Spin-orbit coupling in three-orbital Kanamori impurity model and its relevance for transition-metal oxides

We investigate the effects of spin-orbit coupling (SOC) in a three-orbital impurity model with a Kanamori interaction using the numerical renormalization group method. We focus on the impurity occupancy $N_d=2$ relevant to the dynamical mean-field theory studies of Hund's metals. Depending on the strength of SOC $\lambda$, we identify three regimes: the usual Hund's impurity for $\left|\lambda \right|<{\lambda }_c$, the van Vleck nonmagnetic impurity for $\lambda >{\lambda }_c$, and a $J=2$ impurity for $\lambda <-{\lambda }_c$. They all correspond to a Fermi liquid but with very different quasiparticle phase shifts and different physical properties. The crossover between these regimes is controlled by an emergent scale, the orbital Kondo temperature ${\lambda }_c=T_K^{\mathrm{orb}}$, that drops with increasing interaction strength. This implies that oxides with strong electronic correlations are more prone to the effects of spin-orbit coupling.

Figure 1

(a) Effective local moment. Inset: Effective spin and orbit moment for $\lambda =0$. (b) Impurity entropy. (c) Expectation value $\langle J^2\rangle$. Solid (dashed) lines denote results for $\lambda <0$ ($\lambda >0$).

Figure 2

(a) Kondo temperature. (b) Zero-temperature total angular momentum susceptibility ${\chi }_J$ (solid line) and expectation value $\langle J^2\rangle$ (dashed line). (c) Quasiparticle phase shifts.

Figure 3

Impurity spectral functions in the presence of the spin-orbit coupling for $j=\frac{1}{2}$ (solid lines) and $j=\frac{3}{2}$ (dashed) excitations. The insets are closeups of the low-energy region.