Weyl Semimetal: Topology & Key Properties

• Weyl semimetal is a three-dimensional topological phase where isolated band-touching points create Weyl fermions with linear dispersion.
• Its surface hosts open Fermi arcs connecting projections of opposite-chirality nodes, exemplifying bulk-boundary correspondence.
• Unique transport phenomena—such as the chiral anomaly and negative magnetoresistance—are observed in materials like TaAs.

Searching arXiv for recent and foundational Weyl semimetal papers to ground the article. Using the arXiv search tool to retrieve relevant Weyl semimetal papers. to=arxiv_search code: {"query":"Weyl semimetal Fermi arcs chiral anomaly TaAs review", "max_results": 10} A Weyl semimetal is a three-dimensional topological semimetal in which valence and conduction bands touch at isolated points in the Brillouin zone, the low-energy bulk quasiparticles near those points behave as Weyl fermions, and the surface hosts open electronic contours known as Fermi arcs rather than ordinary closed Fermi surfaces. In the standard modern formulation, the phase is defined by three linked hallmarks: topologically protected Weyl nodes in the bulk, surface Fermi arcs connecting the projections of opposite-chirality nodes, and anomaly-related electromagnetic responses such as negative longitudinal magnetoresistance for $\mathbf{E}\parallel\mathbf{B}$ (Jia et al., 2016).

1. Band-topological definition and low-energy structure

A Weyl semimetal is neither an ordinary metal nor an insulator. In an ordinary metal, the Fermi surface is typically closed and metallicity follows from partially filled bands. In an insulator, valence and conduction bands are separated by a full gap. In a Weyl semimetal, by contrast, conduction and valence bands touch only at isolated points in three-dimensional momentum space, and the dispersion is linear in all three momentum directions near each touching. This distinguishes the phase both from conventional semimetals and from Dirac semimetals, where a Dirac point may be viewed as two superimposed Weyl nodes of opposite chirality stabilized by additional symmetries (Jia et al., 2016).

The universal low-energy form near a Weyl node at momentum $\mathbf{k}_W$ is

$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$

with $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$, Pauli matrices $\sigma_i$ acting in the two-band subspace, effective velocities $v_i$, and chirality $\chi=\pm1$. In the isotropic limit this is often written as

$$H_{\chi}(\mathbf{q})=\chi v\,\boldsymbol{\sigma}\cdot\mathbf{q},$$

with spectrum

$$E_{\pm}(\mathbf{q})=\pm\sqrt{v_x^2 q_x^2+v_y^2 q_y^2+v_z^2 q_z^2}.$$

The absence of a fourth anticommuting matrix in the two-band Weyl problem means that a single node cannot be gapped by a local perturbation; perturbations move the node in momentum space, but removal requires annihilation with a node of opposite chirality (Jia et al., 2016).

This robustness underlies the standard distinction between Weyl and Dirac systems. In a Dirac semimetal, the fourfold point can often be gapped if the symmetry protecting the superposition of opposite chiralities is relaxed. In a Weyl semimetal, once opposite chiralities are separated, the individual nodes are stable until brought back together. The review literature therefore treats the Weyl semimetal as a gapless extension of the topological band-structure paradigm beyond topological insulators, which have fully gapped bulks and closed boundary contours rather than open arcs (Rao, 2016).

2. Chirality, Berry curvature, and symmetry requirements

The topological meaning of a Weyl node is that it is a source or sink of Berry curvature in momentum space. The associated topological charge is obtained by integrating Berry curvature over a closed surface $S$ surrounding the node,

$\mathbf{k}_W$0

with $\mathbf{k}_W$1 for an ordinary Weyl point. Because the Brillouin zone is compact, the net monopole charge must vanish, so Weyl nodes occur in pairs or larger charge-neutral sets. This is the momentum-space content of the Nielsen–Ninomiya no-go constraint and explains why isolated single Weyl nodes do not occur in a lattice (Jia et al., 2016).

