Dirac zero mode and topological index from domain wall mass profile
The Atiyah-Singer index theorem states: ind(D) = dim ker D − dim ker D† = topological invariant (Chern number).
In 1+1D, a Dirac operator with a sign-changing mass m(x) hosts exactly one zero-energy bound state — the zero mode.
The mass profile m(x) = m₀·tanh((x−x₀)/ξ) changes sign at the domain wall, guaranteeing index = 1
regardless of perturbations. Left panel: mass profile. Right panel: energy spectrum showing zero mode (red) isolated from bulk bands.