Topologically Protected Weyl Points

• Topologically protected Weyl points are discrete band-touching singularities with a nonzero topological charge that ensure stability against perturbations.
• They manifest through features like surface Fermi arcs and anomalous transport phenomena, validated in systems such as Weyl semimetals and engineered photonic setups.
• Their creation and annihilation, governed by symmetry constraints and singularity theory, provide actionable pathways for tuning topological phase transitions in quantum devices.

Topologically protected Weyl points are discrete, robust band-touching singularities in parameter-dependent Hamiltonians, characterized by a nonzero topological charge, typically a Chern number or monopole strength. Their stability and physical consequences are central to the understanding of gapless phases in quantum matter, mechanical lattices, photonic systems, superconductors, and engineered circuits. The following sections systematically discuss the mathematical definition, topological invariants, physical realizations, bulk-boundary correspondence, mechanisms of creation and annihilation, the role of symmetry, and experimental detection and manifestations.

1. Mathematical Definition and Topological Charge

A Weyl point is an isolated point in the relevant parameter space (commonly momentum space) where two energy bands become degenerate and cross linearly in all directions. The minimal effective Hamiltonian near a Weyl point at $k_W$ takes the form

$$H(\mathbf{k}) = v_F(\mathbf{k} - \mathbf{k}_W) \cdot \boldsymbol{\sigma}$$

with $v_F$ the Fermi velocity, and $$\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$$ the Pauli matrices.

The defining topological property is encoded by the quantized Berry flux through a sphere $S^2$ that encloses the Weyl point:

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

where $\mathbf{\Omega}(\mathbf{k})$ is the Berry curvature of the occupied band. $C$ is the monopole charge (chirality), most commonly $+1$ or $-1$ for simple Weyl points, but higher for multi-Weyl points (e.g., double or triple Weyl nodes exhibited as quadratic or cubic dispersions) (Xu et al., 2017, Shivam et al., 2017).

Topological protection implies the node is robust under any perturbation that does not annihilate it with a partner of opposite charge.

2. Symmetry Constraints and Protection Mechanisms

The band crossing and the topological protection of Weyl points rely on generic symmetry constraints:

• Broken Time-Reversal (TRS) or Inversion (P) Symmetry: Generic Weyl points require P or TRS breaking to avoid four-fold (Dirac) degeneracies (Zhou et al., 2019).
• Point Group and Nonsymmorphic Symmetry: Rotational (e.g., $C_4$, $C_3$, $C_6$), mirror, or screw symmetries can stabilize higher charge nodes (double, triple Weyl points) or enforce band crossings at high-symmetry points or lines (Shivamoggi et al., 2013, Wang et al., 2019, González-Hernández et al., 2020).

For instance, in superconductors with $C_4h$ point group symmetry, the block-diagonal structure along high symmetry momentum (HSM) lines ensures that only certain terms remain nonzero, prohibiting gap opening except at critical momenta. The topological invariant is computable from symmetry eigenvalue changes along these lines:

$$\nu(\Gamma_a) = -\text{sgn}[d_3(\Gamma_a)], \quad \chi = \prod_a \nu(\Gamma_a)$$

with $\Gamma_a$ denoting HSM points (Shivamoggi et al., 2013). Nonsymmorphic symmetries, such as screw rotations, enforce Weyl points at generic $k$-points in certain hexagonal and trigonal crystals (Wang et al., 2019, González-Hernández et al., 2020).

In synthetic and engineered systems, parameter spaces (e.g., superconducting phases in multi-terminal Josephson junctions or geometric parameters of optical unit cells) furnish analogous protection (Wang et al., 2017, Takemura et al., 5 Oct 2025).

3. Bulk-Surface Correspondence and Physical Manifestations

The nontrivial topology of Weyl points is manifested in observable physical properties:

• Surface Fermi Arcs: The bulk-boundary correspondence dictates that nontrivial Chern number slices in the Brillouin zone enforce open surface states (Fermi arcs) connecting projections of Weyl nodes of opposite chirality (Shivamoggi et al., 2013, Xu et al., 2017). In mechanical (Rocklin et al., 2015) and phononic systems (Wang et al., 2019), this appears as boundary-localized zero modes or surface arcs spanning the surface Brillouin zone.
• Anomalous Transport: Chiral anomaly-induced negative longitudinal magnetoresistance and quantized Hall responses are signatures of Weyl quasiparticles when the chemical potential is pinned between saddle points of the Weyl cones (Xu et al., 2017).
• Topological Thermal and Spin Hall Effects: The quantized topological charge underpins quantized thermal conductance in Weyl superconductors (Pacholski et al., 2017), anomalous spin Hall conductance in materials with multiple Weyl nodes (González-Hernández et al., 2020), and chiral edge transport in gapped 2D systems (e.g. altermagnets characterized by a $\pi$-Berry phase) (Parshukov et al., 14 Mar 2024).
• Localized Interface States: Synthetic Weyl points in photonic or optical parameter space manifest as robust interface states with phase vortices, analogs of Fermi arcs (Wang et al., 2017).

