MAGNETIC LATTICE / BULK

Hofstadter Lattice · Split a Band

Turn the plaquette flux, reveal the magnetic unit cell and collect a broadened energy histogram. Compare exact rational-flux matrices with a finite butterfly atlas and the low-flux Landau approximation.

Reduced flux $\alpha=\frac13$

Recorded sample $M=0$

Single-particle tight binding. The atlas and momentum grid are finite. Collected energies use a declared uniform band mixture and Gaussian resolution; they are simulated records, not measured material spectra.

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 / SELECT 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 / ACQUIRE 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

A mesh is not 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 Aidelsburger et al. · 2013 Tong · Quantum Hall Effect Open the companion experiment

Start with the essentials

Focus question: How much flux does it take to turn one band into several?
One-sentence intuition: At rational flux, the reduced denominator fixes the magnetic periodicity. Bands can touch; the denominator is not the number of 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

The longitudinal plane wave reduces the square lattice to a transverse matrix. The strip omits bonds beyond its walls and adds the selected potential to its left two sites.

Magnetic periodicity

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

Reduce the fraction first. Boundary hoppings are added, including both paths for a two-site cell and the self bond for a one-site cell.

Low-flux reference

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

Expand the dispersion near the band bottom at small positive flux. Finite lattice curvature produces corrections; this is not an exact spectrum.

Common difficulties

Which system?

Typical misconceptionThe butterfly, strip and noisy records are interchangeable views of the same data.

Better mental modelThe atlas samples periodic bulk matrices. The strip has physical open walls. Records use a separately stated ideal preparation and readout protocol.

What is measured?

Typical misconceptionAn edge-colored point or histogram certifies a Hall integer.

Better mental modelThese experiments show model spectra and sampled energy or position. They do not implement transport, filled-band pumping or a Chern-number estimator.

Run the experiment

01
Predict before changing
Compare one third, one half and zero flux. Watch the reduced magnetic cell and the sampled band table.
What to observe: At half flux the central bands touch. A dark patch in a coarse atlas does not prove an open gap.
02
Acquire and save
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 raw rows, which can be exported and recomputed.
03
Challenge the calculation
Refine the momentum mesh. Compare the first three low-flux bands with the Landau reference.
What to observe: Numerical sampling, finite strip width and continuum approximation errors are different effects.