Hidden anomalous Hall effect in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ with chiral superconductivity dominated by the Ru $d_{xy}$ orbital

The polar Kerr effect in superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ implies finite ac anomalous Hall conductivity. Since intrinsic anomalous Hall effect (AHE) is not expected for a chiral superconducting pairing developed on the single Ru $d_{xy}$ orbital, multiorbital chiral pairing actively involving the Ru $d_{xz}$ and $d_{yz}$ orbitals has been proposed as a potential mechanism. Here we propose that AHE could still arise even if the chiral superconductivity is predominantly driven by the $d_{xy}$ orbital. This is demonstrated through two separate models which take into account subdominant orbitals in the Cooper pairing, one involving the oxygen $p_x$ and $p_y$ orbitals in the $\mathrm{Ru}{\mathrm{O}}_2$ plane, and another the $d_{xz}$ and $d_{yz}$ orbitals. In both models, finite orbital mixing between the dominant $d_{xy}$ and the other orbitals may induce interorbital pairing between them, and the resultant states support intrinsic AHE, with Kerr rotation angles that could potentially reconcile with the experimental observation. Our proposal therefore sheds new light on the microscopic pairing in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We also show that intrinsic Hall effect is generally absent for nonchiral states such as $\mathcal{S}+i\mathcal{D}, \mathcal{D}+i\mathcal{P}$, and $\mathcal{D}+i\mathcal{G}$, which provides a clear constraint on the symmetry of the superconducting order in this material.

Figure 1

Lattice and electronic structure of the $dpp$ model. (a) Sketch of the $dpp$ model with Ru-$d_{xy}$ and O-$p_x$ and $p_y$ orbitals on the square lattice of the $\mathrm{Ru}{\mathrm{O}}_2$ plane. (b) Band structure of the $dpp$ model (dotted lines) and the single-orbital model with $d_{xy}$ orbital (solid line). We choose a set of tight-binding parameters to match with the first-principle calculations [26]. The color gradient in dotted lines encrypts the variation of the weights of the three orbitals in the electronic states. The color codes are shown in (d). (c) Fermi surfaces of the $dpp$ (red dashed) and the single-$d_{xy}$-orbital model (black solid). (d) Angular dependence of the individual orbital weights across the Fermi surface.

Figure 2

The real (blue solid line) and imaginary (red dashed line) part of the Hall conductivity for various chiral superconducting states at $T=0$: (a) ${\mathcal{P}}_x+i{\mathcal{P}}_y$, (b) ${\mathcal{D}}_{x^2-y^2}+i{\mathcal{D}}_{xy}$, (c) ${\mathcal{D}}_{xz}+i{\mathcal{D}}_{yz}$. The interorbital pairing between the $d_{xy}$ and the $p$ orbitals is set at one-tenth of the intraorbital pairing on $d_{xy}$. The gap functions of the latter two states are presented in the Supplemental Material [28]. (d) The ${\mathrm{\Delta }}_2$ dependence of the Hall conductance at fixed ${\mathrm{\Delta }}_1=0.35$ meV and $\hslash \omega =3.25\mathrm{eV}$.

Figure 3

(a) Lattice structure of the $d^3$ model. Each site of the square lattice hosts all three Ru $t_{2g}$ orbitals. (b) Real (blue solid line) and imaginary (red dashed line) parts of the Hall conductivity for a chiral $p$-wave state in the $d^3$ model. Details of the model are provided in the Supplemental Material [28].