---
title: "Magnetic Edge · Find the Boundary State Interactive Lab | Sandbox Physics"
description: "Which states move along the boundary instead of filling the sample? The states on edge branches that cross the bulk energy gap."
canonical_url: "https://sandboxphysics.com/labs/magnetic-edge-spectroscopy/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/magnetic-edge-spectroscopy/"
last_modified: "2026-10-28"
---

Magnetic lattice / Boundary

# Magnetic Edge · Find the Boundary State

**Which states move along the boundary instead of filling the sample?** The states on edge branches that cross the bulk energy gap. They stay close to the boundary and travel along it, at a speed set by the slope of energy against momentum. States in the bulk bands spread through the whole sample, and so does an edge branch once it runs into a bulk band.

Pick a state in a magnetic strip with open edges. Move the wall, tune the potential at the edge and measure the position across the strip, preparing a fresh state for each shot. Compare how often the particle lands at the edge with the spectrum and the model group velocity.

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

The strip is infinite and periodic along its length. Each position measurement starts from a freshly prepared, ideal eigenstate. The group velocity is a derivative calculated from the model; it is not inferred from the position counts. No quantized conductance is measured.

OPEN ACROSS · PERIODIC ALONG

$\alpha=\frac13$

Disk area: transverse probability, rescaled per state. Bond hue: hopping phase. Seven vertical copies illustrate a longitudinal plane wave, not seven measured atoms.

$E/J_x=0$

02 / Choose a state

## Find a state on the edge.

Eigenstate at current momentum

$k_y a/\pi$ · $E/J_x$

Direct model probability at the selected momentum. Picking a band changes the prepared state; an eigenstate is stationary.

$\text{—}$

03 / Measure independent preparations

## How often do you find it at an edge?

Prepare the selected eigenstate anew for every shot, then read its transverse site. The edge region is the outer two columns on each side. Counts estimate position probabilities, not velocities or a Chern number.

No records yet.

$M=0$ $\text{—}$

Estimates use records only. Mean uncertainty is one standard error; edge intervals are pointwise 95% Wilson intervals. Neither includes grid error or systematic preparation error.

04 / Check the approximation

## How close is the mesh to the continuum?

| Band | Sampled minimum | Sampled maximum | Landau reference |
| --- | --- | --- | --- |

$\text{—}$

The table uses the clean periodic bulk, excluding the edge potential. Two nested momentum meshes check sampled band extrema. The change is a diagnostic, not a rigorous error bound. Landau values appear only for small positive flux, near the band bottom; the reference is never fitted to records.

## Keep a comparison

| Trial | Transverse site |
| --- | --- |

Latest twelve outcomes shown. CSV retains all rows and preparation settings. Up to six saved samples; select one to restore it.

Model, conventions & sources

Square lattice, one orbital per site, noninteracting and coherent. Horizontal hopping sets the energy unit and lattice spacing sets the length unit. Positive flux means a positive oriented hopping phase around a counterclockwise loop. The periodic cell contains the reduced denominator of the flux; the open strip has a separate width. No disorder, reservoirs, material calibration, atom loss or transport contacts are included.

$\begin{gathered}(E/J_x)u_m=-u_{m-1}-u_{m+1}\\-2(J_y/J_x)\cos(k_y a+2\pi\alpha m)u_m\end{gathered}$

Every displayed probability is from the corresponding matrix eigenvector. Velocity uses the Hellmann–Feynman expectation. A colored edge state alone does not establish a quantized Hall response.

[Hofstadter · 1976](https://doi.org/10.1103/PhysRevB.14.2239) [Aidelsburger et al. · 2013](https://arxiv.org/html/1308.0321v1) [Tong · Quantum Hall Effect](https://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf) [Open the companion experiment](https://sandboxphysics.com/labs/hofstadter-butterfly/)

Physics tutorial

## Magnetic Edge · Find the Boundary State

**Background** Pick a state in a magnetic strip with open edges. Move the wall, tune the potential at the edge and measure the position across the strip, preparing a fresh state for each shot. Compare how often the particle lands at the edge with the spectrum and the model group velocity.

