--- title: "Majorana End Modes Interactive Lab | Sandbox Physics" description: "A gapped wire. Two luminous ends. Move a spectral probe along a superconducting chain. Find the near-zero-energy weight at its ends, close the bulk gap, and watch the endpoint signal disappear. Resolve the finite-size spectrum and collect an ideal local energy distribution." canonical_url: "https://sandboxphysics.com/labs/kitaev-majorana-end-modes/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/kitaev-majorana-end-modes/" last_modified: "2026-10-18" --- KITAEV WIRE / LOCAL SPECTRAL SCANNING # Majorana End Modes Move a spectral probe along a superconducting chain. Find the near-zero-energy weight at its ends, close the bulk gap, and watch the endpoint signal disappear. Resolve the finite-size spectrum and collect an ideal local energy distribution. > Interactive simulation: open this page in a web browser to change the parameters and run the model. The ideal spinless p-wave Kitaev chain, with open ends and prescribed pairing. A local zero-energy peak alone does not establish Majorana modes in a material. No braiding, qubit operation, disorder, interactions or experimental identification is simulated. Spectra use a declared Lorentzian resolution. A SPECTRAL PORTRAIT OF A QUANTUM WIRE LOWEST EXCITATION ENERGY $0$ PROBE AT THE GOLD MARKER $0$ CYAN · ELECTRON CORAL · HOLE HEIGHT · 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 $j=1$ $0$ 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 · linear values in the charts 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. Change the phase. Analytic positive and negative bands of the infinite uniform chain. This is a reference, not a reconstruction of the local records. $k\ \longrightarrow\quad E/J$ TURN THE MODEL INTO RECORDS ## Collect a local spectrum. 0 / 8000 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 $\text{—}$ 95% Wilson interval · shot noise $\text{—}$ Model probability $\text{—}$ All-eigenpair residual $\text{—}$ ### Records in gold. Model in cyan. 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 bulk dispersion are separate finite and infinite models. The topological regime of this uniform ideal chain has nonzero pairing and chemical potential magnitude below twice the hopping. A near-zero spectral peak alone is not a general material diagnostic. Sources: Kitaev’s original wire model and the BdG formulation in the Kwant superconductivity tutorial. - [Kitaev · Unpaired Majorana fermions in quantum wires](https://arxiv.org/abs/cond-mat/0010440) - [Kwant · Superconductors and electron–hole degrees of freedom](https://kwant-project.org/doc/latest/tutorial/superconductors) Physics tutorial ## Majorana End Modes **Background** Move a spectral probe along a superconducting chain. Find the near-zero-energy weight at its ends, close the bulk gap, and watch the endpoint signal disappear. Resolve the finite-size spectrum and collect an ideal local energy distribution. **Why it matters** Connect a visible change to the spectrum of a superconducting system. Start with the essentials - **Focus question**: A gapped wire. Two luminous ends. - **One-sentence intuition**: The ideal topological wire has near-zero modes at its boundaries while its bulk remains gapped. ### 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 at chemical potential magnitude twice the hopping. A finite chain retains a discrete spectrum. ### Common difficulties #### Respect the scope **Typical misconception** Any zero-bias peak proves a Majorana particle. **Better mental model** The ideal spinless p-wave Kitaev chain, with open ends and prescribed pairing. A local zero-energy peak alone does not establish Majorana modes in a material. No braiding, qubit operation, disorder, interactions or experimental identification is simulated. Spectra use a declared Lorentzian resolution. #### A diagnostic is not a photograph **Typical misconception** The raised ribbons are paths followed by individual particles. **Better mental model** Ribbon height encodes a declared probability or spectral density. The two colors share the same physical site coordinate. ### Run the experiment 1. 01 #### Make a prediction Compare the three presets, then change one continuous control. **What to observe:** The endpoint peak vanishes as the wire becomes trivial. 2. 02 #### Inspect the cause Move the probe from an end to the center; scan energy across the gap. **What to observe:** Local spectral weight is not the same at the boundary and in the bulk. 3. 03 #### Acquire evidence Collect repeated records, inspect the histogram and interval, then export the complete CSV. **What to observe:** The model curve is a separate reference. Only recorded events build the gold histogram. Physical conclusion The ideal spinless p-wave Kitaev chain, with open ends and prescribed pairing. A local zero-energy peak alone does not establish Majorana modes in a material. No braiding, qubit operation, disorder, interactions or experimental identification is simulated. Spectra use a declared Lorentzian resolution. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. 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