proposed HQV lattice structures and we conclude with a summary and outlook in section \[sec:conclusion\].

The triplet order parameter {#sec:OP}
===========================

The order parameter of a triplet superconductor takes a matrix form in the spin space[@RevModPhys.75.657; @RevModPhys.63.239]: $$\hat{\Delta} ({\bf k}) \!=\! \left [
\begin{array}{cc}
  \Delta_{\uparrow\uparrow}({\bf k}) & \Delta_{\uparrow\downarrow}({\bf k}) \\
  \Delta_{\downarrow\uparrow}({\bf k}) & \Delta_{\downarrow\downarrow}({\bf k})\\
\end{array}
\right ] \!\equiv\! \left[
  \begin{array}{cc}
  -d_x + id_y & d_z \\
  d_z & d_x + id_y \\
\end{array}
\right ], \label{EQ:dVec}$$ where the spin quantization axis is along the $z$ direction. The triplet pairing requires $\Delta_{\uparrow\downarrow}\!=\!\Delta_{\downarrow\uparrow}$ and a set of three complex functions of ${\bf k}$, namely $(d_x({\bf k}),d_y({\bf k}), d_z({\bf k}))$, were introduced to parameterize the gap matrix. When the three functions are collectively represented using a vector notation, the “unit vector” ${\bf\hat d}({\bf k})$ represents the symmetry direction (zero projection direction) with respect to the rotation of Cooper pair spin. In the presence of the sufficiently high field along the $c$-axis, Zeeman splitting between electrons with opposite spins prohibits pairing, leading to $\Delta_{\uparrow\downarrow}= \Delta_{\downarrow\uparrow}=0$. In the $\bf{d}$-vector notation, this implies that the $\bf{d}$-vector lies in-plane (perpendicular to the applied field). In the rest of the paper, we assume that the field is 