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arXiv:2302.06300

Aharonov--Bohm and Aharonov--Casher effects in meso-scopic physics: A brief review

Y. Avishai, Y. B. Band

Published Feb 13, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2302.06300

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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.
We establish systematic consolidation of the Aharonov–Bohm (AB) and Aharonov–Casher effects including their scalar counterparts. Their formal correspondences in acquiring topological phases are revealed on the basis of the gauge symmetry in non-simply-connected spaces and the adiabatic condition for the state of magnetic dipoles. In addition, investigation of basic two-body interactions between an electric charge and a magnetic dipole clarifies their appropriate relative motions and discloses physical interrelations between the effects. Based on the two-body interaction, we also construct an exact microscopic description of the AB effect, where all the elements are treated on equal footing, i.e., magnetic dipoles are described quantum-mechanically and electromagnetic fields are quantized. This microscopic analysis not only confirms the conventional (semiclassical) results and the topological nature but also allows one to explore the fluctuation effects due to the precession of the magnetic dipoles with the adiabatic condition relaxed.
ABSTRACT\nThe interaction of charged and uncharged particles with intrinsic magnetic moment for the latter with an electromagnetic field in the microworld brings out the actual role of the electromagnetic potentials as emphasized in quantum theory which generate electromagnetic interaction via gauge invariance requirement.\nIn the Aharonov-Bohm (A-B) effect (dealing with two electron coherent beams passing a double slit system), the resulting interference pattern is shifted by the insertion of a very long solenoid between the two electron beams with an influence described by the magnetic vector\npotential A on the electron wave functions. Aharonov and Casher presented the complement of the Aharonov-Bohm effect involving the role of the electric scalar potential co on a beam of neutrons each carrying an internal magnetic dipole moment. A similar shift in the interference pattern is found involving two coherent neutron beam interference, where the solenoid in the A-B arrangement is replaced by an electric line charge, and the electrons by polarized neutrons. The effective Hamiltonian describing the two effects is used to determine the interference pattern.\nThe experiment to verify the existence of the Aharonov-Bohm effect which have been performed using an electron holography with a small toroidal ferromagnetic substance confining a magnetic flux by Tonomura in 1982 is discussed. The Aharonov-Casher effect, which was conducted using a Bonse-Hart single-crystal neutron interferometer by A. Cirnmino in 1989, is also discussed. Both experimental results lead to the same phenomenon: a phase shift in the observed pattern interference quantitatively explained as caused by the influence of the proper component of the electrodynamic potentials.\n\nKey Word: Kuantum, Aharonov- cahser
The Aharonov-Casher effect is a quantum interference arising in mesoscopic rings involving particles with magnetic moments. The wave function acquires a phase due to an electric-field flux F enclosed by the ring and gives rise to flux-dependent oscillations in persistent spin currents. In the ground state of the spin-1/2 Heisenberg chain the period and amplitude of the oscillations are associated with the properties of the Fermi surface of the elementary excitations of the spin chain (spinons). We consider an antiferromagnetic spin-1/2 Heisenberg ring with a concentration x of impurity spins S. The calculations are performed within the framework of Bethe's ansatz. Due to the finite concentration of impurities the ground state consists of two (rather than one) Dirac seas, corresponding to spinons and 2S strings of rapidities. The persistent spin current reveals two periods of oscillation associated with the properties of the two Fermi seas. The interference pattern as a function of F is discussed for several external magnetic fields.
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 investigate anomalous oscillations due to the Aharonov-Bohm(AB) and Aharonov-Casher(AC) effects of the one-dimensional Hubbard ring with flux in the strong coupling limit. By using the exact diagonalization method and the Shiba transformation, we examine the energies of the ground state and few excited states in the presence of the flux producing the AB or AC effect, where the transformation not only reverses the sign of the interaction U, but also exchanges the role between the AB and AC effects in the model Hamiltonian. We classify the number of oscillation of the ground state as a function of the flux systematically and clarify the close relationship between the AB and AC effects. For example, it is shown that the number of the periods are given by N_L - N_e(N_L - N_↑ + N_↓) for the AB(AC) effect in the very strong attraction, where N_L, N_e, N_↑, and N_↓ are the system size, the number of the total electrons, the number of electrons with up-spin, and the number of electrons with down-spin, respectively, in the condition of N_L > N_e > N_↓ > N_↑. More special case, such as the half-filled band and spin balanced case (N_L = N_e and N_↓ = N_↑ ), we find the periodicity of the system to be 0(2) for the AB(AC) effect in the very strong repulsion. These results show us the nature of interesting phenomena originated by the interplay between the strong correlation and the quantum interference effect in the mesoscopic ring.