Nodal Lines and Weyl Points

• Nodal lines and Weyl points are topological band degeneracies where isolated Weyl points act as monopole sources of Berry curvature and continuous nodal lines are stabilized by crystalline symmetries.
• They manifest robust phenomena such as chiral surface Fermi arcs and nearly flat drumhead modes, directly reflecting nontrivial topology and underlying symmetry constraints.
• Their physical realizations span electronic, photonic, and phononic systems, with experiments like ARPES and pump-probe techniques revealing dynamic node conversion and interaction effects.

Nodal lines and Weyl points represent two fundamental classes of topological band degeneracies in crystalline solids, photonic systems, and other engineered lattices. Weyl points are isolated, zero-dimensional band touchings in momentum space that act as monopole sources of Berry curvature, carrying quantized Chern (chirality) charge. Nodal lines, in contrast, are one-dimensional manifolds of two-band degeneracies, often stabilized by crystalline symmetries. Both types of nodes exhibit robust topological phenomena, including protected boundary states, unconventional transport, and non-trivial responses to external fields. Their interplay, mutual conversion, and symmetry constraints define a major theme in the modern classification of gapless topological phases.

1. Topological Structure and Symmetry Protection

Weyl points occur when three independent band crossing conditions are met in three-dimensional momentum space. The minimal two-band Hamiltonian near a Weyl node takes the form

$$H(\mathbf{k}) = v_x (k_x - k_{0x})\, \sigma_x + v_y (k_y - k_{0y})\, \sigma_y + v_z (k_z - k_{0z})\, \sigma_z,$$

with eigenvalues $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$, where $q_i = k_i - k_{0i}$ $1207.0478$. Each Weyl point is stabilized by its topological charge (chirality)

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

rendering it robust against any perturbation that does not connect it to a partner of opposite charge.

Nodal lines typically manifest in the presence of mirror or glide-plane symmetries. They arise when, due to symmetry, only two of three band crossing conditions are independent in 3D, yielding a continuous $1$D locus of degeneracy. The Berry phase around a loop encircling a nodal line is quantized to $\pi$ (mod $2\pi$), providing a $\mathbb{Z}_2$ classification that ensures the existence of the line node unless it is gapped by symmetry breaking or shrunk to a point $2201.06639$. In multi-band settings, non-Abelian invariants (e.g., Wilczek-Zee or quaternion-valued Wilson loops) can further characterize nodal lines' linking and braiding properties.

Mirror symmetry is especially critical: for example, in a system with mirror symmetry $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$0, the relative homotopy group $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$1 classifies the combined node content, enforcing non-trivial conversion rules between Weyl points and nodal lines when the parameters are tuned [$$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$2].

2. Homotopy Theory and Topological Invariants

In the modern approach, the classification of Weyl points and nodal lines employs homotopy groups of mappings into the space of gapped Hamiltonians. For a two-band Hamiltonian, the Bloch vector $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$3 defines a map $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$4 (sphere in $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$5-space surrounding the node to sphere in $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$6-space), and its degree is the Chern number of a Weyl point: $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$7 When point group symmetry $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$8 is present (e.g., mirror, $$E_\pm = \pm \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$$9), classification uses relative homotopy: $q_i = k_i - k_{0i}$0 with $q_i = k_i - k_{0i}$1 the subset of $q_i = k_i - k_{0i}$2 invariant under $q_i = k_i - k_{0i}$3. For mirror $q_i = k_i - k_{0i}$4,

$q_i = k_i - k_{0i}$5

while for $q_i = k_i - k_{0i}$6 (two-fold rotation combined with time reversal),

$q_i = k_i - k_{0i}$7

where the invariants are total chirality $q_i = k_i - k_{0i}$8 and helicity $q_i = k_i - k_{0i}$9 [$1207.0478$0].

In centrosymmetric extensions of the tenfold way, doubly charged (multiply robust) nodal lines are classified by both the Berry $1207.0478$1-phase and a higher homotopy $1207.0478$2 charge, ensuring their pairwise stability unless both charges vanish [$1207.0478$3].

