Topological Phases

In the language of quantum mechanics a pure physical state is represented by a state vector in Hilbert space. From this state one can calculate the probability of finding the particle at a specific place with a certain spin. Furthermore, the state vector carries phase information, property that cannot be measured in an intensity measurement. However, a phase difference between states vectors is a physical quantity, giving rise to the phenomenon of interference. Phase accumulation is, at first sight, a consequence of dynamics of the state evolutions. In addition to and independent of this dynamical phase Sir Micheael V. Berry discovered in the beginning of the eighties of the last century a phase due to the geometrical (or topological) origin of the state evolution. The peculiarity of Berry’s geometric phase phase lies in the fact that it does not depend on the dynamics of the system, but purely on the evolution path of the state in parameter space.