The essence of the Hund's metal

I think one of the most interesting ideas to emerge in the theory of correlated electron materials over the past five years is that of a Hund's metal, particularly how bad metal behaviour is enhanced by the presence of the Hund's rule coupling associated with degenerate d-orbitals and multiple bands.  This is relevant to many transition metal compounds. I recently found the following paper quite helpful. It builds on important earlier work by Luca de Medici.

Electronic correlations in Hund metals
L. Fanfarillo and E. Bascones

A couple of key ideas.

In a single band Hubbard model as one approaches the Mott insulator the probability of double occupancy decreases and so does the local charge fluctuations. This reduces the quasi-particle weight Z, which is the overlap of the ground state with a non-interacting Fermi sea.
A Hund's metal is different.

Hund's coupling polarizes the spin locally. The small Z in a Hund metal is due to the small overlap between the noninteracting states and the spin polarized atomic states [5,7,36]. The suppression of Z is thus concomitant with an enhancement of the spin fluctuations CS; see Fig. 3(a) [below].

Here CS=⟨S2⟩−〈S〉2 with 〈S〉=0 and S=12∑a=1,...,N(na↑−na↓). Arrows in Fig. 3(a) mark J∗H(U) the interaction at which the system enters into the Hund metal defined empirically as the value of JH with the strongest suppression of Z, i.e., the most negative dZ/dJH value, after which Z stays finite; see Fig. S2(c) in SM [35]. Above J∗H, CS reaches a value close to that of the Mott insulator at this filling showing that in the Hund metal state each atom is highly spin polarized, Figs. 3(a) and S3(b) and S3(c) in SM [35].

The three curves (black, green, red) correspond to 5 electrons in 6 orbitals, 3 electrons in 4 orbitals, and 2 electrons in 3 orbitals, respectively.

Unlike in a single band, as the system becomes more correlated (with increasing Hund's coupling J_H) the charge fluctuations can increase, as shown below.

It would be interesting to see how much of this essential physics is captured in a two-site Hubbard-Kanomori model such as this one.

An important open question is whether the signatures of a bad metal (such as thermopower of order k_B/e, no Drude peak, .... are the same for a Hund's metal and a single band system.