Phonon Weyl Points

• Phonon Weyl points are topologically nontrivial singularities in the vibrational spectrum, where two phonon branches cross linearly in three-dimensional momentum space and carry a quantized topological charge.
• They arise from symmetry breaking and branch inversion in materials like CuI and CdTe, leading to ideal type-II phases with clean surface arcs and measurable Berry curvature.
• Emerging research on high-charge, spin-1, and unconventional Weyl phonons highlights potential applications in defect-immune transport and thermal management in advanced phononic systems.

Searching arXiv for recent and foundational papers on phonon Weyl points to ground the article in published work. Phonon Weyl points are topologically nontrivial band-touching singularities in the vibrational spectrum of a crystal or phononic crystal. In the most standard case, a phonon Weyl point is an isolated, twofold crossing between two phonon branches that disperse linearly in three-dimensional momentum space and carry a nonzero topological charge, so that the node acts as a monopole source or sink of Berry curvature and can be removed only by annihilation with a node of opposite chirality (Liu et al., 2019). Research on the subject has expanded the Weyl concept from electrons to bosonic excitations, first through photonic and acoustic analogues and then through first-principles predictions in crystalline solids, yielding type-I, type-II, spin-1, double-, charge-3, charge-4, higher-order, and nodal-surface-partnered phonon Weyl phases (Lu et al., 2015).

1. Topological definition and invariants

A Weyl point is the simplest stable band degeneracy in three-dimensional momentum space: two bands touch at a single point and disperse linearly in all directions away from it. Near such a node, the low-energy structure is described by a Weyl Hamiltonian of the form

$H(\mathbf{q})=\sum_{i,j} v_{ij} q_i \sigma_j,$

with $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$ measured relative to the node; in the isotropic limit this reduces to $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$, where the sign gives the chirality (Lu et al., 2015). In phonon systems, the same topology is defined in terms of vibrational eigenmodes rather than electronic Bloch states.

The topological characterization is expressed through Berry connection, Berry curvature, and a quantized Chern number. For a phonon band $n$ with eigenmode $|u_n(\mathbf{k})\rangle$,

$$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$

$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$

and the topological charge of an isolated node is

$$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$

with $C=\pm 1$ for a conventional Weyl point (Ge et al., 2018). This flux-monopole structure underlies the robustness of the crossing: small perturbations may shift the node in momentum space, but cannot gap it unless an opposite-charge partner is brought into coincidence (Lu et al., 2015).

Phonon studies also use loop invariants. In MgB$_2$, for example, Berry phases on loops encircling phononic topological Weyl nodal lines are $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$0, and the corresponding topological charge is written as $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$1, making explicit that a nodal line can be locally viewed as a continuum of Weyl points aligned along a straight trajectory in momentum space (Xie et al., 2018).

2. Symmetry conditions and formation mechanisms

The occurrence of isolated Weyl points requires the breaking of either inversion symmetry $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$2 or time-reversal symmetry $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$3, because under combined $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$4 symmetry the Berry curvature vanishes (Lu et al., 2015). In phonon systems, the dominant route is the absence of inversion symmetry while preserving time reversal. This is explicit in noncentrosymmetric wurtzite CuI with space group $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$5, where nondegenerate phonon branches can cross accidentally and form Weyl nodes (Liu et al., 2019). It is likewise central in zinc-blende CdTe with space group $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$6 and in chiral NbSi$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$7 with space group $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$8 (Xia et al., 2019).

The literature identifies several distinct microscopic mechanisms for forming phonon Weyl points. One is accidental crossing of nondegenerate branches at generic momenta, as in CuI, where the nodes do not lie on high-symmetry lines and one node at a generic position generates the other eleven by mirror operations, $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$9, and time-reversal symmetry (Liu et al., 2019). A second is phonon-branch inversion. In CdTe, the key mechanism is inversion between the longitudinal acoustic and transverse optical branches, with the crossing protected because the two branches carry opposite eigenvalues $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$0 under a $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$1 rotation (Xia et al., 2019). A third is symmetry-protected parent degeneracy that splits into Weyl points upon symmetry breaking. In centrosymmetric MgB$$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$2, mirror symmetry protects straight phononic topological Weyl nodal lines along $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$3-$$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$4-$$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$5; when the mirror symmetry is broken in MgBC, the line degeneracy lifts into a pair of Weyl points with opposite chirality (Xie et al., 2018).

