}_{i+\delta \sigma }\rangle {|}^2$ and (b) its derivative as functions of $U$ at $\delta =0.02$ in the AFM (red) and PM (black) states obtained by four-site cluster.

Figure 7

(a) The hopping probability between the next nearest neighbors $|\langle c_{i\sigma }^{†}{c}_{i+{\delta }^{\prime }\sigma }\rangle {|}^2$ and (b) its derivative as functions of $U$ at $\delta =0.02$ in the AFM (red and violet) and PM (black) states obtained by four-site cluster.

Figure 8

(a) The AFM energy gap ${\mathrm{\Delta }}_{\text{AFM}}/t$ where the inset shows the same data at small $U$ together with the fitting data (black line) and (b) the staggered magnetization $M$ where the inset shows the same data at small $U$ together with that calculated by Eq. (24) (black line).

Figure 9

(a) The quasiparticle weight $Z$ and the generalized Gutzwiller factor $g_t$, (b) the holon-doublon correlators between the nearest neighbors $C_{12}$ and the next nearest neighbors $C_{14}$, (c) the expectation value of the cluster slave-spin Hamiltonian, and (d) the double occupancy $\langle D\rangle$ in the AFM state as functions of $U$ obtained by the two/four-site clusters (blue, red), respectively, and those in the PM state obtained by the four-site cluster (black).

Figure 10

The AFM gap ${\mathrm{\Delta }}_{\text{AFM}}/t$ (black), ${\mathrm{\Delta }}_{\text{AFM}}^{\mu }/t$ (red), ${\mathrm{\Delta }}_{\text{AFM}}^{\lambda }/t$ (blue) as functions of $U$ obtained from the four-site cluster at (a) $\delta =0.0$ and (b) $\delta =0.02$.

Figure 11

The staggered magnetization $M$ as functions of $U$ and $\delta$ in the AFM states obtained by the four-site cluster.

Figure 12

The AFM gap ${\mathrm{\Delta }}_{\text{AFM}}$ as functions of $U$ and $\delta$ in the AFM states obtained by the four-site cluster.

Figure 13

The compressibility $\kappa$ as functions of $U$ and $\delta$ in the AFM states obtained by the four-site cluster. $\kappa$ in the red and blue region are greater than 0.5 and less than $-0.5$, respectively.

Figure 14

The compressibility $\kappa$ as functions of $\delta$ in the AFM state at $U=5t, 10t$, and $18t$ obtained by four-site cluster. Red lines are the extrapolated curves from the corresponding self-consistent data using ${\kappa }_0+\alpha {\delta }^2+\beta {\delta }^4$. The red circles at $\delta =0$ denotes the discontinuity of $\kappa =0$ at $\delta =0$.

Figure 15

The phase diagram of the Hubbard model in the $U-\delta$ plane obtained by the four-site cluster, where exist the crossover $U_c$ (red dashed line) dividing the weak- and strong-coupling regimes, at which the ${\mathrm{\Delta }}_{\text{AFM}}$ is maximized, and the AFM for $U<U_c$ and $U>U_c$ is advocated by the interaction potential and kinetic energy gain, respectively. The $U_{\text{Mott}}$ for the Mott transition in the PM state is marked by the red triangle. The half-filling case is highlighted by the heavy blue line, in which the system is an AFM insulator with $\kappa =0$. Here the black dashed line represents the boundary ${\delta }_{\text{MF}}\left(U\right)$ between the AFM and PM phases within the HF method.