Symmetry-Enforced Weyl Points

• Symmetry-Enforced Weyl Points are band crossings whose presence and properties are fixed by crystalline symmetry rather than accidental tuning.
• They arise through mechanisms such as non-primitive screw rotations, algebraic anticommutation, and symmetry indicator mismatches, leading to single, double, and spin-1 configurations.
• Their topological character governs Berry flux redistribution and surface-state connectivity, resulting in robust Fermi arcs and distinctive boundary phenomena.

Symmetry-enforced Weyl points are isolated band crossings whose existence, location, and often monopole charge are fixed by crystalline symmetry rather than by parameter tuning. In this sense they differ from merely accidental Weyl nodes: the relevant little-group representations, nonsymmorphic phase factors, or symmetry indicators require a gap closing at specified points, lines, or planes of the Brillouin zone. The concept applies across electronic, phononic, photonic, and even mechanical spectra, and includes not only conventional Weyl points with $C=\pm1$ but also double Weyl points with $|C|=2$, spin-1 crossings, and symmetry-related minimal configurations dictated by time-reversal and lattice symmetry (Wang et al., 2019, Liu et al., 2019, Xue et al., 26 Feb 2026).

1. Topological definition and minimal multiplicity

A Weyl point is a two-band crossing that acts as a monopole of Berry flux in momentum space. For a two-band Hamiltonian $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$, the monopole charge is the Chern number

$$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$

with $C=\pm1$ for an ordinary linear Weyl node and $|C|=2$ for a double Weyl node in the quadratic-linear anisotropic case. This topological charge is the invariant that controls surface-arc connectivity and the redistribution of Berry curvature in the Brillouin zone (Wang et al., 2015, Bai et al., 8 Apr 2026).

In nonmagnetic crystals, time-reversal symmetry maps a Weyl point at $\mathbf k$ to a partner at $-\mathbf k$ with the same chirality. Combined with the Nielsen–Ninomiya constraint that the total monopole charge over the full Brillouin zone vanishes, this implies that the minimal conventional configuration contains four $|C|=1$ Weyl points, not two. A closely related statement appears in time-reversal-invariant photonic crystals, where the minimal number of symmetry-related Weyl points is four and all can be equifrequency (Xue et al., 26 Feb 2026, Wang et al., 2015).

Bosonic systems require a different mechanism because $T^2=+1$ does not produce electronic Kramers degeneracy. In chiral nonsymmorphic space groups, however, two-dimensional little-group irreducible representations can be the only allowed representations at certain Brillouin-zone boundary points, so the degeneracy itself must be a Weyl point. This is the setting of symmetry-enforced Weyl phonons and related bosonic realizations (Liu et al., 2019).

2. Symmetry mechanisms that force Weyl crossings

A central mechanism is the momentum-dependent phase of a screw rotation. In trigonal and hexagonal lattices without inversion symmetry but with time reversal, the non-primitive screw operator

$|C|=2$0

acts on a Bloch phonon state as

$|C|=2$1

Because the screw phase $|C|=2$2 winds by $|C|=2$3 under $|C|=2$4, two branches whose $|C|=2$5 eigenvalues differ by $|C|=2$6 cannot avoid crossing when brought close in frequency on the invariant axis or at high-symmetry points such as $|C|=2$7, $|C|=2$8, $|C|=2$9, and $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$0 (Wang et al., 2019).

In spinful systems with finite spin-orbit coupling, the same basic idea appears as hourglass or accordion connectivity. For a single $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$1-fold screw axis $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$2, the screw eigenvalues along the invariant line interpolate between different Kramers pairings at $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$3 and $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$4. The resulting mismatch forces an “$$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$5-hourglass” connectivity with $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$6 symmetry-enforced crossings in the minimal case (Zeng et al., 2019).

A second mechanism relies on algebraic anticommutation. In the phonon classification of chiral nonsymmorphic space groups, two unitary symmetries $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$7 and $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$8 can anticommute up to a reciprocal-lattice translation at a high-symmetry point, enforcing an isolated twofold degeneracy. In space group 80, a closely related antiunitary mechanism uses $$H(\mathbf q)=\mathbf d(\mathbf q)\cdot \boldsymbol\sigma$$9 with $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$0 to enforce a Kramers-like twofold degeneracy at $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$1 (Liu et al., 2019).

A third mechanism is indicator-based. Qian et al. showed that in time-reversal-invariant, noncentrosymmetric crystals with $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$2 symmetry, a mismatch between the product $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$3 of the two time-reversal-plane $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$4 invariants and the $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$5 symmetry indicator $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$6 forces the bulk to be gapless; the gap-closing objects are Weyl points (Qian et al., 2019).

