Spin-Orbit Coupling and Electronic Correlations in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We investigate the interplay of spin-orbit coupling (SOC) and electronic correlations in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ using dynamical mean-field theory. We find that SOC does not affect the correlation-induced renormalizations, which validates Hund’s metal picture of ruthenates even in the presence of the sizable SOC relevant to these materials. Nonetheless, SOC is found to change significantly the electronic structure at $k$ points where a degeneracy applies in its absence. We explain why these two observations are consistent with one another and calculate the effects of SOC on the correlated electronic structure. The magnitude of these effects is found to depend on the energy of the quasiparticle state under consideration, leading us to introduce the notion of an energy-dependent quasiparticle spin-orbit coupling ${\lambda }^{*}(\omega )$. This notion is generally applicable to all materials in which both the spin-orbit coupling and electronic correlations are sizable.

Figure 1

(a) $\mathrm{DMFT}+\mathrm{SOC}$ quasiparticle dispersions along the $X\text{−}\mathrm{\Gamma }\text{−}M$ path (black dots). The ARPES measurements [31] are displayed as open circles, and the noninteracting $\mathrm{TB}+\mathrm{SOC}$ bands as light gray lines. The SOC-induced splitting at the $\mathrm{\Gamma }$ point is indicated as a vertical bar. The cross at the Fermi level along the $X\text{−}\mathrm{\Gamma }$ path indicates the SOC-induced band splitting near the Fermi level. (b) Same as (a) but for DMFT without SOC. The square indicates the point where the $\beta$ and $\gamma$ sheets are nearly degenerate. The arrows indicate Fermi surface crossings along $\mathrm{\Gamma }$-M at which the effect of SOC is small.

Figure 2

FSs of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ in the DMFT with included SOC (a) and DMFT without SOC (b). The blue, orange, and red lines present $\beta$, $\gamma$, and $\alpha$ sheets. The FS from the ARPES experiment Ref. [51] is also reproduced (gray).

Figure 3

Real (a) and imaginary (b) parts of the real-frequency orbital-diagonal components of the self-energy (with SOC dashed, without SOC plain). Two linear fits $\text{const}+\omega (1-1/Z)$ are also displayed in (a), corresponding to low-frequency (dashed line, slope set by $Z_a^L$) and higher frequency around $\omega =-0.5\mathrm{eV}$ (plain line, slope set by $Z_a^H$). (c) Off-diagonal components of the Matsubara self-energy, presented as an effective SOC (see text). Note the weak frequency dependence. The straight horizontal lines denote average values over the range [0, 2] eV and the bare value $\lambda =0.1\mathrm{eV}$. (d) Energy dependent quasiparticle SOC ${\lambda }_z^{*}(\omega )$ and ${\lambda }_{xy}^{*}(\omega )$ (see text).

Figure 4

(a) Momentum distribution curves (MDC) at zero energy $\omega =0$ along the $\mathrm{\Gamma }\text{−}X$ path, projected on the $xy$, $xz-yz$, and $xz+yz$ orbital components. The arrows emphasize the nonzero contribution of the $xz-yz$ orbital to the $\gamma$ and $\alpha$ bands. (c) Energy distribution curves (EDC) at the $\mathrm{\Gamma }$ point, for states with spin parallel ($ls>0$) and antiparallel ($ls<0$) to the orbital moment, compared to spin-resolved ARPES results [32]. A Gaussian broadening of 0.05 eV was used.