Differentiating Hund from Mott physics in a three-band Hubbard-Hund model: Temperature dependence of spectral, transport, and thermodynamic properties

We study the interplay between Mott physics, driven by Coulomb repulsion $U$, and Hund physics, driven by Hund's coupling $J$, for a minimal model for Hund metals, the orbital-symmetric three-band Hubbard-Hund model (3HHM) for a lattice filling of $1/3$. Hund-correlated metals are characterized by spin-orbital separation (SOS), a Hund's-rule-induced two-stage Kondo-type screening process, in which spin screening occurs at much lower energy scales than orbital screening. By contrast, in Mott-correlated metals, lying close to the phase boundary of a metal-insulator transition, the SOS window becomes negligibly small and the Hubbard bands are well separated. Using dynamical mean-field theory and the numerical renormalization group as real-frequency impurity solver, we identify numerous fingerprints distinguishing Hundness from Mottness in the temperature dependence of various physical quantities. These include ARPES-type spectra, the local self-energy, static local orbital and spin susceptibilities, resistivity, thermopower, and lattice and impurity entropies. Our detailed description of the behavior of these quantities within the context of a simple model Hamiltonian will be helpful for distinguishing Hundness from Mottness in experimental and theoretical studies of real materials.

Figure 1

(a) The zero-temperature phase diagram of the 3HHM has three phases in the $J\text{−}U$ plane: a metallic phase (squares), a coexistence region (circles), and an insulating phase (triangles). These are separated by two-phase transition lines $U_{c1}$ (solid red curve) and $U_{c2}$ (dashed black curve), respectively. The color intensity of the symbols in the metallic and coexistence regions indicates the value of $Z\in [0,1]$: the lower $Z$ the more faded is the red color. The phase diagram is adapted from Ref. [35]. We will present temperature-dependent results for a Hund system H1 far away from the $U_{c1}$ phase transition line deep in the metallic state (cross), a Mott system M1 near the transition (asterisk), an intermediate system I2 having both Hund and Mott features (open diamond), and a weakly correlated system W0 with $J=0$ far from $U_{c1}$ (open square). (b) The local density of states $A\left(\omega \right)$ for M1 (black), H1 (yellow), I2 (red), and W0 (blue). The legend lists the corresponding values of the bare gap, ${\mathrm{\Delta }}_b=U-2J$. Triangles, circles, and crosses mark the bare atomic excitation scales, ${\omega }_h, {\omega }_{e1}$, and ${\omega }_{e2}$ (listed in increasing order), respectively, defined in Sec. 3. The inset zooms into the peaks around the Fermi level $\omega =0$. (c) The spin and orbital Kondo scales, $T_{\text{spin}}$ (solid) and $T_{\text{orb}}$ (dashed), plotted as function of $U$ for $J=0$ (blue), $J=1$ (brown), and $J=2$ (red); these scales are defined as the maxima of the imaginary parts of the dynamic orbital and spin susceptibilities, see Appendix pp1. The SOS window is marked by a vertical yellow (black) bar for H1 (M1).

Figure 2

A Hund system (H1) with parameters ${\mathrm{\Delta }}_b=1$ and $J=1$. [(a),(c),(e)–(h)] The structure factor $A({\epsilon }_k,\omega )$. [(b),(d)] The dispersion relation $E\left({\epsilon }_k\right)$, (i) the spectral function $A\left(\omega \right)$, [(j),(k)] the real and imaginary parts of the self-energy, $\mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$ and $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$, respectively, all plotted for various temperatures. [(a),(c),(e)–(h)] The colored curves highlight the dispersion relation $E\left({\epsilon }_k\right)$ and the white curves show the alternative definition of the dispersion relation $E^{*}\left(\omega \right)$. Panels (a) and (b) are low-energy zooms of panels (c) and (d). The FL regime, ${\omega }_{\text{FL}}^{-}<\omega <{\omega }_{\text{FL}}^{+}$, lies between the dash-dotted red lines, running horizontally in (a) and (b) and vertically in (i)–(k). The thick dashed red line in panel (a) denotes FL behavior of the low-energy dispersion relation. Its slope $Z=m/m^{*}$ reflects the strength of local correlations. The yellow solid horizontal lines in (b) and vertical lines in (i)–(k) denote, for $\omega <0$, the energy scale ${\omega }_{\text{cr}}^{-}$ of the maximum in $\mathrm{Re}{\mathrm{\Sigma }}_{T=0}(\omega <0)$, and for $\omega >0$, the energy scale ${\omega }_{\text{cr}}^{+}$ of the kink in $\mathrm{Re}{\mathrm{\Sigma }}_{T=0}(\omega >0$).

