Possibility of mixed helical p-wave pairings in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

The exact nature of the unconventional superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ remains a mystery. At the phenomenological level, no superconducting order parameter proposed thus far seems able to coherently account for all essential experimental signatures. Among the latter is the prominent polar Kerr effect, which implies a nonzero ac anomalous Hall conductivity. Assuming the Kerr effect is intrinsic, it can be accounted for by a bulk chiral Cooper pairing with nonzero orbital angular momentum, such as $p+ip$ or $d+id$, which, however, has difficulties in being reconciled with other experimental results. Given the situation, in this paper we propose alternative possibilities with complex mixtures of distinct helical p-wave order parameters, namely $A_{1u}+i{A}_{2u}$ and $B_{1u}+i{B}_{2u}$ in group theory nomenclature. These states essentially consist of two copies of chiral p-wave pairings with opposite chirality and different pairing amplitudes, and therefore they support intrinsic Hall and Kerr effects. We further show that these states exhibit salient features that may explain several other key observations in this material, including the absence of spontaneous edge current, a substantial Knight shift drop, and possibly signatures in uniaxial strain and ultrasound measurements.

Figure 1

Illustration of the Cooper pairing in the mixed helical p-wave $A_{1u}+i{A}_{2u}$ state in a single-orbital model. The state consists of two copies of the chiral p-wave model with opposite chirality. The Cooper pairs are drawn with different sizes to reflect their different pairing amplitudes. Green arrows indicate the direction of the Cooper pair orbital angular momentum $L_z$. Another candidate state $B_{1u}+i{B}_{2u}$ has a similar pairing configuration.

Figure 2

Real (blue solid line) and imaginary (red dashed line) parts of the ${\sigma }_{\text{H}}\left(\omega \right)$ as a function of $\omega$ for $T=0$. Band parameters are [32] $t=\mu =0.4\mathrm{eV}$, $t^{\prime }=0.05t$ and $\tilde{t}=0.1t$. ${\mathrm{\Delta }}_{A_{1u}}=2{\mathrm{\Delta }}_{A_{2u}}=\frac{2}{3}{\mathrm{\Delta }}_{max}$, with ${\mathrm{\Delta }}_{max}=|{\mathrm{\Delta }}_{A_{1u}}|+|{\mathrm{\Delta }}_{A_{2u}}|=0.23\mathrm{meV}$ [32]. We choose $|{\mathrm{\Delta }}_{A_{1u}}|\ne |{\mathrm{\Delta }}_{A_{2u}}|$ so that the gap magnitudes are nonzero for both spin up and down; otherwise, one spin component would remain normal at zero temperature, which is not observed.

Figure 3

Distribution of zero-temperature spontaneous edge current in our two-orbital model with a mixed helical p-wave $A_{1u}+i{A}_{2u}$ pairing. The $A_{1u}$ component is held fixed at ${\mathrm{\Delta }}_{A_{1u}}=0.05t$. Inset: a comparison of the spontaneous current of the mixed helical and chiral p-wave pairing states with similar gap amplitudes. These calculations follow those in Refs. [39, 40, 41], and the lattice constant is defined to be unity.

Figure 4

Temperature dependence of the spin susceptibility in the $A_{1u}+i{A}_{2u}$ (solid curves) and $B_{1g}$ (dashed curves) superconducting states, in the presence of a small Zeeman field in the $x$-direction. Here, $\eta$ denotes the strength of SOC.