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在本文中,我们研究了在存在Aharonov–Casher效应的情况下,拓扑,非惯性和自旋效应对二维Dirac振荡器的影响。 接下来,我们确定两组分狄拉克旋子和束缚态的相对论能谱。 我们观察到,该自旋是根据合流超几何函数编写的,并且该光谱明确取决于量子数n和$$ m_1 $$ ml,参数s和$$ \ eta $$η与拓扑和自旋效应相关, 量子相位$$ \ varPhi _ {AC} $$ΦAC,以及与旋转框架的非惯性效应相关的角速度$$ \ varOmega $$Ω。 在非相对论极限下,我们获得了具有两种耦合类型的量子谐波振荡器:自旋轨道耦合和自旋旋转耦合。 我们注意到相对论和非相对论谱的绝对值随着$$ \ eta $$η,$$ \ varOmega $$Ω和$$ \ varPhi _ {AC} $$ΦAC的函数而增长,并且其周期性由于 旋转框架。 最后,我们将我们的问题与其他作品进行了比较,我们证实了我们的研究结果概括了某些特殊的平面案例。
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Eur. Phys. J. C (2019) 79:725
https://doi.org/10.1140/epjc/s10052-019-7237-y
Regular Article - Theoretical Physics
Topological, noninertial and spin effects on the 2D Dirac oscillator
in the presence of the Aharonov–Casher effect
R. R. S. Oliveira
a
Departamento de Física, Universidade Federal do Ceará, Campus do Pici, Fortaleza, CE 60455-760, Brazil
Received: 1 July 2019 / Accepted: 20 August 2019 / Published online: 29 August 2019
© The Author(s) 2019
Abstract In the present paper, we investigate the influence
of topological, noninertial and spin effects on the 2D Dirac
oscillator in the presence of the Aharonov–Casher effect.
Next, we determine the two-component Dirac spinor and the
relativistic energy spectrum for the bound states. We observe
that this spinor is written in terms of the confluent hypergeo-
metric functions and this spectrum explicitly depends on the
quantum numbers n and m
l
, parameters s and η associated to
the topological and spin effects, quantum phase Φ
AC
, and of
the angular velocity Ω associated to the noninertial effects
of a rotating frame. In the nonrelativistic limit, we obtain the
quantum harmonic oscillator with two types of couplings:
the spin-orbit coupling and the spin-rotation coupling. We
note that the relativistic and nonrelativistic spectra grow in
absolute values as functions of η, Ω, and Φ
AC
and its peri-
odicities are broken due to the rotating frame. Finally, we
compared our problem with other works, where we verified
that our results generalizes some particular planar cases of
the literature.
1 Introduction
Since that the relativistic version of the quantum harmonic
oscillator (QHO) for spin-1/2 particles was formulated in the
literature in 1989 by M. Moshinsky and A. Szczepaniak, the
so-called Dirac oscillator (DO) [1], several works on this
model have been and continue being performed in different
areas of physics, such as in thermodynamics [2,3], nuclear
physics [4–6], quantum chromodynamics [7,8], quantum
optics [9,10], and graphene physics [11–13]. Besides that,
the OD also is studied in other interesting physical contexts,
such as in quantum phase transitions [14,15], noncommuta-
tive spaces [16,17], minimal length scenario [18,19], super-
symmetry [20,21], etc. To model the DO, it is necessary
inserted into free Dirac equation (DE) the following nonmin-
a
e-mail: rubensfq1900@gmail.com
imal coupling p → p−im
0
ωβ r, where m
0
is the rest mass of
the oscillator with angular frequency ω>0 and r is the posi-
tion vector [1]. In 2013, the DO was verified experimentally
by J. A. Franco-Villafañe et al [22]. Recently, the DO was
studied in the context of the position-dependent mass [23], in
the presence of the Aharonov–Bohm–Coulomb system [24],
and of topological defects [25,26].
On the other hand, the study of noninertial effects due
to rotating frames have been widely investigated in the lit-
erature since to 1910 decade [27], where the best-known
effects are the Sagnac [28], Barnett [29], Einstein-de Hass
[30] and Mashhoon [31] effects. In the last years, nonineral
effects have also been investigated in some condensed matter
systems, such as in the quantum Hall effect [32,33], Bose-
Einstein condensates [34,35], fullerene molecules [36,37],
and in atomic gases [38,39]. In addition, the study of non-
inertial effects in relativistic quantum systems also gained
relevance and focus of investigations in recent years [40–
44]. In particular, the DE in a rotating frame has several
interesting applications, for instance, is applied in physi-
cal problems involving spin currents [45,46], Sagnac and
Hall effects [47,48], chiral symmetry [49], external magnetic
fields [50,51], fullerene molecules [52–54], nanotubes and
carbon nanocons [55,56], and so on.
In literature, the first formal approach on nonrelativis-
tic neutral quantum particles with magnetic dipole moment
(MDM) interacting with external electric fields was made by
Y. Aharonov and A. Casher [57], where was verified theoret-
ically that the wave function of the neutral particle acquires a
topological quantum phase due to interaction with the field,
even the force of Lorentz being null. This peculiar quantum
effect is known currently as Aharonov–Casher (AC) effect
[58,59]. Next, some researchers formulated the relativistic
AC effect via DE [
60–62]. Moreover, the DE for spin-1/2
particles in the presence of external electric fields is study
in connection with the graphene [63,64], fermions pair pro-
duction [65,66], noncommutative quantum electrodynamics
123
725 Page 2 of 8 Eur. Phys. J. C (2019) 79 :725
[67], vacuum instability [68], magnetic fields in conical and
flat spacetimes [69–71], electric dipole moment [72,73], and
so on. It is important to mention that the DO was originated
through of neutral Dirac particles with MDM interacting with
external electric fields [74].
