Majorana Fermions — Kitaev Chain

Topological edge modes & phase diagram

Phase: TOPOLOGICAL — Majorana edge modes present
Kitaev's 1D p-wave superconductor hosts Majorana zero modes at its ends when in the topological phase. The Hamiltonian (in Majorana basis): H = iΣ[t·γ_{2j}γ_{2j+1} + (t+Δ)γ_{2j-1}γ_{2j+2}/2 - (μ/2)γ_{2j-1}γ_{2j}]
Topological phase: |μ/t| < 2, any Δ ≠ 0
Pairs of Majorana operators (γ₂ⱼ₋₁, γ₂ⱼ) represent each physical site. In the topological phase, γ₁ and γ₂ₙ remain unpaired — they are the non-local edge modes. These modes are topologically protected and are candidates for fault-tolerant quantum computing.