When a quantum system's Hamiltonian is slowly (adiabatically) transported around a closed loop in parameter space, the eigenstate acquires a geometric phase γ = i∮⟨n|∇_R|n⟩·dR, independent of how fast the loop is traversed.
For a spin-1/2 in a slowly rotating magnetic field, the Berry phase equals half the solid angle subtended by the loop on the Bloch sphere: γ = -Ω/2.
This holonomy is measurable via interference and underlies topological phenomena: the quantum Hall effect, topological insulators, and the Aharonov-Bohm effect.
The Berry connection A = i⟨n|∂/∂R|n⟩ is the geometric vector potential; its curl is the Berry curvature (analogous to magnetic field).