Realizing separated Weyl nodes requires lifting the conditions that force opposite chiralities to coincide. In practice, either time-reversal symmetry $\mathbf{k}_W$2 or inversion symmetry $\mathbf{k}_W$3 must be broken. A standard consequence is that time-reversal-invariant but inversion-breaking Weyl semimetals require at least four Weyl points, whereas a magnetic Weyl semimetal can in principle realize only one pair. This symmetry logic is central both to inversion-breaking materials such as TaAs and TaIrTe$\mathbf{k}_W$4, and to magnetic systems such as Co$\mathbf{k}_W$5Sn$\mathbf{k}_W$6S$\mathbf{k}_W$7 (Belopolski et al., 2016).

Several minimal lattice constructions made this structure explicit before the first material discoveries. A layered two-band model built from coupled quantum anomalous Hall planes realizes a Weyl semimetal phase between a normal insulator and a $\mathbf{k}_W$8 quantum Hall insulator, with Weyl points at

$\mathbf{k}_W$9

in the simplest regime (Delplace et al., 2012). A complementary analytically simple construction uses a honeycomb array of topological-insulator nanowires threaded by half flux quantum; there, inversion breaking splits each valley Dirac point into a Weyl pair at

$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$0

illustrating that separated Weyl nodes do not by themselves guarantee a nonzero net anomalous Hall response when multiple node pairs cancel (Vazifeh, 2012).

A third route is the topological-insulator/normal-insulator superlattice with broken time-reversal symmetry. In that framework, inversion breaking generically shifts the two Weyl nodes to different energies, and a magnetic field along the growth direction yields an equilibrium dissipationless current

$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$1

with $$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$2 the node energy separation (Zyuzin et al., 2012). Other theoretical work showed that a nonmagnetic pyrochlore Dirac semimetal can become a Weyl semimetal through spontaneous breathing-mode inversion breaking, producing 12 Weyl nodes and termination-dependent arc connectivity (Bzdušek et al., 2015).

3. Surface Fermi arcs and bulk-boundary correspondence

The most distinctive surface manifestation of a Weyl semimetal is the Fermi arc: an open contour in the surface Brillouin zone terminating at the projections of Weyl nodes of opposite chirality. The topological explanation is a dimensional-reduction argument. If the three-dimensional Brillouin zone is sliced into two-dimensional planes, then the Chern number of a given slice changes when the slice crosses a Weyl node, because the node injects or removes one quantum of Berry flux. Each gapped slice with nonzero Chern number supports a chiral edge mode, and these edge modes assemble into a surface band whose Fermi-level contour is open rather than closed (Jia et al., 2016).

In the layered lattice model, this logic is explicit: the interval in $$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$3 between the two Weyl points is precisely the range in which the corresponding two-dimensional slices are topological, so surface chiral bands exist only there. At zero energy, the resulting surface Fermi surface is an open arc connecting the projected Weyl nodes. The same model also shows how the arc evolves at phase boundaries: it shrinks and disappears at the Weyl-semimetal-to-normal-insulator transition, but stretches into a closed chiral surface Fermi line at the transition to the quantum Hall insulator (Delplace et al., 2012).

Surface arcs are topologically robust as to existence, but not as to detailed geometry. In pyrochlore $$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$4 films, for example, their connectivity depends strongly on termination, and a surface potential can induce a Lifshitz-like reconnection of the arcs, termed a Weyl-Lifshitz transition in that context (Bzdušek et al., 2015). The same sensitivity to surface conditions appears in first-principles surface calculations of TaAs-family compounds, where the topological existence of arcs is fixed by the bulk Weyl structure while the detailed arc morphology may vary between surfaces (Weng et al., 2014).

Experimental confirmation of a Fermi arc requires more than arc-like shape. In TaAs, ARPES established the topological character by counting surface-state crossings along a closed loop and obtaining an odd total, which is impossible for a set of only closed Fermi contours. The same work and related measurements showed that the arc terminations coincide with projected Weyl nodes, tying surface and bulk topology directly together (Lv et al., 2015).

