Scan the local spectrum of a Kitaev wire

Majorana End Modes

Why does a superconducting wire with a gapped interior show zero-energy states at its ends? Because the wire is in its topological phase. Picture each electron site as two Majorana halves. Hopping and pairing bind each half to one on the next site. One half is left unpaired at each end, at zero energy, while the interior keeps its gap. Raise the chemical potential past twice the hopping and the gap closes; beyond that the wire is an ordinary superconductor and the end states are gone.

Move a spectral probe along a superconducting chain and find the spectral weight near zero energy at its ends. Close the bulk gap and watch the end signal disappear. Then look at the discrete spectrum of the finite chain and collect an ideal record of local energies.

The model is the ideal spinless p-wave Kitaev chain with open ends and fixed pairing. A local peak at zero energy is not, on its own, proof of Majorana modes in a material. Braiding, qubit operations, disorder, interactions and experimental identification are not simulated. The spectra are broadened by a stated Lorentzian resolution.

A spectral portrait of a quantum wire

Lowest excitation energy

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Probe at the gold marker

0

CYAN · ELECTRONCORAL · HOLEHEIGHT · LOCAL SPECTRAL DENSITY

Both ribbons share the same physical sites. Their separation is a display convention. Height is linear spectral density at the selected energy; gain is fixed. Drag the probe strip below.

Move the probe

j=1

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01 / POSITION × ENERGY

Find the hidden endpoint weight.

Electron spectral density: position left to right, energy −3.5 at the bottom to 3.5 at the top. Drag the crosshair to set both probe and energy. Square-root color, fixed scale from 0 to 3 per hopping energy.

Dark 0 → bright 3 · charts below use linear values

02 / A LOCAL LOOK

The edge and the middle tell different stories.

Cyan: electron spectrum at your probe. Dashed gray: middle site. Gold: selected energy. Lorentzian half-width is 0.06; its tails extend beyond the displayed window.

$E/J\ \longrightarrow\quad J\rho_j(E)$

03 / FINITE WIRE

Keep the actual levels in view.

Every dot is a finite-chain BdG eigenvalue. Gold marks the two closest to zero; gray shading is the infinite-chain bulk gap. A finite-chain excitation energy is not a many-body parity measurement.

$n\ \longrightarrow\quad E_n/J$

04 / INDEPENDENT BULK REFERENCE

Close the gap and the phase changes.

Analytic positive and negative bands of the infinite uniform chain. This is a reference, not a reconstruction of the local records.

$k\ \longrightarrow\quad E/J$

Sample records from the model

Collect a local spectrum.

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Ideal samples from the normalized electron spectral measure at the probe, with Lorentzian energy resolution. These are simulated energy records, not measured tunnel currents. The selected display energy does not filter the sample. Fixed reproducible seed; all tails retained in CSV.

Recorded near-zero fraction

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95% Wilson interval · shot noise

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Model probability

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All-eigenpair residual

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Gold is the record; cyan is the model.

The central window is −0.15 to 0.15. Histogram bins cover −3.6 to 3.6; the outer bins include all overflow. Each bar is a fraction of every recorded trial.

Latest 20 records · export every trial

Trial	Site	$E/J$

Model, units and sources

Real nearest-neighbor hopping, open ends, uniform spinless p-wave pairing, no disorder. The full real symmetric BdG matrix is diagonalized; all eigenpairs enter the spectra. Broadening is a resolution kernel, not a fitted lifetime. Excitation energies below one ten-billionth of the hopping are reported as unresolved.

Position increases to the right in lattice-spacing units. Hopping sets the energy unit; time is measured in reduced Planck constant divided by hopping. Ribbons are a diagnostic representation, with schematic spacing between components.

H=\begin{pmatrix}h&D\\-D&-h\end{pmatrix},\quad h_{jj}=-\mu,\quad h_{j,j+1}=-J,\quad D_{j,j+1}=-D_{j+1,j}=\Delta_p

The real-space matrix and the bulk dispersion are separate finite and infinite models. This uniform ideal chain is topological when the pairing is nonzero and the chemical potential is smaller in magnitude than twice the hopping. A spectral peak near zero is not, on its own, a general test for materials.

Sources: Kitaev’s original wire model and the BdG formulation in the Kwant superconductivity tutorial.

Start with the essentials

Focus question: Why does a superconducting wire with a gapped interior show zero-energy states at its ends?
One-sentence intuition: In the ideal topological wire, the bulk keeps its energy gap, yet modes near zero energy sit at the two ends.

Core mathematical model

Electron and hole amplitudes

i\hbar\partial_t\begin{pmatrix}u\\v\end{pmatrix}=H_{\mathrm{BdG}}\begin{pmatrix}u\\v\end{pmatrix}

These are two components of a quasiparticle, not independent physical paths.

Infinite-chain spectrum

E(k)^2=(\mu+2J\cos k)^2+4\Delta_p^2\sin^2 k

The bulk gap closes when the magnitude of the chemical potential equals twice the hopping. A finite chain still has a discrete spectrum.

Common difficulties

What the model can and cannot claim

Typical misconceptionAny peak at zero bias proves there is a Majorana particle.

Better mental modelThe model is the ideal spinless p-wave Kitaev chain with open ends and fixed pairing. A local peak at zero energy is not, on its own, proof of Majorana modes in a material. Braiding, qubit operations, disorder, interactions and experimental identification are not simulated. The spectra are broadened by a stated Lorentzian resolution.

The ribbons are not particle paths

Typical misconceptionThe raised ribbons are paths followed by individual particles.

Better mental modelThe ribbon height shows a probability or spectral density, as labeled. Both colors are plotted against the same lattice-site coordinate.

Run the experiment

01
Make a prediction
Compare the three presets, then change one continuous control.
What to observe: The end peak disappears when the wire becomes topologically trivial.
02
Find the cause
Move the probe from an end to the center, then scan the energy across the gap.
What to observe: The local spectral weight at the ends differs from that in the bulk.
03
Collect data
Collect repeated records, inspect the histogram and interval, then export the complete CSV.
What to observe: The model curve is shown separately as a reference; the gold histogram is built only from recorded events.