Dirac spectrum in magnetic field — adiabatic gauge change pumps chiral charge
A massless Dirac fermion in a magnetic field has Landau levels E_n = ±√(2neBℏ). Under adiabatic threading of magnetic flux (θ: 0→2π), the zero mode flows from the left-handed to right-handed sector — this is spectral flow. Net chiral charge is not conserved: ∂_μj^μ₅ = (e²/2π²)E·B.