Berryphase

Berryphase, a scaled dragon-knight, coils a glowing violet helix beside him.

The Physics

Carry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ whose magnetic field direction is swept round a cone, the geometric phase is half the solid angle the field traces out.

|γ| = Ω/2

• γ: geometric (Berry) phase (rad)
• Ω: solid angle traced by the field direction (sr)

Classical cousins include the Foucault pendulum's precession and parallel transport on a sphere. Berry phases underlie the quantum Hall effect and topological materials.

Honest caveat: The ½ is for spin-½; a state with spin projection m picks up −mΩ. The adiabatic limit and a non-degenerate state are assumed.

On the sticker

Berryphase stands beside a glowing violet helix of light. Printed text: BERRYPHASE with the line GAMMA = OMEGA/2 (γ = Ω/2 spelled out; it gives the size of the phase, and the sign depends on whether the spin is aligned or anti-aligned with the field: γ = ∓Ω/2). Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.