Quantum Anomalous Hall Effect — k-space Berry Curvature

Film Geometry

Magnetization

Band Structure

Exchange gap Δ 28 meV

Fermi level E_F 0 meV

Transport

Gapped · C = +1 · one chiral edge channel

Chern number C

σ_xy (e²/h)

+1.00

σ_xx (longitudinal, schematic)

0.00 — dissipationless

Single-cone Berry integral ∫Ω d²k/2π (numeric)

−0.500

How it works

Each surface of the magnetized topological-insulator film is a massive Dirac fermion:

H(k) = ħv_F(k_x σ_y − k_y σ_x) + Δ σ_z
E±(k) = ±√((ħv_F k)² + Δ²)

The lower band carries a Berry curvature concentrated near k=0 that this simulator computes directly on a discretized 2D (k_x,k_y) grid — this is the actual quantity being visualized, not a stand-in animation:

Ω(k) = −Δ(ħv_F)² / [2((ħv_F k)² + Δ²)^(3/2)]

Integrating Ω over the full plane gives exactly ±1/2 per Dirac cone (shown live as the numeric Riemann-sum estimate above, converging to the analytic result as the k-grid is refined). A thin film hybridizes its top and bottom surfaces so both pick up the same sign of Δ from the magnetization, and their half-integer contributions add to an integer Chern number:

C = (1/2)[sign(Δ_top) + sign(Δ_bottom)] = sign(Δ)   (thin film)
σ_xy = C·e²/h,   σ_xx = 0 (dissipationless)

In real space, a nonzero Chern number forces a chiral edge state whose wavefunction decays exponentially into the bulk with penetration depth ξ = ħv_F/Δ — plotted below as a density profile across the sample, in place of the animated 3D loop.

Thin / Thick film — only a thin film hybridizes enough to open the topological gap; a thick film has decoupled, ungapped surfaces and no quantization.
↑M / ↓M — flips the sign of Δ, flips C, and reverses the curvature map's polarity and the edge current's circulation.
Δ slider — exchange-gap size; Δ = 0 collapses the curvature peak (gapless Dirac cone).
E_F slider — once |E_F| exceeds Δ/2 the edge channel mixes with diffusive bulk states and σ_xy drops off its plateau.

k-space Berry curvature Ω(k_x,k_y) Real-space edge/bulk density profile