The origin of strong correlations and superconductivity in Na${}_x$CoO${}_2$

We propose a minimal model resolving a puzzle of enigmatic correlations observed in sodium-rich Na${}_x$CoO${}_2$ where one expects a simple, free motion of the dilute $S=1\mathrm{/}2$ holes doped into a band insulator NaCoO${}_2$. The model also predicts singlet superconductivity at experimentally observed compositions. The model is based on a key property of cobalt oxides – the spin-state quasidegeneracy of CoO${}_6$ octahedral complex – leading to an unusual physics of, {\it e.g.}, LaCoO${}_3$. We show that correlated hopping between $t_{2g}$ and $e_g$ states leads to the spin-polaron physics at $x\sim 1$, and to an extended s-wave pairing at larger doping when coherent fermionic bands are formed.