Superconducting Symmetries of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ from First-Principles Electronic Structure

Although correlated electronic-structure calculations explain very well the normal state of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, its superconducting symmetry is still unknown. Here we construct the spin and charge fluctuation pairing interactions based on its correlated normal state. Correlations significantly reduce ferromagnetic in favor of antiferromagnetic fluctuations and increase interorbital pairing. From the normal-state Eliashberg equations, we find spin-singlet $d$-wave pairing close to magnetic instabilities. Away from these instabilities, where charge fluctuations increase, we find two time-reversal symmetry-breaking spin triplets: an odd-frequency $s$ wave, and a doubly degenerate interorbital pairing between $d_{xy}$ and ($d_{yz},d_{xz}$).

Figure 1

Partial in-plane spectral weight of Ru $t_{2g}$ orbitals on the FS with $k_z=0$ (left) and $k_z=\pi$ (right) obtained from the $\mathrm{LDA}+\mathrm{DMFT}$ calculation at $T=100\mathrm{K}$. Here $d_{xy}$ is blue, $d_{yz}$ is green, and $d_{xz}$ is red. The two planes are next to each other because of the face-centered nature of the Brillouin zone. The principal nesting vectors are labeled. Important segments of the FS on the $d_{yz}$ orbital are encircled and numbered. Inset: nearly ferromagnetic vector ${\mathit{q}}_{\mathrm{nFM}}$ around the $M$ point.

Figure 2

Comparison between LDA (left panel), QP (middle panel) and $\mathrm{LDA}+\mathrm{DMFT}$ (right panel) components of the bubble $p\text{−}h$ susceptibility $[{\mathit{\chi }}_{ph}^0(\mathbf{q},{\nu }_n=0){]}_{l_1{l}_2;l_1{l}_2}$ of SRO at $T=100\mathrm{K}$. Each panel shows the intra-orbital (interorbital) components with full (dashed) lines. Dominant nesting vectors of Fig. 1 are labeled.

Figure 3

Real part of the bubble pairing susceptibility at the lowest fermionic frequency along a high-symmetry path for QP (left panel) and $\mathrm{LDA}+\mathrm{DMFT}$ (right panel).

Figure 4

Phase diagram of the leading superconducting instabilities. A lower $J_s/U_s$ implies more charge fluctuations, while the magnetic Stoner factor $S^m$ quantifies the proximity to a magnetic instability.