Topology change from a monopole to a dipole in Berry’s phase

Abstract

The smooth topology change of Berry's phase from a Dirac monopolelike configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection ${\mathcal{A}}_{k}$ is that the geometrical center of the monopolelike configuration and the origin of the Dirac string are displaced in the parameter space. Gauss's theorem ${\ensuremath{\int}}_{S}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{S}={\ensuremath{\int}}_{V}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})dV=0$ for a volume $V$ which is free of singularities shows that a combination of the monopolelike configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge ${e}_{M}=2\ensuremath{\pi}\ensuremath{\hbar}$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.