4. Creation, Annihilation, and Trajectories: Singularity Theory Perspective

Weyl points are not destroyed by small perturbations but can undergo pair creation or annihilation when parameters are varied so that two nodes of opposite charge merge. This occurs at phase boundaries in the space of control parameters (e.g., strain, gate voltages, transmission probability), associated with singularities classified by singularity theory (Frank et al., 2023):

• Fold Singularities: Pairwise creation/annihilation (locally described by $f(x) = x^2$) is the generic process for a single control parameter.
• Cusp and Swallowtail Singularities: More intricate mergers of three or four Weyl points arise when multiple parameters are finely tuned, with corresponding alterations in the phase diagram topology.
• Trajectory Classification: In synthetic spaces, Weyl point trajectories exhibit closed loops, open lines, and possible pair exchanges as control parameters (such as transmission through a quantum point contact in Josephson junctions) are varied. The number of Weyl points and associated Chern numbers change discretely at critical values, encoding topological phase transitions (Takemura et al., 5 Oct 2025).

These singularities are universal and stable against generic perturbations, and they govern the patterns of topological phase transitions across broad classes of systems.

5. Generalizations: Non-Abelian Invariants and Exotic Complexes

Beyond the conventional Abelian (Chern-charge) topology, several works highlight richer structures:

• Non-Abelian Topological Invariants: In systems with antiunitary symmetries squaring to $+1$ (e.g., $C_2T$), Weyl points can carry frame-rotation (Euler-class) invariants valued in the quaternion group. Braiding of Weyl nodes inside symmetry planes may yield noncommutative phase factors and prevent conventional annihilation, instead resulting in conversions to nodal lines under strain (Bouhon et al., 2019).
• Triangular Weyl Complexes: In materials respecting certain screw symmetries, unconventional groupings emerge (e.g., one double and two single Weyl phonons forming a triangle in $k$-space) with corresponding surface arcs that do not simply connect pairs of nodes (Wang et al., 2019).
• Parameter-Space Weyl Points: In quantum-dot and Josephson-junction systems, Weyl points appear as point degeneracies of energy levels in synthetic parameter space (magnetic fields, superconducting phases), with topological properties anchored in a parameter-space Berry curvature and quantized Chern numbers (Scherübl et al., 2018, Stenger et al., 2019, Takemura et al., 5 Oct 2025).

6. Experimental Realizations and Detection

Topologically protected Weyl points have been experimentally observed or predicted in a range of platforms:

• Condensed Matter: Weyl semimetals (TaAs, NbP, TaP), with ARPES mapping of bulk Weyl cones and surface Fermi arcs (Xu et al., 2017, Jiang et al., 2018).
• Superconducting Devices: Double quantum dots under magnetic fields reveal Weyl points through Kondo resonances and ground-state degeneracy, with transport signatures robust to spin-orbit coupling (Scherübl et al., 2018).
• Mechanical and Photonic Metamaterials: Maxwell lattices and topological circuits exhibit Weyl points in vibrational or electromagnetic spectra; synthetic parameter tuning enables clear observation and control (Rocklin et al., 2015, Li et al., 2019, Wang et al., 2017).
• Phononic Systems: Surface phonon arcs of the triangular Weyl complex are extremely long and dominate iso-frequency contours, facilitating detection through surface-resolved spectroscopies (Wang et al., 2019).
• Neutron Scattering: Magnon Weyl points in ordered magnets are directly probed by characteristic anisotropies and singularities in dynamical structure factor measurements (Shivam et al., 2017).

7. Outlook and Applications

The universal mechanism of topological protection via monopole charge, and its generalizations to non-Abelian invariants and synthetic parameter spaces, provides a robust foundation for designing materials and devices with quantized, dissipation-tolerant edge, surface, and interface states. Applications range from spintronics, where large intrinsic spin Hall conductivities are predicted due to multiple topological crossings (González-Hernández et al., 2020), to quantum computation platforms leveraging robust degeneracies and the interplay with Majorana physics (Stenger et al., 2019), to technologically relevant phononic and photonic structures with engineered surface transport (Wang et al., 2017, Wang et al., 2019).

Advances in singularity theory and the classification of Weyl point trajectories further enrich the understanding of phase diagrams and the dynamical manipulation of topological nodes, which may be exploited for field-tunable topological transitions and robust quantum operations in multi-terminal quantum devices (Frank et al., 2023, Takemura et al., 5 Oct 2025).