Geometric Phases

Berry phase on the Bloch sphere — parallel transport around a loop acquires geometric holonomy

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Berry phase γ (rad)

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Solid angle Ω

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Loop colatitude θ

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Phase shift (degrees)

Colatitude θ: 60° Loop radius: 40° Animation: speed

The Berry phase γ = -Ω/2 is a purely geometric quantity — half the solid angle enclosed by the loop on the Bloch sphere. It is independent of the rate of traversal. This holonomy underlies the quantum Hall effect (Chern number = TKNN invariant) and topological insulators.