Electronic correlations in Hund metals

To clarify the nature of correlations in Hund metals and its relationship with Mott physics we analyze the electronic correlations in multiorbital systems as a function of intraorbital interaction $U$, Hund's coupling $J_H$, and electronic filling $n$. We show that the main process behind the enhancement of correlations in Hund metals is the suppression of the double occupancy of a given orbital, as it also happens in the Mott insulator at half-filling. However, contrary to what happens in Mott correlated states the reduction of the quasiparticle weight $Z$ with $J_H$ can happen in spite of increasing charge fluctuations. Therefore, in Hund metals the quasiparticle weight and the mass enhancement are not good measurements of the charge localization. Using simple energetic arguments we explain why the spin polarization induced by Hund's coupling produces orbital decoupling. We also discuss how the behavior at moderate interactions, with correlations controlled by the atomic spin polarization, changes at large $U$ and $J_H$ due to the proximity to a Mott insulating state.

Figure 1

(a) Quasiparticle weight $Z$ vs intraorbital interaction $U$ and Hund's coupling $J_H$ for six electrons in five orbitals, the filling of undoped iron superconductors. $U$ and $J_H$ are in units of the bare bandwidth $W$ and $U$. A strongly correlated metallic region with small $Z$, in violet, appears in a wide range of parameters. The Mott insulating state is in black. The region in yellow-orange is metallic with moderate correlations. (b) $Z$ vs $J_H$ for the system in (a) and selected $U$. (c) $Z$ vs electronic filling $n$ and $J_H$ with $U=W$ for a five-orbital system. The strong suppression of $Z$ with $J_H$ seems connected to the $n=5$ half-filled Mott insulator.

Figure 2

Quasiparticle weight $Z$ vs intraorbital interaction $U$ and Hund's coupling $J_H$ for a three-orbital system with electronic filling (a) $n=0.5$, (b) $n=1.0$, (c) $n=1.5$, (d) $n=2.0$ (e) $n=2.5$, and (f) $n=3.0$ half-filling. The Mott transition, in black, is found for the commensurate values $n=1,2,3$ with different dependence on $J_H$. Extended metallic regions with strongly reduced $Z$ are only found for filling close to half-filling. $U$ and $J_H$ are respectively given in units of the nonrenormalized bandwidth $W$ and of $U$. The system shows particle hole symmetry; results are also valid for electronic filling $2N-n$.

Figure 3

(a) Enhancement of spin fluctuations $C_S$ and suppression of $Z$ with $J_H$. $U=1.5W$ for two electrons in three orbitals (red), $U=W$ for six electrons in five orbitals (black), and for three electrons in four orbitals (green). $C_S$ and $Z$ are renormalized to their $J_H=0$ value at the given $U$. Arrows mark $J_H^{*}\left(U\right)$. The reduction of $Z$ is concomitant to the enhancement of $C_S$. (b) Charge fluctuations $C_{n_T}$ and quasiparticle weight $Z$ vs Hund's coupling $J_H$ renormalized to $C_{n_T}^0$ and $Z^0$, their value at $J_H=0$, and the corresponding $U$; see legend. The enhancement of $C_{n_T}$ while $Z$ is suppressed differs from the behavior of Mott correlated states. The region of interaction parameters for which $C_{n_T}$ is suppressed or enhanced is shown in Fig. S4 in SM [35]. (c) $C_{n_T}/C_{n_T}^0$ vs $J_H$ for five orbitals with $U=W$ and different electronic fillings $n$. $Z$ decreases with $J_H$ for all values in this figure and vanishes in the Mott state at $n=5$ (not shown). (d) Intraorbital $C_n^{\text{intra}}$ and interorbital $C_n^{\text{inter}}$ charge fluctuations vs $U$ and $J_H$ for six electrons in five orbitals. With increasing $J_H$, both $C_n^{\text{intra}}$ and $C_n^{\text{inter}}$ decrease in absolute value. In the Hund metal $C_n^{\text{intra}}$ quickly saturates to its value in the Mott state, while $C_n^{\text{inter}}$ decreases towards zero with $J_H$.

Figure 4

Charge fluctuations $C_{n_T}$ for a (a) five-orbital system with six electrons and (b) a three-orbital system with two electrons. $C_{n_T}$ is normalized to its noninteracting value ($U=0$ and $J_H=0$) to facilitate comparison between its suppression and that of $Z$ in Figs. 1 and 2. $C_{n_T}$ and $Z$ show different dependence on interactions. This is true not only in the Hund metal region discussed in the text, but also in the low $J_H$ region. For small $J_H/U, C_{n_T}$ are strongly suppressed with $U$, even for interactions at which $Z$ is only slightly renormalized. This suppression is due to the enhancement of the (negative) $C_n^{\text{inter}}$.

Figure 5

Comparison between quasiparticle weight $Z$ of computed with DMFT and the slave spin method described. We show for a model with three equivalent orbitals with semicircular density of states with $n=1,2,3$ electrons per site. A semicircular density of states with half-bandwidth $D$ is used in both calculations. The DMFT data, taken from [9], were computed using the complete interacting Hamiltonian including the spin-flip and the pair-hopping terms. Only the density-density part (Ising approximation) is included in the slave-spin calculation.