Universal Signature of Hundness and Its Quantification

Abstract: Hund's coupling $J$ induces fundamentally different correlation effects from Hubbard $U$. This leads to a violation of the Brinkman--Rice scenario and the emergence of a Janus-faced phase owing to its low-energy effectiveness, in which band renormalization is confined below a characteristic energy scale. We propose a quantitative framework to capture low-energy effectiveness through two correlation factors: $z_L$ for low-energy quasiparticle renormalization and $z_H$ for high-energy charge fluctuations, newly introduced in this study. The discrepancy between $z_L$ and $z_H$ reflects the Hund character of the correlation. By establishing a one-to-one correspondence between correlation factors and the spin and charge susceptibilities, we identify the spin-degree-of-freedom effectiveness as the microscopic origin of low-energy effectiveness. Our framework, validated across multiorbital models and real materials, provides a universal and quantitative measure of Hundness.

• The paper introduces normalized low- and high-energy correlation factors, z_L and z_H, showing that z_H > z_L universally identifies Hundness while z_H ≈ z_L recovers the Brinkman–Rice limit.
• It explains low-energy effectiveness through spin-degree-of-freedom effectiveness: Hund coupling suppresses Kondo coherence and quasiparticle weight while largely preserving charge fluctuations and high-energy spectral weight.
• DFT+DMFT results place iron-based superconductors, most Sr transition-metal oxides, pressurized NiS₂, and BaOsO₃ in the Hund regime, while proposing specific heat and XPS/XAS as experimental probes of z_L and z_H.

Overview

The paper "Universal Signature of Hundness and Its Quantification" (2608.14480) addresses a long-standing deficiency in the quantitative characterization of Hund-driven correlations in multiorbital systems. In the single-band Hubbard model, dynamical mean-field theory (DMFT) confirms the Brinkman–Rice (BR) picture: as $U$ grows, the quasiparticle renormalization factor $Z$ and the overall conduction-band renormalization remain equivalent, with spectral weight transferring uniformly into Hubbard bands. In multiorbital systems with finite Hund's coupling $J$, however, this equivalence breaks down. Only the narrow energy window near the Fermi level associated with quasiparticles is strongly renormalized, while the broader conduction band is weakly affected—a phenomenon the authors term the low-energy effectiveness (LEE) of Hund correlations, underlying the "Janus-faced" physics of Hund metals. The central contribution is a system-independent, quantitative measure of this dichotomy, built from two correlation factors whose discrepancy constitutes a universal signature of Hundness.

Two correlation factors: $z_L$ and $z_H$

The low-energy correlation factor $z_L$ is the conventional quasiparticle renormalization factor, $$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$, i.e., the inverse quasiparticle mass enhancement $1/m^*$.

The key methodological innovation is the high-energy correlation factor $z_H$, constructed from the probability $P(N_{\mathrm{occ}})$ of finding the local ground state in atomic multiplets whose occupation equals the nominal occupancy. The authors first establish, for half-filled two-orbital models with equal $Z$0 but different $Z$1, that the Hund-correlated case retains substantially more spectral weight within the bare bandwidth and exhibits a notably smaller $Z$2, indicating larger charge fluctuations and ill-developed Hubbard bands. Conversely, systems with equal $Z$3 show nearly identical conduction-band spectral weight loss, establishing a direct correspondence between $Z$4 and high-energy spectral structure.

Because the raw $Z$5 has occupancy- and orbital-dependent bounds, the authors normalize it between two well-defined limits: the atomic limit where charge fluctuations are fully suppressed ($Z$6) and the non-interacting limit where all multiplets are equally probable, $Z$7 for $Z$8 orbitals. Linear interpolation between these limits defines

$Z$9

A crucial asymmetry distinguishes the two factors: $J$0 signals an insulator, whereas $J$1 corresponds to an isolated atom, and $J$2 remains finite even in insulating states due to residual charge fluctuations. Comparisons across the Mott transition in half-filled two-orbital systems for $J$3 from 0 to 0.4 show that the Hundless case follows $J$4 throughout—reinterpreting the BR picture as the limiting case where low- and high-energy renormalizations coincide—while finite $J$5 produces a clear separation, with larger $J$6 yielding higher $J$7 at fixed $J$8. The supplement confirms the same dichotomy in the 1/3-filled three-orbital case, so the framework is not specific to half-filling.

Microscopic origin: spin-degree-of-freedom effectiveness

The paper goes beyond spectral phenomenology by identifying the microscopic mechanism behind LEE. Hubbard $J$9 suppresses charge fluctuations, raising the energy cost of hopping and thereby reducing the Kondo exchange; Hund's coupling instead lifts intra-atomic degeneracy, locking electrons into high-spin states. This spin locking suppresses the Kondo coherence scale while leaving charge fluctuations essentially unaffected. The authors call this spin-degree-of-freedom effectiveness (SDFE).

