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Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry

We perform a complete classification of two-band $\bk\cdot\mathbf{p}$ theories at band crossing points in 3D semimetals with $n$-fold rotation symmetry and broken time-reversal symmetry. Using this classification, we show the 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 a double-Weyl metal protected by $C_4$ symmetry. Furthermore, if the direction of the ferromagnetism is shifted away from the [001]- to the [111]-axis, the double-Weyl node splits into four single Weyl nodes, as dictated by the point group $S_6$ of that phase. Finally, we discuss experimentally relevant effects including splitting of multi-Weyl nodes by applying $C_n$ breaking strain and the surface Fermi arcs in these new semimetals.