Formation of non-unitary state near the upper-critical field of Sr₂RuO₄

We have studied the superconducting state of Sr$_2$RuO$_4$ under a magnetic field parallel to the superconducting plane. We show that due to a weak spin-orbit coupling, non-unitary k$_y(\hat{\mathbf{z}}-i\alpha\hat{\mathbf{y}})$ state is formed right at H$_{c2}$, then changes to unitary k$_y\hat{\mathbf{z}}$ state, as a magnetic field is lowered. In terms of this crossover, we address the origin of the observed double peaks of specific heat and the disappearance of the double peaks at low fields.