Weyl Semimetals: Chiral Fermions and the Fermi Arc

What a Weyl Semimetal Is and Why It Matters

A Weyl semimetal is a crystalline conductor in which the bulk electronic bands touch at isolated, non-degenerate points in the three-dimensional Brillouin zone. Near each such Weyl node the low-energy electrons obey the Weyl equation — the massless, single-handed limit of the Dirac equation that Hermann Weyl wrote in 1929. What the Standard Model forbids for real particles (neutrinos turned out to have mass, and gauge invariance ties left- and right-handed sectors together), a crystal permits as an emergent quasiparticle.

The deep reason these nodes are stable is topology, not fine-tuning. Each node is a quantized source or sink of Berry curvature — a monopole in momentum space carrying a Chern number C = ±1, its chirality. A monopole cannot be removed by any small perturbation; it can only be annihilated by meeting an antimonopole of opposite chirality. This makes the Weyl phase robust and gives it experimentally sharp fingerprints: exotic surface states and anomalous transport that ordinary metals cannot mimic. It was the first solid-state platform where a genuinely relativistic, chiral field-theory phenomenon — the chiral anomaly — became a table-top measurement.

The Mechanism: Nodes, Chirality, and Berry Monopoles

Consider two bands crossing at a point in k-space. Expanding the Bloch Hamiltonian to linear order gives H(k) = ℏ Σ_ij v_ij (k_i − k₀_i) σ_j, where σ are Pauli matrices spanning the two bands. Because there are exactly three Pauli matrices and three components of k, a generic crossing in 3D is fully gapped by any perturbation only if you can shift it — never open it. That is the topological protection: you can move a Weyl node around the Brillouin zone but cannot destroy it in isolation.

The sign of det(v_ij) defines the chirality χ = ±1. The Berry curvature Ω(k) surrounding the node behaves like the field of a magnetic monopole of charge χ, so the flux of Ω over any enclosing surface is quantized: (1/2π) ∮ Ω·dS = χ. The Nielsen–Ninomiya theorem (1981) — a lattice no-go result — forces the total chirality in any Brillouin zone to vanish: Weyl nodes always come in pairs of opposite handedness. Breaking either inversion symmetry (as in TaAs) or time-reversal symmetry (in a magnet) separates a Dirac point into two Weyl nodes; without one of those symmetries broken, the two would sit on top of each other and the topological charges would cancel.

The Key Equations, Scales, and the Chiral Anomaly

The minimal model is the two-band Weyl Hamiltonian H = ±ℏ v_F σ·(k − k₀), giving a linear cone E = ±ℏ v_F |k − k₀|. In TaAs the Fermi velocity is v_F ≈ 1–3 × 10⁵ m/s and the 24 Weyl nodes (12 pairs) lie within roughly 10–100 meV of the Fermi level. The Berry curvature near a node scales as Ω(k) ≈ χ k / (2|k|³).

The signature dynamical effect is the chiral (Adler–Bell–Jackiw) anomaly. Apply parallel electric and magnetic fields E ∥ B: charge is pumped from one chirality to the other at a rate dn₅/dt = (e²/4π²ℏ²c) E·B (in the lowest Landau level, via the chiral zeroth level). Because intervalley scattering only slowly relaxes this imbalance (relaxation time τ_inter), a net chiral charge builds up and enhances conduction along B. The result is a negative longitudinal magnetoresistance — resistance that drops as B grows when E ∥ B — the transport smoking gun predicted by Nielsen and Ninomiya (1983) and measured in TaAs and Na₃Bi. Separately, the node separation b in k-space produces an intrinsic anomalous Hall conductivity σ_xy = (e²/2πh)·2b.

How It Is Realized and Measured: TaAs and Fermi Arcs

The theoretical breakthrough came in 2011, when Wan, Turner, Vishwanath, and Savrasov predicted Weyl nodes and Fermi arcs in pyrochlore iridates. The experimentally decisive prediction was that the noncentrosymmetric monopnictides TaAs, TaP, NbAs, and NbP host 24 Weyl nodes (Weng et al. and Huang et al., early 2015). Within months, in July 2015, Su-Yang Xu, M. Zahid Hasan and collaborators used angle-resolved photoemission spectroscopy (ARPES) on TaAs to image both the bulk Weyl cones (with soft-X-ray ARPES) and the surface states (with VUV ARPES).

