When a quantum system is carried slowly around a closed loop in the space of its Hamiltonian's parameters, its state returns with an extra phase that depends only on the geometry of the loop, not on how long the trip took or on any force acting along the way. A student learns to read this phase as a holonomy: wavefunctions are sections of a vector bundle over the parameter space, the Berry connection defines parallel transport between neighbouring Hilbert spaces, and the phase acquired around a closed path is the integral of the Berry curvature over the enclosed surface, quantized by Chern numbers when the surface is closed. The same construction explains the Aharonov-Bohm phase of an electron encircling shielded magnetic flux, the Pancharatnam phase of light cycled around the Poincare sphere, the Zak phase of Bloch electrons crossing the Brillouin zone, the quantized Hall conductance, the modern theory of electric polarization, and anyonic exchange statistics in two dimensions.
A geometric phase is the extra phase factor a quantum state acquires when the parameters of its Hamiltonian are carried slowly around a closed loop: a phase fixed by the geometry of the path rather t…