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

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