Figure 3

Refined schematic depiction of the two-stage Kondo screening process of SOS at filling $n_d=2$ (based on Fig. 13 of Ref. [35]). With decreasing energy orbital screening sets in first, roughly at the orbital Kondo scale $T_{\text{orb}}$. This involves the formation of an orbital singlet by building a large effective Hund's-coupling-induced $3/2$ spin including a bath spin degree of freedom. $|{\omega }_{\text{cr}}^{\pm }|$ approximately marks the completion of orbital screening. Below $|{\omega }_{\text{cr}}^{\pm }|$ the $3/2$ spin is gradually screened by the three effective channels of the 3HHM. Well below the spin Kondo scale $T_{\text{spin}}$, full screening of both orbital and spin degrees of freedom is reached at the FL scale $|{\omega }_{\text{FL}}^{\pm }|$, below which FL behavior occurs in frequency-dependent quantities. Our schematic sketch ignores the effects of particle-hole asymmetry on the crossover scales, $|{\omega }_{\text{cr}}^{-}|\ne |{\omega }_{\text{cr}}^{+}|$ and $|{\omega }_{\text{FL}}^{-}|\ne |{\omega }_{\text{FL}}^{+}|$.

Figure 4

Same quantities as in Fig. 2 for a Mott system (M1) with parameters ${\mathrm{\Delta }}_b=4.5$ and $J=1$.

Figure 5

Same quantities as in Fig. 2 for an intermediate system (I2) with parameters ${\mathrm{\Delta }}_b=3.5$ and $J=2$.

Figure 6

A weakly coupled system W0 with parameters ${\mathrm{\Delta }}_b=3.5$ and $J=0$. [(a),(c)–(f)] The structure factor $A({\epsilon }_k,\omega )$. (b) The dispersion relation $E\left({\epsilon }_k\right)$, (g) the spectral function $A\left(\omega \right)$, [(h),(i)] the real and imaginary parts of the self-energy, $\mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$ and $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$, respectively, all plotted for various temperatures. (h) Note that the difference in $\mathrm{Re}\mathrm{\Sigma }(\omega =0)$ between $T=0$ and $T>0$ arises from a $4\%$ deviation of $n_d(T=0)$ from $n_d=2$. FL and crossover scales are not shown. Note that the latter do not exist for $J=0$.

Figure 7

The static local orbital (dashed) and spin (solid) susceptibilities, [(a),(b),(e),(f)] $T{\chi }^{\text{orb},\text{spin}}$ and [(c),(d),(g),(h)] ${\chi }_0^{\text{orb},\text{spin}}$, all plotted as functions of temperature on a linear (left) and a logarithmic (right) scale, for M1 (black), H1 (yellow), I2 (red), and W0 (blue). In addition, the quasiparticle weight (dotted) $Z\left(T\right)$ is shown in (a), (b), (e), and (f). The squares mark the onset of orbital screening $T_{\text{orb}}^{\mathrm{onset}}$ below which $T{\chi }_0^{\text{orb}}$ deviates from a constant value, i.e., from Curie-like behavior. Note that $Z\left(T\right)$ diverges for $T>T_{\text{orb}}^{\mathrm{onset}}$. The triangles mark the maxima of ${\chi }_0^{\text{orb}}$ and also signal the onset of spin screening $T_{\text{spin}}^{\mathrm{onset}}$ below which $T{\chi }_0^{\text{spin}}$ deviates from Curie-like behavior. The crosses denote the FL scale $T_{\text{FL}}$ below which FL behavior is found. In M1, we observe that $T_{\text{spin}}^{\mathrm{onset}}\approx T_{\text{orb}}^{\mathrm{onset}}=T_{\mathrm{M}}$. In H1, we find $T_{\text{orb}}^{\mathrm{onset}}\gg T_{\text{spin}}^{\mathrm{onset}}$, as discussed in Ref. [7]. The data for the static susceptibilities shown in panels (a)–(d) are adapted from Ref. [7]. A Curie-Weiss analysis of the data of panel (c) is presented in Appendix pp2.