4. Experimental identification and material realizations

The decisive probe of the Weyl semimetal state has been angle-resolved photoemission spectroscopy. The criteria established in TaAs are stringent: in the bulk, one must observe isolated band-touching points with linear dispersion away from the crossing along all three momentum directions and at generic locations in the Brillouin zone; on the surface, one must observe nonclosed Fermi contours, verify topological edge-mode counting or Chern number on suitable momentum-space loops, and show that arc terminations line up with projected bulk Weyl nodes (Jia et al., 2016).

The theoretical prediction that made the first material realization possible was the identification of TaAs, TaP, NbAs, and NbP as stoichiometric nonmagnetic inversion-breaking Weyl semimetals. First-principles work found 12 pairs of Weyl points in each of these compounds, with nodal rings in mirror planes without spin-orbit coupling and Weyl points displaced off the mirror planes once spin-orbit coupling gaps the nodal rings (Weng et al., 2014). Experimental ARPES then established TaAs as a Weyl semimetal through direct observation of surface Fermi arcs and, in subsequent high-photon-energy measurements, linear bulk dispersions across Weyl points (Lv et al., 2015). A parallel ARPES study reported the complete band structure of TaAs, including the unique Fermi-arc Fermi surface and linear bulk dispersion across Weyl points, with theory indicating 12 pairs of Weyl nodes distributed among the $$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$5 and $$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$6 planes (Yang et al., 2015).

A second major materials branch is the type-II Weyl semimetal, where the Weyl point appears at the touching of electron and hole pockets. Orthorhombic MoTe$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$7 was established as a type-II Weyl semimetal through ARPES, photon-energy dependence, and spin-resolved measurements of nondegenerate surface arcs, with the candidate Weyl points predicted slightly above the Fermi level (Jiang et al., 2016). WTe$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$8 was presented as spectroscopic evidence for a type-II Weyl state through reconstruction of the full low-energy electronic structure and identification of a key surface state connecting electron and hole sectors in agreement with slab calculations (Wang et al., 2016). TaIrTe$$H_{\chi}(\mathbf{k})=\chi\, v_x q_x \sigma_x+v_y q_y \sigma_y+v_z q_z \sigma_z,$$9 was predicted as a type-II Weyl semimetal with only four Weyl points, the minimum allowed by symmetry for a time-reversal-invariant inversion-breaking Weyl semimetal (Koepernik et al., 2016), and pump-probe ARPES subsequently demonstrated that its Weyl points and large Fermi arc lie above the Fermi level, establishing the first minimal inversion-breaking “hydrogen atom” Weyl semimetal with four nodes (Belopolski et al., 2016).

Magnetic Weyl semimetals form the complementary symmetry class. In Co$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$0Sn$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$1S$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$2, ARPES visualized both triangular surface Fermi arcs and linear bulk dispersions through Weyl points, establishing a magnetic Weyl semimetal in a ferromagnetic kagome crystal with three pairs of Weyl points per bulk Brillouin zone according to theory (Liu et al., 2019). In CeAlSi, surface-sensitive vacuum-ultraviolet ARPES and bulk-sensitive soft-X-ray ARPES, together with DFT, identified surface Fermi arcs and a bulk Weyl cone, while Ce $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$3 spectral weight near the Fermi level marked the material as a correlated non-centrosymmetric magnetic Weyl platform (Sakhya et al., 2022).

5. Variants beyond the canonical Weyl cone

The canonical Weyl Hamiltonian admits several extensions that have become major subtopics. The most important is the type-II Weyl semimetal, described by a tilted Hamiltonian

$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$4

where the tilt term $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$5 is large enough to overtip the cone, so the node occurs at the touching of electron and hole pockets rather than at a pointlike Fermi surface (Jia et al., 2016).

