Aharonov-Bohm effect and geometric phases -- Exact and approximate topology

TL;DR

This paper investigates the topological nature of geometric phases, specifically contrasting the trivial topology of adiabatic Berry's phase with the non-trivial topology of the Aharonov-Bohm effect, using an exactly solvable model.

Contribution

It provides a unified analysis of adiabatic and non-adiabatic geometric phases and clarifies their topological differences through an exactly solvable model.

Findings

Berry's phase topology is trivial in the model.

Aharonov-Bohm effect topology is non-trivial and exact.

Topological distinctions depend on the nature of the gauge fields.

Abstract

By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.