A fourth route is dynamic symmetry lowering driven by lattice vibrations themselves. In NbNiTe$$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$6, optical-phonon-induced atomic displacements lower the symmetry from space group 53, Pmna, to space group 28, Pma2, removing a glide plane along $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$7, the associated rotation and screw axis, and the inversion center. This converts a full gap between two electronic band manifolds into a Weyl semimetal state with 20 Weyl points, including four type-II Weyl points along $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$8-X (Wang et al., 2017). Although this is an electronic Weyl transition, it establishes a direct role for optical phonons as a control knob for Weyl topology.

3. Canonical material realizations and ideal type-II phases

A major theme in the phonon literature is the search for “ideal” Weyl phases, meaning that the Weyl nodes dominate the relevant frequency window and are not obscured by strong bulk-state overlap. Wurtzite CuI is a leading example: it hosts an ideal type-II phonon Weyl phase with six symmetry-related pairs of phonon Weyl points in the $$H(\mathbf{q})=\pm v\,\mathbf{q}\cdot\boldsymbol{\sigma}$$9 plane at about $n$0 (Liu et al., 2019). The nodes are type-II because the cone is strongly tilted, and they are described as ideal because the crossings are not buried in a large overlapping bulk phonon continuum; the band overlap with the continuum vanishes or is very small, which yields clean open surface arcs. The separation between each opposite-chirality pair connected by an arc is about $n$1, roughly ten times larger than in TaAs, and on the I-terminated surface the arcs are especially clean (Liu et al., 2019).

CdTe provides a symmetry-protected ideal type-II realization in a conventional semiconductor. There, 12 Weyl points arise from the LA–TO branch inversion at about $n$2, remain pinned to the $n$3-$n$4 boundary lines by combined $n$5 and $n$6 symmetries, and carry chiralities $n$7 (Xia et al., 2019). The Weyl points are well separated in momentum space, with $n$8 in the reported coordinate set, and produce long surface arcs on both the (001) and (111) surfaces (Xia et al., 2019).

Engineered phononic crystals realize the same ideality in a more controlled setting. A three-dimensional layer-stacked phononic crystal with triangular scatterers supports the minimal number of four ideal type-II Weyl points in the first Brillouin zone, all symmetry related and therefore at the same frequency (Huang et al., 2019). In that platform, the Weyl semimetal phase exists for

$n$9

and is separated from two distinct valley-insulating phases by Weyl-node annihilation transitions (Huang et al., 2019).

System	Weyl-phonon content	Distinguishing feature
wurtzite CuI	12 type-II WPs in $|u_n(\mathbf{k})\rangle$0 at about $|u_n(\mathbf{k})\rangle$1	ideal type-II phase with clean open arcs (Liu et al., 2019)
CdTe	12 symmetry-protected type-II WPs near $|u_n(\mathbf{k})\rangle$2	LA–TO inversion on BZ-boundary high-symmetry lines (Xia et al., 2019)
layer-stacked phononic crystal	4 ideal type-II WPs	minimal-number ideal type-II phase and Weyl–valley transitions (Huang et al., 2019)

These cases clarify an important distinction. In phonon systems, “ideal type-II” does not merely refer to a tilted cone; it also denotes a frequency window in which the Weyl crossings remain spectroscopically identifiable rather than being overwhelmed by trivial bulk pockets.

4. Unconventional, high-charge, and beyond-pairing Weyl phonons

The phonon literature extends far beyond the conventional two-band, charge-$|u_n(\mathbf{k})\rangle$3 node. In chiral acoustic crystals and related tight-binding models, a spin-1 Weyl point is formed by three bands touching at one momentum point, with two cone-like dispersions and one flat band. Such nodes carry topological charge $|u_n(\mathbf{k})\rangle$4 and admit a linearized three-band Hamiltonian of the form

$|u_n(\mathbf{k})\rangle$5

as shown in a chiral phononic crystal based on a 3D Lieb-lattice construction (Shi et al., 2020). An acoustic spin-1 Weyl semimetal realizes the same topology at the $|u_n(\mathbf{k})\rangle$6 and $|u_n(\mathbf{k})\rangle$7 points, with monopole charges $|u_n(\mathbf{k})\rangle$8 and $|u_n(\mathbf{k})\rangle$9 for the three bands and double Fermi arcs in the gaps between the first and second bands and between the second and third bands (Deng et al., 2020).

