Q064 · A loop that moves particles

Thouless Transport Pump

How can slowly cycling a chain's parameters back to their start still move particles across it? The loop has to wind around the point where the energy gap closes. Cycle the Rice-Mele parameters slowly around that point, and a filled band moves a whole number of particles across per cycle, even though the parameters end where they began. This is a Thouless pump. A partly filled band, fast driving, edges or disorder spoil the exact count.

Drag a Rice–Mele parameter loop around the point where the gap closes. Compare a completely filled lower band with partial filling, and find out why speed, boundaries and disorder change the transported charge.

Recorded trials $0$

Model reference $\text{—}$

The chain is finite, open and noninteracting, and the particles start in the lowest single-particle orbitals. The topology of the clean infinite band is shown separately as a reference. Position records are samples of the normalized one-body density, not snapshots of all particles at once. The transported charge is never rounded to an integer.

Prepare → cycle → compare

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Horizontal $\delta/J$ · Vertical $\Delta/J$

Within the marked cell

v/J

Between cells

w/J

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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.

Cycle progress

t/T

0

Ready. No recorded trials.

02 / Simulated acquisition

Estimate the displacement from the records.

Colored bars: recorded frequencyDashed: model reference

M=0

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Each trial draws one starting position and one final position, each from its own preparation. The displacement is estimated from their sample means, and the uncertainty shown is one standard error. Positions are given in unit-cell lengths.

03 / Model references, shown separately

Three quantities answer different questions.

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Keep a comparison

The table shows the last twelve trials; the CSV keeps every record and setting. You can save up to six samples and click one to restore it.

Model, limits & primary sources

The model is a nearest-neighbor Rice–Mele chain with positive hopping and no interactions or heat bath. At the start, the lowest orbitals are filled up to the chosen fraction of one band. Bond disorder and onsite disorder follow one fixed pattern. The Wannier center and curvature integral are computed for the clean, infinite band and serve as reference diagnostics, even when the finite chain is disordered. An open finite chain can carry charge across its center while its center of mass moves by less than one cell. Pumping slowly compared with the bulk gap does not remove the edge dynamics, and slower is not always better.

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

This is a dimensionless teaching model. It does not reconstruct an experimental apparatus or published data.

Start with the essentials

Focus question: How can slowly cycling a chain's parameters back to their start still move particles across it?
One-sentence intuition: The quantized transport through the bulk and the shift of a finite cloud's center are different observables.

Core mathematical model

One chain model throughout

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})

Hopping amplitudes are positive, and the sites run A then B from left to right. Hopping inside a cell is the reference coupling plus the dimerization; hopping between cells is the reference minus the dimerization. Both ends are open.

Transported charge is the integrated current

Q(t)=\int_0^t I_{\mathrm{cut}}(s)\,ds=\langle N_R(t)\rangle-\langle N_R(0)\rangle

The rightward current through the central bond is integrated on its own and checked against the change in right-half population. It is a model reference; it is not inferred from pairs of independent position samples.

Common difficulties

Model or measurement?

Typical misconceptionEvery smooth curve was reconstructed from the recorded data.

Better mental modelOnly the histogram frequencies, sample means and their sampling errors come from records. Spectra, winding, Wannier centers and currents are computed directly from the model.

Finite size and timing still matter

Typical misconceptionTopology removes all finite-size and timing effects.

Better mental modelThe chain is finite, open and noninteracting, and the particles start in the lowest single-particle orbitals. The topology of the clean infinite band is shown separately as a reference. Position records are samples of the normalized one-body density, not snapshots of all particles at once. The transported charge is never rounded to an integer.

Run the experiment

01
Make a prediction
Start with the loop enclosing the origin and predict the direction of transport.
What to observe: At first the screen shows only model references; nothing has been recorded yet.
02
Record a comparison
Finish a cycle, record positions, save the sample, then reverse the direction.
What to observe: Estimate the displacement from the initial and final position samples alone.
03
Find where it breaks down
Move the loop off the origin, run it faster or lower the filling. Check the bulk and finite-chain references separately.
What to observe: The conditions behind each reference matter as much as the picture.

Thouless Transport Pump

Estimate the displacement from the records.

Three quantities answer different questions.

Keep a comparison