Observation of Double Weyl Phonons in Parity-Breaking FeSi

Condensed matter systems have now become a fertile ground to discover emerging topological quasiparticles with symmetry protected modes. While many studies have focused on fermionic excitations, the same conceptual framework can also be applied to bosons yielding new types of topological states. Motivated by Zhang et al.’s recent theoretical prediction of double Weyl phonons in transition metal monosilicides [Phys. Rev. Lett. 120, 016401 (2018)], we directly measure the phonon dispersion in parity-breaking FeSi using inelastic x-ray scattering. By comparing the experimental data with theoretical calculations, we make the first observation of double Weyl points in FeSi, which will be an ideal material to explore emerging bosonic excitations and its topologically nontrivial properties.

Figure 1

Two types of double Weyl points are predicted in FeSi [18]. (a),(b) Cubic unit cell and the Brillouin zone (BZ) of FeSi, respectively. The purple (101) plane corresponds to the exposed sample surface in our measurement. The high symmetry points in the 3D BZ and the projected 2D BZ are shown in black and light blue, respectively. (c),(d) Schematic 3D view of the spin-1 Weyl point and charge-2 Dirac point, respectively. Their corresponding 2D view projections and Chern numbers are shown in (e),(f), respectively.

Figure 2

Effective spin-1 Weyl acoustic phonons. (a) DFPT calculated phonon dispersion along the $X\text{−}\mathrm{\Gamma }\text{−}R\text{−}M\text{−}\mathrm{\Gamma }$ direction. The Chern numbers for the longitudinal mode at higher energy and the two transverse modes at lower energy are 0 and $\pm 2$, respectively. Because of the parity symmetry ($\mathcal{P}$) breaking in FeSi, the two transverse acoustic modes are split with the largest energy separation located in the red dashed square at the $M$ point. (b) IXS measured phonon dispersion along the $M(4.5,0.5,0)\text{−}\mathrm{\Gamma }(4,0,0)\text{−}R(4.5,0.5,0.5)$ direction. The dispersion along the $\mathrm{\Gamma }\text{−}M$ and $R\text{−}\mathrm{\Gamma }$ directions is symmetrized from the raw data with respect to the $M$ and the $R$ points, respectively. (c) High-statistic IXS data provide evidence for splitting of the two transverse acoustic phonons at the $M$ point.

Figure 3

Effective charge-2 Dirac points. (a),(b) Phonon dispersion and its second derivative plot near the $R$ point. The green and red curves are the same rescaled DFPT calculations. High-statistic IXS spectra at $\mathbf{Q}=(4.26,0.26,0.26)$ and (4.5,0.5,0.5) are shown in (c),(d), respectively. The dashed blue curves are the fitted result. The number of phonon peaks in the fitting is based on the DFPT calculation (see the Supplemental Material [25]). (e) Rescaled phonon dispersion near the high-energy effective charge-2 Dirac point. The high-statistic IXS spectra at ${\mathbf{Q}}^{\mathrm{\Gamma }R}$, ${\mathbf{Q}}^R$, and ${\mathbf{Q}}^{RM}$ are shown in (f). The peak positions extracted from the IXS spectra are slightly higher than the rescaled DFPT calculation, suggesting different rescale factors for the low- and high-energy phonons.

Figure 4

Topological surface modes. (a) Momentum resolved phonon dispersion projected on the (110) surface. The definition of the surface high symmetry points is shown in Fig. 1. (b) Phonon DOS at the surface $M^{\prime }$ and ${\mathrm{\Gamma }}^{\prime }$ points. The high surface DOS at the $M^{\prime }$ point makes this point the best place to identify topological edge modes.