change of field and temperature. Indeed, the interaction $f_{in}^{(4)}$ plays this role. In particular, we numerically find that for $\gamma>0$, then this term tends to favor HQV lattices. A positive sign of $\gamma$ occurs when $\rho_{sp},\rho_s$ and this is to be expected in spin-triplet superconductors [@chung:197002]. Note, that unlike the other contributions in $\tilde{\beta}_3$, the contribution from $f_{in}^{(4)}$ vanishes as $T \rightarrow T_c$. Consequently, this term can drive a field and temperature dependent transition between a HQV lattice and a full quantum lattice. Figures 1 and 2 show the phase diagram for $\gamma=0$. Note the similarity between the calculated phase diagram and that found in the context of two-component Bose condensates [@PhysRevLett.88.180403; @barnett:240405].

![Transition between different vortex lattice structures when $\gamma=0$. (a)-(e) label the vortex lattice structure shown in Fig. (\[fig:1a\]).[]{data-label="fig:1b"}](lattice_phase.pdf){width="45.00000%"}

The effects of the Zeeman term {#subsec:zeeman}
------------------------------

For simplicity, we consider here the effect of Zeeman field alone ignoring screening and setting $\gamma=0$ (ignoring $f_{in}^{(4)}$). In the presence of Zeeman field, the free energy Eq.  would have different coefficients for two quartic terms: $$\langle \tilde{f} \rangle = -\tilde{\alpha_1}
C^2_{\uparrow\!\uparrow} -\tilde{\alpha_2}
C^2_{\downarrow\!\downarrow} +
\tilde{\beta}(C_{\uparrow\!\uparrow}^4 +
C_{\downarrow\!\downarrow}^4) + \tilde{\beta_3}
C_{\uparrow\!\uparrow}^2 C_{\downarrow\!\downarrow}^2
\label{eq-free-1}$$ where $\tilde{\alpha}_1=\alpha+\frac{K}{l^2}+H\tilde{\kappa}$, $\tilde{\alpha}_2=\alpha+\frac{K}{l^2}-H\tilde{\kappa}$, $\tilde{\beta}=\beta_1\langle |\phi_0|^4\rangle$, and $\tilde{\beta}_3=-\beta_2\langle|\phi_0|^2|\tilde{\phi}_0|^2|^2\rangle$. The 