---
title: "Hofstadter Lattice · Split a Band Interactive Lab | Sandbox Physics"
description: "How much flux does it take to turn one band into several? Any rational flux that is not a whole number of flux quanta per cell."
canonical_url: "https://sandboxphysics.com/labs/hofstadter-butterfly/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hofstadter-butterfly/"
last_modified: "2026-10-27"
---

Magnetic lattice / Bulk

# Hofstadter Lattice · Split a Band

**How much flux does it take to turn one band into several?** Any rational flux that is not a whole number of flux quanta per cell. With p over q quanta per cell, in lowest terms, the magnetic unit cell becomes q times larger and the band splits into q subbands. Some subbands can touch, so q does not tell you how many gaps are open.

Turn the magnetic flux through each lattice square, see the magnetic unit cell it creates and collect a histogram of energies blurred by finite resolution. Compare the exact matrices at rational flux with a finite butterfly plot and the low-flux Landau approximation.

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

The model is single-particle tight binding. The butterfly plot and the momentum grid have finitely many points. Collected energies are drawn evenly from all bands and blurred by a Gaussian resolution. They are simulated data, not spectra measured on a real material.

MAGNETIC UNIT CELL

$\alpha=\frac13$

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

## Watch the band split.

Eigenstate at current momentum

$\alpha=p/q$ · $E/J_x$

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

$\text{—}$

03 / Measure independent preparations

## Can your resolution separate the bands?

For this acquisition only: choose a momentum-grid point and a band uniformly, then read its energy with Gaussian noise. The histogram estimates this specified density of states; it does not use the selected single eigenstate.

No records yet.

$M_{\rm window}=0$

The shaded window counts recorded energies. Move its center in advanced controls; existing records stay. An empty window is not proof of a true gap.

$M=0$ $\text{—}$

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 | Energy in horizontal-hopping units |
| --- | --- |

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/magnetic-edge-spectroscopy/)

Physics tutorial

## Hofstadter Lattice · Split a Band

**Background** Turn the magnetic flux through each lattice square, see the magnetic unit cell it creates and collect a histogram of energies blurred by finite resolution. Compare the exact matrices at rational flux with a finite butterfly plot and the low-flux Landau approximation.

**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**: How much flux does it take to turn one band into several?
- **One-sentence intuition**: At a rational flux, the denominator of the reduced fraction sets the magnetic period. Bands can touch, so the denominator does not count the open gaps.

### 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.

#### Low-flux reference

$$
E_n\simeq-2(J_x+J_y)+4\pi\alpha\sqrt{J_xJ_y}(n+\tfrac12)
$$

Expanding the dispersion near the band bottom gives this result for small positive flux. The curvature of the lattice band adds corrections, so it is not the exact spectrum.

### 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

   Compare flux one third, one half and zero. Watch the magnetic cell set by the reduced fraction and the table of sampled bands.

   **What to observe:** At half flux the central bands touch. A dark patch in the coarse butterfly plot does not prove that a gap is open there.

2. 02

   #### Collect and save data

   Collect energies at one third flux. Save the sample, worsen the energy resolution 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 model is single-particle tight binding. The butterfly plot and the momentum grid have finitely many points. Collected energies are drawn evenly from all bands and blurred by a Gaussian resolution. They are simulated data, not spectra measured on a real material.

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)