Spin-triplet superconductivity in Sr2RuO4 due to orbital and spin fluctuations: Analyses by two-dimensional renormalization group theory and self-consistent vertex-correction method

We study the mechanism of the triplet superconductivity (TSC) in Sr$_2$RuO$_4$ based on the multiorbital Hubbard model. The electronic states are studied using the recently developed renormalization group method combined with the constrained random-phase-approximation, called the RG+cRPA method. Thanks to the vertex correction (VC) for the susceptibility, which is dropped in the mean-field-level approximations, strong orbital and spin fluctuations at $Q \approx(2\pi/3,2\pi/3)$ emerge in the quasi one-dimensional Fermi surfaces (FSs) composed of $d_{xz}+d_{yz}$ orbitals. Due to the cooperation of both fluctuations, we obtain the triplet superconductivity in the $E_u$ representation, in which the superconducting gap is given by the linear combination of $(\Delta_x(k),\Delta_y(k))\sim (\sin 3k_x,\sin 3k_y)$. Very similar results are obtained by applying the diagrammatic calculation called the self-consistent VC method. Thus, the idea of "orbital+spin fluctuation mediated TSC" is confirmed by both RG+cRPA method and the self-consistent VC method. We also reveal that a substantial superconducting gap on the $d_{xy}$-orbital FS is induced from the gaps on the quasi one-dimensional FSs, in consequence of the large orbital-mixture due to the 4$d$ spin-orbit interaction.