Berry Phase

Parameter space loop

Cone half-angle θ: 45° Loop speed: 1.0 Show solid angle: yes

Loop angle φ0°

Cone angle θ45°

Solid angle Ω—

Berry phase γ—

γ = −Ω/2—

State |ψ⟩ phase—

γ = i∮⟨n|∇_R|n⟩·dR
For spin-½ in magnetic field: Berry phase = −Ω/2 where Ω is the solid angle subtended by the loop on parameter space (Bloch sphere).

Key insight: The phase is purely geometric — it depends only on the path, not the speed of traversal.