Q061 · Why does it stay at the edge?
Is a particle trapped at the edge of an SSH chain always in a topological state? Not necessarily. A topological SSH edge mode sits at zero energy and exists because the bulk bands have a nontrivial winding. An ordinary defect on a single site can trap a particle at the edge without any winding. Measuring the local spectrum tells the two apart.
Swap which bonds are strong and weak, choose where the particle starts and record where it is found. Then probe the local spectrum to tell a topological edge mode from a state trapped by an ordinary defect.
Horizontal · Vertical
Marked pairs define the unit cell. The winding curve is the clean bulk reference; disorder and the local defect belong to the finite chain below.
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02 / Simulated acquisition
The position and energy modes use separate preparations; they do not measure position and energy together. The local spectral probe acts at the chosen site, and the evolution time affects only the position measurement.
03 / Model references, shown separately
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.
The Lab diagonalizes a finite open SSH chain exactly and evolves a localized initial state through its spectrum. Each A/B pair forms one unit cell; swapping how the chain ends exchanges the two hopping values. A fixed multiplicative bond disorder keeps chiral symmetry, and the chosen onsite defect breaks it. The spectral probe samples the local weights of the eigenstates with a Gaussian energy resolution. In a finite chain the edge modes hybridize and split, so they are not guaranteed to sit exactly at zero energy.
This is a dimensionless teaching model. It does not reconstruct an experimental apparatus or published data.
Physics tutorial
BackgroundSwap which bonds are strong and weak, choose where the particle starts and record where it is found. Then probe the local spectrum to tell a topological edge mode from a state trapped by an ordinary defect.
Why it mattersA picture can look convincing. Check it against a consequence you can measure.
Start with the essentials
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.
The probe samples energies by their ideal local spectral weight, then blurs them with a Gaussian energy resolution. It is an idealized probe, not a calibrated STM current.
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.
Typical misconceptionTopology removes all finite-size and timing effects.
Better mental modelThe chain is finite, open at both ends and noninteracting. The winding diagram describes the clean periodic bulk for the unit cell shown. A local onsite defect breaks chiral symmetry, and localization by itself does not prove topology. All recorded data are simulated.
Launch the particle at the left end, where the first bond is weak.
What to observe: At first the screen shows only model references; nothing has been recorded yet.Record positions, then the local spectrum, and compare with the ordinary-defect preset.
What to observe: A defect state can be localized and still sit away from zero energy.Swap the termination, close the gap of the clean chain and add a defect that breaks the symmetry.
What to observe: The conditions behind each reference matter as much as the picture.