Dynamic Properties of Weyl Points

• The paper demonstrates that time-dependent perturbations induce a dynamic chiral magnetic effect, enabling controlled topological currents in Weyl semimetals.
• It shows that tuning system parameters can shift, merge, or annihilate Weyl nodes, triggering phase transitions and altering Fermi arc connectivity.
• The study reveals that dynamic correlations and octonionic diagnostics offer a local, basis-free approach for tracking Weyl point topology across diverse physical platforms.

A Weyl point is a two-fold band degeneracy in a three-dimensional parameter space (often momentum), characterized by a quantized topological charge (chirality), with a conical energy dispersion in its vicinity and a singular distribution of Berry curvature. The dynamic properties of Weyl points refer to their behavior under time-dependent fields, tunable interactions, parameter variations, as well as their responses in non-equilibrium and driven scenarios. These properties not only manifest in electronic systems but reach across photonic, mechanical, magnonic, and even synthetic parameter spaces, enabling unique transport, optical, and wave phenomena. The interplay of symmetry, topology, and dynamics around Weyl points forms an essential foundation for new phases of matter and applications in quantum and classical systems.

1. Chiral Dynamics and Non-Equilibrium Responses

A defining dynamic property of Weyl points in inversion symmetry-breaking systems is the dynamic chiral magnetic effect (DCME), which arises due to an energy mismatch—known as the chiral chemical potential—between left- and right-handed Weyl nodes. In such systems, a time-dependent magnetic field $B(t)$ generates an electrical current $\mathbf{j}(t)$ parallel to $\mathbf{B}(t)$, a phenomenon absent in equilibrium with static fields. The general dynamic chiral magnetic conductivity $\sigma_{\mathrm{ch}}(\omega)$ obeys

$$\sigma_{\mathrm{ch}}(\omega) \sim \frac{128e^2 T^2}{\hbar^4 \omega^2}$$

in the high-frequency limit. This current is mediated via the Berry curvature of the Weyl fermions and relies crucially on the presence of the chiral chemical potential—without which the coupling vanishes. The DCME is directly probed via measurements of the natural optical activity (rotary power), as the induced gyrotropic current results in different propagation constants for left and right circularly polarized light. As predicted, rotary power values in materials such as TaAs in the infrared far exceed those of conventional optically active compounds such as tellurium (Goswami et al., 2014).

Non-equilibrium dynamics in non-Hermitian systems exhibit additional unique features: for example, in point-gap Weyl semimetals, Weyl points shifted along the imaginary energy axis (i.e., with complex energies) yield a time-dependent current along an applied magnetic field without any electric field. This is due to unequal dissipation rates of chiral Landau levels and is quantified by topological invariants in the complex energy plane, such as the three-winding number $W_3$. These systems also present distinctive boundary–skin modes—corner-localized states arising from the interplay of topological surface Fermi arcs and non-Hermitian skin effects—that are dynamically revealed via wave-packet propagation (Hu et al., 2021).

2. Parametric Motion, Creation, and Annihilation of Weyl Points

Weyl points are robust to local perturbations due to their nonzero topological charge (Chern number). However, upon continuous tuning of system parameters (hopping amplitudes, external fields, strain, chemical substitution, etc.), Weyl nodes can move in the Brillouin zone or more abstract parameter space. In paradigmatic lattice models, explicit control parameters (e.g., $\alpha$ and $\beta$ in cubic optical lattices) determine the positions of Weyl nodes via analytical expressions such as

$$W_{j} = (k_x^{(j)},~k_y^{(j)},~\pm \arccos(\alpha \pm \beta))$$

By varying these parameters, Weyl points of opposite chirality can merge and annihilate at critical values, accompanied by a topological phase transition to a trivial insulator where the Fermi arc surface states also vanish (Hou et al., 2015).

Singularity theory provides a rigorous description of creation and annihilation events: as control parameters traverse phase boundaries, the number of Weyl points changes via universal singularities (fold, cusp, swallowtail) of the projection map from the Weyl locus in configuration space to control space. Fold lines describe pairwise mergers, cusps capture triple-point processes, and a swallowtail singularity governs the merger of four nodes (Frank et al., 2023). This topological classification governs phase diagrams for Weyl semimetals under external tuning and extends to all band-structure models, including electronics, photonics, magnonics, and mechanical structures.

3. Types, Symmetry, and Generalized Weyl Points

Dynamical manipulations enable continuous transitions between distinct Weyl point types:

• Type-I: Upright (Lorentz-invariant) cones with a vanishing Fermi surface at the node.
• Type-II: Strongly tilted cones such that electron and hole pockets touch at the node; both bands have group velocities of the same sign in certain directions, resulting in finite Fermi surfaces (Shastri et al., 2016, Wu et al., 2017, Li et al., 2019).
• Type-III/IV (in interacting, especially nonrelativistic systems): Weyl points characterized by certain components of the emergent metric $g_{\mu\nu}$ becoming negative, admitting regions of complex energy (dynamical instability) or even closed timelike curves, with transitions between these phases interpreted as Lifshitz transitions (Nissinen et al., 2017).