The unmistakable fingerprint is the Fermi arc: a surface Fermi contour that is an open ribbon rather than a closed loop, with its two ends terminating exactly at the surface projections of a Weyl node pair. No ordinary two-dimensional Fermi surface can have a loose end, so an arc is only possible if it dives into the bulk at a Berry monopole. Xu et al. found every arc termination matched a projected bulk node within resolution — direct proof of the bulk-boundary correspondence. Transport groups simultaneously reported the chiral-anomaly negative magnetoresistance in TaAs, providing an independent confirmation.

Where It Operates: Types, Materials, and Distinctions

Weyl semimetals split into families. Type-I nodes (TaAs, TaP, NbAs, NbP) have upright cones and a point-like Fermi surface. Type-II nodes (WTe₂, MoTe₂) are strongly tilted so the cone overtips, producing touching electron and hole pockets and a Lorentz-symmetry-violating dispersion impossible for real relativistic particles. Magnetic Weyl semimetals such as Co₃Sn₂S₂ and Mn₃Sn break time-reversal instead of inversion, giving giant intrinsic anomalous and topological Hall responses.

It is essential to distinguish a Weyl semimetal from its relatives. A Dirac semimetal (Na₃Bi, Cd₃As₂) has four-fold-degenerate nodes — two coincident Weyl nodes of opposite chirality protected by crystal symmetry; break P or T and each Dirac point splits into a genuine Weyl pair. A topological insulator is fully gapped in the bulk with a single closed Dirac cone on the surface, whereas a Weyl semimetal is a gapless bulk metal with open arcs. Nodal-line semimetals extend the touching from points to loops. The organizing principle across all of them is momentum-space topology of the Bloch bands, captured by Berry-phase geometry.

Applications, Open Questions, and Significance

Weyl semimetals matter first as a laboratory for high-energy physics: the chiral anomaly, once an abstruse feature of quantum field theory tied to π⁰ → 2γ decay, becomes a resistance you can plot. They also probe the mixed axial-gravitational anomaly, seen in NbP as a magnetic-field-dependence of the thermoelectric response tracking the gravitational anomaly coefficient.

Technologically, the linear dispersion and topological protection promise high carrier mobilities (up to ~10⁶ cm²/V·s in TaAs), strong nonlinear optical responses (a colossal bulk photovoltaic and second-harmonic response from the Berry curvature), and candidate low-dissipation interconnects and catalysts. The intrinsic anomalous Hall effect in magnetic Weyl systems is attractive for spintronics and Hall sensors.

Open questions remain sharp: how robust is arc transport against disorder and surface reconstruction; can interactions drive a Weyl system into an axionic charge-density-wave or Weyl-Mott insulator; are Weyl nodes present in correlated and superconducting materials (Weyl superconductivity); and can strain act as a synthetic axial gauge field to engineer pseudo-Landau levels? Weyl matter turned momentum-space topology from a classification scheme into a source of new, measurable physics.

Weyl semimetals versus related topological and gapless phases

Property	Weyl semimetal	Dirac semimetal	Topological insulator
Band touching	Isolated points, 2-fold degenerate (Weyl nodes)	Isolated points, 4-fold degenerate (Dirac nodes)	No bulk touching — full gap
Required broken symmetry	Inversion OR time-reversal must be broken	Both P and T preserved (crystal-symmetry protected)	T-invariant (Z₂ class)
Topological charge	Chern number C = ±1 (Berry monopole)	Net zero (two overlapping Weyl nodes)	Z₂ invariant ν = 1
Surface Fermi surface	Open Fermi arcs joining node projections	Arcs that can merge / are non-robust	Closed spin-momentum-locked Dirac cone
Bulk state	Gapless metal (semimetal)	Gapless metal	Insulating bulk, conducting surface
Signature transport	Chiral-anomaly negative magnetoresistance	Large linear magnetoresistance	Quantized surface / QH conduction