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.

Reduced flux $\alpha=\frac13$

Recorded sample $M=0$

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 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 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 misconceptionThe butterfly, the strip and the noisy data are just different pictures of the same data.

Better mental modelThe 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 misconceptionA colored edge point or a histogram tells you the Hall integer.

Better mental modelThese 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

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.
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.
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.