Theory of Chiral $p$-Wave Superconductivity with Near Nodes for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We use the functional renormalization group method to study a three-orbital model for superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Although the pairing symmetry is found to be a chiral $p$ wave, the atomic spin-orbit coupling induces near nodes for quasiparticle excitations. Our theory explains a major experimental puzzle between a $d$-wavelike feature observed in thermal experiments and the chiral $p$-wave triplet pairing revealed in nuclear-magnetic resonance and the Kerr effect.

Figure 1

(a) Band dispersion along high-symmetry cuts. (b) Fermi surface (lines) and the spectral weight of the $d_{xy}$-orbital (color scale) thereon.

Figure 2

One-loop diagrams contributing to $\partial {\mathrm{\Gamma }}_{1234}/\partial \mathrm{\Lambda }$, quadratic in $\mathrm{\Gamma }$ itself (wavy lines, fully antisymmetrized with respect to incoming or outgoing fermions, labeled by the numerical indices). The color of the wavy line signifies the scattering of fermion bilinears in the pairing (blue), crossing (red), and direct (green) channels.

Figure 3

(a) Flow of negative leading eigenvalue (among all momenta), $S_{\mathrm{PH}}$ in the PH channel, shown as $1/S_{\mathrm{PH}}$ for clarity. The inset shows $-S_{\mathrm{PH}}(\mathbf{q})$ in the momentum space at the divergence scale $\mathrm{\Lambda }={\mathrm{\Lambda }}_c$. (b) Flow of NLE’s $S_{\mathrm{PP}}(\mathbf{q}=0)$. The thick line denotes the two eventually diverging $p$-wave pairing modes. Arrows indicate level crossing for $\mathbf{Q}/\pi$ in the PH channel (a) and the pairing symmetries (b).

Figure 4

(a) FRG-derived $p_x+i{p}_y$-wave gap function on the FS (thin black lines). The arrow represents ($\mathrm{Re}{\mathrm{\Delta }}_{\mathbf{k}n}$, $\mathrm{Im}{\mathrm{\Delta }}_{\mathbf{k}n}$) for $n\in (\alpha ,\beta ,\gamma )$. (b) The solid lines show the gap amplitude (up to a global scale) on the FS versus the Fermi angle $\theta$ in a quadrant of the respective pocket. The dashed line shows the gap on the $\gamma$ pocket if SOC is switched off artificially, showing the effect of SOC in generating deeper near node along $G\text{−}X$ ($\theta =0$) and local minimum along $G\text{−}M$ ($\theta =45°$).

Figure 5

The calculated physical properties in the SC state (lines), in comparison with experimental data (symbols). The solid lines are for our chiral $p$ wave, while the dashed lines are $d$-wave fits. (a) The electronic specific heat $C$ versus the temperature $T$. Here, ${\gamma }_n$ is the (constant) value of $C/T$ in the normal state. The symbols are extracted from Ref. [11], where $T_c=1.48\mathrm{K}$. (b) Superfluid density $\rho$ versus $T$, with symbols from Ref. [14]. (c) Spin-lattice relaxation rate $1/T_1$ versus $T$, normalized with respect to the value at $T_c$. The symbols are from Ref. [15], where $T_c=1.48\mathrm{K}$. (d) The direction-resolved Knight-shift $K$ versus $T$. (e) Thermal conductivity $\kappa$ versus $T$. The symbols are from Ref. [18]. Here, we use $\zeta /T_c=0.26$ according to the experimental $T_c=1.2\mathrm{K}$. (f) Low temperature limit of $\kappa /T$ as a function of impurity scattering rate $\zeta$. The temperature is fixed at $T_0=T_c/30$. The symbols are from Ref. [16] (circles), Ref. [17] (triangles), and Ref. [18] (squares). The numerical results are normalized by an assumed non-SC scale ${\epsilon }_n$, the value of $\kappa /T$ at $T=T_0$, $\zeta =0.6T_c$, and zero gap.