Quantum Spin Hall Effect — Helical Edge States

Z₂ topological insulator: spin-↑ and spin-↓ edge modes counterpropagating

1.50
0.80
0.00
H(k) = A·k·σₓ + (M − Bk²)σᵤ + λ·σᵧ·sᵤ  |  Z₂ = 1 when M/B > 0 (topological phase)
Phase: Topological | Z₂ = 1 | Gap = 3.00
The Quantum Spin Hall insulator (Kane-Mele 2005, BHZ model) has a Z₂ topological invariant. When M > 0 the bulk gap is topological: helical edge states cross the gap — spin-↑ travels right, spin-↓ travels left. Time-reversal symmetry protects them from backscattering by non-magnetic disorder.