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The non-Abelian phase, computed end to end

Kitaev honeycomb + field: the Ising-anyon phase — Chern numbers, a chiral Majorana edge, vortex-bound zero modes — computed exactly, then braided numerically with a clean null control.

What was measured

Adding the field-induced term κ to the Kitaev model opens its non-Abelian phase. In the exact free-Majorana (BdG) solution: the B phase gaps into Chern number ±1 (sign following the field), a cylinder hosts exactly one chiral Majorana edge branch (the c = 1/2 signature), and vortices bind Majorana zero modes — the carriers of Ising-anyon statistics. The zero modes were then braided: parallel transport of the 4-mode subspace along vortex-exchange paths, with the holonomy read out in the vortex-localized basis.

Headline numbers

• Analytic anchor: gap at κ=0.1 equals Kitaev's 6√3·κ to 4 digits; Chern +1/0/−1 exactly across the phase diagram
• Edge: 88 in-gap cylinder states, 100% edge-localized vs 0 in the trivial phase; velocity ±1.83 flips with the field sign
• Vortex zero modes: E₀ down to 3×10⁻³ vs bulk gap 2.0, mode weight ~1.9/2.0 on the two vortices; toric control: nothing
• Braiding: exchanges sharing a vortex give ‖M−I‖ = 2.57 matching the predicted Ivanov pattern at 0.975; disjoint exchanges commute to 0.005 — a ~500× contrast

Method & discipline

Self-diagnosing construction: a wrong sign prescription would read Chern 0, and the flux-counting assertion guarantees exactly two vortices. The commutator-loop observable is gauge-free (the gauge field returns exactly to its start). Two build bugs were caught and corrected by the controls — the vortex string direction and the zero-mode subspace dimension — and are documented in the run log.

Part of a ~30-experiment program run under one hard rule: no result is recorded before its run completes. Methods and logs: github.com/dwatces.