A geometric phase for m=0 spins

A mod jm) spin state in an adiabatically-cycled magnetic field acquires a geometric phase of m times the solid angle described by B, so that for m=0 states the geometric phase vanishes. However, if B is not cycled, but is made to reverse direction, an m=0 state returns to itself and in so doing acquires a geometric phase factor of (-1) j . This phase is of a topological character; parameter space is the real projective plane, in which the phase distinguishes trivial from non-trivial cycles.