suppress HQV formation by introducing a length scale beyond which the HQV’s cannot exist (this length scale diverges as the spin-orbit coupling vanishes). This implies that the vortex lattice spacing must be less than this length scale for the HQV lattice to appear. However, we show that for spin-triplet hexagonal materials (specifically the two-dimensional $\Gamma^-_6$ and $\Gamma^-_5$ representations in the notation of Sigrist and Ueda [@RevModPhys.63.239]), even large spin-orbit coupling still allows for the existence of a fractional vortex lattice. This consideration may apply to Na$_x$CoO$_2\cdot$yH$_2$O.

Among the quartic terms, $f^{(4)}_{hom}$ represents the usual set of homogeneous terms. As is shown in Appendix \[app:weakcoupling\], certain quartic terms vanish in the weak-coupling theory. The terms that vanish are those that lift the energy degeneracy between the full quantum vortex (QV) and the HQV lattice. For this reason, we also include the inhomogeneous quartic term $f^{(4)}_{in}$. This term accounts for the difference between the spin phase stiffness $\rho_{sp}$ and the overall phase stiffness $\rho_s$. Not only does this difference play an important role in the stability of isolated HQV’s as it was shown in Ref.  it plays the role of tuning parameter for the vortex lattice structure.

With multiple systems in mind, 