Symmetry protection (or its breaking) governs many dynamic phenomena. For example, quadratic (charge-2) Weyl points at high-symmetry points (e.g., the $\Gamma$ point in non-symmorphic photonic crystals) can be split into pairs of linear (charge-1) Weyl nodes by tuning symmetry-breaking perturbations, with the nodes moving along high-symmetry lines as the perturbation strength increases (Jörg et al., 2021). Additionally, “hidden” antiunitary symmetry operators can protect Weyl nodes at arbitrary momentum points even when explicit spatial symmetries are broken; this is evidenced by antiunitary operators with square $-1$ (Kramers-like degeneracies), providing robust band crossings as parameters evolve (Hou et al., 2015).

4. Surface States and Fermi Arcs: Dynamics and Experimental Signatures

The bulk–boundary correspondence guarantees that Weyl points act as sources and sinks of Berry curvature, enforcing the existence of open surface states (Fermi arcs) that connect the surface projections of Weyl nodes of opposite chirality. Parametric motion, merging, or transmutation of Weyl nodes induces corresponding changes in the connectivity and length of Fermi arcs. Dynamic symmetry breaking can be used to control the arc structure and surface transport properties, with clear experimental signatures visible in ARPES, photonic, elastic, and mechanical analogs (Lu et al., 2015, Wang et al., 2017, Shi et al., 2019, Takahashi et al., 2018).

Beyond momentum space, synthetic Weyl points can be realized in parameter space—e.g., in photonic crystals with tunable geometric parameters—where the winding of the reflection phase (a vortex) around such a point ensures the existence of robust interface (surface) states, directly analogous to Fermi arcs. The trajectory of these states in the synthetic parameter space mimics the arc connectivity in conventional Weyl systems (Wang et al., 2017).

5. Dynamic Correlations and Topological Response in Correlated and Engineered Systems

Dynamical (frequency-dependent) electronic correlations, captured via methods such as dynamical mean field theory (DMFT), significantly renormalize the positions, energies, and dispersion of Weyl points in half-metallic ferromagnets. For instance, in VAs, the inclusion of dynamic correlations alters exchange splitting and effective masses, resulting in adjusted Weyl point configurations. Changes in magnetization direction, when combined with spin–orbit coupling, further modulate Weyl node positions due to the interplay between magnetic and relativistic effects, although the position sensitivity may remain small when SOC is weak (Ding et al., 4 Mar 2024).

In engineered materials such as (Mn$_{1-\alpha}$Fe$_\alpha$)$_3$Ge, chemical doping shifts Weyl points away from the Fermi level, which is directly observed via suppression of the anomalous Hall effect, longitudinal magnetoconductivity, and planar Hall effect. These quantities scale with the proximity of the Weyl nodes to the Fermi surface and are reduced with increasing dopant concentration, demonstrating direct chemical control over the dynamic manifestation of Weyl topology (Rai et al., 6 Mar 2024).

6. Basis-Free Local Diagnostics and High-Throughput Computation

Conventional methods for determining the chirality (topological charge) of a Weyl point—such as integrating Berry curvature on a small sphere or analyzing $k\cdot p$ expansions—require nonlocal, gauge-dependent procedures. The octonionic Weyl point criterion introduces a fully local, basis-independent scalar density constructed from the G$_2$-invariant three-form on $\mathbb{R}^7$. For a smooth two-band projector $P_2(k)$, a unit octonion $u(k)$ and its associated connection are constructed, with the pseudoscalar density

$$\rho_\mathbb{O}(k) = \frac{1}{6}\varepsilon^{ijk}\,\varphi(A_i, A_j, A_k)$$

whose sign yields the Weyl chirality. In the linear regime, this method is rigorously equivalent to the sign of the determinant of the velocity matrix and to the conventional Berry charge. The approach incorporates a built-in self-consistency check: a vanishing associator norm certifies that the local physics closes inside a quaternionic subspace, while a nonzero associator signals the breakdown of the two-band Weyl description (e.g., near multi-fold touchings or band entanglement). This diagnostic is well-suited for high-throughput, automatable tracking of Weyl point dynamics in realistic band-structure calculations, as well as for dynamical studies under perturbations (Tantardini, 23 Sep 2025).

7. Emergent Properties Across Physical Platforms

Dynamic properties of Weyl points are deeply intertwined with their physical realization:

• In photonic and optical systems, Weyl points yield conical diffraction, unidirectional wave propagation, nonlinear and quantum optical phenomena, and collimated emission patterns. The presence of type-II Weyl points leads to strictly positive group velocities, topologically robust Fermi arc edge states, and field-tunable directionality (Noh et al., 2016, García-Elcano et al., 2020).
• In magnonic systems, neutron scattering measurements near magnon Weyl points reveal characteristic hedgehog pseudospin textures and singularities in the dynamic structure factor, producing distinctive intensity arcs directly related to the monopole “charge” (Shivam et al., 2017).
• In mechanical and elastic structures, the interplay between geometry, symmetry, and dynamic evolution of Weyl points governs the formation, motion, and transmutation of monopole charges, as well as the emergence of robust, directional surface wave states for energy routing and harvesting (Takahashi et al., 2018, Shi et al., 2019).

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In summary, the dynamic properties of Weyl points encompass their nontrivial electromagnetic and optical responses, parameter-dependent mobility and annihilation, symmetry-protected and tunable topological transitions, and their encoding of robust surface (and interface) states. Developments in local, structure-intrinsic diagnostics (e.g., the octonionic criterion), alongside controllable platforms in solid-state, photonic, mechanical, and synthetic systems, provide access to both fundamental insights and technological possibilities rooted in the dynamics of topological band degeneracies.