Thouless Pump: Adiabatic Charge Pumping

Adiabatic cycling of a 1D periodic potential pumps exactly one charge per cycle — quantized by topology.

0.010
δ = 0.50
N = 12
Pumped charge Q = 0.00  |  Cycle: 0  |  Phase θ: 0.00π
The SSH (Su-Schrieffer-Heeger) chain with time-varying hopping: H(θ) = Σ [t₁(θ) c†_{2i} c_{2i+1} + t₂(θ) c†_{2i+1} c_{2i+2}] + h.c.
As θ cycles through [0,2π], the Wannier center winds around the chain → one electron pumped per cycle.
The pumped charge equals the Chern number C = (1/2π) ∫ F(θ,k) dθ dk = ±1.
In-gap edge states (red) cross the gap during pumping — the spectral flow is the topological signature.