Aharonov-Anandan phase and topological vector potentials underlying an Akhmediev breather

We investigate the Aharonov-Anandan phase and dynamical phase of a well-known Akhmediev breather. We find that the Aharonov-Anandan phase has two different evolution forms with an identical modulation period, while the dynamical phase has only one form and its value can be time independent. To understand the two different Aharonov-Anandan phase forms, we uncover that there are three different topological vector potentials defined in various spaces by following the zeros of the density using Dirac's theory. The singularities of vector potentials are always located at the zeros of the density in each parameter space. Our analysis indicates that the root multiplicity of a zero of density is closely related to the topological charge of the vector potential. These results indicate some interesting connections among zeros of the density, singularities, topological properties, and geometric phases.