Figure 1

(Color online) Electron density $n$ per orbital per spin computed as a function of chemical potential $\mu$ for different values of the interaction parameter $J$ at $U/t=12$ and temperature $\beta t=50$. The orbital symmetry of the Hamiltonian is unbroken $\left({\Delta }_a=0\right)$ and orbital and spin symmetries were enforced in the calculation. Plateaus in $n\left(\mu \right)$ correspond to Mott insulating states. Open (full) symbols correspond to metallic (insulating) solutions.Reuse & Permissions

Figure 2

(Color online) Metal-insulator phase diagram presented in the space of chemical potential $\mu$ and interaction strength $U$ (measured in units of the quarter bandwidth $t$) for ${\Delta }_a=0$ and $\beta t=50$ at Hund coupling $J=0$ (upper panel) and $J=U/6$ (lower panel). Orbital and spin symmetries were enforced in the calculation. Error bars are on the order of the symbol size. The numerals in the lobes indicate the electron concentration per site in the insulating phases. In the lower panel the solid diamonds indicate the boundary of a spin freezing transition discussed in Ref. 28, while the line with squares plots the locus of $\mu$ and $U$ corresponding to the density $n=2$.Reuse & Permissions

Figure 3

Phase diagram in the plane of Slater-Kanamori parameters $U$ and $J$ calculated for the orbitally symmetric model $\left({\Delta }_a=0\right)$ at $\beta t=50$ and half filling $\left(n=3\right)$. The hashed region $\left(U<3J\right)$ corresponds to an effectively attractive Coulomb interaction; this situation does not normally occur in transition-metal oxides and is not considered here.Reuse & Permissions

Figure 4

Orbital filling as a function of crystal-field splitting $\Delta$ in the symmetric case: ${\Delta }_1=-\Delta$, ${\Delta }_2=0$, and ${\Delta }_3=\Delta$. The parameters are $U/t=7<U_{c2}$, $J/U=1/6$, and $\beta t=50$ and the density at $\Delta =0$ corresponds to one electron. As the crystal-field splitting is increased, band 1 (which is raised) empties out and undergoes a metal-band insulator transition near $\Delta /t\approx 0.4$. At the higher value $\Delta \approx 0.8t$ the second band empties out, leaving what is effectively a one-orbital model for which $U>U_{c2}$, so the state is insulating.Reuse & Permissions

Figure 5

Effect of a 1 up, 2 down crystal-field splitting on the two-electron insulating phase. Heavy black line: boundary of the two-electron Mott insulating state in the space of interaction $U$ and chemical potential $\mu$ computed for ${\Delta }_1=t$, $J=U/6$, and $\beta t=50$. The crystal field splits the threefold degenerate level into a doublet and a singlet, with the singlet lying higher. Dashed lines: metal-insulator phase boundary for the same interaction parameters and ${\Delta }_a=0$ for comparison.Reuse & Permissions

Figure 6

(Color online) Orbital filling as function of crystal-field splitting ${\Delta }_1$ computed for the two-electron insulating state with $U/t=12$, $\beta t=50$, and indicated values of $J/t$. In order to display all of the curves on the same figure the crystal-field coordinate is chosen to be $\mu -{\Delta }_1$ where the value of $\mu$ corresponds to $\mu /t=16,14$ for $J/t=0,1$, respectively. The ground state is insulating for all points shown. The red lines with circles correspond to the occupancy of orbital 1 and the blue lines with stars correspond to the occupancy of orbitals 2 and 3. The conventions are such that increasing ${\Delta }_1$ to positive values (moving to the left on the plot) shifts the nondegenerate orbital up (decreasing its occupancy). The offset between the curves for different $J$ values arises because of the $J$ dependence of the location of the Mott lobes. Dashed horizontal lines are shown at fillings 1/4 and 1/2.Reuse & Permissions

Figure 7

(Color online) $\mu$ dependence of the orbital occupancy per spin $n_{\sigma }\left(\mu \right)$ for $U/t=12$ and $J/t=1$ at $\beta t=50$ and in the presence of a cubic-tetragonal crystal field ${\Delta }_1=\pm 0.25t$ (top) and ${\Delta }_1=\pm t$ (bottom). The crystal field splits the threefold degenerate $d$ level into a doublet and a singlet, with the doublet lying higher or lower according to the sign of ${\Delta }_1$. The singlet orbital is labeled as “orbital 1” and is denoted by open circles (red) for ${\Delta }_1<0$ or diamonds (blue) for ${\Delta }_1>0$; the doublet orbitals are labeled as orbitals 2 and 3 and are denoted by stars (red) for $\Delta <0$ or squares (blue) for $\Delta >0$. Magnetic ordering was suppressed by averaging the Green’s function over spin and additional ordering of orbitals 2 and 3 was suppressed by averaging the Green’s functions in orbitals 2 and 3. The chemical-potential range runs from the $n=1$ Mott phase $\left(\mu \sim 6t\right)$ to the $n=3$ Mott phase $\left(\mu \sim 20t\right)$. Insulating phases are visible as plateaus in all three densities and occur only at integer total density. Orbitally selective Mott phases are visible as plateaus in one density with the other(s) varying with $\mu$.Reuse & Permissions

Figure 8

(Color online) Orbital filling as a function of crystal-field splitting computed for the two-electron state with $U/t=8$ and $\beta t=50$ and indicated values of $J/t$. In order to display all of the curves on the same figure the crystal-field coordinate is chosen to be $\mu -{\Delta }_1$. The red lines with circles correspond to the occupancy of orbital 1 and the blue lines with stars correspond to the occupancy of orbitals 2 and 3. While a metal-insulator transition is evident in the curves for $J/t=0$ and those for $J/t=1,{\Delta }_1>0$, the $J/t=1,{\Delta }_1<0$ curves exhibit a transition to an orbital-selective Mott state (band 1 is insulating and bands 2 and 3 are metallic).Reuse & Permissions

Figure 9

(Color online) Average energy $E$ of the 2/6 filled state at $U/t=8$ and $J=U/6$ as a function of crystal-field splitting ${\Delta }_1$ measured relative to the energy of the ${\Delta }_1=0$ state. To compensate the asymmetry produced by raising/lowering one orbital while leaving the other two in place, we have subtracted $2{\Delta }_1/3t$. The results with ${\Delta }_1/t\lesssim -0.5$ are in an orbital-selective Mott state with the lower orbital insulating and the other two metallic, those in the range $-0.5\lesssim \Delta /t<0.5$ are metallic in all bands and the solutions at $\Delta /t\ge 0.5$ are insulating. In these phases we observe a linear behavior with slopes 1/3 and $-2/3$.Reuse & Permissions