Note on Dirac monopole theory and Berry geometric phase

TL;DR

This paper establishes a fundamental connection between Dirac monopoles and Berry geometric phases by extending monopole theory into parameter space, providing a unified approach to understanding geometric phases and monopoles in quantum systems.

Contribution

It introduces a novel framework linking Dirac monopoles to Berry phases through explicit visualization and rigorous derivation, especially in systems where Berry's framework is inadequate.

Findings

Dirac strings with endpoints correspond to degeneracy points in parameter space.

Berry connection and curvature arise from non-integrable phase factors induced by Dirac strings.

The approach unifies monopole theory and geometric phases, applicable beyond Berry's original framework.

Abstract

This work reveals the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be derived rigorously via Dirac's monopole framework. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. This establishes a unified and effective approach to study monopoles and geometric phases, particularly applicable when Berry's framework fails.