Figure 8

(a) Schematic sketch of different temperature regimes in a Hund metal. For $T>T_{\text{FL}}$, H1 is a NFL up to temperatures in the order of bare energy scales, where also mixed-valence physics becomes important. The NFL regime, which we dub Hund metal regime, reflects the complex SOS screening process of Fig. 3. First orbitals get screened with decreasing temperature for $T_{\text{orb}}^{\mathrm{cmp}}<T<T_{\text{orb}}^{\mathrm{onset}}$. In this regime transport is governed by HQPs, which are characterized by gradually screened orbitals coupled to quasi-free spins. Only when the orbital screening process is completed spins get screened below $T_{\text{spin}}^{\mathrm{onset}}\approx T_{\text{orb}}^{\text{cmp}}$, i.e., in this regime, the HQPs get gradually dressed to form heavier Landau QPs. For $T<T_{\text{FL}}=T_{\text{spin}}^{\mathrm{cmp}}$, H1 is a FL and both orbital and spin degrees of freedom are fully screened. (b) Schematic sketch of different temperature regimes in a multiorbital Mott-correlated metal. In a Mott system, a pseudogap governs the physics in an extended temperature regime, $T>T_{\text{spin}}^{\mathrm{onset}}\approx T_{\text{orb}}^{\mathrm{onset}}=T_{\text{M}}$. For temperatures below $T_{\text{M}}$, both orbital and spin degrees of freedom get screened simultaneously with the onset of a Kondo resonance, which is driven by the DMFT self-consistency condition. The NFL regime for $T_{\text{FL}}<T<T_{\mathrm{M}}$ is followed by a low-temperature FL regime, $T<T_{\text{FL}}<T_{\text{spin}}^{\mathrm{cmp}}\approx T_{\text{orb}}^{\mathrm{cmp}}$, where both orbital and spin degrees of freedom are fully screened.

Figure 9

[(a),(b)] The quasiparticle weight $Z/Z\left(0\right)$ (replotted from Fig. 7 for reference), [(c),(d)] the scattering rate at the Fermi level $\mathrm{Im}\mathrm{\Sigma }(\omega =0)$, [(e),(f)] the coherence scale ${\mathrm{\Gamma }}^{*}/T$, and [(g),(h)] the resistivity $\rho$, all plotted as functions of temperature on a linear (left) and a logarithmic (right) scale for M1 (black), H1 (yellow), I2 (red), and W0 (blue). Symbols are defined as in Fig. 7. [(d),(h)] The dashed grey guide-to-the-eye lines indicate FL behavior. [(e),(f)] The horizontal dashed grey lines mark ${\mathrm{\Gamma }}^{*}/T^{*}=1$. [(g),(h)] The horizontal solid grey line marks the MRI limit defined via $k_{\text{F}}{l}_{\text{min}}\approx 2\pi$.

