# Topology change from a monopole to a dipole in Berry's phase

## TL;DR

This paper analyzes the smooth topological transition of Berry's phase from a monopole to a dipole configuration in an exactly solvable model, revealing novel displacement features and the role of gauge invariance in the topology change.

## Contribution

It introduces a model showing the topology change of Berry's phase from monopole to dipole, highlighting displacement of the monopole center and Dirac string, and the emergence of half-monopoles.

## Key findings

- Berry's phase topology changes smoothly from monopole to dipole.
- Displacement of monopole center and Dirac string in parameter space.
- Half-monopole appears with observable Dirac string during transition.

## Abstract

The smooth topology change of Berry's phase from a Dirac monopole-like 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 ${\cal A}_{k}$ is that the geometrical center of the monopole-like configuration and the origin of the Dirac string are displaced in the parameter space. Gauss' theorem $\int_{S}(\nabla\times {\cal A})\cdot d\vec{S}=\int_{V} \nabla\cdot (\nabla\times {\cal A}) dV=0$ for a volume $V$ which is free of singularities shows that a combination of the monopole-like configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge $e_{M}=2\pi\hbar$ 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.