Stabilizing even-parity chiral superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

Strontium ruthenate (${\mathrm{Sr}}_2{\mathrm{RuO}}_4$) has long been thought to host a spin-triplet chiral $p$-wave superconducting state. However, the singletlike response observed in recent spin-susceptibility measurements casts serious doubts on this pairing state. Together with the evidence for broken time-reversal symmetry and a jump in the shear modulus $c_{66}$ at the superconducting transition temperature, the available experiments point towards an even-parity chiral superconductor with $k_z(k_x\pm i{k}_y)$-like $E_g$ symmetry, which has consistently been dismissed based on the quasi-two-dimensional electronic structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Here, we show how the orbital degree of freedom can encode the two-component nature of the $E_g$ order parameter, allowing for a local orbital-antisymmetric spin-triplet state that can be stabilized by on-site Hund's coupling. We find that this exotic $E_g$ state can be energetically stable once a complete, realistic three-dimensional model is considered, within which momentum-dependent spin-orbit coupling terms are key. This state naturally gives rise to Bogoliubov Fermi surfaces.

Figure 1

(a) Phase diagram showing the stability of $A_{1g}$ and $E_g$ pairing states as a function of the SOC parameters $\eta$ and $t_{56z}^{\text{SOC}}$. The vertical dashed lines indicate the minimum distance between two Fermi surfaces. Percentages are defined as fractions of $2\pi /a$. For small $\eta$, the separation between the $\beta$ and $\gamma$ bands becomes too small, in view of the ARPES data [48]. The thin solid lines indicate the maximum variation of the Fermi surface along the $k_z$ direction. For large $t_{56z}^{\text{SOC}}$, the Fermi surfaces become too dispersive. The blue and magenta dots denote the parameter choices for $E_g$ and $A_{1g}$ stable solutions used in (b). (b) Fermi-surface shapes, projected onto the $k_x{k}_y$ plane, for representative points in the $A_{1g}$ (red) and $E_g$ (blue) regions in (a). For $A_{1g}$, $\eta =57\mathrm{meV}$ and $t_{56z}^{\text{SOC}}=10\mathrm{meV}$, while for $E_g$, $\eta =40\mathrm{meV}$ and $t_{56z}^{\text{SOC}}=12\mathrm{meV}$.

Figure 2

Projected gaps at the Fermi surfaces for a representative (a) $A_{1g}$ and (b) chiral $E_g$ state in the first Brillouin zone. Parameters are the same as in Fig. 1. The color code is normalized to the maximum value of the $A_{1g}$ gap.

Figure 3

BFSs for the chiral $E_g$ state. The Fermi surfaces in red, green, and blue correspond to inflated nodes stemming from the $\alpha$, $\beta$, and $\gamma$ band, respectively.