Twofold quadruple Weyl nodes in chiral cubic crystals

Unlike conventional Weyl nodes, unconventional ones carry a quantized monopole charge $\mathcal{C}>1$, and their existence needs the protection of crystalline symmetries in addition to translation symmetry. There have been many studies on unconventional Weyl nodes, yet we have so far missed one, which is the twofold Weyl node with $\mathcal{C}=4$. In this paper, we study the relationship between the winding number and pseudospin texture in all twofold Weyl nodes, and offer an intuitive way to understand the complex pseudospin texture of the twofold quadruple Weyl node. We not only list all the possible space groups and corresponding momenta, where the twofold quadruple Weyl node can be stabilized, but also propose a series of LaIrSi-type materials that have the Weyl node with $\mathcal{C}=4$ in both the electronic band structure and the phonon spectra. In the electronic band structure, the twofold quadruple Weyl node will evolve into a fourfold quadruple Weyl node and change both the chirality and the monopole charge after considering spin-orbit coupling, which is uncommon in the known Weyl semimetals. In the phonon spectra, we propose a platform having a Weyl node with $\mathcal{C}=4$.

Figure 1

(a–c) Three kinds of Weyl nodes with a monopole charge of $+4$ and (c–f) four kinds of twofold Weyl nodes with a different Chern number. (a) Sixfold double spin-1 Weyl node, which is a composition of two identical spin-1 Weyl nodes. (b) Fourfold spin-$\frac{3}{2}$ Weyl node, with Chern number of $+1,+3,-1,-3$ for each band. (c) Twofold quadruple Weyl node with Chern number of $+4,-4$ for each band. (d–f) Three other kinds of twofold Weyl nodes that can exist in crystalline materials, which are called single Weyl nodes, double Weyl nodes, and triple Weyl nodes, respectively.

Figure 2

(a–c) Pseudospin structure for twofold Weyl nodes with $\mathcal{C}=+1,+2,+3$ on a sphere, by using the Hamiltonian of ${\mathrm{H}}_N\left(\mathbf{k}\right)$ with $N=1,2,3$. (d) Pseudospin structure for a twofold quadruple Weyl node with $\mathcal{C}=+4$ on a sphere, by using the Hamiltonian of ${\mathrm{H}}_{n\text{SOC}}\left(\mathbf{k}\right)$. (e) $\widehat{S_z}$ component for the twofold quadruple Weyl node at eight momenta along diagonal lines. (f) Pseudospin on a half sphere along the diagonal direction, which contributes a half wrapping number. Four insets in the upper right of panels (a)–(d) show the top views of pseudospin structures on the circle located at the $k_z=0$ plane, and the inset in (f) shows the side view along the [111] direction. The pseudospin wraps the unit sphere differently for systems with different Chern numbers.

Figure 3

(a) Crystal structure and (b) Brillouin zone of BaIrP. (c) Distribution of Weyl nodes in BaIrP. (d–e) Band structure of BaIrP with and without SOC.

Figure 4

(a–c) Band structures of BaIrP along the $X\text{−}\mathrm{\Gamma }\text{−}{k}_w$ direction by DFT calculations with different ${\lambda }_{\text{SOC}}$, where $k_w$ is the momentum of Weyl node $W_1^{″}$ generated from the quadruple Weyl node at $\mathrm{\Gamma }$. (d–g) Calculations of Fermi arcs at 0.16 eV along the [001] direction, for different ${\lambda }_{\text{SOC}}$.

Figure 5

(a) Crystal structure and (b) phonon spectra of BaPtGe. (f) Dynamic structural factor calculated by DFT along $X\text{−}\mathrm{\Gamma }\text{−}X$, crossing the first and second Brillouin zone, where the unit for the momentum is $\pi$/$a$.