弹性介质中垂直线螺旋变形下Aharonov-Bohm效应的分析

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本文探讨了在垂直线发生向垂直螺旋形态的几何畸变背景下的Aharonov-Bohm效应（Aharonov-Bohm-type effect）。Aharonov-Bohm效应是量子力学中一个重要的现象，它描述了无旋量子粒子如何受到磁场的影响，即使在磁场被屏蔽的情况下也能感知到。在这个特殊情况下，研究者考虑了一种物理情境，即无旋粒子处于弹性介质中，原本的垂直线形被转化为螺旋形。作者们利用了一个理论模型，将无旋粒子限制在一个圆柱形盒子里，这个盒子模拟了实验条件下的约束。在这个有限空间里，由于几何畸变产生的拓扑缺陷，系统的能量状态发生了改变，从而展现出类似于自由粒子Aharonov-Bohm效应的束缚态特征。这种现象反映了几何形态变化导致的拓扑效应对角动量量子数的影响，即粒子在螺旋结构中的运动受到几何学性质的调控。文章进一步分析了旋转效应的影响。在有旋转的情况下，角动量与角速度之间的关系与标准的Aharonov-Bohm效应有所相似，表明旋转不仅改变了粒子自身的动力学行为，而且与几何畸变共同作用，产生了新的量子效应。结果显示，即使在旋转环境中，束缚态的Aharonov-Bohm效应依然存在，并且其特性在几何学和旋转之间建立了新的耦合关系。这篇研究深化了我们对Aharonov-Bohm效应的理解，尤其是在非平凡几何结构（如垂直线变为垂直螺旋）下，揭示了拓扑效应在量子系统中的新表现形式。这一发现对于理解量子粒子在复杂几何背景下的行为、以及可能的应用在量子信息处理等领域具有重要意义。

Eur. Phys. J. C (2019) 79:559

https://doi.org/10.1140/epjc/s10052-019-7073-0

Regular Article - Theoretical Physics

Aharonov–Bohm-type effect in the background of a distortion of a

vertical line into a vertical spiral

W. C. F. da Silva, K. Bakke

Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, PB 58051-900, Brazil

Received: 7 March 2019 / Accepted: 25 June 2019 / Published online: 3 July 2019

Abstract It is analysed Aharonov–Bohm-type effects when

a spinless quantum particle is in an elastic medium with the

distortion of a vertical line into a vertical spiral. By conﬁning

the spinless particle to a cylindrical box, the analogue of the

Aharonov–Bohm effect for bound states is observed due to

the inﬂuence of the topological defect on the allowed ener-

gies of the system. It corresponds to the shift in the angular

momentum quantum number yielded by the effects of the

topology of the distortion of a vertical line into a vertical

spiral. In addition, it is analysed the effects of rotation. It

is shown that the Aharonov–Bohm effect for bound states

exists. Besides, there is an analogue of the coupling between

the angular momentum and angular velocity of the rotating

frame.

1 Introduction

Based on the elastic theory in solids, dislocations arise from

a break of the translational symmetry [1,2]. Besides, it char-

acterizes the presence of torsion in the elastic medium.

At present days, it is well-known that dislocations can be

described by the differential geometry, i.e., the deformation

of the elastic medium that corresponds to a dislocation can

be described by the Riemann–Cartan geometry. This pro-

posal of using the Riemann–Cartan geometry was proposed

by Katanaev and Volovich [2], and hence, the process of “cut”

and “glue” of the elastic medium (Volterra process [1]) was

connected with the geometrical description of the a topolog-

ical defect in the context of general relativity. Dislocations

can modify the electronic properties of the medium, there-

fore, there is a great interest in studies of semiconductors

[3–5] and quantum dots [6]. This kind of elastic deforma-

tion in a solid is called as a linear topological defect. Exam-

ples of it are the screw dislocation and the spiral dislocation

[7,8]. In recent decades, the Katanaev–Volovich approach

e-mail: kbakke@ﬁsica.ufpb.br

[2] has been widely used in the literature with the purpose of

investigating topological effects associated with screw dis-

locations in quantum rings [9–11], electrons subject to the

deformed Kratzer potential [12], electron gas in a cylindrical

shell [13], quantum holonomies [14] and with an electron in

a uniform magnetic ﬁeld [15–18]. Furthermore, topological

effects associated with a spiral dislocation have been made

with geometric quantum phases [19] and the harmonic oscil-

lator [20].

In this work, we explore this line of research and investi-

gate the topological effects associated with a screw disloca-

tion on a spinless particle conﬁned to a hard-wall conﬁning

potential. The most known kind of screw dislocation in the

literature is the distortion of a circular curve into a verti-

cal spiral. As shown in Refs. [10,16], the main effect of the

presence of the distortion of a circular curve into a vertical

spiral is the shift in the angular momentum quantum num-

ber that gives rise to an analogue effect of the Aharonov–

Bohm effect for bound states [21,22]. With the interface

between general relativity and relativistic quantum mechan-

ics, spacetime with dislocations have drawn attention to rel-

ativistic analogue effects of the Aharonov–Bohm effect [23–

26]. In the present work, we deal with another type of screw

dislocation that corresponds to the distortion of a vertical

line into a vertical spiral [7]. Then, we analyse Aharonov–

Bohm-type effects that stem from the presence of the dis-

tortion of a vertical line into a vertical spiral in the elastic

medium. In addition, we investigate effects of rotation on this

system.

This paper is structured as follows: in Sect. 2, we introduce

the line element of the distortion of a vertical line into a ver-

tical spiral. Then, we obtain the Schrödinger equation in the

presence of this topological defect and analyse the conﬁne-

ment of a spinless quantum particle to a hard-wall conﬁning

potential; in Sect. 3, we investigate effects of rotation; in Sect.

4, we present our conclusions.

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