Symmetry-Protected Topological Triangular Weyl Complex

Weyl points are often believed to appear in pairs with opposite chirality. In this work, we show by first-principles calculations and symmetry analysis that single Weyl phonons with linear dispersion and double Weyl phonons with quadratic dispersion are simultaneously present between two specific phonon branches in realistic materials with trigonal or hexagonal lattices. These phonon Weyl points are guaranteed to locate at high-symmetry points due to the screw rotational symmetry, forming a unique triangular Weyl complex. In sharp contrast to conventional Weyl systems with surface arcs terminated at the projections of a pair of Weyl points with opposite chirality, the phonon surface arcs of the unconventional triangular Weyl complex connect the projections of one double Weyl point and two single Weyl points. Importantly, the phonon surface arcs originating from the triangular Weyl complex are extremely long and span the entire surface Brillouin zone. Furthermore, there are only nontrivial phonon surface states across the isofrequency surface, which facilitates their detection in experiments and further applications. Our work not only offers the promising triangular phonon Weyl complex but also provides guidance for exploring triangular Weyl bosons in both phononic and photonic systems.

Figure 1

(a) A Weyl pair with a surface arc and (b) the corresponding surface states. (c) A special Weyl complex (i.e., two single WPs with $\mathcal{C}=-1$ and one double WP with $\mathcal{C}=+2$ form a triangle denoted as green dashed lines) with surface arcs and (d) the corresponding surface states.

Figure 2

Crystal and phonon spectra of $\alpha \text{−}{\mathrm{SiO}}_2$. (a) Side and (b) top views of $\alpha \text{−}{\mathrm{SiO}}_2$. (c) The phonon spectra of $\alpha \text{−}{\mathrm{SiO}}_2$ along high-symmetry lines. Three nontrivial phonon branches 16, 17, and 18 are highlighted in red, green, and black, respectively. The lower panels show the zoom-in regions 1 and 2 marked by boxes in the upper panel of (c), respectively.

Figure 3

Two phonon branches around the $\mathrm{\Gamma }$ point form (a) quadratic dispersion in the $k_x\text{−}{k}_y$ plane and (b) linear dispersion along the $k_z$ axis. (c) The distribution of Berry curvature in the $k_x\text{−}{k}_z$ plane. The triangular Weyl complex is denoted as the red-dashed triangle.

Figure 4

The phonon surface states of $\alpha \text{−}{\mathrm{SiO}}_2$. The phonon LDOS projected on the (a) (001) surface and (b) (010) surface. The isofrequency surfaces at $\omega =16.22\mathrm{THz}$ projected on the (c) (001) and (d) (010) surfaces. In (c) and (d), the first BZ of (001) and (010) are marked by white-dashed lines, and high-symmetry lines for LDOS in (a) and (b) are marked in green-dashed lines. The yellow and blue dots indicate the projections of WPs with $\mathcal{C}=+2$ and $\mathcal{C}=-1$, respectively.