Majorana Zero Modes in a Topological Nanowire (2D)

Topological Qubit (2D)

— computing phase —

Kitaev Chain Parameters

Chemical potential μ / t 0.00

Pairing Δ / t 1.00

Hopping t 1.00

Chain length N 14

Live Readouts (diagonalized)

Lowest |E| / t (BdG)

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Bulk gap E_gap/t

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Localization ξ (sites)

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Winding invariant

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End-weight fraction

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γ_L–γ_R overlap

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How it works

This is the same Kitaev chain as the 3D version, but instead of reading the phase off a closed-form root formula, this engine builds the real 2N×2N Bogoliubov–de-Gennes Hamiltonian in the Nambu basis Ψ=(c₁..c_N,c₁†..c_N†):

H_BdG = [ h   Δ ]
        [-Δ  -h ]
h_jk = -μδ_jk - t(δ_j,k+1+δ_j,k-1)
Δ_jk = Δ(δ_j,k+1 - δ_j,k-1)

Because t, μ, Δ are real, this matrix is real and exactly symmetric, so it is diagonalized in-browser with the classical cyclic Jacobi eigenvalue algorithm — no library, no lookup table. Its 2N real eigenvalues always come out in exact ±E pairs (particle-hole symmetry (u,v)↔(v,u)); the chart plots every one of them, and the bar plot below it is the site-resolved probability density |u_j|²+|v_j|² of whichever eigenvector has the smallest |E|.

Whenever |μ| < 2t and Δ≠0 that mode's energy is numerically pinned near zero and its weight piles up at the two chain ends — a genuinely computed, not scripted, signature of the topological phase. Push |μ| past 2t (or set Δ=0) and the same diagonalization instead produces a bulk-delocalized mode at finite energy.

μ, Δ, t sliders — rebuild H_BdG and re-diagonalize every time they move.
N slider — chain length; the matrix re-diagonalized is 2N×2N. A shorter wire lets the two end modes overlap and split away from exactly zero energy even inside the topological region.
Measure fermion parity — the two far-separated end Majoranas combine into one non-local fermion d=(γ_L+iγ_R)/2. Its occupation number (0 or 1) is the topological qubit; measuring it collapses that single shared quantum number at both ends at once.

This is the mechanism behind proposed topological qubits in InSb/InAs nanowires with induced superconductivity (Microsoft, Delft, and others) — information stored non-locally in Majorana zero modes is, in principle, protected from local perturbations.

Near-zero mode density Excited BdG levels ±E pair (particle-hole)