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Physical Review A

Equivalence between Aharonov-Bohm and Aharonov-Casher effects, and motive forces

Sangchul Oh, C. M. Ryu, S. H. Suck Salk

Published Dec 1, 1994• 20 citations
Publisher

Abstract

Based on a gauge-theoretic approach, an equivalence between the Aharonov-Bohm effect of a solenoid and the Aharonov-Casher effect of a charged rod is explicitly demonstrated. It is shown that a time-dependent Aharonov-Casher phase can induce a motive force via the SU(2${)}_{\mathrm{spin}}$ gauge field. The intrinsic non-Abelian nature of the SU(2${)}_{\mathrm{spin}}$ gauge field predicts the non-Abelian Faraday law, subject to further experimental verification.

DOI
10.1103/physreva.50.5320

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Based on a gauge-theoretic approach, an equivalence between the Aharonov-Bohm effect of a solenoid and the Aharonov-Casher effect of a charged rod is explicitly demonstrated. It is shown that a time-dependent Aharonov-Casher phase can induce a motive force via the SU(2)spin{)}_{\mathrm{spin}})spin​ gauge field. The intrinsic non-Abelian nature of the SU(2)spin{)}_{\mathrm{spin}})spin​ gauge field predicts the non-Abelian Faraday law, subject to further experimental verification.
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.
We briefly review the theoretical formulations and applications of the Aharonov--Bohm effect and the Aharonov--Casher effect with emphasis on mesoscopic physics. Topics relating to the Aharonov--Bohm effect include: locality, periodicity, non-integrable phase factors, Abelian gauge theory, interference, the spectrum and persistent current of electrons on a ring pierced by a magnetic field, Onsager reciprocity relations, and Aharonov--Bohm interferometer. Topics relating to the Aharonov--Casher effect include: a magnetic dipole in an electric field, locality, periodicity, non-Abelian gauge invariance, SU(2) non-integrable phase factors, spin-orbit coupling, Pauli equation, Rashba Hamiltonian, Aharonov--Casher interferometer, conductance and polarization in two-channel systems due to the Aharonov--Casher effect.
Whether the time-dependent Aharonov-Bohm (AB) effect even exists or not has been the subject of long-standing debate. There are two factors complicating the problem. First, in the closed spacetime line integral of the vector potential that is thought to give the AB-phase shift, how to treat the time-varying vector potential is highly nontrivial. Second, the time-varying magnetic flux generates induced electric field even outside the solenoid. In the present paper, motivated by a recent work by Gao, we re-investigate the role of the induced electric field with the utmost care. This analysis reveals a highly nontrivial effect of the induced electric field, which turns out to be useful for verifying the very existence of the time-dependent AB-effect.
We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.