the $\beta_1$ term the interaction energy between two fractional vortex species each forming lattices. Specifically, the quartic term $-\beta_2\langle |\phi_0|^2|\tilde{\phi}|^2\rangle C_{\uparrow\uparrow}^2C_{\downarrow\downarrow}^2$ determines the stability of the HQV lattice. If $\beta_2<0$, then a HQV lattice is the ground state. If $\beta_2>0$, then full quantum vortex lattice is the ground state. In Appendix \[app:weakcoupling\], we show that $\beta_2=0$ in weak-coupling theories, so that the two lattice structures are degenerate. In the rest of the paper, we will focus on aspects that are unique to triplet superconductors in subsections \[subsec:screening\] and \[subsec:zeeman\]: the effects of the screening fields $f_{in}^{(4)}$, and the Zeeman-field.

The effects of screening and $f_{in}^{(4)}$ {#subsec:screening}
-------------------------------------------

![Real space vortex lattice structure[]{data-label="fig:1a"}](lattice_structure.pdf){width="48.00000%"}

Here we look into the effect of screening. Ignoring the Zeeman field (in weak-coupling theories, the Zeeman field is vanishing in the clean limit), we have symmetry between $C_{\uparrow\!\uparrow}$ and $C_{\downarrow\!\downarrow}$, giving us $$\tilde{\alpha_1} = \tilde{\alpha_2} \equiv
\tilde{\alpha} = \frac{2\pi K(H_{c2}-H)}{\Phi_0}\langle |\phi_0|^2 \rangle
\label{EQ:quadCoeff}$$ and $$\tilde{\beta}_1=\tilde{\beta}_2 \equiv
\tilde{\beta} = \left(\frac{\beta}{2} - \frac{8\pi^3 K^2}{\Phi_0^2}\right)\langle |\phi_0|^4 \rangle,
\label{EQ:quartCoeff}$$ which means that the free energy in Eq. takes a simpler form: $$\langle \tilde{f} \rangle = -\tilde{\alpha} (C^2_{\uparrow\!\uparrow} + C^2_{\downarrow\!\downarrow}) +
\tilde{\beta}(C^4_{\uparrow\!\uparrow} + C^4_{\downarrow\!\downarrow}) + \tilde{\beta_3} C^2_{\uparrow\!\uparrow} C^2_{\downarrow\!\downarrow},
\label{EQ:freeScreen}$$ with $$\tilde{\beta_3} = 