calculated by assuming $|\hat{\Delta}| \propto (H_{c2} - H)^{1/2}$ near the second order phase transition at $H = H_{c2}$. Here we will deal only with $O(|\hat{\Delta}|^4)$ (or equivalently, $O(1-H/H_{c2})^2$) and ignore higher order terms. This allows us to calculate $\bf{j}$, and consequently $h_s$, solely from quadratic terms. In the lowest Landau level, this yields the screening field: $$\begin{aligned}
h_s &=& \frac{8\pi^2 K}{\Phi_0} (|\Delta_{\uparrow\!\uparrow}|^2 + |\Delta_{\downarrow\!\downarrow}|^2)
-4\pi \tilde{\kappa}(|\Delta_{\uparrow\!\uparrow}|^2 -|\Delta_{\downarrow\!\downarrow}|^2)\nonumber\\
&=&\!\left(\!\frac{8\pi^2 K}{\Phi_0}\!-\!4\pi\tilde{\kappa}\!\right)\!C^2_{\uparrow\!\uparrow}\!|\phi_0|^2\!
  +\!\left(\!\frac{8\pi^2 K}{\Phi_0}\!+\!4\pi\tilde{\kappa}\!\right)\!C^2_{\downarrow\!\downarrow}\!|\tilde{\phi}_0|^2.\nonumber\\
\label{EQ:screenField}\end{aligned}$$

Inserting Eq. (\[EQ:screenField\]) into Eq. (\[EQ:freeExpect\]), the free energy takes the following form: $$\langle \tilde{f} \rangle = -\tilde{\alpha_1} C_{\uparrow\!\uparrow}^2 -\tilde{\alpha_2} C_{\downarrow\!\downarrow}^2
+\tilde{\beta_1} C_{\uparrow\!\uparrow}^4 +\tilde{\beta_2} C_{\downarrow\!\downarrow}^4 + \tilde{\beta_3} C_{\uparrow\!\uparrow}^2 C_{\downarrow\!\downarrow}^2
\label{eq:simplef}$$ with terms quadratic or quartic in $C_{\uparrow\!\uparrow}$ and $C_{\downarrow\!\downarrow}$ with coefficients that are independent in general (we will further specify these coefficients is the next two subsections). In the absence of screening fields, Zeeman-fields and the term proportional to $\gamma$, the form of the free energy in Eq.  is similar to that examined in Ref.  in the context of two-component Bose condensates (spin-half spinor condensate).In that case, the vortex lattice structure is solely determined by the competition between the $\beta_1$ and $\beta_2$ terms of Eq. (\[EQ:freeLLL1\]); the $\beta_1$ term determines the interaction energy within each vortex lattices, and 