Topological Insulator — Surface States & Dirac Cone

Quantum Topology at the Surface

A topological insulator is bulk-insulating but hosts metallic surface states protected by time-reversal symmetry. These states form a Dirac cone — a conical band crossing at the Γ point where the Fermi level intersects.

The surface electrons are spin-momentum locked: the spin direction is perpendicular to the momentum, winding once around the Fermi surface (Z₂ invariant ν=1).

Backscattering is forbidden by topology — impurities cannot scatter electrons backward without flipping spin.

E(k) = ±ℏvF|k| · √(1 + λk²/vF²)
Spin texture: ⟨σ⟩ ⊥ k (helical)