Symmetry-enforced Weyl phonons

Topological phonons—quantized vibrational excitations of atoms—have attracted growing interest due to their unique properties and analogies with electronic topological quasiparticles. Prior studies predicted or observed various topological phonons in 3D crystals, including Weyl (monopole) phonons, Dirac phonons, spin-1 Weyl phonons, and charge-2 Dirac phonons, revealing exotic surface states such as noncontractible arcs and double-helicoid structures. In contrast to electrons, phonons are not constrained by the Pauli principle, enabling unusual transport behaviors across broad energies. According to the Nielsen–Ninomiya theorem, Weyl points (WPs) must appear in pairs of opposite chirality and act as sources/sinks of Berry curvature. While generic WPs lack symmetry protection and often require exhaustive searches, in spinful electronic chiral crystals, time-reversal (TR) symmetry enforces Kramers degeneracies at TR-invariant momenta to be WPs. This work shows that symmetry-enforced WPs can also arise in bosonic systems due to nonsymmorphic symmetries. A symmetry analysis across all 230 space groups (with TR symmetry, spinless single-valued reps) reveals that, for certain chiral nonsymmorphic space groups, some high-symmetry k-points on the BZ boundary can host only twofold WPs for phonons—termed symmetry-enforced Weyl phonons (SEWPs). These SEWPs are pinned by nonsymmorphic symmetries, making their surface arcs robust and long. The k-points hosting SEWPs are summarized in Table 1. As a concrete example, first-principles calculations and symmetry analysis identify K2Sn2O3 (space group 199) with twofold degeneracies at P that are monopole WPs. Time-reversal symmetry relates two nonequivalent P points, yielding WPs of the same chirality. To satisfy the net Chern number constraint in the 3D BZ, a spin-1 Weyl phonon appears at H around ~17.5 THz. The large separation between P and H leads to long, clearly visible surface arcs.

The paper situates its contribution within extensive work on topological phonons and related bosonic/photonic topological states. Prior research established topological boundary modes in mechanical and phononic systems and demonstrated experimental realizations of Weyl points and associated surface states in photonic and phononic crystals. Studies predicted/observed spin-1 Weyl and charge-2 Dirac phonons (e.g., in CoSi), double Weyl phonons, and triply degenerate points with double Fermi arcs, revealing phenomena like noncontractible arcs and double-helicoid surface states. In chiral electronic crystals, time-reversal symmetry can enforce WPs at TR-invariant momenta when improper symmetries are absent, but such WPs may be buried due to weak SOC. The present work extends the concept of symmetry-enforced WPs to spinless bosonic systems via nonsymmorphic symmetries, cataloging high-symmetry k-points in specific chiral space groups where only twofold Weyl phonons occur (SEWPs).

Symmetry analysis: The authors surveyed all 230 space groups (with time-reversal symmetry, spinless single-valued representations) to identify high-symmetry k-points whose little groups admit only two-dimensional irreducible representations for phonons, indicative of twofold degeneracies. They excluded space groups with improper rotations or twofold screw symmetries satisfying C2T = −1 to avoid line/plane degeneracies intersecting the k-points. They analyzed unitary anticommutation relations {A, B} = 0 or antiunitary constraints (e.g., combined TR and screw symmetries) that enforce twofold degeneracies at boundary k-points, and computed associated Chern numbers (±1, ±2, ±4) depending on dispersion character. Example derivations (e.g., SG 199 at P) show explicit nonsymmorphic operations leading to {A, B} = 0 and hence enforced twofold degeneracy; no higher-dimensional irreps occur at P and no protected degeneracies along intersecting lines/planes were found. First-principles calculations: Density functional theory (VASP) with GGA-PBE exchange–correlation was used. Structural optimization minimized interionic forces below 10 eV/Å with a plane-wave cutoff of 520 eV and a 3×3×3 k-point grid. Phonon dispersions were obtained via density functional perturbation theory using Phonopy, with force constants from a 2×2×2 supercell. To analyze topological properties, a phononic tight-binding model was constructed, and surface local density of states computed using WannierTools and surface Green’s functions. Irreducible representations of phonon bands were obtained with ir2tb on the phononic TB Hamiltonian. Wilson loop calculations determined Chern numbers/topological charges for monopole and spin-1 Weyl phonons. Effective k·p models at high-symmetry points (e.g., SG 199 P point) were derived consistent with symmetry constraints, showing linear Weyl Hamiltonians with fitted coefficients from ab initio data.