3. Effective Hamiltonians and allowed charges

Near an enforced crossing, the low-energy theory is typically a symmetry-constrained two-band $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$7 model. For the trigonal phonon problem, the general form near a Weyl point is

$$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$8

with $$C=\frac{1}{2\pi}\oint_{S^2}\mathbf\Omega\cdot d\mathbf S,$$9. At the Brillouin-zone corners $C=\pm1$0 and $C=\pm1$1, where only the screw $C=\pm1$2 is present, the symmetry constraint forces $C=\pm1$3 to be linear in $C=\pm1$4, yielding the single-Weyl Hamiltonian

$C=\pm1$5

with $C=\pm1$6 (Wang et al., 2019).

At $C=\pm1$7 and $C=\pm1$8, the combined antiunitary symmetry $C=\pm1$9 protects a quadratic touching in $|C|=2$0 while remaining linear in $|C|=2$1. The lowest-order Hamiltonian becomes

$|C|=2$2

and the Berry curvature integrates to $|C|=2$3 over a small sphere (Wang et al., 2019).

The same quadratic-linear structure reappears in other symmetry settings. For the classification of exactly four double-Weyl points in nonmagnetic crystals, the generic $|C|=2$4-protected model is

$|C|=2$5

with linear dispersion along $|C|=2$6 and quadratic dispersion in the transverse plane. When the ratio of the two $|C|=2$7 eigenvalues is $|C|=2$8, the enclosed Chern number is $|C|=2$9 (Bai et al., 8 Apr 2026).

Photonic charge-2 Weyl points provide an experimentally controlled realization of the same symmetry logic. In the chiral woodpile photonic crystal with space group $\mathbf k$0, $\mathbf k$1 symmetry forbids linear in-plane terms, so the leading Hamiltonian near $\mathbf k$2 is quadratic in $\mathbf k$3 and linear in $\mathbf k$4. Breaking the protecting screw symmetry by making one rod layer inequivalent lowers the symmetry to $\mathbf k$5 and introduces a linear perturbation such as $\mathbf k$6, which splits the charge-2 Weyl point into two charge-1 Weyl points (Jörg et al., 2021).

4. Surface-state connectivity and reciprocal-space complexes

Because a Weyl point is a source or sink of Berry flux, projected chiral charge controls the topology of boundary modes. In conventional two-node pictures, a single surface arc terminates at the projections of a pair of opposite-chirality Weyl points. Symmetry-enforced settings can be qualitatively different (Wang et al., 2019).

The clearest example is the “triangular Weyl complex” in trigonal or hexagonal phonon systems. In $\mathbf k$7-quartz, a double Weyl point at $\mathbf k$8 and two single Weyl points at symmetry-related $\mathbf k$9 points form the vertices of an equilateral triangle in reciprocal space. On a (001) surface, these nodes project to one double-charged point $-\mathbf k$0 and two single-charged points $-\mathbf k$1. Two distinct phonon-surface arcs emanate from the double-charged projection and terminate at the two single-charged projections; because the triangular network fills each half of the surface Brillouin zone, these arcs are forced to span the entire first surface Brillouin zone and remain spectrally isolated because no trivial bulk modes lie at the same frequency (Wang et al., 2019).

Other enforced configurations display equally distinctive connectivity. In K$-\mathbf k$2Sn$-\mathbf k$3O$-\mathbf k$4, the two $-\mathbf k$5-point Weyl phonons have the same chirality, so a spin-1 Weyl phonon at $-\mathbf k$6 supplies the compensating charge; on the (001) surface the arcs connect $-\mathbf k$7, forming double helicoids as frequency is swept (Liu et al., 2019). In THRLN-C$-\mathbf k$8, every $-\mathbf k$9 double Weyl point emits or absorbs two Fermi arcs, giving closed-loop or extended arcs on the (100) surface and compact Fermi rings on the (110) surface (Bai et al., 8 Apr 2026). In minimal four-Weyl boron allotropes, projected charges $|C|=1$0 can force two parallel arcs or a “Z”-shaped double-arc on selected terminations (Xue et al., 26 Feb 2026).

5. Classifications and representative realizations

Symmetry-enforced Weyl points now form a broad classification problem spanning multiple families of space groups and quasiparticles. Representative systems include the following.

