Energy levels of 1D Dirac Hamiltonian flow through zero as gauge field winds: ∂_μ j^μ_5 = e²/(2π²) E·B
Gauge & Fermion
Topology
Zero crossings—
Chiral charge Q_5—
Net flow ΔN—
1D Dirac: H(k) = v_F k σ_z + m σ_x
With flux: k → k − eA
Spectral flow: E_n(A) crosses zero once
per unit Chern number as A winds by Φ_0
Chiral anomaly: ∂_t ρ_5 = eE/π (1D)
Index theorem: flow = Chern number