Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry

We perform a complete classification of two-band $\mathbf{k}·\mathbf{p}$ 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 ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ 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] axis 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 the splitting of multi-Weyl nodes by applying a $C_n$ breaking strain and the surface Fermi arcs in these new semimetals.

Figure 1

(a), (d) Band structure along $\Gamma Z$ ($\Gamma L$) in 3D FM ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ with magnetization $\mathbf{M}$ parallel to [001] ([111]) directions obtained from an eight-band tight-binding model fitted from LDA results. The $C_4$ ($C_3$) eigenvalues of the conduction and the valence bands are also shown. (b), (e) The schematic showing the two double-Weyl points (four Weyl points) in 3D BZ with $\mathbf{M}$ parallel to [001] ([111]). Red/darker grey (blue/lighter gray) means nodes with positive (negative) monopole charge(s). (c), (f) The schematic of the Chern number as a function of $k_z$ ($k_z^{\prime }$) along $\Gamma Z$ ($\Gamma L$) with $\mathbf{M}$ parallel to [001] ([111]). (g) The conduction and the valence band dispersion at $k_z=k_c=0.43$ with $\mathbf{M}$ parallel to [001]. (h) and (i) The conduction and the valence band dispersion at $k_z^{\prime }=k_{c1}=0.43$ and $k_z^{\prime }=k_{c2}=0.15$ with $\mathbf{M}$ parallel to [111], respectively.

Figure 2

(a) Momentum resolved spectral weight, $\propto \mathrm{Im}[G(k_y,k_z,E)]$, calculated on the surface of FM ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ after the system is cut perpendicular to the $k_x$ direction. The energy is varied as a function of $k_z$ to ensure that it is inside the bulk gap. The intensity is normalized such that unity means the state is completely localized on the surface. (b) and (c) Schematics showing that a double- and a triple-Weyl node breaks down to two and three Weyl nodes under an applied strain ${ϵ}_{xx}$, respectively.