under these circumstances, the order parameter symmetry takes the $U(1)\times U(1)$ form, which can allow for HQV’s with $h/4e$ vorticity associated with $\pi$ orbital phase winding and $\pi$ ${\bf d}$-vector winding.

The Gibbs free energy {#sec:GibbsFree}
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In order to identify stable vortex type and the lattice structure itself, we start with the Gibbs free energy including all the terms allowed by symmetry up to quartic order. As usual, the quartic terms determine the vortex lattice structure. Due to additional spin degrees of freedom, the full expression for the Gibbs free energy involves a number of additional terms compared to singlet superconductor case and it is instructive to consider different contributions separately: $$f = f_{mag} + f^{(2)}_0 + f^{(2)}_{Z} + f^{(2)}_{in} +
f^{(2)}_{SO} +  f^{(4)}_{hom} + f^{(4)}_{in}. \label{EQ:freeTot}$$ where $f_{mag} = h^2/8\pi - hH/4\pi$ is the magnetic energy (the field $h$ is the sum of the external field $H$ and the screening field), the superscript (2) indicates terms quadratic in OP and (4) quartic in OP

Other than the conventional quadratic term $f^{(2)}_0$: $$f^{(2)}_0 = -\alpha \sum_i |\Delta_i|^2 \label{EQ:f0}$$ the remaining quadratic terms in Eq. are consequences of additional spin degree of freedom for the triplet superconductors. The Zeeman coupling 