Strongly Correlated Materials from a Numerical Renormalization Group Perspective: How the Fermi-Liquid State of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ Emerges

The crossover from fluctuating atomic constituents to a collective state as one lowers temperature or energy is at the heart of the dynamical mean-field theory description of the solid state. We demonstrate that the numerical renormalization group is a viable tool to monitor this crossover in a real-materials setting. The renormalization group flow from high to arbitrarily small energy scales clearly reveals the emergence of the Fermi-liquid state of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We find a two-stage screening process, where orbital fluctuations are screened at much higher energies than spin fluctuations, and Fermi-liquid behavior, concomitant with spin coherence, below a temperature of 25 K. By computing real-frequency correlation functions, we directly observe this spin-orbital scale separation and show that the van Hove singularity drives strong orbital differentiation. We extract quasiparticle interaction parameters from the low-energy spectrum and find an effective attraction in the spin-triplet sector.

Figure 1

(a) Dynamic spin and orbital susceptibilities, ${\chi }_{\mathrm{sp}}^{\prime \prime }(\omega )$ and ${\chi }_{\mathrm{orb}}^{\prime \prime }(\omega )$, showing spin-orbital scale separation. Inset: Static spin susceptibility as a function of temperature. (b) NRG flow diagram, showing the rescaled eigenenergies (with quantum numbers given in the legend) as a function of the energy scale [50], for the impurity model at self-consistency. The spin and orbital Kondo temperatures (maximum of ${\chi }^{\prime \prime }$) and the Fermi-liquid scale, $T_{\mathrm{FL}}$, are marked by vertical lines.

Figure 2

Orbital-resolved, dynamic spin and angular-momentum susceptibilities, ${\chi }^{\prime \prime }(\omega )$. Inset: Temperature dependence of ${\partial }_{\omega }{\chi }^{\prime \prime }{|}_{\omega =0}$ in the spin sector, with $T_{\mathrm{FL}}$ marked as a dashed line.

Figure 3

Main panels: Real-frequency correlation functions at zero temperature. Insets: Characteristic values as a function of temperature, converging below $T_{\mathrm{FL}}$ (dashed line). (a) Local spectral function, ${\mathcal{A}}_{\mathrm{loc}}(\omega )$, from $\mathrm{DFT}+\mathrm{DMFT}$ (solid lines) and DFT (dotted lines). Inset: ${\max }_{\omega }{\mathcal{A}}_{\mathrm{loc}}(\omega )$. (b) Imaginary part of the self-energy, $\mathrm{Im}\mathrm{\Sigma }(\omega )$. Inset: $\mathrm{Im}\mathrm{\Sigma }(\omega =0)$. (c) Real part of the self-energy, $\mathrm{Re}\mathrm{\Sigma }(\omega )$, with the two linear regimes for $\omega <0$ and the low-energy, positive slope for $\omega >0$ highlighted. Inset: $Z=(1-{\partial }_{\omega }\mathrm{Re}\mathrm{\Sigma }{|}_{\omega =0}{)}^{-1}$; thick horizontal lines show the $T=0$ result for $Z$ calculated via renormalized parameters.

Figure 4

(a) Spectral function of the hybridization ${\mathcal{A}}_{\mathrm{\Delta }}(\omega )$ in $\mathrm{DFT}+\mathrm{DMFT}$ (solid lines) and DFT (dotted lines). Inset: temperature dependence of the van Hove dip in ${\mathcal{A}}_{\mathrm{\Delta },xy}(\omega )$. (b) Quasiparticle weight $Z$ as a function of Hubbard $U$, for a simple two-orbital model with identical half bandwidth $D$. The hybridization functions at $U=0$ are shown in the inset.