with $J_z=-1/2, +3/2$ can be discarded, and the Hamiltonian projected onto the two level subspace of low energy states. We introduce Pauli matrices in this subspace, so that $\tau^z =
|\tfrac{1}{2}\rangle\langle\tfrac{1}{2}| -
|-\tfrac{3}{2}\rangle\langle-\tfrac{3}{2}|$, $\tau^+ = (\tau^x + i
\tau^y)/2 = |\tfrac{1}{2}\rangle\langle-\tfrac{3}{2}| =
(\tau^-)^\dagger$, and $\tau^0 =
|\tfrac{1}{2}\rangle\langle\tfrac{1}{2}| +
|-\tfrac{3}{2}\rangle\langle-\tfrac{3}{2}|$, which is the identity matrix in the $2\times 2$ subspace. We define two energy parameters, $\epsilon_{-3/2} = H (\frac{3}{2}\cos(\theta) +
\frac{27}{8} \sin(\theta))$ and $\epsilon_{1/2} =-H
(\frac{1}{2}\cos(\theta) + \frac{1}{8} \sin(\theta)) $. Then the reduced Hamiltonian becomes $$\begin{aligned}
&&\mathcal{H}_{2} = \epsilon_{+} \tau^0+ (\epsilon_{-}+d_5(k)) \tau^z +d_4(k) \tau^x - d_3(k) \tau^y, \nonumber \\
&&\epsilon_{\pm}=(\epsilon_{-3/2}\pm\epsilon_{1/2})/2.\end{aligned}$$ There are level crossing points at $k_x=k_y=0$ and $k_z=\pm
K$, with $K=\sqrt{\epsilon_-}$. We expand around these points, letting $k_x=p_x$, $k_y=p_y$ and $k_z =\pm
K+ p_z$, which gives $\mathcal{H}_2(\pm K\hat{z}+\vec{p}) = \epsilon_+ \tau^0 + %EG
\mathcal{H}_{2}^\pm(\vec{p})$, with, to leading order in $\vec{p}$, $$\begin{aligned}
\mathcal{H}_{2}^\pm = \mp v\, p_z \tau^z +d_4(p) \tau^x + d_3(p) \tau^y \ , \nonumber\end{aligned}$$ with $v=2\sqrt{\epsilon_-}$. The energy spectrum is $$\begin{aligned}
E(p)=\pm \sqrt{v^2 p_z^2 + \frac{3}{4}(p_x^2+p_y^2)^2}. \nonumber\end{aligned}$$ We see that the electrons disperse linearly along the field and quadratically orthogonal to it, near the touching point. This can be understood as a consequence of 4-fold rotational symmetry around the 