3. Conversion Rules and Symmetry Constraints

The conventional scenario—pairwise annihilation of Weyl points of opposite chirality—is modified by crystalline symmetries. If two Weyl points are mirror-related, annihilation is strictly forbidden: upon collision in the mirror-invariant plane, they convert into a nodal loop (the "mirror-enforced conversion rule") [$1207.0478$4]:

• Construction: Bring two $1207.0478$5 chirality mirror-paired Weyl points together within the mirror plane. Instead of annihilating, they merge into a 1D nodal loop lying entirely in that plane.
• Significance: The process is fully captured by the behavior of the relative homotopy invariant $1207.0478$6, which remains nontrivial until the loop is gapped or recombined via further symmetry-breaking or parameter tuning.

With multiple perpendicular mirrors, pairs of Weyl points can convert into multiple nodal loops, which may further combine into nodal chains—linked or touching loops forming complex line-degeneracy structures. These higher connectivities are enforced by corresponding higher-dimensional invariants (quadrupole, octupole) in the relative homotopy sequence.

$1207.0478$7-symmetric systems introduce the concept of Weyl-point helicity. Even when two Weyl points have opposite chirality, their annihilation requires cancellation of both chirality $1207.0478$8 and helicity $1207.0478$9, so pairs with equal helicity cannot annihilate [$$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$0].

4. Minimal Model Realizations and Material Platforms

Minimal two-band tight-binding models explicitly demonstrate these conversion rules and topological features:

• Single mirror, cubic lattice:

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

For $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$2, the model hosts two mirror-related Weyl points; as $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$3, they merge into a nodal ring in $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$4 (the mirror plane) [$$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$5].

• Multiple mirrors:

$$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$6

with mirrors $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$7, $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$8. For $$C = \frac{1}{2\pi} \oint_{S^2} \mathbf{F} \cdot d\mathbf{S} = \pm 1,$$9, four Weyl points exist, converting pairwise into two loops in $1$0 and $1$1 as $1$2, which at their merger form a nodal chain.

• $1$3 symmetric version:

$1$4

$1$5 vanishes in $1$6; Weyl points with nontrivial helicity appear in this plane and can annihilate only if both chirality and helicity sum to zero.

These models have been applied to electronic systems (Sun et al., 2018), photonic crystals with gyroid or metallic-mesh structures [$1$7] [$1$8], and phononic crystals (e.g., straight phononic Weyl nodal lines in MgB$1$9) [$\pi$0]. Real material examples include transition-metal tellurides (TaIrTe$\pi$1 hosts both Weyl points and mirror-protected nodal lines) [$\pi$2], rhombohedral IV–VI semiconductors under strain [$\pi$3], and pentagonal 2D materials where nodal lines appear along Brillouin zone boundaries [$\pi$4].

5. Surface States: Fermi Arcs and Drumhead Modes

The bulk-node topology manifests at surfaces in the following characteristic ways:

• Weyl points: Each pair of opposite-chirality Weyl points is connected on a given crystal facet by a chiral surface state ("Fermi arc") that terminates at their projections. The arc is open (not enclosing a Fermi surface) and reflects the topological Chern charge of the bulk [$\pi$5] [$\pi$6].
• Nodal lines: The projection of a bulk nodal line onto a surface Brillouin zone encloses a region bounded by a "drumhead" surface band. These surface states are nearly flat (dispersionless) and localized within the projected loop [$\pi$7] [$\pi$8].
• Nodal chains: Merging multiple mirror-enforced nodal loops leads to even richer surface state phenomena, including quadratic drumhead touchings at chain points [$\pi$9].

The nature of these surface states, their dispersion, and their robustness directly mirror the symmetry and topology of the bulk nodes. For example, in systems with mirror or glide-plane symmetries, drumhead states can be split or gapped only by breaking these symmetries.

