Geometric Phase, Bundle Classification, and Group Representation

TL;DR

This paper explores the deep connections between geometric phase, bundle classification, and group representation theory, introducing topological charges for Lie groups and relating non-adiabatic phases to Berry's adiabatic phase.

Contribution

It establishes a link between geometric phase and the classification of complex line bundles, and introduces a method to relate non-adiabatic phases to Berry's phase for cranked Hamiltonians.

Findings

Topological charges for compact semisimple Lie groups are defined.

A procedure to reduce non-adiabatic phase to Berry's phase is proposed.

Relation between parameter space geometry and Berry's connection is elucidated.

Abstract

The line bundles which arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the representation theory. The remarkable relation of the geometric phase to the classification of complex line bundles provides the necessary tools for establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry's connection. The results about the fibre bundles and group theory are used to introduce a procedure to reduce the problem of the non-adiabatic (geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-abelian monopoles is pointed out.