Condensed-matter systems with inherent topological orders have received a great deal of attention lately. On the one hand, since quasiparticle excitations in realistic materials provide analogs of relativistic fermions or bosons in quantum field theory, these topological systems offer exotic platforms to study elementary particles and their related phenomena in high-energy physics. However, non-trivial topology, which is characterized by topological invariants, gives rise to topological quasiparticles in crystalline solids [1], providing an intriguing way to study symmetry-protected topological orders. Additionally, the crystal symmetry rather than the Poincare symmetry constraints quasiparticles in crystalline solids. There is therefore a possibility of discovering unusual topological quasiparticles [2–10] without high-energy physics counterparts in condensed-matter physics in addition to the traditional Dirac, Weyl, and Majorana particles in the standard model.

Numerous conventional and non-conventional topological quasiparticles have been proposed up to this point. For instance, intense research focuses on topological bosons in crystalline solids and various non-trivial fermions in topological semimetals [11–29]. Weyl-type excitations stand out among these non-trivial quasiparticles as being particularly significant. A quantized chiral charge, also known as the Chern number C, is what defines the topology of a Weyl point (WP). WPs are present in a system by shattering either the time-reversal or inversion symmetry because of the twofold-degenerate feature.

However, the crystal symmetries of crystalline solids are more intricate and may contain unusual Weyl-type quasiparticles. For example, the screw rotational symmetry can protect double WPs or WPs with the higher Chern number C. The 4-fold or 6-fold rotational symmetry can protect quadratic-double or cubic-triple WPs [28, 30–39].

This work identifies a Weyl complex composed of 12 C-1 WPs and 6 C-2 WPs in the phonon dispersion for P4₃32 BaSi_2. Note that P4₃32 BaSi₂ is a prepared experiential material [40]. BaSi₂ crystallizes in the cubic P4₃232 space group. Ba²⁺ is bonded in an 8-coordinate geometry to eight equivalent Si¹⁻ atoms. Si¹⁻ is bonded in a 7-coordinate geometry to four equivalent Ba²⁺ and three equivalent Si¹⁻ atoms. The optimized lattice constants for P4₃32 BaSi₂ are a = b = c = 6.771 Å, which are in good agreement with the experimental data, i.e., a = b = c = 6.715 Å [40]. The crystal structure of the relaxed BaSi₂ is shown in Figure 1A.

Using the Vienna ab initio Simulation Package [41] and the DFT framework, computations for the realistic material BaSi₂ were carried out. The calculation’s energy and force convergence conditions were set to 10⁻⁶ eV and −0.01 eV/Å, respectively. A 5 × 5 × 5 Monkhorst-Pack grid was used to sample the whole BZ after the plane-wave expansion was truncated at 500 eV. We used the density functional perturbation theory to obtain the force constants for phonon spectrum calculations, and then we used the PHONOPY package [42] to calculate the phonon dispersion spectrum. We obtained the phonon Hamiltonian of the tight-binding model and the surface local DOSs with the open-source software WANNIER TOOLS [50] and surface Green’s functions.

We determine the phonon spectra using first-principles calculations and verify that the two obvious phonon crossing points are present in the optical phonon branches of P4₃32 BaSi₂. The absence of an imaginary frequency in the phonon spectrum, as seen in Figure 1C, demonstrates the P4₃32 BaSi₂’s dynamical stability. We mainly focus on the frequencies around 8 THz and find two obvious phonon crossing points, P1 and P2, on Γ-X and Γ-M, respectively (see Figure 1B).

Figures 2A, B display the three-dimensional plot of the twofold degenerate phonon bands around the P1 and P2 points, respectively. From Figure 2A, one finds that the WP at P1 is a Charge-two WP. The charge-2 Weyl point (C-2 WP) is a topologically charged 0D two-fold band degeneracy with a charge of 2. It has a quadratic energy splitting in the plane perpendicular to the Γ-X direction and a linear dispersion in one direction (Γ-X). On a high-symmetry line or at a high-symmetry point in the BZ, the C-2 WP can happen. Figure 2B shows that the WP at P2 is a Charge-one WP. The charge-1 Weyl point (C-1 WP) is a degeneracy of the 0D two-fold band. It can occur at a generic k point in BZ and features a linear energy splitting in any direction in momentum space.

Note that C-2 WP and C-1 WP are also named as double WP and single WP, respectively. In the scientific literature, double-Weyl points with greater topological ordering and emerging in particular crystals with particular symmetries have been discovered. Because double-Weyl points result from the coalescence of two single-Weyl points, their topological charge values are equal to 2 and −2.

In order to determine the chirality of Weyl phonons, we employ the Wilson-loop method within the evolution of the average position of Wannier centers. Figures 2C, D show the evolution of the average position of the Wannier centers for the P1 WP with positive chirality and the evolution of the average position of the Wannier centers for the P2 WP with negative chirality, respectively. These findings suggest that the WP at P1 (or P2) forms a double phonon WP with chiral charge 2 (or a single WP with chiral charge −1).

As shown in Figure 3A, one finds a total of 12 single WPs with C = −1 and 6 double WPs with C = +2 in the 3D BZ. The positions for all these multiple C-1 and C-2 WPs are shown in Figure 3B. These 12 C-1 WPs and 6 C-2 WPs will form a Weyl complex, which has a zero net chiral charge and obeys the Nielsen-Ninomiya no-go theorem [51, 52]. Note that the Weyl compelx has also be predicted by series research groups [53–56].

Therefore, our findings present ideal candidates for C-1 and C-2 WP phonons to form phononic Weyl complexes. Moreover, our findings are applicable to fermionic systems.

Unique non-trivial surface states are associated with the exotic C-2 and C-1 WP phonons. We build a phonon tight-binding Hamiltonian in the Wannier representation using second-order interatomic force constants to demonstrate this. The iterative Green’s function method is used to calculate phonon surface states in this model. Figure 4 depicts the local phonon density of states (LDOS) projected on a semi-infinite (001) surface of P4₃32 BaSi₂. As anticipated, there are two visible phonon surface states, each of which begins at the projection of the double WP and ends at the projections of two single WPs. The lack of trivial bulk states on the (001) surface of P4₃32 BaSi₂ substantially simplifies experimental detection and subsequent applications [43–49].

We demonstrated that in real material P4₃32 BaSi₂, there are 12 single WPs with C = −1 and 6 double WPs with C = +2 in the 3D BZ. These Weyl phonons create a Weyl complex with zero net charge number, and their non-trivial surface states connect the projections of phonon WPs are visible.