Spin-Orbit Coupling and Symmetry of the Order Parameter in Strontium Ruthenate

Determination of the orbital symmetry of the pairing state in the spin triplet superconductor Sr$_2$RuO$_4$ is a challenge of considerable importance. Most of the experiments show that a chiral state of the $\hat{z} (k_x \pm ik_y)$ type is realized and remains stable on lowering the temperature. Here we have studied the stability of various superconducting states of Sr$_2$RuO$_4$ in the presence of spin-orbit coupling. Numerically we found that the chiral state is never the minimum energy. Alone among the five states studied it has $<{\bf \hat L}.{\bf \hat S} >=0$ and is therefore not affected to linear order in the coupling parameter $\lambda$. We found that stability of the chiral state requires spin dependent pairing interactions. This imposes a strong constraint on the pairing mechanism.