field limit we are considering, it is possible to re-scale lengths in two directions perpendicular to applied field such that $\tilde{K}_i=\tilde{K}_j$ for $i\ne j$ (where $\tilde{K}_i$ refers to the new coefficient in the re-scaled coordinates). We will therefore ignore the difference between the $K_i$ and assume that for orthorhombic point groups we are working in re-scaled co-ordinates. The term $f_{in}^{(4)}$ is not the most general such term allowed by symmetry. However, it is this term that allows the GL theory to give the same physics as in Ref. . Indeed, one can gain more insight into the vortex lattice solutions that minimizes Eq. and Eqs.(\[EQ:f2hom-NX\]-\[EQ:f4-NX\]) by relating the coefficient of the inhomogeneous quartic term $\gamma$ to the stiffness ratio $\rho_{sp}<\rho_s$ which controls the energetic stability of a pair of HQV’s[@chung:197002]. Within the London approximation, the gradient terms in Eq. and Eq. amounts to phase bending energy which will be proportional to $(\rho_{s}+\rho_{sp})$ and $(\rho_{s}-\rho_{sp})$ respectively for $\Delta_{\uparrow\!\uparrow}$ and $|\Delta_{\downarrow\!\downarrow}|$. Combining Eq. and Eq. with the homogeneous solution $|\Delta_{\uparrow\!\uparrow}|^2=|\Delta_{\downarrow\!\downarrow}|^2 = \alpha/(\beta_1-\beta_2)$, we obtain the following relation between $\gamma$ and $\rho_{sp}/\rho_s$: $$\gamma = \frac{K_1(\beta_1-\beta_2)}{\alpha}
\frac{1-\rho_{sp}/\rho_s}{1+\rho_{sp}/\rho_s}.$$ Hence $\gamma >0$ would imply stability of HQV’s and double transitions into two possible vortex phases: a 