Thouless Pump
In an adiabatic pump, slowly varying a periodic potential through a full cycle transports an integer number of charges. This integer is a topological invariant — the Chern number of the occupied band.
Q = e·C₁
C₁ = (1/2π)∮ Ω dk dt
Ω = ∂ₖAₜ - ∂ₜAₖ
C₁ = (1/2π)∮ Ω dk dt
Ω = ∂ₖAₜ - ∂ₜAₖ
Controls
Statistics
0.00
Phase φ/π
1
Chern #
0.0
Pumped Q
0
Cycles
Top panel: Lattice potential (wavy curve) and charge density (colored dots). Bottom panel: Accumulated charge pumped vs time — note the quantized staircase.