Note on Dirac monopole theory and Berry geometric phase

Abstract: We discuss the intrinsic relations between Dirac monopole theory and Berry geometric phases. We demonstrate that the existence of Dirac strings with endpoints brings non-integrable phase factors in the parameters space. We choose one of the simplest two-mode Hamilton model to visualize Dirac string and its endpoint of a wave function, based on its eigenstates. The geometric phase variation around an arbitrary circle can be calculated explicitly according to Dirac's picture, where the well-known Berry connection and curvature can be derived directly by performing Dirac monopole theory in the parameters space. The correspondence between the endpoints of Dirac strings and the accident degenerated points of eigenvalues are clearly shown for the Hermitian systems. These results suggest that Berry phase can be seen as the non-integrable phase factor induced by Dirac strings with endpoints in the parameters space, and would motivate more studies on geometric phase by performing or extending Dirac monopole theory.