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Competing electronic states in high temperature phase of NaTiO$_2$

First principle density functional theory (DFT) calculations on the high temperature phase of layered triangular lattice system NaTiO$_2$ have revealed that there exists a collective electronic state energetically close to the ground state but with competing transport properties: the latter is metallic with partially occupied doubly degenerate e$_g^{\prime}$ orbitals whereas the former is insulating with a$_{1g}$ orbital fully occupied. Significant occupation of this excited state is possible at non zero temperature either thermally or thanks to very soft (large amplitude) oxygen vibrations. Possible explanations of the experimental low conductivity based on competing orbital transport and of the specific heat jump at a structural transition based on orbital entropy are discussed.