Multiple Weyl and double-Weyl points in the phonon dispersion of P4332 BaSi2

Weyl semimetals, classified as solid-state crystals and whose Fermi energy is accurately situated at Weyl points (WPs), have received much attention in condensed matter physics over the past 10 years. Weyl quasiparticles have been observed in the electronic and bosonic regimes, in addition to the extensive amount of theoretical and numerical predictions for the Weyl semimetals. This study demonstrates that 12 single Weyl phonons with linear dispersion and six double Weyl phonons with quadratic dispersion coexist between two specific phonon branches in real material P4332 BaSi2. The 12 single Weyl phonons and the six double Weyl phonons can form a Weyl complex phonon, which hosts a zero net chirality.


Introduction
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.
This work identifies a Weyl complex composed of 12 C-1 WPs and 6 C-2 WPs in the phonon dispersion for P4 3 32 BaSi 2. Note that P4 3 32 BaSi 2 is a prepared experiential material [40]. BaSi 2 crystallizes in the cubic P4 3 232 space group. Ba 2+ is bonded in an 8-coordinate geometry to eight equivalent Si 1− atoms. Si 1− is bonded in a 7coordinate geometry to four equivalent Ba 2+ and three equivalent Si 1− atoms. The optimized lattice constants for P4 3 32 BaSi 2 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 BaSi 2 is shown in Figure 1A.

Methods
Using the Vienna ab initio Simulation Package [41] and the DFT framework, computations for the realistic material BaSi 2 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.

Results and discussion
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 P4 3 32 BaSi 2 . The absence of an imaginary frequency in the phonon spectrum, as seen in Figure 1C, demonstrates the P4 3 32 BaSi 2 '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 highsymmetry 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 Frontiers in Physics frontiersin.org 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][54][55][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 P4 3 32 BaSi 2 . As anticipated, there are two visible phonon surface states, each of which begins at the projection of the double WP and

Summary
We demonstrated that in real material P4 3 32 BaSi 2 , 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.

Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions
Investigations and writing: YL.

Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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