**Why it matters** Follow one magnetic phase from the Hamiltonian matrix to the shape of a quantum state and on to data you can record.

Start with the essentials

- **Focus question**: Which states move along the boundary instead of filling the sample?
- **One-sentence intuition**: Edge branches join the bulk bands, but not every state, and not every momentum, is localized at an edge.

### Core mathematical model

#### Harper matrix

$$
(Hu)_m=-J_x(u_{m-1}+u_{m+1})-2J_y\cos(k_ya+2\pi\alpha m)u_m
$$

Taking a plane wave along the strip turns the square lattice into a matrix in the transverse coordinate. In the strip, bonds that would cross a wall are dropped, and the chosen potential is added to the two leftmost sites.

#### Magnetic period

$$
\alpha=p/q,\qquad u_{m+q}=e^{i\theta_x}u_m
$$

Reduce the fraction first. Hoppings across the cell boundary are added in: a two-site cell gets both paths, and a one-site cell gets its bond to itself.

#### Group velocity

$$
v_y=\frac{1}{\hbar}\frac{\partial E}{\partial k_y}=\frac{2aJ_y}{\hbar}\sum_m|u_m|^2\sin(k_ya+2\pi\alpha m)
$$

This is the model's Hellmann–Feynman expectation value. Counting the sites where the particle lands gives no independent measurement of it.

### Common difficulties

#### Which system is shown?

**Typical misconception** The butterfly, the strip and the noisy data are just different pictures of the same data.

**Better mental model** The butterfly plot samples matrices of the periodic bulk. The strip has real open walls. The noisy data come from an ideal preparation and readout procedure that is described separately.

#### What is measured?

**Typical misconception** A colored edge point or a histogram tells you the Hall integer.

**Better mental model** These experiments show model spectra and sampled energies or positions. They do not simulate transport or the pumping of a filled band, and they do not estimate a Chern number.

### Run the experiment

1. 01

   #### Predict first

   Click a colored edge point in the spectrum and see where the state's weight sits across the strip. Move the wall with the handle or the arrow keys.

   **What to observe:** The chosen eigenstate may mix with the opposite edge or spread into the bulk.

2. 02

   #### Collect and save data

   Collect positions, save them, select a bulk-like state and repeat.

   **What to observe:** The bars and estimates are computed from the raw measurements alone. You can export those and redo the calculation.

3. 03

   #### Test the calculation

   Refine the momentum mesh. Compare the first three low-flux bands with the Landau reference.

   **What to observe:** Coarse momentum sampling, the finite strip width and the continuum approximation each cause a different kind of error.

Physical conclusion

The strip is infinite and periodic along its length. Each position measurement starts from a freshly prepared, ideal eigenstate. The group velocity is a derivative calculated from the model; it is not inferred from the position counts. No quantized conductance is measured.

## 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 a quantum Hall plateau became a resistance standard**](https://sandboxphysics.com/stories/condensed-matter/integer-quantum-hall/)
2. [Lab **Chern Insulator · Follow the Edge** Chern insulator](https://sandboxphysics.com/labs/chern-insulator/)
3. [Article **How a one-third Hall plateau led to fractional charge**](https://sandboxphysics.com/stories/condensed-matter/fractional-quantum-hall/)
4. [Article **How wave geometry fixes an integer Hall response**](https://sandboxphysics.com/stories/condensed-matter/thouless-hall-topology/)
5. [Lab **Thouless Transport Pump** Thouless quantum pump](https://sandboxphysics.com/labs/thouless-quantum-pump/)
6. [Lab **Topological Photonic Edge State** Topological photonic edge state](https://sandboxphysics.com/labs/topological-photonic-edge-state/)

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)