--- title: "Thouless Transport Pump Interactive Lab | Sandbox Physics" description: "How can slowly cycling a chain's parameters back to their start still move particles across it?" canonical_url: "https://sandboxphysics.com/labs/thouless-quantum-pump/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/thouless-quantum-pump/" last_modified: "2026-10-28" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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 $\text{—}$ Horizontal $\delta/J$ · Vertical $\Delta/J$ Within the marked cell $v/J$ Between cells $w/J$ $\text{—}$ Open chain · A then B · position in cell units Circle 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 frequency Dashed: model reference $M=0$ $\text{—}$ 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. $\text{—}$ $\text{—}$ ## 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](https://arxiv.org/html/1509.02295v1) [Nakajima et al. · 2016](https://doi.org/10.1038/nphys3622) This is a dimensionless teaching model. It does not reconstruct an experimental apparatus or published data. Physics tutorial ## Thouless Transport Pump **Background** 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. **Why it matters** A picture can look convincing. Check it against a consequence you can measure. 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 misconception** Every smooth curve was reconstructed from the recorded data. **Better mental model** Only 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 misconception** Topology removes all finite-size and timing effects. **Better mental model** 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. ### Run the experiment 1. 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. 2. 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. 3. 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. Physical conclusion 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. ## Related experiments and articles Condensed matter · Reading path ### Magnetism, disorder and topological response Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **How wave geometry fixes an integer Hall response**](https://sandboxphysics.com/stories/condensed-matter/thouless-hall-topology/) 2. [Lab **Chern Insulator · Follow the Edge** Chern insulator](https://sandboxphysics.com/labs/chern-insulator/) 3. [Lab **Magnetic Edge · Find the Boundary State** Magnetic edge spectra](https://sandboxphysics.com/labs/magnetic-edge-spectroscopy/) 4. 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