Observable Dirac-type singularities in Berry's phase and the monopole

A three-dimensional generalization of the sign-change ( π phase shift) rule for adiabatic cycles of spin-1/2 or two-state wave functions encircling a degeneracy in the parameter space of the Hamiltonian yields a Dirac-type singularity wherein any closed circuit of the adiabatic cycle in which the degeneracy is “looped” results in an observable ±2π phase shift. It is concluded that an interferometer loop similarly taken around a magnetic monopole of strength n/2 yields an observable ±2nπ phase shift, n being an integer