Two quantitative correspondences substantiate the claim. First, $z_L$0 and $z_L$1 show a one-to-one correspondence independent of $z_L$2 throughout the metallic regime, and $z_L$3 is demonstrably $z_L$4-independent at fixed charge-transfer energy $z_L$5—evidence that $z_L$6 does not act on the charge sector. Second, $z_L$7 maps one-to-one onto the inverse spin susceptibility normalized by the effective unscreened spin square, $z_L$8, with $z_L$9 for $z_H$0 and $z_H$1 for $z_H$2. The implication is direct and, relative to the BR tradition, contrarian: quasiparticle renormalization in Hund systems is governed by spin suppression, not charge suppression. The resemblance of the $z_H$3-vs-$z_H$4 trajectories to the $z_H$5-vs-$z_H$6 phase diagram, in both half-filled two-orbital and 1/3-filled three-orbital systems, closes the logical chain linking SDFE to LEE.

A methodological point worth noting: the authors deliberately work in the $z_H$7 parameter space rather than $z_H$8, since $z_H$9 for the half-filled two-orbital case but $z_L$0 for the three-orbital case; using $z_L$1 disentangles $z_L$2 from charge suppression, which is essential for the occupancy-independent characterization they seek.

Application to real materials

The framework is validated with charge-self-consistent DFT+DMFT calculations for iron-based superconductors (Fe-SC), strontium transition-metal oxides Sr(TM)O$z_L$3, NiS$z_L$4 under 5.9 GPa pressure, and BaOsO$z_L$5. For non-degenerate $z_L$6 shells, a spectral-weight-averaged low-energy factor $z_L$7 is used, with $z_L$8 the Fermi-level spectral weight of orbital $z_L$9.

The resulting $$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$0-vs-$$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$1 phase diagram places all materials except SrVO$$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$2 and $$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$3-less NiS$$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$4 in the Hund region ($$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$5). SrVO$$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$6, with $$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$7, shows $$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$8, consistent with prior model studies contrasting it with SrCrO$$z_L = Z = (1 - \partial_\omega \mathrm{Re}\,\Sigma(\omega)|_{\omega=0})^{-1}$$9 ($1/m^*$0 in $1/m^*$1), the canonical Janus-faced Hund metal. The controlled NiS$1/m^*$2 comparison—switching $1/m^*$3 off at fixed $1/m^*$4 eV—leaves $1/m^*$5 unchanged while strongly suppressing $1/m^*$6, and the same behavior holds for BaOsO$1/m^*$7 with strong spin-orbit coupling, extending the framework to the $1/m^*$8-basis regime.

The paper also proposes experimental access: $1/m^*$9 relates to the average mass enhancement measurable via specific heat, while $z_H$0 encodes the local charge-state distribution $z_H$1, accessible through core-level spectroscopies such as XPS/XAS. This is a concrete, testable claim rather than a vague call for measurement.

Limitations and open questions

Several caveats are stated plainly in the paper. The SDFE–LEE correspondence is strictly valid only in the metallic phase; the one-to-one mappings between correlation factors and susceptibilities are established in the correlated metal, not across the insulating state. In systems with strong SOC where the spin-orbit splitting exceeds the bandwidth (e.g., iridates), the framework's validity is explicitly excluded: whether a one-to-one correspondence holds between $z_H$2 and the total-moment susceptibility $z_H$3 "needs further verification." The two-orbital model is a reduced version of a $z_H$4-shell-parametrized interaction rather than an independently parametrized Kanamori Hamiltonian, and the parameter regime is motivated by, but does not literally reproduce, $z_H$5-shell couplings. Finally, the framework is developed at low temperature ($z_H$6 eV); the authors acknowledge that at high temperatures quasiparticles become ill-defined and suggest—without demonstrating—that the decay rate of the dynamical impurity susceptibility, which encodes the Kondo scale $z_H$7, could serve as a proxy for $z_H$8. Extension to non-equilibrium conditions is likewise left open.

Conclusion

The paper converts the qualitative, spectral distinction between Hund and Mott–Hubbard correlations into a quantitative, system-independent diagnostic: the discrepancy between a conventional low-energy factor $z_H$9 and a newly normalized high-energy factor $P(N_{\mathrm{occ}})$0, with $P(N_{\mathrm{occ}})$1 serving as the universal signature of Hundness and $P(N_{\mathrm{occ}})$2 recovering the Brinkman–Rice limit. By anchoring $P(N_{\mathrm{occ}})$3 to charge susceptibility and $P(N_{\mathrm{occ}})$4 to normalized spin susceptibility, the authors identify spin-degree-of-freedom effectiveness as the microscopic origin of low-energy effectiveness, and demonstrate the diagnostic across multiorbital models and DFT+DMFT results for Fe-SCs, strontium oxides, NiS$P(N_{\mathrm{occ}})$5, and BaOsO$P(N_{\mathrm{occ}})$6. The main open issues—validity beyond the metallic phase, the SOC-dominated $P(N_{\mathrm{occ}})$7 regime, and finite-temperature generalization—define the boundaries within which this framework currently applies.