Spectral Flow, Chern-Simons Theory & Adiabatic Pumping

Topological charge pumping, level crossing, and the Atiyah-Patodi-Singer index theorem

System Parameters

Number of levels: 8 Coupling strength g: 0.50 Chern number C: 1 Pump speed (cycle/s): 0.30 Temperature (gap units): 0.10

Topological Invariants

Spectral flow sf(A)—

Chern number C₁—

Pumped charge (e)0.00

Berry phase γ—

APS index—

Cycle count0

Legend

Filled circles = occupied levels (Fermi sea). Color = energy eigenvalue. The spectral flow counts net crossings of zero energy per cycle.

Spectral Flow and the APS Index Theorem

Spectral flow quantifies how many eigenvalues of a Dirac-like operator cross zero as a parameter θ is adiabatically varied through a full cycle. The Atiyah-Patodi-Singer (1975) index theorem relates this to a topological index: sf(A) = ind(D) = C₁, the first Chern number of the Berry curvature bundle over parameter space.

sf(D_θ) = C₁ = (1/2π) ∫∫ F dθ dφ

In condensed matter, this manifests as the Thouless charge pump (1983): adiabatically cycling through one period of a parameter transports exactly C₁ electrons across the system per cycle, regardless of details. This is topologically protected by the integer Chern number. The TKNN formula for the quantum Hall conductance is the 2D analog: σ_xy = C₁(e²/h).

The visualization shows eigenvalue evolution (spectral flow), Berry phase accumulation, and the pumped charge counter — which always increments by exactly an integer per cycle.