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arXiv:2302.07050 · Physical Review Research

Unified description of the Aharonov-Bohm effect in isotropic multiband electronic systems

R. Németh, József Cserti

Published Jun 6, 2023• 1 citation
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Abstract

We present a unified treatment of the Aharonov-Bohm (AB) effect for two-dimensional multiband electronic systems possessing isotropic band structures. We propose an integral representation of the AB scattering state of an electron scattered by an infinitely thin solenoid. Moreover, we derive the asymptotic form of the AB scattering state and obtain the differential cross section from that. We found a remarkable result, namely, that this cross section is the same for all isotropic systems and agrees with that obtained first by Aharonov and Bohm for spinless free-particle systems. To demonstrate the generality of our theory, we consider several specific multiband systems relevant to condensed matter physics.

DOI
10.1103/physrevresearch.5.023154

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We present a unified treatment of the Aharonov-Bohm (AB) effect for two-dimensional multiband electronic systems possessing isotropic band structures. We propose an integral representation of the AB scattering state of an electron scattered by an infinitely thin solenoid. Moreover, we derive the asymptotic form of the AB scattering state and obtain the differential cross section from that. We found a remarkable result, namely, that this cross section is the same for all isotropic systems and agrees with that obtained first by Aharonov and Bohm for spinless free-particle systems. To demonstrate the generality of our theory, we consider several specific multiband systems relevant to condensed matter physics.
We develop a unified treatment of the non-Abelian Aharonov-Bohm (AB) effect in isotropic multiband systems, namely, the scattering of particles on a gauge field corresponding to a noncommutative Lie group. We present a complex contour integral representation of the scattering states for such systems, and using their asymptotic form, we calculate the differential scattering cross section. The angular dependence of the cross section turns out to be the same as that obtained originally by Aharonov and Bohm in their seminal paper [Phys. Rev. 115, 485 (1959)], but this time, it depends on the polarization of the incoming plane wave. As an application of our theory, we perform the contour integrals for the wave functions explicitly and calculate the corresponding cross section for three nontrivial isotropic multiband systems relevant to condensed matter and particle physics. To have a deeper insight into the nature of the scattering, we plot the probability and current distributions for different incoming waves. This paper is a generalization of our recent results on the Abelian AB effect providing an extension of exactly solvable AB scattering problems.
The Aharonov--Bohm effect in a model system described by the generalized Schrödinger equation is considered. The scattering cross section is calculated in the standard formulation: an electron beam impinges on a long impenetrable solenoid encompassing a magnetic field. It is shown that the incident wave is scattered regardless of whether the magnetic flux through the solenoid is an integer of flux quanta whereby the scattering cross section becomes a continuously nonzero function of the total magnetic flux. The problem may find its application in investigations of the soliton-magnon interaction in the low-dimensional magnetism.
Abstract In this paper we review some aspects of the scattering Aharonov–Bohm effect and Berry’s phase. Specifically, the problem of scattering of free 2d electrons on the system of an arbitrary number of parallel, infinitely thin and infinitely long coils (non-interacting, point magnetic fluxes) is modeled as free semi-classical particle propagation on a topologically non-trivial background. We show that in this case it is possible to obtain Berry’s phase by integration. First the case of a single coil is analyzed, upon which a particular solution for the case of scattering on an arbitrary number of coils is found. Considering the solution in the momentum representation, an analytic continuation of the angle coordinate in the momentum space is introduced, in accord with the non-trivial geometry of the configuration space. Finally, some simple experiments with possible applications for quantum computing are proposed.
The non-stationary Aharonov-Bohm effect (scattering of electron in the field of a narrow solenoid with alternating current) is considered. Using the eikonal approximation, the wave function of electron, the differential and total scattering cross sections are found. Unlike the case of direct current, the total cross section in the case of alternating current turns out to be finite. An oscillating asymmetry in the differential scattering cross section is discovered. The possibility of experimental observation of the effect is discussed.
We briefly review the scattering theory for the Schrödinger operator related to the Aharonov-Bohm effect, by exhibiting some explicitly solvable models; an infinitesimally thin solenoid in mathbb{R}^{2}, two infinitesimally thin solenoids with the quantized magnetic fluxes in mathbb{R}^{2}, and an infinitesimally thin ring solenoid with the quantized magnetic flux in mathbb{R}^{3}. We give explicit formulas for the incoming wave solution and the scattering amplitude for these models in terms of some special functions, with the graphs obtained by a numerical method. We also review known high-energy asymptotic formulas related to these examples.
We study the Aharonov-Bohm effect (the AB effect) in quantum resonances for scattering by three solenoids in two dimensions under the situation that the centers of the solenoids are placed almost in line.The resonances are shown to be generated near the real axis by the trapping trajectories when the centers are largely separated from one another.We analyze how the AB effect is reflected in the location of these resonances.The results are described by use of the backward scattering amplitude by one solenoid and depend on the ratio of the distances between the solenoids as well as on the magnetic fluxes.The method makes a full use of the information from the scattering system by one solenoid, which is known to be exactly solvable in quantum mechanics.We also discuss the case of four solenoids.