Spin-orbit coupling and correlations in three-orbital systems

We investigate the influence of spin-orbit coupling $\lambda$ in strongly-correlated multiorbital systems that we describe by a three-orbital Hubbard-Kanamori model on a Bethe lattice. We solve the problem at all integer fillings $N$ with the dynamical mean-field theory using the continuous-time hybridization expansion Monte Carlo solver. We investigate how the quasiparticle renormalization $Z$ varies with the strength of spin-orbit coupling. The behavior can be understood for all fillings except $N=2$ in terms of the atomic Hamiltonian (the atomic charge gap) and the polarization in the $j$ basis due to spin-orbit induced changes of orbital degeneracies and the associated kinetic energy. At $N=2, \lambda$ increases $Z$ at small $U$ but suppresses it at large $U$, thus eliminating the characteristic Hund's metal tail in $Z\left(U\right)$. We also compare the effects of the spin-orbit coupling to the effects of a tetragonal crystal field. Although this crystal field also lifts the orbital degeneracy, its effects are different, which can be understood in terms of the different form of the interaction Hamiltonian expressed in the respective diagonal single-particle basis.

Figure 1

Energy levels of the considered models. For both SOC and tetragonal crystal-field splitting, the orbitally threefold degenerate $t_{2g}$ level splits into a twofold degenerate and a onefold degenerate level. Each level has an additional spin degeneracy. In case of the crystal field, the $d_{xy}$ orbital is higher in energy, whereas it is the $j=1/2$ orbital in case of SOC. The respective orbitals are plotted left (crystal field) and right (SOC) of the energy levels. The color denotes the spin. The fact that the interaction matrix elements in the $j$ basis differ from the ones in the cubic $t_{2g}$ basis is also indicated in the figure.

Figure 2

Real (a) and imaginary (b) part of the self-energy for the parameters $N=1, \lambda =0.1, U=3$, and $J_H=0.1U$. The green squares display the results without SOC for comparison. The lines show a polynomial fit of degree four through the first six Matsubara frequencies.

Figure 3

Influence of the spin-orbit coupling for a filling of $N=1$ (right column) and $N=5$ (left column) for $U=3$. (a) Quasiparticle weight $Z$ of the $j=3/2$ orbitals (for $N=1$) and of the $j=1/2$ orbitals (for $N=5$). (b) Electron density $n$ of the $j=3/2$ orbitals ($N=1$) and hole density of the $j=1/2$ orbitals ($N=5$) to allow for a better comparability. The green dotted line displays the respective noninteracting results. (c) Atomic charge gap ${\mathrm{\Delta }}_{a}t$.

Figure 4

Quasiparticle weight $Z_{3/2}$ of the $j=3/2$ orbital as a function of $U$ for $J_H=0.1U$ and a total filling of $N=3$.

Figure 5

Quasiparticle weight $Z_{3/2}$ (a) and filling $n_{3/2}$ (b) of the electrons in the $j=3/2$ orbitals as functions of $\lambda$ for $U=2$. The green dotted line displays the respective noninteracting results. (c) Atomic charge gap ${\mathrm{\Delta }}_{a}t$.

Figure 6

Quasiparticle weight $Z$ of the $j=3/2$ orbital as a function of $U$ for $J_H=0.2U$ and $N=2$. The dashed line shows the corresponding $Z$ of the $d_{xz}$ orbital in the case of an infinite tetragonal crystal-field splitting.

Figure 7

The Mott insulator occurs for values of $U$ indicated by bars for a Hund's coupling of $J_H=0.2U$. The left plot (a) shows the case without SOC, the right (b) with infinite SOC. Note that in the latter case no Mott insulator occurs for $N=4$ since this case is a band insulator. The critical values for $\lambda =0$ are taken from Ref. [26]. The red crosses indicate the critical $U$ in the case where a tetragonal crystal field is applied instead of the SOC.

Figure 8

Quasiparticle weight of the electrons (a), filling (b), and the atomic charge gap (c) for $N=2$ and $U=2$. Solid lines correspond to the SOC case, and $j=3/2$ quantities are plotted as functions of $\lambda$. Dashed lines are the results for a crystal-field splitting, where we plot $d_{xz/yz}$ quantities as functions of ${\mathrm{\Delta }}_{c}f$.

Figure 9

Quasiparticle weight $Z_{3/2}$ of the $j=3/2$ orbital as a function of $U$ for $\lambda \to \infty$ and a total filling of $N=2$. The inset shows the respective impurity spectral functions for $U=3$ and $J_H=0$ (blue) and $J_H=0.2U$ (black). As the Hund's coupling $J_H$ increases, the quasiparticle weight (= area of the quasiparticle peak) stays the same, whereas the position of the Hubbard bands changes due to different charge gaps. To obtain the spectral functions, imaginary-time data has been analytically continued using a maximum entropy method [62] with an alternative evidence approximation [63] and the preblur formalism [64].

Figure 10

Quasiparticle renormalization (a), filling (b), and atomic charge gap (c) of the orbitals as functions of spin-orbit coupling (full lines) and crystal-field splitting (dashed lines) for $N=4, U=2, J_H=0.2U$. Full dots indicate insulating phases. In the case of SOC, all calculations with $\lambda \ge 0.7$ are insulating, whereas in the case of a crystal field only the last point shown (${\mathrm{\Delta }}_{c}f=1.5$) is insulating. The green dotted lines shows the orbital fillings in the noninteracting case. Then, crystal field and SOC are equivalent.

Figure 11

Difference of the self-energies ${\mathrm{\Sigma }}_d={\mathrm{\Sigma }}_{\frac{1}{2}}-{\mathrm{\Sigma }}_{\frac{3}{2}}$ for $N=4, \lambda =0.1$, and $U=2$. Subplots (a) and (b) show ${\mathrm{\Sigma }}_d$ as a function of Matsubara frequencies ${\omega }_n$ for Hund's couplings $J_H=0.2U$ and $J_H=0.1U$, respectively. The dashed lines are the corresponding Hartree-Fock values. Subplot (c) shows $Re{\mathrm{\Sigma }}_d\left(i{\omega }_0\right)\approx Re{\mathrm{\Sigma }}_d(i{\omega }_n\to 0)$ (full line) and the Hartree-Fock values ${\mathrm{\Sigma }}_d^{H}F$ equivalent to ${\mathrm{\Sigma }}_d(i{\omega }_n\to \infty )$ (dashed) as a function of $J_H$. While the Hartree-Fock value strongly decreases with $J_H, {\mathrm{\Sigma }}_d\left(i{\omega }_0\right)$ is hardly influenced.

Figure 12

Increase of the first Matsubara self-energy ${\mathrm{\Sigma }}_d\left(i{\omega }_0\right)\approx {\mathrm{\Sigma }}_d(\omega =0)$ with the SOC for $U=2, J_H=0.1U$, and all integer fillings. For $N=3$ and $\lambda <0.3$, the system is a Mott insulator, and for $N=4$ and $\lambda >0.3$ a band insulator. The data points are not shown for these parameters.