• Symmetry-enforced Weyl phonons (SEWPs) exist in spinless bosonic systems due to nonsymmorphic symmetries in certain chiral space groups. These twofold WPs are pinned at high-symmetry k-points on the 3D BZ boundary and are protected by unitary anticommutation relations or antiunitary TR–screw combinations.
• Table 1 enumerates space groups and k-points where only twofold monopole WPs occur for phonons (SEWPs), including SGs 24, 80, 98, 199, 210, and 214, with the protecting symmetry relations specified.
• Case study: K2Sn2O3 (space group 199, body-centered cubic). All phonon bands are doubly degenerate at P, forming Weyl phonons between the 39th and 40th bands. Two nonequivalent P points (P1, P2) related by TR host WPs of the same chirality, each with Chern number C = −1 (defined for the lower band), verified by Wilson loops on enclosing spheres.
• To satisfy the total Chern number constraint in the BZ, a spin-1 Weyl phonon appears at H near ~17.5 THz formed by bands 39–41, with Chern numbers +2 (lower), 0 (middle), and −2 (upper). The H point acts as a source and the P points as sinks of Berry curvature.
• Effective k·p model at P (SG 199) yields a linear Weyl Hamiltonian H = V1 σx kx + V2 σy ky + V3 σz kz; fitting yields V1 = 2.19 THz, V2 = 2.19 THz, V3 = −2.19 THz (sign convention as in the text).
• Surface states: Large momentum-space separation between P and H produces long, robust surface arcs. (001) and (110) surface calculations show arc connectivity between the projected WPs at M and the spin-1 WP at Γ, double-helicoid surface states, and a Lifshitz transition in the evolution of arc-like surface contours with frequency.
• Robustness: The spin-1 Weyl phonon at H lies on the BZ boundary and is robust against LOTO splitting modifications in the phonon spectrum.

The study addresses whether Weyl points in bosonic systems can be symmetry enforced, analogous to Kramers-enforced WPs in spinful electronic chiral crystals. By systematically analyzing space group symmetries with time-reversal, the authors show that nonsymmorphic symmetries can pin twofold phonon degeneracies at BZ-boundary k-points and enforce them to be Weyl points with nontrivial Chern numbers. This provides a direct and symmetry-based route to predict Weyl phonons without exhaustive 3D searches. The case of K2Sn2O3 (SG 199) exemplifies SEWPs: monopole WPs at two P points of identical chirality, complemented by a spin-1 Weyl at H to balance the net Chern number. The large separation of sources and sinks of Berry curvature yields long, observable surface arcs and double-helicoid surface states, enhancing experimental detectability. The compiled list of SEWPs across space groups offers guidance for identifying and engineering topological phonons in realistic materials and may extend to other bosonic platforms like photonics.

This work establishes that nonsymmorphic crystal symmetries in chiral space groups can enforce twofold Weyl phonons at specific high-symmetry BZ-boundary points in spinless bosonic systems. A catalog of such symmetry-enforced Weyl phonons (SEWPs) is provided. First-principles calculations on K2Sn2O3 (SG 199) confirm two monopole WPs at P (C = −1 each) and a spin-1 Weyl at H (C = +2), producing long surface arcs and double-helicoid surface states. These results offer a symmetry-driven strategy to discover Weyl phonons and promising material candidates for experimental studies of monopole and spin-1 Weyl excitations. Potential future work includes experimental verification of the predicted surface arcs and helicoid states, exploration of additional materials in the listed space groups, and extension to other bosonic systems (e.g., photonics) leveraging analogous nonsymmorphic symmetries.