The present paper has as its goal to study the influence
of topological, noninertial and spin effects on the relativistic
and nonrelativistic quantum dynamics of the 2D DO in the
presence of the AC effect. To include the topological and
noninertial effects in our problem, we rotate our system in
the form ϕ → ϕ + Ωt , where Ω is the constant angular
velocity of the rotating frame, and we use the cosmic string
background (gravitational topological defect) described by
a deficit angle η, where η = 1 − 4M and M is the linear
mass density of the cosmic string [75–77]. Now, to include
the spin effects, we modify a of the Dirac matrices in the
form γ
2
→ sγ
2
, where the parameter s (spin parameter)
characterizes the two spin states of the DO, with s =+1for
spin “up” and s =−1 for spin “down”, respectively [78,79].
Besides that, it is important to report that the first papers that
studied the DO interacting with a topological defect (cosmic
string) in the presence of the Aharonov–Bohm and AC effects
are found in Refs. [80,81]. Last but not least, the topological
and noninertial effects analogous to the of this paper were
recently applied in nonrelativistic quantum dots [82].
This work is organized as follows. In Sect. 2, we introduce
the cosmic string background and the configuration of the
electric field in the rotating frame. In Sect. 3, we investigate
the influence of topological, noninertial and spin effects on
the relativistic quantum dynamics of the 2D DO in the pres-
ence of the AC effect. Next, we determine the Dirac spinor
and the relativistic energy spectrum for the bound states of
the system. In Sect. 4, we analyze the nonrelativistic limit of
our results. Finally, in Sect. 5 we present our conclusions. In
this work, we use the natural units where
¯
h = c = G = 1
and the spacetime with signature (+−−−).
2 The rotating electric cosmic string background
In this section, we configure a curved spacetime in a rotating
frame, whose spacetime chosen is the cosmic string space-
time. In cylindrical coordinates, the line element for this topo-
logical object is written as [75–77]
ds
2
= dT
2
− dR
2
− η
2
R
2
dΦ
2
− dZ
2
, (1)
where the deficit angle η is defined in the range 0 <η<1.
In particular, the geometry characterized by the line element
(1) has a conical singularity that gives rise to a curvature cen-
tered on the cosmic string axis (Z -axis), however, in all other
places the curvature is null [75,76]. This conical singularity
is represented by the following curvature tensor
R
ρ,ϕ
ρ,ϕ
=
1 − η
4η
δ
2
(r), (2)
where δ
2
(r) is the Dirac delta. For instance, in condensed
matter physics, it is already well known that linear topo-
logical defects such as disclinations and dislocations can be
described by a line element in the same that a topological
defect in general relativity [83,84]. It is noteworthy that in
the case of the cosmic string, the spatial part of its line ele-
ment corresponds to the line element of a disclination. In this
way, when we take the nonrelativistic limit of the DE, we can
extend this formalism to the solid state physics context [85].
As we are interested in working in a rotating frame, we
must use the following coordinate transformations
T =t, R =ρ, Φ=ϕ +Ωt, Z = z, (3)
where Ω is the constant angular velocity (not a angular fre-
quency) of the rotating frame and must satisfy the condition
Ωρ 1. As we will see shortly, this condition will allow us
to obtain exact solutions for the system. With this transfor-
mations, the line element (1) becomes [75,76,86]
ds
2
= (1−κ
2
)
dt −
κηρ
(1−κ
2
)
dϕ
2
−dρ
2
−
η
2
ρ
2
(1 − κ
2
)
dϕ
2
−dz
2
, (4)
where κ ≡ Ωηρ > 0.
With the line element (4), we need to construct a local ref-
erence frame where the observers will be placed; this case, we
can to define the Dirac matrices in the curved spacetime. So,
a local reference frame can be built through of an noncoordi-
nate basis given by
ˆ
θ
a
= e
a
μ
(x)dx
μ
, which its components
e
a
μ
(x) satisfy the following relation [75,76,85]
g
μν
(x) = e
a
μ
(x)e
b
ν
(x)η
ab
,(μ,ν= 0, 1, 2, 3), (5)
where g
μν
(x) is the rotating curved metric tensor, η
ab
is the
Minkowski metric tensor and the indices a, b = 0, 1, 2, 3
indicate the local reference frame. The components of the
noncoordinate basis e
a
μ
(x) are called tetrads, and whose
inverse is defined as dx
μ
= e
μ
a
(x)
ˆ
θ
a
, where e
a
μ
(x)e
μ
b
(x) =
δ
a
b
and e
μ
a
(x)e
a
ν
(x) = δ
μ
ν
(x) must be satisfied. Our interest
is to build a rotating frame where there is no torque. Thus,
we choose the tetrads and its inverse in the form [75,85]
e
a
μ
(x) =
⎛
⎜
⎜
⎜
⎝
√
1 − κ
2
0 −
κηρ
√
1−κ
2
0
0100
00
ηρ
√
1−κ
2
0
0001
⎞
⎟
⎟
⎟
⎠
,
e
μ
a
(x) =
⎛
⎜
⎜
⎜
⎝
1
√
1−κ
2
0
κ
√
1−κ
2
0
0100
00
√
1−κ
2
ηρ
0
0001
⎞
⎟
⎟
⎟
⎠
. (6)
123
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