Dynamical Mean-Field Theory Plus Numerical Renormalization-Group Study of Spin-Orbital Separation in a Three-Band Hund Metal

We show that the numerical renormalization group is a viable multi-band impurity solver for dynamical mean-field theory (DMFT), offering unprecedented real-frequency spectral resolution at arbitrarily low energies and temperatures. We use it to obtain a numerically exact DMFT solution to the Hund metal problem for a three-band model on a Bethe lattice at $1/3$ filling. The ground state is a Fermi liquid. The one-particle spectral function undergoes a coherence-incoherence crossover with increasing temperature, with spectral weight being transferred from low to high energies. Further, it exhibits a strong particle-hole asymmetry. In the incoherent regime, the self-energy displays approximate power-law behavior for positive frequencies only. The spin and orbital spectral functions show “spin-orbital separation”: spin screening occurs at much lower energies than orbital screening. The renormalization group flows clearly reveal the relevant physics at all energy scales.

Figure 1

Benchmark comparison of NRG and CTQMC for the three-band SCAHM. (a) Imaginary and (b) real part of the self-consistently converged self-energy as a function of Matsubara frequencies. Grey lines in (a) are power-law fits to low and intermediate-frequency data, respectively. The inset of (a) shows $\mathrm{Im}\mathrm{\Sigma }(i{\omega }_n)$ on a linear scale.

Figure 2

(a) The local spectral function $A(\omega )$ and (b) the imaginary part of the retarded self-energy, $\mathrm{Im}{\mathrm{\Sigma }}^R(\omega )$, for the SCAHM, plotted versus frequency for four temperatures. Insets show a larger frequency range for $T={10}^{-8}$. (c)–(d) Same as in (a) and (b), but for an IAHM.

Figure 3

(a)–(f) Spin-orbital separation in real-frequency, ground state correlators. The left column uses the same parameters and color code as Fig. 2 for the SCAHM (black) and IAHM with $(\mathrm{\Gamma },D)=(0.778,0.5)$ (red). For comparison, the right column shows IAHM results with $(\mathrm{\Gamma },D)=(0.200,1.0)$ (blue), yielding smaller crossover scales. (a) and (d) The local spectral functions, (b) and (e) the local self-energy, and (c) and (f) the spin and orbital susceptibilities, ${\chi }_{\text{sp}}^{\prime \prime }$ (solid) and ${\chi }_{\text{orb}}^{\prime \prime }$ (dashed). We use a logarithmic frequency scale, with thick (thin) lines for $\omega <0$ ($\omega >0$). Insets show data on a linear scale. In all panels, solid (dashed) vertical lines mark the spin (orbital) Kondo scale, $T_{\mathrm{K}}^{\text{sp}}$ ($T_{\mathrm{K}}^{\text{orb}}$). Grey guide-to-the-eye lines indicate Fermi-liquid power laws (solid) or apparent fractional power laws (dashed). Inset to (f): Kondo scales $T_{\mathrm{K}}^{\text{sp}}$ (solid) and $T_{\mathrm{K}}^{\text{orb}}$ (dashed) for the IAHM, plotted as a function of $n_d$. (g) and (h) show NRG eigenlevel flow diagrams for the SCAHM and IAHM of panels (a)–(c) and (d)–(f), respectively: the rescaled energies of the lowest-lying eigenmultiplets of a Wilson chain of (even) length $k$ are plotted versus its characteristic level spacing ${\omega }_k\propto {\mathrm{\Lambda }}^{-k/2}$ (see text). Numbers above lines give multiplet degeneracies, $Q$ their symmetry labels.