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Sandbox Physics

Q064 · A LOOP THAT MOVES PARTICLES

Thouless Transport Pump

Drag a Rice–Mele cycle around the gap-closing point. Compare a complete lower band with partial filling, and test why speed, boundaries and disorder change the transported charge.

Interactive modelThouless Transport Pump
Recorded trials00
Model reference—\text{—}
PREPARE → CYCLE → COMPARE—\text{—}

Horizontal δ/J\delta/J · Vertical Δ/J\Delta/J

Within the marked cell v/Jv/JBetween cells w/Jw/J
—\text{—}
Open chain · A then B · position in cell unitsCircle area: model occupation

The loop is a control-space diagram. The chain shows model density, not particle trajectories. The vertical gold cut is where current is integrated.

Ready. No recorded trials.

02 / SIMULATED ACQUISITION

Estimate motion from the records.

Colored bars: recorded frequencyDashed: model reference
M=0M=0—\text{—}

Each trial draws one initial and one final position from separate preparations. The displacement estimate uses their sample means; the uncertainty is one standard error. Positions are in unit-cell lengths.

03 / SEPARATE MODEL REFERENCES

Three quantities. Different questions.

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Keep a comparison

Last twelve trials shown. CSV retains every record and setting. Save up to six samples; click one to restore it.

Model, limits & primary sources

Nearest-neighbor Rice–Mele chain, positive hopping, no interactions or bath. Initially occupy the lowest orbitals up to the chosen fraction of one band. Bond and onsite disorder share a fixed pattern. The clean infinite-band Wannier center and curvature integral are reference diagnostics, even when the finite chain is disordered. An open finite chain can move charge across its center while its center-of-mass shift remains smaller than one cell. Slow compared with the bulk gap does not eliminate edge dynamics; still slower is not always better.

Asbóth, Oroszlány & Pályi · Chapters 1–5Nakajima et al. · 2016

Dimensionless teaching model; not a reconstruction of experimental apparatus or published data.

Physics tutorial

Thouless Transport Pump

BackgroundDrag a Rice–Mele cycle around the gap-closing point. Compare a complete lower band with partial filling, and test why speed, boundaries and disorder change the transported charge.

Why it mattersCompare a convincing picture with a measurable consequence.

Start with the essentials

Focus question
Return the controls. Move the particles.
One-sentence intuition
Quantized bulk transport and finite-cloud center displacement are different observables.

Core mathematical model

The same chain throughout

H=∑j[v∣j,B⟩⟨j,A∣+w∣j+1,A⟩⟨j,B∣+h.c.]+Δ∑j(nj,A−nj,B)H=\sum_j[v|j,B\rangle\langle j,A|+w|j+1,A\rangle\langle j,B|+\mathrm{h.c.}]+\Delta\sum_j(n_{j,A}-n_{j,B})

Positive hopping, A then B from left to right. Intracell hopping is the reference coupling plus dimerization; intercell hopping is the reference minus dimerization. Open ends.

Transport is an integral

Q(t)=∫0tIcut(s) ds=⟨NR(t)⟩−⟨NR(0)⟩Q(t)=\int_0^t I_{\mathrm{cut}}(s)\,ds=\langle N_R(t)\rangle-\langle N_R(0)\rangle

Rightward current across the central bond is integrated independently and checked against right-half population. It is a model reference, not obtained from independent position pairs.

Common difficulties

Model or measurement?

Typical misconceptionAll smooth curves were reconstructed from the records.

Better mental modelOnly histogram frequencies, sample means and their sampling errors use records. Spectra, winding, Wannier centers and currents are direct model references.

Boundaries

Typical misconceptionTopology removes all finite-size and timing effects.

Better mental modelFinite open noninteracting chain, initially occupying the lowest single-particle orbitals. The clean infinite-band topology is a separate reference. Position records sample the normalized one-body density; they are not simultaneous many-particle snapshots. No integer rounding is applied to transport.

Run the experiment

  1. 01

    Predict

    Start with the loop surrounding the origin. Predict the direction of transport.

    What to observe: The initial screen contains model references and no recorded data.
  2. 02

    Acquire a contrast

    Finish a cycle, record positions, save the sample, then reverse the direction.

    What to observe: Estimate displacement using only initial and final position samples.
  3. 03

    Find the failure

    Move the loop off the origin, accelerate it, or reduce filling. Inspect the bulk and finite-chain references separately.

    What to observe: The reference conditions matter as much as the picture.