Hund Physics Landscape of Two-Orbital Systems

Motivated by the recent discovery of superconductivity in infinite-layer nickelates ${\mathrm{RE}}_{1-\delta }{\mathrm{Sr}}_{\delta }{\mathrm{NiO}}_2$ ($\mathrm{RE}=\mathrm{Nd}$, Pr), we study the role of Hund coupling $J$ in a quarter-filled two-orbital Hubbard model, which has been on the periphery of the attention. A region of negative effective Coulomb interaction of this model is revealed to be differentiated from three- and five-orbital models in their typical Hund metal active fillings. We identify distinctive regimes including four different correlated metals, one of which stems from the proximity to a Mott insulator, while the other three, which we call “intermediate” metal, weak Hund metal, and valence-skipping metal, from the effect of $J$ being away from Mottness. Defining criteria characterizing these metals is suggested, establishing the existence of Hund metallicity in two-orbital systems.

Figure 1

Atomic limit phase diagrams at $T=0$, $U>0$, and $J/U\ge 0$ for three different models ($M=2$, 3, and 5 systems) with $n_d=M+1$. The lowest-energy configurations are indicated at each region. The VS phase is highlighted in the sky blue region.

Figure 2

(a) $\mathrm{\Delta }{T}^{\text{onset}}$ as a function of $U_{\mathrm{eff}}$ obtained from $2\le U/D\le 8$ and $0\le J/U\le 0.5$. The red dotted line indicates the approximate value of $U_{\mathrm{eff}}$ below which $\mathrm{\Delta }{T}^{\text{onset}}>0$. The black dotted line denotes $U_{\mathrm{eff}}=0$. We used $T=0.05$ to stabilize the VS insulator phase [84]. The diamond symbols denote the region in which $T_{\mathrm{orb}}^{\text{onset}}>1$. Inset: $T_{\mathrm{orb}}^{\text{onset}}/T_{\mathrm{spin}}^{\text{onset}}$ as a function of $U_{\mathrm{eff}}$. (b)–(d) $C_{\text{intra}}$ (green square) and $C_{\text{inter}}$ (magenta circle) at $U=4$ for (b) Mott metal ($J/U=0.15$; $U_{\mathrm{eff}}=2.2$), (c) WH metal ($J/U=0.3$; $U_{\mathrm{eff}}=0.4$), and (d) VS metal ($J/U=0.37$; $U_{\mathrm{eff}}=-0.44$). (e)–(f) ${\chi }_{\mathrm{s}/\mathrm{o}}$, $\mathrm{\Gamma }/T$, and $C_{\mathrm{s}/\mathrm{o}}(\beta /2)/T$ plotted as a function of $T$ at $U=4$ for (e) Mott metal ($J/U=0.15$; $U_{\mathrm{eff}}=2.2$) and (f) WH metal ($J/U=0.3$; $U_{\mathrm{eff}}=0.4$). $T_{\mathrm{spin}/\mathrm{orb}}^{\mathrm{peak}}$ are marked with magenta (spin) and green (orbital) arrows. The gray dashed lines are guides to the eye to indicate quasilinearity of $\mathrm{\Gamma }/T$ (i.e., $\mathrm{\Gamma }\sim T^2$).

Figure 3

(a) $Z$ as a function of $U_{\mathrm{av}}(=U-J)$ for several $J/U[=(J/U_{\mathrm{av}})/(1+J/U_{\mathrm{av}})]$. (b) Phase diagram with color scheme representing $Z$. The red and black dotted lines indicate the same as in Fig. 2. The green dotted line is an estimated boundary of $J/U$ above which the Janus effect emerges: $(\partial Z/\partial J{)}_{U_{\mathrm{av}}}<0$, while $d{U}_c(J)/dJ>0$. Green diamonds are actual crossing points where $(\partial Z/\partial J{)}_{U_{\mathrm{av}}}=0$, whereas blue ones are $(\partial Z/\partial J{)}_U=0$ plotted for comparison. A set of $U$ and $J/U$ values belonging to the same $U_{\mathrm{av}}$ are connected with a white solid line. (c) Characteristic features of different correlated metals. Here, $+/-$ denote the sign of the corresponding quantity.