A second extension is the multi-Weyl node. SrSi$\mathbf{q}=\mathbf{k}-\mathbf{k}_W$6 was proposed as a Weyl semimetal already without spin-orbit coupling and, after including spin-orbit coupling, some of its Weyl nodes remain stuck together on $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$7 axes and form quadratic double-Weyl fermions with chiral charge $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$8. These nodes are linear along the symmetry axis, quadratic in the transverse directions, and split into pairs of ordinary Weyl points when the protecting $\mathbf{q}=\mathbf{k}-\mathbf{k}_W$9 symmetry is lowered by uniaxial strain (Huang et al., 2015).

A third proposed generalization is the type-III Weyl semimetal. In $\sigma_i$0, first-principles work identified double Weyl points whose nodal Fermi surfaces consist of two electron pockets or two hole pockets touching at a multi-Weyl point, with chiral charges $\sigma_i$1, four-fold helicoidal surface states, and strain-driven transitions among type-III, type-II, and type-I regimes (Li et al., 2019). Closely related theoretical work also introduced a hybrid Weyl semimetal in which one node is type-I and the other type-II, together with an analogous “type-1.5” Weyl phase in an optical-lattice construction, showing that chirality and node type are distinct attributes and can be tuned independently (Li et al., 2016).

Real materials often depart from the isolated-node limit even when their topology remains Weyl-like. In NbP, magneto-infrared spectroscopy and theory showed that opposite-chirality nodes can hybridize into coupled Weyl points, qualitatively modifying Landau quantization, optical selection rules, and the fate of the zeroth Landau level. In that picture, the relevant low-energy object is not always an isolated Weyl cone but a hybridized two-node structure with parameters $\sigma_i$2, $\sigma_i$3, and $\sigma_i$4 controlling velocity, hybridization gap, and spin splitting (Jiang et al., 2018).

6. Chiral anomaly, transport, and control

One of the central reasons Weyl semimetals attracted attention is that they realize the chiral anomaly of relativistic field theory in a condensed-matter setting. When an electric field is applied parallel to a magnetic field, charge is pumped between opposite-chirality Weyl nodes. The anomaly equation is commonly written as

$\sigma_i$5

with $\sigma_i$6 and $\sigma_i$7 the axial charge and current. In Weyl semimetals this pumping enhances conduction along the field direction and can produce negative longitudinal magnetoresistance (Jia et al., 2016).

Transport evidence, however, is not self-sufficient. The TaAs review emphasized that negative magnetoresistance alone is not definitive, since current jetting, magnetic giant magnetoresistance, or more generic Berry-curvature mechanisms can mimic it. More compelling is the systematic link between transport and Weyl-node topology. In TaAs, the chiral coefficient was observed to scale as $\sigma_i$8 as the chemical potential approached the Weyl-node energy, consistent with the singular Berry-curvature profile near a momentum-space monopole (Jia et al., 2016).

Hall responses provide a second major transport signature. In a simple Weyl semimetal with node separation $\sigma_i$9, the Hall conductivity is proportional to the separation in momentum space. In the tunable optical-lattice model, the type-I phase exhibits

$v_i$0

whereas overtilting into type-II or type-1.5 regimes reduces the Hall response as electron and hole pockets appear at zero chemical potential (Kong et al., 2016). In the TI/NI superlattice with inversion breaking, the additional possibility of energy-separated nodes leads to the magnetic-field-induced equilibrium current already noted,

$v_i$1

which directly ties an axial energy offset to an electromagnetic response (Zyuzin et al., 2012).

Control of Weyl physics has therefore become a major theme. Proposed directions include discovering simpler magnetic Weyl semimetals with minimal node counts, nonlocal transport devices exploiting slow internode relaxation, thickness-dependent quantum oscillations built from bulk propagation and surface arcs, Floquet control of node positions under circularly polarized light, and large photogalvanic responses in inversion-breaking systems (Jia et al., 2016). A plausible implication is that the Weyl semimetal should be regarded not only as a topological classification, but also as a tunable platform in which symmetry breaking, surface termination, correlation effects, and external fields reorganize both the node geometry and the accessible observables.