as observed in HgTe [@zhang]. The situation in applied Zeeman field is more interesting. Notice that for $\vec{k} = k\hat{z}$, $J_z$ is a good quantum number, and there is no level repulsion between bands of different $J_z$. This allows (non-degenerate) bands to cross along this axis, which indeed occurs when $|\Delta|$ is not too large. Further analysis in the Supplementary Material shows that these crossings correspond to a pair of [*double Weyl points*]{}, with linear dispersion along the $z$ axis and quadratic dispersion normal to it. These points are strength $\pm 2$ monopoles in momentum space. Away from the $k_z$ axis, electron and hole pockets may accidentally cross the Fermi energy. If this does not occur, one has a pristine double Weyl semimetal, which occurs for the angular range $\theta_1 \le \theta \le \theta_2$, where $\theta_1 =
-\tan^{-1}(\frac{8+4 \sqrt{3}}{7 \sqrt{3} +26})$ and $\theta_2 =
\tan^{-1}(\frac{8-4 \sqrt{3}}{7 \sqrt{3} -26})$ for $\delta=0$, as shown in horizontal axis of Fig. \[fig1\]. When $0<|\Delta|<\infty$, we observe insulating, double Weyl semimetal, and Weyl metal (with coexisting electron-hole pockets) phases, as shown in Figure \[fig1\]. Note that in all these phases, the Coulomb interactions become either unimportant (in the insulator), screened (in the metal), or marginally 