Figure 10

[(a),(b)] The effective chemical potential ${\mu }_{\mathrm{eff}}$, [(c),(d)] the thermopower $\alpha$, and [(e),(f)] the lattice entropy $S_{\text{latt}}$ (solid) and the impurity contribution to the entropy $S_{\text{imp}}$ (dashed), all plotted as functions of temperature on a linear (left) and logarithmic (right) scale for M1 (black), H1 (yellow), I2 (red), and W0 (blue). Symbols are defined as in Fig. 7. In (f), the grey dash-dotted curves indicate FL behavior for $S_{\text{imp}}$ and $S_{\text{latt}}$, respectively. We remark that wiggles in $S_{\text{latt}}$ are an artefact due to few data points used in its computation.

Figure 11

Overview of important SOS features in the 3HHM for $n_d=2$ at $T=0$. Features are described as functions of decreasing frequency.

Figure 12

Overview of important Hund and Mott features in the temperature dependence of various physical quantities. Hund-related features are marked yellow, Mott-related features are marked grey. Common features are on white background. Note that the temperature scale is only schematic. Features are described as functions of decreasing temperature.

Figure 13

[(a),(b)] The imaginary part $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ and [(c),(d)] the real part $\mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$ of the self-energy; [(e),(f)] the local spectral function $A\left(\omega \right)$; [(g),(h)] the imaginary part ${\chi }^{\prime \prime }\left(\omega \right)$ and [(i),(j)] the real part ${\chi }^{\prime }\left(\omega \right)$ of the spin (solid) and orbital (dashed) susceptibilities are plotted versus frequency for I2 (${\mathrm{\Delta }}_b=3.5, J=2$) at $T=0$. Left panels are zooms into the FL regime, whereas their insets show the quantities on a large frequency range. The SOS window is presented in the right panels. Dashed red fits reveal FL behavior for $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right), \mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$, and $A\left(\omega \right)$ in the asymmetric range, ${\omega }_{\text{FL}}^{-}<\omega <{\omega }_{\text{FL}}^{+}$, with ${\omega }_{\text{FL}}^{+}\approx \frac{1}{3}{\omega }_{\text{FL}}^{-}$ (indicated by vertical dash-dotted red lines) and for the orbital and spin susceptibilities in the symmetric range, $\left|\omega \right|<{\omega }_{\text{FL}}^{+}$. The vertical solid yellow line at $\omega <0$ denotes the energy scale ${\omega }_{\text{cr}}^{-}$ of the maximum in $\mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$ at $\omega <0$. In (b), ${\omega }_{\text{cr}}^{+}=-\frac{1}{3}{\omega }_{\text{cr}}^{-}$ marks the kink in $\mathrm{Re}\mathrm{\Sigma }\left(\omega \right)$ at $\omega >0$. Filled dots and open squares mark the orbital and spin Kondo scales, respectively. The grey area in (a) indicates a systematic error in $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ (cf. Sec. 3.2 of Ref. [73] for details).

Figure 14

Testing the applicability of a Curie-Weiss (CW) form for various susceptibilities by replotting the data from Fig. 7 as $1/{\chi }_0\left(T\right)$ vs $T$. The top row shows the spin susceptibilities of M1 (left) and H1 (right) using solid lines, the bottom row the same for the orbital susceptibilities, using dashed lines. Dotted lines show Curie-Weiss fits to those data points (shown using crosses) for temperatures higher than the temperature at which ${\chi }_0\left(T\right)$ is maximal.

Figure 15

[(a),(b)] The optical conductivity $\sigma \left(\omega \right)$ and the kinetic energy $K\left(\mathrm{\Omega }\right)$ are plotted for various temperatures on [(a),(c)] a linear and [(b),(d)] a logarithmic frequency scale for I2 (${\mathrm{\Delta }}_b=3.5, J=2$). In addition, data for W0 (${\mathrm{\Delta }}_b=3.5, J=0$) at $T=0.15$ is shown in black. [(b),(d)] $|{\omega }_{\text{FL}}^{\pm }|$, below which FL behavior should set in, is marked by vertical dash-dotted red lines. The vertical solid yellow lines denote $|{\omega }_{\text{cr}}^{\pm }|$. Filled dots and open squares mark the orbital and spin Kondo scales, respectively.