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 in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω/2. The anti-aligned state (m = −½) picks up +Ω/2.

γ = −Ω/2

• γ: geometric (Berry) phase of the aligned state (rad)
• Ω: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr)
• m: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)

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: In general a state with spin projection m picks up γ = −mΩ (defined modulo 2π), so the sign flips between aligned and anti-aligned states. 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: the phase for the spin aligned with the field, m = +½; the anti-aligned state gets +Ω/2). Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.