MAGNETIC LATTICE / BOUNDARY

Magnetic Edge · Find the Boundary State

Choose a state in an open magnetic strip. Move its wall, tune the edge potential and read the transverse position one preparation at a time. Compare measured edge weight with the spectrum and model group velocity.

Reduced flux $\alpha=\frac13$

Recorded sample $M=0$

The strip is infinite and periodic along its length. Each position comes from a fresh ideal eigenstate preparation. Group velocity is a model derivative, 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 / SELECT 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 / ACQUIRE 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

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	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 Aidelsburger et al. · 2013 Tong · Quantum Hall Effect Open the companion experiment

Start with the essentials

Focus question: Which states move along the boundary instead of filling the sample?
One-sentence intuition: An edge branch joins bulk bands, but not every state or 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

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.

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)

A Hellmann–Feynman model expectation. Site counts do not independently measure this velocity.

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
Select an edge-colored spectral point and inspect the spatial weight. Move the wall using the handle or arrow keys.
What to observe: The selected eigenstate can hybridize with the other edge or become bulk-like.
02
Acquire and save
Collect positions, save them, select a bulk-like state 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.