Higher-charge twofold nodes have also been identified. Transition-metal monosilicides $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$0Si ($$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$1 Fe, Co, Mn, Re, Ru) host spin-1 Weyl points at $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$2 and charge-2 Dirac points at $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$3, with double-helicoid surface phonon sheets and a non-contractible loop in the surface Brillouin zone (Zhang et al., 2017). A hybrid-Weyl phononic crystal with space group $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$4 realizes a quadruple Weyl point of charge $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$5, described as the highest topological charge a twofold degenerate node can carry in the reported spinless setting, together with conventional, quadratic, and spin-1 Weyl points in the same platform (Luo et al., 2022).

Charge-3 Weyl phonons appear in still more unconventional forms. In $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$6-LiIO$$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$7, two Weyl points with $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$8 occur at the neck crossing-points of an hourglass-type band enforced by a sixfold screw rotation in space group 173, producing hourglass charge-3 Weyl phonons and triple- or sextuple-helicoid surface arc states depending on projection (Wang et al., 2022). In $$\mathbf{A}_n(\mathbf{k}) = i\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle,$$9-quartz, a symmetry-protected triangular Weyl complex combines one double Weyl phonon with two single Weyl phonons, so that the charge balance follows $$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$0 or its opposite-sign counterpart (Wang et al., 2019).

Several works explicitly revise the conventional expectation that Weyl nodes must appear only as simple opposite-chirality pairs. “Beyond no-go theorem” isolated Weyl phonons are defined as charge-$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$1 or charge-$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$2 nodes in acoustic spectra, protected by time-reversal symmetry and point-group symmetries and accompanied by nodal walls on Brillouin-zone boundaries; notably, they do not form surface arcs in the surface Brillouin zone (Liu et al., 2022). A related extension is the single-pair-Weyl-points phase with the same odd chiral charge $$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$3 in nonmagnetic spinless crystals, compensated by a charged nodal surface; explicit phonon realizations are reported for Zr$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$4O and NaPH$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$5NO$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$6 (Ding et al., 2023). Y(OH)$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$7 adds a higher-order variant, an unconventional topological Weyl-dipole phonon in which one charge-$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$8 Weyl point and three charge-$$\mathbf{\Omega}_n(\mathbf{k})=\nabla_{\mathbf{k}}\times \mathbf{A}_n(\mathbf{k}),$$9 Weyl points form a Weyl dipole with a $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$0 higher-order charge protected by a quantized quadrupole moment (Wang et al., 24 Feb 2025).

5. Surface arcs, drumheads, helicoids, and higher-order boundary states

Bulk-boundary correspondence is central to the identification of phonon Weyl phases. For conventional isolated nodes, the characteristic boundary signature is an open surface arc connecting the surface projections of opposite-chirality Weyl points. CuI exhibits clean, open surface arcs, especially on the I-terminated surface; on the Cu-terminated surface, trivial closed-state contours also appear, but the topological arcs remain distinguishable (Liu et al., 2019). CdTe displays two long arcs on the (001) surface and six symmetry-related arcs on the (111) surface (Xia et al., 2019). In the 2018 acoustic Weyl phononic crystal, the experimentally measured acoustic Fermi arcs connect the projections of the Weyl points at $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$1 and $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$2 and support collimated one-way surface transport (Ge et al., 2018).

When the parent phase is a nodal line rather than isolated points, the surface signature changes. MgB$$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$3 supports drumhead-like nontrivial phononic surface states inside the rectangular region bounded by the projected straight nodal lines of opposite chirality on the $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$4 surface; when the nodal lines split into Weyl points in MgBC, these drumhead states evolve into open phonon arcs (Xie et al., 2018).

Higher-charge nodes produce multiply wound surface states. In transition-metal monosilicides the surface states form double-helicoid sheets globally describable by a double-periodic Weierstrass elliptic function (Zhang et al., 2017). The charge-4 node in a hybrid-Weyl phononic crystal produces quadruple-helicoid surface states (Luo et al., 2022). In $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$5-LiIO$$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$6, a single charge-3 Weyl point yields a triple-helicoid surface structure, while the projection of two time-reversal-related charge-3 nodes onto the same point yields a sextuple-helicoid surface structure (Wang et al., 2022).

