Quantum Spin Hall & Helical Edge States (Z₂ Topological Insulator)

The quantum spin Hall insulator has a Z₂ topological invariant ν=1. Its bulk is gapped, but its edges host pairs of counter-propagating states: spin-up moves right, spin-down moves left (helical). Time-reversal symmetry forbids backscattering by non-magnetic impurities — Kramers theorem protects the edge states.

Z₂ = 1 — Topological

Model Parameters (BHZ)

Mass M/B: -1.0 Spin-orbit coupling A: 1.0 Magnetic perturbation (breaks TR): 0.0

BHZ model: H(k) = M(k)σ_z + A·k_x·σ_x + A·k_y·σ_y. M(k) = M - Bk². Topological phase: M/B < 0 (or 0 < M/B < 4 for lattice). Edge states form helical Dirac cone at k=0.