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To reveal the origin of the "nematic electronic fluid phase" inSr$_3$Ru$_2$O$_7$, we apply the self-consistent vertex correction analysis tothe ($d_{xz},d_{yz}$)-orbital Hubbard model. It is found that theAslamazov-Larkin type vertex correction causes the strong coupling between spinand orbital fluctuations, which corresponds to the Kugel-Khomskii spin-orbitalcoupling in the local picture. Due to this mechanism, orbital nematic orderwith $C_2$ symmetry is induced by the magnetic quantum criticality inmultiorbital systems, while this mechanism is ignored in therandom-phase-approximation. The present study naturally explains the intimaterelation between the magnetic quantum criticality and the nematic state inSr$_3$Ru$_2$O$_7$ and Fe-based superconductors.

Spin-Fluctuation-Driven Orbital Nematic Order in Ru-Oxides: Self-Consistent Vertex Correction Analysis for Two-Orbital Model