A further extension is higher-order Weyl topology. A phononic higher-order Weyl semimetal in a chiral crystal with uniaxial screw symmetry exhibits both chiral Fermi arcs on surfaces and hinge arc states on hinge boundaries; the higher-order Weyl points simultaneously carry the usual chiral charge $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$7 and an additional higher-order charge $$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$8 tied to a quadrupole topological number (Luo et al., 2020). Y(OH)$$C = \frac{1}{2\pi}\oint_S \mathbf{\Omega}(\mathbf{k})\cdot d\mathbf{S},$$9 similarly combines 2D sextuple-helicoid surface arcs with 1D hinge states connecting the two Weyl dipoles along side hinges, with the hinge states guaranteed by the quantized quadrupole moment (Wang et al., 24 Feb 2025).

NbSi$C=\pm 1$0 shows that projected multiplicity can produce unconventional arc connectivity. Around the surface $C=\pm 1$1 point, three bulk Weyl points with $C=\pm 1$2 coexist with two projected $C=\pm 1$3 nodes that make $C=\pm 1$4 behave as an effective $C=\pm 1$5 point; the resulting Fermi arcs connect $C=\pm 1$6 and/or $C=\pm 1$7 in a frequency-dependent manner (Mahraj et al., 18 Jul 2025). This directly contradicts the oversimplified view that surface arcs always attach one-to-one to isolated projected $C=\pm 1$8 nodes.

6. Experimental platforms, observability, and physical consequences

The broader bosonic context of phonon Weyl points was established through photonic and acoustic experiments. In an inversion-breaking double-gyroid photonic crystal, Weyl points were observed as linear point touchings in angle-resolved microwave transmission, confirming that Weyl topology is a general wave phenomenon in three-dimensional band theory (Lu et al., 2015). Optical Weyl points of light were then observed at optical frequencies in a 3D array of laser-written waveguides, including type-II Weyl points diagnosed by conical diffraction and Fermi-arc surface states (Noh et al., 2016). Acoustic Weyl points and topological surface states were subsequently realized experimentally in a 3D phononic crystal with broken inversion symmetry, using angle-resolved transmission measurements and direct observation of acoustic Fermi arcs and step-barrier-immune transport (Ge et al., 2018).

Direct phononic experiments have since accessed more elaborate Weyl structures. Acoustic spin-1 Weyl semimetals were observed through bulk and surface field mapping, including robust propagation against multiple joints and topological negative refraction of surface arc waves (Deng et al., 2020). A chiral phononic crystal realized spin-1 Weyl points with nearly straight acoustic Fermi arcs and robust propagation around corners or defects without reflection (Shi et al., 2020). A higher-order phononic Weyl semimetal was observed with near-field spectroscopies that resolved both surface chiral Fermi arcs and hinge arc states (Luo et al., 2020).

For intrinsic crystalline solids, the proposed probes are largely spectroscopic. CuI was identified as favorable for Raman scattering, infrared spectroscopy, inelastic x-ray scattering, and high-resolution electron energy-loss spectroscopy; the $C=\pm 1$9 phonon mode is both IR- and Raman-active (Liu et al., 2019). Transition-metal monosilicides were proposed for neutron scattering and electron energy-loss spectroscopy (Zhang et al., 2017), while the unusually long and bulk-unobscured arcs of the triangular Weyl complex in $_2$0-quartz were argued to facilitate surface-sensitive detection (Wang et al., 2019).

The predicted physical consequences are predominantly transport-related. In CuI, opposite Berry curvature at opposite-chirality nodes is used to propose a Weyl phonon Hall effect in which circularly polarized light selectively excites phonons and deflects them transversely under a temperature gradient, analogous to a valley Hall response; the same work emphasizes possible consequences for novel thermal transport and phonon-dissipation control (Liu et al., 2019). CdTe is discussed in relation to one-way phonon transport, suppressed backscattering, and topological thermal devices (Xia et al., 2019). In engineered acoustic systems, the experimentally demonstrated consequences include collimated one-way transport, defect-immune or corner-immune surface propagation, and boundary-condition-tunable yet topologically persistent surface channels (Ge et al., 2018).

Taken together, the literature establishes phonon Weyl points as a broad topological category within bosonic band theory rather than a single band-structure motif. The category includes isolated twofold nodes, line-derived Weyl structures, multifold and high-charge crossings, ideal type-II phases, same-chirality single-pair nodes compensated by charged nodal surfaces, and higher-order Weyl-dipole complexes with hinge responses. This suggests that phonon Weyl physics is best understood not as a direct copy of electronic Weyl semimetals, but as a wider topological framework in which the dynamical matrix, crystal symmetry, and surface projection geometry jointly determine the allowed monopole charges, node connectivity, and boundary transport channels (Liu et al., 2022).