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Physical review. B./Physical review. B

Aharonov-Casher phase: Considerations regarding force, time dependence, and Berry phase

Igor Kuzmenko, Y. B. Band, Y. Avishai

Published Nov 19, 2025• 1 citation
Publisher

Abstract

The relation of the Aharonov-Casher (AC) effect and the force on a particle having a magnetic moment is explored. The general form of the AC Hamiltonian is derived using the Foldy-Wouthuysen transformation to the Dirac equation. Geometries in which an analytic expression for the phase can be obtained are examined, as well as the relation of the AC phase to the Berry phase. The AC phase is determined for an arbitrary homogeneous electric field; it is quadratic (linear) in the field strength for small (large) electric field strengths.

DOI
10.1103/r1f4-6tfp

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The relation of the Aharonov-Casher (AC) effect and the force on a particle having a magnetic moment is explored. The general form of the AC Hamiltonian is derived using the Foldy-Wouthuysen transformation to the Dirac equation. Geometries in which an analytic expression for the phase can be obtained are examined, as well as the relation of the AC phase to the Berry phase. The AC phase is determined for an arbitrary homogeneous electric field; it is quadratic (linear) in the field strength for small (large) electric field strengths.
The relation of the Aharonov-Casher (AC) effect and the force on a particle having a magnetic moment is explored. The general form of the AC Hamiltonian is derived using the Foldy-Wouthuysen transformation to the Dirac equation. Geometries in which an analytic expression for the phase can be obtained are examined, as well as the relation of the AC phase to the Berry phase. The AC phase is determined for an arbitrary homogeneous electric field; it is quadratic (linear) in the field strength for small (large) electric field strengths.
In this paper we give a covariant expression for Aharonov–Casher phase. This expression is a combination of the canonical electric field, Aharonov–Casher phase plus a magnetic field phase shift. We use this covariant expression for the Aharonov–Casher phase to investigate the case of a neutral particle with a non-zero magnetic moment moving in the time dependent electric and magnetic fields of a plane electromagnetic wave background. We focus on the case where the magnetic moment of the particle is oriented so that both the electric and magnetic fields lead to non-zero phases, and we look at the interplay between these electric and magnetic phases.
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).
The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two special orientations of the magnetic moment (same as in the quantum case) the motion is effectively free and similar to the AB motion of a charged particle. Thus, the AC effect is not really topological. Presence of higher multipoles totally destroys the possible AB-AC duality.
We provide arguments which demonstrate that the Berry phase effects involving particle interference in the presence of external electric and magnetic fields—namely the Aharonov–Bohm (AB), Aharonov–Casher (AC) and the Röntgen (R) effects—cannot be unambiguously explained (as has been controversially argued) through the use of any classical mechanical force. In fact, in the three scenarios which we consider, the only means to reclaim the standard quantum mechanical results is to use the expressions for the force that fail to satisfy the necessary conditions of gauge invariance. Our arguments explain the foundation of the controversy and confirm the standard idea that the Berry phase effects in question are non-local phenomena which only arise in the application of quantum theory. We lay special emphasis on the use of the concept of a canonical force, this being the quantity which ensures the agreement between the classical and quantum formulations. It is shown that if the Aharonov–Bohm effect is interpreted as being due to a force, through the use of the Bohmian formulation of quantum mechanics, then this force is equal in magnitude to the canonical force.
The Aharonov-Casher (AC) phase is calculated in relativistic wave equations of spin 1. The AC phase has previously been calculated from the Dirac-Pauli equation using a gauge-like technique. In the spin-1 case, we use Kemmer theory (a Dirac-like particle theory) to calculate the phase in a similar manner. However, the vector formalism, the Proca theory, is more widely known and used. In the presence of an electromagnetic field, the two theories are `equivalent' and may be transformed into one another. We adapt these transformations to show that the Kemmer theory results apply to the Proca theory. Then we calculate the Aharonov-Casher phase for spin-1 particles directly in the Proca formalism.