The Aharonov-Casher phase is geometrical and not topological

TL;DR

This paper clarifies that the Aharonov-Casher phase is a geometric phase dependent on the particle's path details, not a topological invariant, with examples illustrating both path-dependent and path-independent cases.

Contribution

It provides a counterexample demonstrating the path dependence of the AC phase and distinguishes between Abelian and non-Abelian cases regarding topological properties.

Findings

AC phase depends on the path details in general

Paths with Abelian AC phase can be path-independent

Paths with non-Abelian AC phase may be path-dependent

Abstract

It is demonstrated that the Aharonov-Casher (AC) phase is a geometric phase that, in general, depends on the details of the closed path taken by a particle with a magnetic moment that is subject to an electric field. Consequently, it is not a topological phase. The proof of this statement is obtained by developing a counterexample that elucidates the dependence of the AC phase on the details of the path. Furthermore, we demonstrate that, in the particular example considered here, paths having an Abelian AC phase factor, also have an AC phase that is path-independent, whereas paths having a non-Abelian AC phase factor may have an AC phase that is path-dependent (i.e., not topological).