Double-Weyl Phonons in Transition-Metal Monosilicides

We employed ab initio calculations to identify a class of crystalline materials of $M\mathrm{Si}$ ($M=\mathrm{Fe}$, Co, Mn, Re, Ru) having double-Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed “spin-1 Weyl point” at the Brillouin zone center and “charge-2 Dirac point” at the zone corner. The corresponding gapless surface phonon dispersions are two helicoidal sheets whose isofrequency contours form a single noncontractible loop in the surface Brillouin zone. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions.

Figure 1

Single Weyl point and two types of double-Weyl points. (a) A spin-$1/2$ Weyl point of Chern number of $\pm 1$. (b) The 2-band quadratic Weyl point with Chern numbers of $\pm 2$. (c) The 3-band spin-1 Weyl point with Chern numbers of 0, $\pm 2$. (d) The 4-band charge-2 Dirac point with Chern numbers $\pm 2$. Here we refer to quadratic Weyl point, spin-1 Weyl point and charge-2 Dirac points as double Weyl points. We focus on the 3-band and 4-band double Weyl points in this paper.

Figure 2

$M\mathrm{Si}$ crystal structure and phonon bands. (a) Cubic unit cell contains 4 $M$ and 4 Si atoms. (b) Bulk and (001) surface BZs. (c) Phonon dispersion of FeSi along high-symmetry directions. The acoustic bands have Chern numbers of $0,\pm 2$. The orange boxes are centered around the spin-1 Weyl points at $\mathrm{\Gamma }$ and the purple boxes are centered around the charge-2 Dirac points at $R$. (d) $\mathrm{\Gamma }$ point is a spin-1 Weyl point. (e) $R$ point is a charge-2 Dirac point.

Figure 3

Double-helicoid surface states and noncontractable surface arcs. (a) The surface LDOS for the (001) surface along high-symmetry directions. The corresponding surface arcs for different frequency are showed in (b)–(d), plotted in the log scale. There are two arcs rotates around the two double Weyl points as the frequency decreases from f1 to f3, which demonstrates the double-helical surface states.

Figure 4

The double-helicoid surface state is described by the Weierstrass elliptic function. The red spin-1 Weyl point corresponds to the double zero point and the blue charge-2 Dirac point corresponds to the double pole point. Two yellow surfaces rotate around these two double-Weyl points as double helicoids.