Q061 · WHY DOES IT STAY?

SSH Chain & Edge States

Swap the alternating bonds, choose a launch site, and record where the particle is found. Probe the local spectrum to distinguish a topological edge mode from an ordinary defect.

Recorded trials $0$

Model reference $\text{—}$

Finite open, noninteracting chain. The winding diagram describes the clean periodic bulk in the displayed unit cell. A local onsite defect breaks chiral symmetry; localization alone does not establish topology. All records are simulated.

INJECT → EVOLVE → PROBE

\text{—}

Horizontal $d_x/J$ · Vertical $d_y/J$

Within the marked cell

v/J

Between cells

w/J

\text{—}

Open chain · A then B · position in cell unitsCircle area: model occupation

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.

Evolution time

tJ/\hbar

0

Probe

Ready. No recorded trials.

02 / SIMULATED ACQUISITION

What did the probe actually see?

Colored bars: recorded frequencyDashed: model reference

M=0

\text{—}

Position and energy modes are separate preparations, not a joint position–energy measurement. The local spectral probe acts at the selected site; time affects the position protocol only.

03 / SEPARATE MODEL REFERENCES

Localization needs an energy check.

\text{—}

\text{—}

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

Exact diagonalization of a finite open SSH chain and spectral evolution of a localized initial state. A/B pairs define the unit cell; swapping termination exchanges the two hopping values. Fixed multiplicative bond disorder preserves chiral symmetry; the selected local onsite defect breaks it. The spectral probe samples local eigenstate weights with Gaussian energy resolution. Finite chains have hybridized, split edge modes, not guaranteed exact zero modes.

Asbóth, Oroszlány & Pályi · Chapters 1–5

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

Start with the essentials

Focus question: Trapped at the edge. But is it topological?
One-sentence intuition: An onsite defect can trap a particle without a nontrivial bulk winding.

Core mathematical model

The same chain 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})

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.

What the local probe sees

p(E|j)=\sum_n|\langle j|n\rangle|^2\frac{e^{-(E-E_n)^2/(2\sigma_E^2)}}{\sqrt{2\pi}\sigma_E}

Ideal local spectral-weight sampling followed by Gaussian energy resolution. This is a defined probe protocol, not a calibrated STM current.

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. The winding diagram describes the clean periodic bulk in the displayed unit cell. A local onsite defect breaks chiral symmetry; localization alone does not establish topology. All records are simulated.

Run the experiment

01
Predict
Launch at the left end with a weak first bond.
What to observe: The initial screen contains model references and no recorded data.
02
Acquire a contrast
Record positions and then the local spectrum. Compare the ordinary-defect preset.
What to observe: A defect state can be localized but away from zero energy.
03
Find the failure
Swap the termination, close the clean gap, and add a symmetry-breaking defect.
What to observe: The reference conditions matter as much as the picture.