Hund physics landscape of two-orbital system

Motivated by the recent discovery of superconductivity in infinite-layer nickelates RE$_{1-\delta}$Sr$_\delta$NiO$_2$ (RE$=$Nd, Pr), we study the role of Hund's 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's 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's metal, and valence-skipping metal, from the effect of $J$ being away from Mottness. Defining criteria characterizing these metals are suggested, establishing the existence of Hund's metallicity in two-orbital systems.