Kitaev's 1D p-wave superconductor H = Σ[−t(c†c+h.c.) − μc†c + Δcc + h.c.] hosts a topological phase with Majorana zero modes at its ends when |μ| < 2t. The topological invariant is the Pfaffian Z₂ invariant. The energy spectrum shows a gap that closes at the phase boundary μ = ±2t.
Kitaev Hamiltonian (k-space):
H(k) = (−2t cos k − μ)τᶻ + 2Δ sin k τʸ
Topological phases:
|μ| < 2t : Topological (Z₂ = −1)
|μ| > 2t : Trivial (Z₂ = +1)
Gap closes at:
μ = ±2t (phase transition)
Majorana modes:
γ₁ = c + c†, γ₂ = i(c − c†)
Zero-energy edge states
Pfaffian invariant:
ν = sgn[Pf H(0) · Pf H(π)]