---
title: "SSH Chain & Edge States Interactive Lab | Sandbox Physics"
description: "Is a particle trapped at the edge of an SSH chain always in a topological state? Not necessarily."
canonical_url: "https://sandboxphysics.com/labs/ssh-topological-chain/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ssh-topological-chain/"
last_modified: "2026-10-27"
---

Q061 · Why does it stay at the edge?

# SSH Chain & Edge States

**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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The 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.

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 units Circle 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 frequency Dashed: model reference

$M=0$ $\text{—}$

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

## Seeing localization is not enough; check the energy too.

$\text{—}$

$\text{—}$

## 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 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.

[Asbóth, Oroszlány & Pályi · Chapters 1–5](https://arxiv.org/html/1509.02295v1)

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

Physics tutorial

## SSH Chain & Edge States

**Background** 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.

**Why it matters** A picture can look convincing. Check it against a consequence you can measure.

Start with the essentials

- **Focus question**: Is a particle trapped at the edge of an SSH chain always in a topological state?
- **One-sentence intuition**: An onsite defect can trap a particle even when the bulk winding is trivial.

### 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.

#### What the local probe sees

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

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.

### 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 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.

### Run the experiment

1. 01

   #### Make a prediction

   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.

2. 02

   #### Record a comparison

   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.

3. 03

   #### Find where it breaks down

   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.

Physical conclusion

The 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.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)