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We perform a complete classification of two-band $\bk\cdot\mathbf{p}$theories at band crossing points in 3D semimetals with $n$-fold rotationsymmetry and broken time-reversal symmetry. Using this classification, we showthe existence of new 3D topological semimetals characterized by$C_{4,6}$-protected double-Weyl nodes with quadratic in-plane (along $k_{x,y}$)dispersion or $C_6$-protected triple-Weyl nodes with cubic in-plane dispersion.We apply this theory to the 3D ferromagnet HgCr$_2$Se$_4$ and confirm it is adouble-Weyl metal protected by $C_4$ symmetry. Furthermore, if the direction ofthe ferromagnetism is shifted away from the [001]- to the [111]-axis, thedouble-Weyl node splits into four single Weyl nodes, as dictated by the pointgroup $S_6$ of that phase. Finally, we discuss experimentally relevant effectsincluding splitting of multi-Weyl nodes by applying $C_n$ breaking strain andthe surface Fermi arcs in these new semimetals.

Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry