Motivation and learning objective
Why can the net transport over a full cycle remain quantized while the instantaneous curvature changes continuously?
A Rice–Mele-type alternating lattice can transport an integer number of particles per adiabatic cycle when its control loop winds around an isolated gap-closing point. Compute the full momentum–cycle Berry curvature and its integrated pump index while moving the control loop into and out of the topological regime.
Recommended prerequisites: the relevant mathematical formulation in the English lecture notes, complex amplitudes, expectation values, and a basic understanding of quantum operators. Read the derivation below before interpreting the plots as physical observables.
Predictions and benchmarks
Live model outputs
Trajectory in the dimerization–onsite plane
The gap closes at the origin. Moving a continuous closed loop across the origin changes its topological winding only when the bulk gap closes.
Pump loop in dimerization delta and staggered potential Delta parameter plane
The red point marks the current cycle position. The center/radius criterion determines whether the loop encloses the bulk degeneracy.
Momentum-integrated Berry curvature
Curvature is distributed unevenly around the cycle, so the transported quantity grows nonuniformly with phase even when the full winding is quantized.
Instantaneous momentum-integrated Berry curvature versus cycle phase
Define F(φ)=(1/2π)∫BZ Ω₋(k,φ)dk. The full cycle integral ∫F dφ approaches the integer Chern pump index.
Cumulative geometric pump index
Accumulating Berry flux reveals that quantization pertains to the complete adiabatic cycle rather than intermediate partial paths.
Berry-flux accumulated pump charge index versus cycle phase
The final index follows the orientation and lower-band Berry convention A=i〈u|∂u〉. In this model an origin-enclosing counterclockwise loop yields C=−1.
Derivation and assumptions
Reconstruct the physical calculation
Use the Rice–Mele family H(k,φ)=d(k,φ)·σ with d=(t₁+t₂cosk,t₂sink,Δ). Set t₁=t₀+δ, t₂=t₀−δ, t₀=1, δ(φ)=δ₀+Rcosφ and Δ(φ)=Δ₀+Rsinφ. The control point traces a circle in the δ–Δ plane, encircling the critical point (0,0) whenever √(δ₀²+Δ₀²)<R.
For the occupied lower band and Berry convention A=i〈u|∂u〉, the two-dimensional integral over momentum and cyclic parameter is C=(1/2π)∫₀²πdφ∫BZdk Ω₋(k,φ). This is a Chern number on the momentum–time torus whenever the occupied band stays separated from the empty band over the entire cycle.
The discrete numerical model computes d and both derivatives analytically and integrates their triple product by a periodic midpoint quadrature. This avoids finite-difference gauge discontinuities. A control loop enclosing the gap-closing point yields C=−1 for the stated Fourier, pump orientation and lower-band conventions; a displaced loop that excludes the origin yields C=0. Reversing pump direction reverses the index.
Define the cumulative Berry-flux charge index Q(φ)=(1/2π)∫₀φdφ′∫BZdk Ω₋(k,φ′). It changes smoothly during the cycle and becomes quantized at φ=2π. Its physical sign relative to an electrical charge current depends on carrier charge and transport orientation, so the display reports the convention-fixed geometric index, not an unqualified signed electron current.
Check exact and singular limits
For δ₀=Δ₀=0 and R>0, the loop winds once and the lower-band index approaches −1. When the circle lies entirely outside the origin, it approaches zero. If R equals √(δ₀²+Δ₀²), the path crosses a bulk degeneracy at k=π and the isolated-band Chern number is undefined.
Numerical and physical validity conditions
Noninteracting two-band Rice–Mele lattice at half filling, infinite periodic bulk, isolated occupied lower band, slow adiabatic motion, fixed baseline hopping t₀=1, and a closed pump cycle. Finite-duration corrections, boundary charge redistribution and charge-specific current signs are beyond the model.
Active understanding
Three independent investigations
- Show quantization. Choose a centered loop, change momentum mesh N, and verify that the final integrated index approaches −1.
- Leave the topological region. Shift Δ₀ beyond the radius. Examine the loop and determine why the indexed winding falls to zero without requiring changes in the instantaneous signal formula.
- Approach a singular cycle. Bring the loop close to the degeneracy. Inspect the minimum spectral gap, Berry peaks and slower numerical convergence; explain why the exactly critical cycle has no isolated-band invariant.
Self-check: When does an isolated occupied band have a well-defined quantized Thouless pump index for this loop?
Extension: reproduce one model with a separate implementation or an AI-assisted notebook. Check numerical convergence, normalizations, phase conventions, or independent limiting-case results before trusting the answer.
Physical conclusions
Interpret the invariant or approximation carefully
A Thouless pump is a dynamical manifestation of band topology: smoothly varying instantaneous responses integrate to a robust integer across a closed, gapped adiabatic cycle. Numerical integration, loop geometry and the spectral gap provide independent checks of the result.
Lecture notes: 05_quantum_pumping.tex ↗.