suppose the spin is initially aligned with the magnetic field, i.e. the particle is in an eigenstate of the system’s Hamiltonian. Then the spin will remain aligned with the magnetic field vector (the particle remains in the instantaneous eigenstate) when the magnetic field vector is made to rotate adiabatically. Upon completion of some closed trajectory, the final state of the particle differs from its initial one by an overall phase factor, which is part dynamical, part geometric in origin. Now, it turns out that the geometric phase is in fact equal in size to precisely half the solid angle subtended at the origin by the magnetic field vector’s trajectory in parameter space. This result not only serves to highlight in a particularly acute way the geometric character of Berry’s phase, but it also gives an indication of the origin of Berry’s phase: the spin’s state is degenerate when the magnetic field is switched off, and the geometric phase can thus be interpreted as providing a measure of the view of the circuit as seen from that point of degeneracy at the origin of the parameter space.

In fact, in any general setting, the emergence of geometric phases can always be 