In quantum mechanics, many of the most surprising phenomena result from the fact that particles can behave like waves. One such wave phenomenon is the Berry phase.
This phase, also known as Pancharatnam-Berry or geometric phase, was named after Sir Michael Berry. It arises in many different kinds of wave systems (particularly in quantum mechanics because of wave-particle duality, but also e.g. in classical optics), when one can externally control at least two parameters affecting the wave. Waves are characterized by amplitude and phase, and both may vary as a function of said parameters.
The Berry phase is the case in which both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could e.g. involve rotations but also translations of particles, which are apparently undone at the end. Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursion is a cyclic loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases.
This phase difference is called Berry phase, and its occurrence typically indicates that the system's parameter dependence is singular (undefined) for some combination of parameters.
To measure the Berry phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle.
For the connection to mathematics, see curvature tensor.
