Weyl Semimetal Chiral Anomaly — Landau-Channel Pumping (2D)

Applied Fields

Electric field E 0.60

Magnetic field B 0.60

Angle θ between E and B 0°

Intervalley relax. time τ_v 1.20

Live Readouts

Active chiral channels

dN₅/dt (counted crossings)

0.000

N₊ (source pool)

0.000

N₋ (sink pool)

0.000

N₅ = N₊ − N₋ (chiral imbalance)

0.000

Δσ_xx/σ₀ (neg. magnetoresistance)

0.000

Legend

χ = +1 node — source, N₊ grows

χ = −1 node — sink, N₋ falls

Electron in a chiral (n=0) Landau channel

How this 2D model differs from the 3D scene

The 3D version renders the Weyl cones and Berry-curvature monopoles, then evaluates the anomaly's closed-form rate equation directly. This 2D companion instead counts individual chiral electrons: a Weyl semimetal's n=0 Landau level is degenerate — it hosts one conducting channel per unit of transverse flux, so the number of channels threading the sample is proportional to B (Landau degeneracy g = eB/2πħ per unit area). Each channel is drawn here as one horizontal lane carrying one electron.

Nch  = round(60 · B)              (rendered Landau channels, ∝ B)
dk/dt = −eE/ħ                     (semiclassical Bloch acceleration)
each channel's electron sweeps the Brillouin zone and, on
crossing the zone boundary, is annihilated at the χ=−1 (sink)
node and re-created at the χ=+1 (source) node — one quantum
of chiral charge pumped per crossing, the literal picture
behind spectral flow / the Adler–Bell–Jackiw anomaly.

dN₅/dt (measured) = (crossings per second) · (charge per crossing)
                   ≈ e²(E·B)/(4π²ħ²v)     — same closed form as the 3D sim,
                                            now an emergent statistic instead
                                            of a typed-in formula
N₅ relaxes as  dN₅/dt = pump − N₅/τᵥ
Δσ_xx ∝ B² cos²θ · τᵥ                     (Son–Spivak negative magnetoresistance)

Only the component of E along B (E·B = EB cos θ) sets each channel's sweep speed, so θ = 90° halts every channel — visibly, no electrons drift — even though the channel count (set by B alone) is unchanged. Longer τ_v lets more pumped charge survive intervalley scattering before it relaxes, building a larger steady-state N₅ and a bigger magnetoresistance signal, exactly as in the 3D sim, but here you can watch the individual pumping events (the flashes at each edge) that add up to it.

E, B sliders — B sets how many chiral lanes exist; E (times cos θ) sets how fast electrons sweep along each one.
θ slider — only the E·B component drives pumping; at 90° the lanes stay populated but every electron sits still.
τ_v slider — the memory time of the imbalance: how long a pumped electron survives before intervalley scattering sends it back.
The strip chart at the bottom traces N₅(t) and the resulting Δσ_xx/σ₀(t) live, so you can watch the anomaly signal build up and decay as you move the sliders.

Source node (χ=+1) Sink node (χ=−1) Chiral electrons