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.

Reduced flux $\alpha=\frac13$

Recorded sample $M=0$

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 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: 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 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
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.
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.
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.

Magnetic Edge · Find the Boundary State

Find a state on the edge.

How often do you find it at an edge?

How close is the mesh to the continuum?

Keep a comparison