A Weyl semimetal's low-energy bands touch at isolated points in momentum space, the Weyl nodes, where the dispersion is a linear double cone E(k) = ±ħv|k − kW| — a single 2×2 Weyl Hamiltonian H = χħv σ·(k − kW) with Pauli matrices σ and chirality χ = ±1. Each node acts as a point source (χ = +1) or sink (χ = −1) of Berry curvature Ω(k) = χ(k − kW)/2|k − kW|³ — a momentum-space "magnetic monopole" of charge χ. By the Nielsen–Ninomiya theorem, Weyl nodes on a lattice must always come in pairs of opposite chirality so the total monopole charge cancels.
In a real magnetic field B, each node's lowest Landau level (n = 0) is chiral: it disperses only along B, with a group velocity whose sign is fixed by χ, so unlike every higher Landau level it cannot backscatter. Turning on an electric field E with a component along B accelerates every state in this channel via the semiclassical equation ħ dk/dt = −eE, continuously sweeping filled states through one node and away from the other. This is the (3+1)D chiral anomaly: the separate particle numbers at the two nodes are not conserved, only their sum is —
dn₅/dt = e²(E·B)/(4π²ħ²v) − n₅/τᵥ
n₅ = n₊ − n₋ (chiral charge imbalance)
Δσ_xx ∝ B² cos²θ · τᵥ (Son–Spivak negative magnetoresistance)
The pumping term e²(E·B)/4π²ħ² is exactly the QED chiral anomaly ∂μj₅μ ∝ E·B, realized here in a crystal instead of the vacuum. Intervalley scattering (rate 1/τv) relaxes the imbalance back to zero once the fields are removed. Because more current-carrying chiral channels open as B grows, the anomaly makes conductivity along B increase with B — a signature negative longitudinal magnetoresistance seen in real Weyl semimetals (TaAs, Na₃Bi, ZrTe₅).
- E, B sliders — field magnitudes; the pump rate scales with their product.
- θ slider — angle between E and B; only the E·B = EB cos θ component drives the anomaly, so θ = 90° switches pumping off entirely.
- τv slider — how quickly intervalley scattering erases the imbalance; longer τv lets N₅ (and the magnetoresistance signal) build up further.
- The streaming particles trace the chiral n = 0 Landau channel carrying charge from the sink node (blue) to the source node (red); the thin radiating spokes are the Berry-curvature monopole field of each node.