Majorana Zero Modes in a Topological Nanowire

Topological Qubit

— 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

Bulk gap E_gap/t

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Localization ξ (sites)

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Winding invariant

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γ_L–γ_R overlap

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How it works

The Kitaev chain is the minimal model of a 1D topological p-wave superconductor / spinless nanowire with proximity-induced pairing:

H = -μΣc_j†c_j - Σ[t(c_j†c_j+1+h.c.) + Δ(c_jc_j+1+h.c.)]

Writing each ordinary fermion as two Majorana operators, c_j = (γ_A,j+iγ_B,j)/2, the on-site term pairs γ_A,j with γ_B,j (strength ∝ μ) while hopping+pairing links γ_B,j to γ_A,j+1 of the next site (strength ∝ t+Δ). At the solvable point μ=0, Δ=t, every Majorana pairs with its neighbor across the site boundary, leaving γ_A,1 and γ_B,N completely unpaired — two spatially separated, exactly zero-energy Majorana modes.

More generally the chain is topological whenever |μ| < 2t (with Δ≠0). The end-mode decay is governed by the roots of

(t+Δ)λ² + μλ + (t-Δ) = 0

with localization length ξ = -1/ln|λ_dominant|; the bulk excitation gap is min_k √[(2t cos k - μ)² + 4Δ²sin²k], plotted live above.

μ, Δ, t sliders — tune the Hamiltonian; bond brightness in the 3D view directly shows the on-site (μ) vs inter-site (t+Δ) Majorana pairing strength computed from these values.
N slider — chain length; a shorter wire lets the two end modes overlap and split away from exactly zero energy even in the topological phase.
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, illustrating why local noise can't dephase it.

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.

Majorana γ_A Majorana γ_B End zero mode

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