\frac{2\gamma}{l^2} \langle |\phi_0|^2 (|\tilde{\phi}_0|^2- |\tilde{\phi}_1|^2)\rangle
- \left(\frac{16\pi^3 K^2}{\Phi_0^2} + \beta_2 \right) \langle |\phi_0|^2 |\tilde{\phi}_0|^2 \rangle.
\label{EQ:quartCoeff2}$$ Now the free energy of Eq. can be minimized by choosing $C^2_{\uparrow\!\uparrow} = C^2_{\downarrow\!\downarrow} = \tilde{\alpha}/(2\tilde{\beta} + \tilde{\beta_3})$ (which gives $|\hat{\Delta}| \propto (H_{c2} - H)^{1/2}$ as mentioned), giving us the free energy expectation value $$\langle f \rangle = -\frac{H^2}{8\pi} -
\frac{\tilde{\alpha}^2}{2\tilde{\beta}+\tilde{\beta_3}}.
\label{EQ:freeFormulaLLL}$$ Eqs. and allows for understanding the role of both screening and $f_{in}^{(4)}$. The screening affect the vortex lattice structure through the dependence of terms proportional to $K^2$ in Eqs. (\[EQ:quartCoeff\]) and (\[EQ:quartCoeff2\]) on the lattice structure parameters $\varsigma$, $\sigma$ and ${\bm
\tau}$. Since all quartic expectation values depend on the lattice structure, the lattice structure will be determined through minimizing $(2\tilde{\beta}+\tilde{\beta_3})$. Since the magnitude of $K^2$ term in Eq. is larger for full quantum vortices (Eq. is not affected), screening tend to disfavor HQV lattices, in line with earlier observation for isolated HQV’s[@chung:197002]. However, since weak-coupling theories lie near the point $\tilde{\beta_3}=0$ we expect the screening effect will put the physical system at a fine balance between interlacing lattices of HQV’s and the ordinary Abrikosov vortex lattice. Hence it should be possible to observe the transition between the two phases upon small 