System	Enforced feature	Symmetry mechanism
$|C|=1$1-quartz	single Weyl phonons at $|C|=1$2 and double Weyl phonons at $|C|=1$3; triangular Weyl complex	threefold screw $|C|=1$4 and $|C|=1$5
K$|C|=1$6Sn$|C|=1$7O$|C|=1$8	symmetry-enforced Weyl phonons at $|C|=1$9 and a spin-1 Weyl phonon at $T^2=+1$0	chiral nonsymmorphic little-group algebra
Modified double-gyroid photonic crystal	four equifrequency Weyl points	inversion breaking with preserved $T^2=+1$1 and $T^2=+1$2
THRLN-C$T^2=+1$3	exactly four symmetry-protected double-Weyl points near the Fermi level	$T^2=+1$4 or fourfold screw protection
P6-B$T^2=+1$5 and TBIN-B$T^2=+1$6	exactly four conventional Weyl points near the Fermi level	minimal nonmagnetic four-Weyl configuration
$T^2=+1$7-$T^2=+1$8	double-, Type-I, and Type-II Weyl points coexisting with a nodal surface	$T^2=+1$9, $|C|=2$00, and $|C|=2$01

The phonon classification identifies seven chiral nonsymmorphic space groups in which certain Brillouin-zone boundary points can host only Weyl points. The electronic classification of minimal double-Weyl semimetals finds exactly 28 space groups admitting precisely four symmetry-protected double-Weyl points. The corresponding classification for conventional $|C|=2$02 Weyl points gives 76 space groups in the spinless limit and 83 in the spinful case that allow exactly four Weyl points. Orthorhombic and tetragonal classifications add a different minimality notion: certain band pairs can carry only two Weyl points in the entire Brillouin zone, notably in orthorhombic space groups 18, 36, 44, 45, and 46 and in tetragonal space groups 119 and 120 (Liu et al., 2019, Bai et al., 8 Apr 2026, Xue et al., 26 Feb 2026, Leonhardt et al., 2021, Hirschmann et al., 2021).

Concrete materials extend well beyond the canonical examples above. Cu$|C|=2$03ZnGeSe$|C|=2$04 realizes the $|C|=2$05-indicator route with eight Weyl nodes in a mirror plane, AgF$|C|=2$06 and AuF$|C|=2$07 realize screw-enforced electronic Weyl points in hexagonal lattices together with large intrinsic spin-Hall conductivity, and CoSi in space group 198 exhibits a symmetry-enforced network in which multifold points, Weyl points, and nodal planes collectively satisfy fermion doubling (Qian et al., 2019, González-Hernández et al., 2020, Huber et al., 2021).

6. Misconceptions, conversions, and stability

A common misconception is that Weyl points “appear in pairs with opposite chirality” in any direct and local sense. The lattice constraint is global charge neutrality over the full Brillouin zone, but symmetry can organize the charges into less conventional local structures. The triangular Weyl complex is one explicit example: the relevant surface-arc connectivity is set by one projected double Weyl point and two projected single Weyl points, not by a single isolated opposite-chirality pair (Wang et al., 2019).

Another misconception is that opposite-chirality Weyl points necessarily annihilate when they meet. In a two-band model with mirror symmetry and opposite mirror eigenvalues for the two orbitals, a mirror-related pair cannot gap out upon collision in the mirror plane; instead it converts into a nodal loop in that plane. For $|C|=2$08-symmetric systems, in-plane Weyl points carry an additional integer “helicity,” and annihilation requires cancellation of both total chirality and total helicity (Sun et al., 2018).

Symmetry breaking can also transmute enforced nodes rather than simply removing them. In the near-infrared photonic crystal experiment, reducing $|C|=2$09 to $|C|=2$10 splits a charge-2 Weyl point into two charge-1 Weyl points with a symmetry-selected direction of motion. In THRLN-C$|C|=2$11, hydrostatic compression drives annihilation into a trivial insulator, $|C|=2$12-preserving tensile strain yields two three-terminal Weyl complexes, and weak $|C|=2$13-axis strain breaks $|C|=2$14 so that each double Weyl point degenerates into two conventional $|C|=2$15 Weyl points (Jörg et al., 2021, Bai et al., 8 Apr 2026).

Finally, perturbative many-body stability can itself be symmetry-enforced. If a spatial symmetry $|C|=2$16 anticommutes with the free nodal Hamiltonian, $|C|=2$17, and leaves the interaction invariant, $|C|=2$18, then the diagrammatic self-energy at the nodal point vanishes to all orders for generic two-body interactions. The result applies to type-I and type-II Weyl semimetals, higher-charge Weyl points, and related Dirac systems (Carlström et al., 2017).

Symmetry-enforced Weyl points therefore occupy a precise niche within topological band theory: they are not merely robust crossings, but crossings whose existence is encoded in crystalline representation theory, nonsymmorphic algebra, or symmetry-indicator mismatch. That distinction is what makes their multiplicity, chirality, motion under perturbation, and boundary signatures unusually constrained—and unusually diagnostic.