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arXiv:2411.04589 · Physical review. B./Physical review. B

The Aharonov-Casher phase is geometrical and not topological

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

Published Jun 17, 2025• 5 citations
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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).

DOI
10.1103/rm8w-8rmm

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Modern Physics Letters B•Feb 10, 2006•0 citations
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.
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.
We consider two two-dimensional (2D) electronic systems in the presence of a perpendicular homogeneous electric field that generates a Rashba spin-orbit interaction (RSOI): a system of non-interacting electrons in a 2D conductor, modeled using the 2D Schrödinger equation (SE), and a single-layer graphene system, modeled using a 2D Dirac equation (DE) for massless fermions. In both cases the RSOI is expressed via an SU(2)SU(2)SU(2) Rashba vector potential AR{\bf A}_{R}AR​. We demonstrate that AR{\bf A}_{R}AR​ cannot be eliminated from either the 2D SE or the 2D DE via a gauge transformation. Nevertheless, for a plane wave solution, an SU(2)SU(2)SU(2) matrix exists that eliminates AR{\bf A}_{R}AR​ from the resulting 1D SE. This unitary matrix is an Aharonov-Casher (AC) phase factor, and facilitates the calculation of the AC phase in the Schrödinger scheme. The plane wave solution for the DE contains two components of AR{\bf A}_{R}AR​: AR,kA_{R, k}AR,k​ in the direction of the wave vector k{\bf k}k, and AR,nA_{R, n}AR,n​ normal to k{\bf k}k. The latter generates an effective electron mass that cannot be eliminated from the DE. The former generates an AC phase that can be eliminated by a time-dependent unitary transformation. Thus, the Dirac AC phase is time-dependent, i.e., it is a dynamical phase. This is in contradistinction to the Schrödinger AC phase which is geometrical.
We have studied the spin motive force induced by the time-dependent nonadiabatic Aharonov–Casher (AC) phase. It has been shown that the spin motive force has the new term explicitly dependent on the Aharonov–Anandan (AA) phase. This AA phase dependent term could be obtained both in the canonical and path-integral formulations. In path-integral formulations this AA dependent term originates from the nontrivial path-ordering nature of a non-Abelian spin gauge group. We also have shown that this term can be understood as the rotating frame effect in the classical equation of motion for the spin angular momentum. With this term the duality between the Aharonov–Bohm and the AC effect reaches the Newton's third law in the classical limit.