Theoretical study of insulating mechanism in multiorbital Hubbard models with a large spin-orbit coupling: Slater versus Mott scenario in ${\text{Sr}}_2$${\text{IrO}}_4$

To examine the insulating mechanism of 5$d$ transition metal oxide ${\text{Sr}}_2$${\text{IrO}}_4$, we study the ground state properties of a three-orbital Hubbard model with a large relativistic spin-orbit coupling on a square lattice. Using a variational Monte Carlo method, we find that the insulating state appearing in the ground state phase diagram for one hole per site varies from a weakly correlated to a strongly correlated antiferromagnetic (AF) state with increasing Coulomb interactions. This crossover is characterized by the different energy gain mechanisms of the AF insulating state, i.e., from an interaction-energy-driven Slater-type insulator to a band-energy-driven Mott-type insulator with increasing Coulomb interactions. Our calculations reveal that ${\text{Sr}}_2$${\text{IrO}}_4$ is a “moderately correlated” AF insulator located in the intermediate coupling region between a Slater-type and a Mott-type insulator.

Figure 1

(a) Fermi surfaces and (b) energy dispersions of the noninteracting tight-binding energy band with electron density $n=5$. The SOC is set to be $\lambda =1.028t$ ($\sim 0.37$ eV). The other parameters are given in Eq. (14). Dashed lines in (a) represent the folded AF Brillouin zone. $E_{\mathrm{F}}$ stands for Fermi energy.

Figure 2

Same as Fig. 1 but the SOC is set to be $\lambda =1.4t$ ($\sim 0.50$ eV).

Figure 3

Hole momentum distribution functions $n_{\alpha }\left(\mathit{k}\right)$ for (a) $U/t=6.5$ and (b) $U/t=7.5$ with $(\lambda /t,J/U)=(1.028,0.0)$ and $L=20$. For comparison, the unprojected momentum distribution function $n_{\alpha }^0\left(\mathit{k}\right)$ (see the text for definition) for $U/t=7.5$ is also shown in (c). The momentum pass taken in the first Brillouin zone is indicated in the upper panel.

Figure 4

Charge structure factor $N\left(\mathit{q}\right)$ for $U/t=6.5$ and $7.5$ with $(\lambda /t,J/U)=(1.028,0.0)$ and $L=20$. For comparison, the unprojected charge structure factor $N_0\left(\mathit{q}\right)=\frac{1}{N}\langle \Phi \left|n_{-\mathit{q}}{n}_{\mathit{q}}\right|\Phi \rangle /\langle \Phi |\Phi \rangle$ for $U/t=7.5$ is also shown. Inset: $N\left(\mathit{q}\right)/{\left|\mathit{q}\right|}^2$ and $N_0\left(\mathit{q}\right)/{\left|\mathit{q}\right|}^2$ for $\left|\mathit{q}\right|\sim 0$ are plotted.

Figure 5

Distance $r$ dependence of charge Jastrow factor $v\left(r\right)$ in $P_{{\mathrm{J}}_{\mathrm{c}}}$ [Eq. (19)] for $(\lambda /t,J/U)=(1.028,0.0)$ and $L=20$. The values of $U/t$ used are indicated in the figure.

Figure 6

Ground state phase diagrams within a paramagnetic state in the (a) $U/t$-$J/U$ and (b) $U^{\prime }/t$-$J/U$ planes for $\lambda /t=1.028$ (solid lines) and $1.4$ (dashed lines). PM and PI denote a paramagnetic metal and a paramagnetic insulator, respectively. Note that $U=U^{\prime }+2J$ is imposed.

Figure 7

$U/t$ dependence of the generalized double occupancy of hole $D^{\mathrm{hole}}$ for different sets of parameters $(\lambda /t,J/U)$ indicated in the figures. The results for $L=16$ and 20 are denoted by triangles and circles, respectively, where open (solid) symbols represent the metallic (insulating) state. The results for $L=10$ are also plotted by crosses, from which the transition point is difficult to determine. The renormalization factor $z$, estimated from the jump of the momentum distribution function $n\left(\mathit{k}\right)={\sum }_{\alpha }{n}_{\alpha }\left(\mathit{k}\right)$ for $L=20$, is indicated for the largest $U$ in the metallic phase.

Figure 8

Evolution of the FS (red solid lines) in the one-body part $|\Phi \rangle$ of the paramagnetic wave functions for $(\lambda /t,J/U)=(1.028,0.0)$ and $L=20$. The momentum $\mathit{k}=(k_x,k_y)$ inside (outside) the FS is denoted by red solid (black open) points. The Coulomb interaction $U/t$ is indicated in the figures. “MIT” denotes the metal-insulator transition. Only the first quadrant of the Brillouin zone is shown.

Figure 9

Same as Fig. 8 but for $(\lambda /t,J/U)=(1.4,0.0)$. Green solid squares (and the symmetrically equivalent points) represent the $\mathit{k}$ points which are only half occupied due to the degeneracy (i.e., open shell).

Figure 10

$U/t$ dependence of $\Delta E$, $\Delta E_{\mathrm{band}}$, and $\Delta E_{\mathrm{int}}$ for (a) $(\lambda /t,J/U)=(1.028,0.0)$, (b) $(\lambda /t,J/U)=(1.4,0.0)$, and (c) $(\lambda /t,J/U)=(1.4,0.1)$. Blue-shaded regions indicate the intermediate region where $\Delta E_{\mathrm{band}}$ and $\Delta E_{\mathrm{int}}$ are both negative.

Figure 11

Total hole momentum distribution function $n\left(\mathit{k}\right)$ for the paramagnetic and the AF states with $(\lambda /t,J/U)=(1.4,0.1)$. As typical examples of the interaction-energy-driven and the band-energy-driven AF insulators, we choose (a) $U/t=4$ and (b) $U/t=8$. PM, PI, and AFI denote paramagnetic metal, paramagnetic insulator, and AF insulator, respectively. Momentum pass in the horizontal axis is the same as in Fig. 3. Notice that, unlike $n_{\alpha }\left(\mathit{k}\right)$ shown in Fig. 3, the total hole momentum distribution is fourfold rotational symmetric.

Figure 12

$U/t$ dependence of the energy difference between the paramagnetic state and the AF state for different terms in $H$. $\Delta E_{\mathrm{kin}}$, $\Delta E_{\mathrm{SO}}$, $\Delta E_U$, $\Delta E_{U^{\prime }}$, $\Delta E_J$, and $\Delta E_{J^{\prime }}$ are indicated in Eqs. (1) and (4). The parameter used here is $(\lambda /t,J/U)=(1.4,0.1)$.

Figure 13

The ground state phase diagram including the AF state. PM, w-AFI, and s-AFI stand for paramagnetic metal, weakly correlated AF insulator, and strongly correlated AF insulator, respectively. The intermediate region corresponds to a region where $\Delta E_{\mathrm{band}}$ and $\Delta E_{\mathrm{int}}$ are both negative. The blue shaded region indicates the values of $U/t$ relevant for ${\text{Sr}}_2$${\text{IrO}}_4$. The SOC is set to be $\lambda /t=1.028$.

Figure 14

Same as Fig. 13 but $\lambda /t=1.4$.