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
a
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
© The Author(s) 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
a
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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