Eigenvalues crossing zero as a parameter varies — net spectral flow equals the topological index
Spectral Flow Definition
Consider a family of self-adjoint operators D(m) on a circle S¹ with a mass kink — a spatially varying mass term m(x) with winding number n.
The spectral flow sf(D) counts the net number of eigenvalues crossing zero from below as m varies:
sf(D) = #{λ_i: λ(m_i)→0⁺} − #{λ_i: λ(m_i)→0⁻}
Atiyah-Singer Index Theorem:
sf(D) = ind(D) = dim ker D₊ − dim ker D₋
The spectral flow is a topological invariant — it equals the winding number of the mass kink and cannot change under smooth deformations.
Connection to Topology & Anomalies
The Atiyah-Singer index theorem (1963) is one of the deepest results in mathematics: the analytic index (spectral data) equals a topological/geometric invariant computed from Chern classes and the Â-genus:
ind(D) = ∫_M Â(M) ∧ ch(E)
Chiral anomaly: in QFT, the current ∂_μJ^5_μ = spectral flow of the Dirac operator, related to the topological charge of the gauge field configuration.
Zero modes: each zero-crossing corresponds to a topologically protected zero-energy mode — a Majorana bound state in condensed matter (topological superconductors), protected by the bulk-boundary correspondence.