lattice of ordinary Abrikosov vortices and a lattice of HQV’s. This transition is determined by the $\beta_2$ term of Eq. (\[EQ:f2hom-NX\]) and the $\gamma$ term of Eq. (\[EQ:f4-NX\]).

Chiral Triplet Superconductor
-----------------------------

With the ruthenate Sr$_2$RuO$_4$ in mind, we consider a chiral triplet $p$-wave superconductor with square symmetry for which $\tilde{f}(k_\sigma) = k_x + i\sigma k_y$ in the Eq. with $$\hat{\Delta}({\bf k})=  \sum_{\sigma=\pm}(k_x + i\sigma k_y)\left[
 \begin{array}{cc}
  \Delta_{\uparrow\uparrow,\sigma}& 0 \\
  0 & \Delta_{\downarrow\downarrow,\sigma}
  \end{array}
\right], \label{EQ:spinChiral1}$$ where $\Delta_{s,\sigma}$ ($s =\uparrow\!\uparrow,\downarrow\!\downarrow$ and $\sigma=\pm$) form expansion parameters for the Gibbs free energy. In terms of the $\bf{d}$-vector notation ${\bf d} \equiv {\bf \hat{x}}(\eta_{xx} k_x + \eta_{xy} k_y) + {\bf \hat{y}}(\eta_{yx} k_x + \eta_{yy} k_y)$, $$\begin{aligned}
\Delta_{\uparrow\!\uparrow,+} = &-& (\eta_{xx} - i\eta_{xy} - i\eta_{yx} - \eta_{yy})/2,\nonumber\\
\Delta_{\uparrow\!\uparrow,-} = &-& (\eta_{xx} + i\eta_{xy} - i\eta_{yx} + \eta_{yy})/2,\nonumber\\
\Delta_{\downarrow\!\downarrow,+} = &\,& (\eta_{xx} - i\eta_{xy} + i\eta_{yx} + \eta_{yy})/2,\nonumber\\
\Delta_{\downarrow\!\downarrow,-} = &\,& (\eta_{xx} + i\eta_{xy} +
i\eta_{yx} - \eta_{yy})/2. \label{EQ:spinChiral2}\end{aligned}$$

Formally, without spin-orbit coupling, this order parameter is a direct product of a $E_u$ orbital representation of the tetragonal point group and the in-plane vector representation for spin rotations. When spin-orbit is included the order parameter contains the four different one-dimensional representations of the tetragonal point 