|\Delta_{\uparrow\!\uparrow,\sigma}|^2
|\Delta_{\downarrow\downarrow,-\sigma}|^2)\nonumber\\
&-&\beta_3[(\Delta_{\uparrow\!\uparrow,+}\Delta_{\downarrow\!\downarrow,-})
(\Delta_{\uparrow\!\uparrow,-}\Delta_{\downarrow\!\downarrow,+})^*
+ {\rm c.c.}]. \label{EQ:int1}\end{aligned}$$ $\beta_2$, $\beta'_2$ and $\beta_3$ terms originate from interaction between spin up-up pairs and down-down pairs. Again, for simplicity, we written the free energy in the limit of a cylindrical Fermi surface. Lastly, we have $$\begin{aligned}
f^{(4)}_{in} &=& \gamma \sum_{\sigma=\pm} [\Delta^*_{\uparrow\uparrow,\sigma} \Delta_{\downarrow\downarrow,\sigma} ({\bm D}\Delta_{\uparrow\uparrow,\sigma})\cdot ({\bm D} \Delta^*_{\downarrow\downarrow,\sigma}) + {\rm c.c.}]\nonumber\\
&+& \gamma' \sum_{\sigma=\pm} [\Delta^*_{\uparrow\uparrow,\sigma} \Delta_{\downarrow\downarrow,-\sigma} ({\bm D}\Delta_{\uparrow\uparrow,\sigma})\cdot ({\bm D} \Delta^*_{\downarrow\downarrow,-\sigma}) + {\rm c.c.}].\nonumber\\
\label{EQ:int2}\end{aligned}$$ Note that the form of Eq. (\[EQ:int2\]) is consistent with the form of the interaction terms in Eq. (\[EQ:int1\]). Again, this is not the most general term allowed by symmetry, but it is the minimal term that captures the physics in the London limit described in Ref. .

Determining the vortex lattice structure - general considerations {#sec:G}
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We consider the vortex lattice phases near the upper critical field to map out the stability condition for HQV lattice phases. As usual, the first step towards determining the vortex lattice structure is to identify the eigenstates of the linearized Ginzburg-Landau (GL) equations. In order to obtain a linearized GL equation we take a variation of the quadratic terms in the free energy, for example: $$\begin{aligned}
f^{(2)}_0 + f^{(2)}_{in} + f^{(2)}_Z 