Multiorbital Correlations and Hund's Metals

Hund exchange favors high-spin atomic multiplets and can suppress the coherence scale of a multiorbital metal even when the system is not closest to a Mott gap. A Hund-metal claim therefore requires a rotationally consistent interaction, filling- and orbital-resolved multiplets, local moments, and coherence scales—not merely large resistivity or mass enhancement.

Required background. Use effective multiorbital Hamiltonians and the DMFT impurity mapping. Helpful background. The Anderson impurity model clarifies screening and coherence.

Kanamori interaction and multiplets

For degenerate orbitals $a$, a rotationally invariant local interaction can be written

$$\begin{aligned} H_{\mathrm K}={}&U\sum_a n_{a\uparrow}n_{a\downarrow} +U'\sum_{a\ne b}n_{a\uparrow}n_{b\downarrow} +(U'-J_H)\sum_{a<b,\sigma}n_{a\sigma}n_{b\sigma}\\ &-J_H\sum_{a\ne b} \left(c_{a\uparrow}^\dagger c_{b\downarrow}^\dagger c_{b\uparrow}c_{a\downarrow} +c_{a\uparrow}^\dagger c_{a\downarrow}^\dagger c_{b\uparrow}c_{b\downarrow}\right), \end{aligned}$$

with $U'=U-2J_H$ for the ideal rotationally invariant shell. Spin-flip and pair-hopping terms are required by that symmetry; dropping them changes multiplets.

For two electrons in different orbitals, the high-spin triplet has energy $U-3J_H$, while the interorbital singlet has energy $U-J_H$. Thus positive $J_H$ lowers the high-spin state by $2J_H$. Crystal fields, spin–orbit coupling, and nondegenerate screened matrix elements split this simple spectrum.

Coherence hierarchy

Hund coupling suppresses orbital fluctuations and builds a large local moment over an intermediate temperature window. The eventual coherent Fermi liquid must screen that moment, so its scale can become much smaller than the bare bandwidth. Different orbital bandwidths and crystal fields can then produce orbital-selective coherence or localization.

This mechanism differs from a half-filled single-band Mott trend: increasing $J_H$ can strengthen correlations away from half filling while sometimes moving a particular atomic charge gap in the opposite direction. Georges, de’ Medici, and Mravlje explain this Janus behavior in Georges, de’ Medici, and Mravlje 2013, §§ 2–4.

Validity tests

Report the orbital basis, full interaction tensor, filling, double-counting choice, crystal field, spin–orbit coupling, impurity solver, and analytic-continuation uncertainty. Compare calculations with $J_H$ varied at fixed physically matched parameters, and verify atomic multiplets plus orbital-resolved self-energies. A generic multiorbital bad metal is not automatically a Hund metal.

Exercises

Compute the triplet–singlet splitting for two electrons in distinct degenerate orbitals.

Solution

The triplet energy is $U'-J_H=U-3J_H$ and the interorbital singlet energy is $U'+J_H=U-J_H$. Therefore $E_S-E_T=2J_H$, so positive Hund exchange favors the triplet.

References

• Antoine Georges, Luca de’ Medici, and Jernej Mravlje, “Strong Correlations from Hund’s Coupling,” Annual Review of Condensed Matter Physics 4 (2013) 137–178, doi:10.1146/annurev-conmatphys-020911-125045, Open PDF.