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The crossover from fluctuating atomic constituents to a collective state asone lowers temperature or energy is at the heart of the dynamical mean-fieldtheory description of the solid state. We demonstrate that the numericalrenormalization group is a viable tool to monitor this crossover in areal-materials setting. The renormalization group flow from high to arbitrarilysmall energy scales clearly reveals the emergence of the Fermi-liquid state ofSr$_2$RuO$_4$. We find a two-stage screening process, where orbitalfluctuations are screened at much higher energies than spin fluctuations, andFermi-liquid behavior, concomitant with spin coherence, below a temperature of25 K. By computing real-frequency correlation functions, we directly observethis spin--orbital scale separation and show that the van Hove singularitydrives strong orbital differentiation. We extract quasiparticle interactionparameters from the low-energy spectrum and find an effective attraction in thespin-triplet sector.

Strongly correlated materials from a numerical renormalization group perspective: How the Fermi-liquid state of Sr$_2$RuO$_4$ emerges