6. Floquet Engineering, Nonlinearity, and Interactions

Beyond equilibrium phases, Weyl points and nodal lines exhibit rich structural evolution under external drives and interactions:

• Floquet Engineering: Periodic driving with circularly polarized light (CPL) can convert a static nodal line into a Weyl semimetal by breaking the protecting symmetries and opening gaps except for two isolated Weyl points. The separation, position, and chirality of these nodes are tunable by the amplitude, frequency, and handedness of the light [$2\pi$0] [$2\pi$1]. For intersecting straight nodal lines, one can further generate novel Floquet phases with linked Fermi arcs that wrap around the entire surface Brillouin zone [$2\pi$2].
• Multi-Weyl Points: Driving crossing nodal lines by CPL can result in multi-Weyl points (with monopole charge $2\pi$3), realized by combining or annihilating multiple Weyl monopoles at symmetry-dictated locations [$2\pi$4].
• Nonlinear Effects: Onsite nonlinearities (e.g., mean-field interactions in photonic/atomic systems) can split Weyl points into extended nodal lines and nodal surfaces, distributing the original monopole charge over a higher-dimensional manifold while preserving global topological invariants [$2\pi$5].
• Interactions: Bulk interactions (Hubbard-$2\pi$6 terms) can shift, shrink, or annihilate Weyl points and nodal lines by favoring spontaneous order parameters (e.g., ferromagnetic moment), eventually gapping out formerly protected nodes when symmetry is broken and the order parameter is large [$2\pi$7].

7. Experimental Realizations and Probes

Numerous systems spanning electronics, photonics, phononics, and synthetic materials demonstrate these phenomena:

• Photonic Crystals: Gyroid and metallic-mesh photonic crystals realize both Weyl points and nodal lines/chain structures, detectable via angle-resolved transmission, Fourier-transformed field scans, and local density of states measurements [$2\pi$8] [$2\pi$9].
• Material Candidates: Recognized Weyl and nodal-line semimetals include TaIrTe$\mathbb{Z}_2$0, SnTe/GeTe (in suitable phases), elemental tellurium, and pentagonal 2D and 3D compounds. Observables include surface Fermi arcs in ARPES, drumhead surface bands in STS, anomalous magnetotransport, and quantum oscillations with Berry-phase signatures [$\mathbb{Z}_2$1] [$\mathbb{Z}_2$2] [$\mathbb{Z}_2$3].
• Floquet Experiments: Pump-probe ARPES and light-induced Hall measurements provide direct probes of dynamically generated Floquet Weyl points and their associated surface Fermi arcs [$\mathbb{Z}_2$4] [$\mathbb{Z}_2$5].
• Phononic/Topolectrical Platforms: Detection of nodal structures via the photonic and phononic density of states, impedance spectroscopy in topolectrical circuits, and synthetic band-structure engineering [$\mathbb{Z}_2$6] [$\mathbb{Z}_2$7].
• Surface State Protection: Topological protection of surface states, including Fermi arcs and drumhead bands, can be reformulated in terms of exceptional points and branch cut topology in a complexified momentum plane [$\mathbb{Z}_2$8].

These experimental avenues collectively establish nodal lines and Weyl points as cornerstones of contemporary topological band theory—underpinning a growing taxonomy of gapless matter and its engineered analogs.

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Key References:

• Conversion rules and homotopy classification: (Sun et al., 2018)
• Surface state characterization: (Lu et al., 2012, Xie et al., 2018, Yan et al., 2017)
• Floquet Weyl semimetals and node conversion: (Yan et al., 2016, Yan et al., 2017, Liu et al., 6 Jul 2025)
• Nonlinear and interaction effects: (Tuloup et al., 2022, Kang et al., 2019)
• Material realizations: (Zhou et al., 2017, Lau et al., 2018, Bravo et al., 2020)

For comprehensive computational and experimental demonstrations, see also studies on symmetry-enforced band crossings and classification in electronic, photonic, and phononic materials [$\mathbb{Z}_2$9] [$2201.06639$0] [$2201.06639$1].