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Recent experiments have shown that the reduced multilayer rare-earth (RE)nickel oxides of form RE$_{p+1}$Ni$_p$O$_{2p+2}$ may belong to the novel familyof superconducting lanthanide nickelates. Here, the correlated electronicstructure of Pr$_{4}$Ni$_3$O$_{8}$ and Nd$_{6}$Ni$_5$O$_{12}$ is studied bymeans of an advanced realistic many-body framework. It is revealed that thelow-energy physics of both systems is dominated by an interplay ofNi-$d_{x^2-y^2}$ and Ni-$d_{z^2}$ degrees of freedom. Whilst theNi-$d_{x^2-y^2}$ orbitals are always highly correlated near an(orbital-selective) Mott-insulating regime, the Ni-$d_{z^2}$ orbitals give riseto intriguing non-dispersive features. At low temperature, the Pr compoundstill displays QP-like Ni-$d_{x^2-y^2}$-derived states at the Fermi level, butthe interacting fermiology of the Nd compound is outshined by an emergentNi-$d_{z^2}$-controlling flat band. These findings translate well to theprevious characterization of doped infinite-layer nickelates, and hence furthermake the case for a mechanism of unconventional superconductivity which isdistinct from the one in high-$T_{\rm c}$ cuprates.

Emergent flat-band physics in $d^{9-\delta}$ multilayer nickelates