# Topological, noninertial and spin effects on the 2D Dirac oscillator in the presence of the Aharonov–Casher effect

## 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 hypergeometric functions and this spectrum explicitly depends on the quantum numbers *n* and \(m_l\), parameters *s* and \(\eta \) associated to the topological and spin effects, quantum phase \(\varPhi _{AC}\), and of the angular velocity \(\varOmega \) 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 \(\eta \), \(\varOmega \), and \(\varPhi _{AC}\) and its periodicities 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, 5, 6], quantum chromodynamics [7, 8], quantum optics [9, 10], and graphene physics [11, 12, 13]. Besides that, the OD also is studied in other interesting physical contexts, such as in quantum phase transitions [14, 15], noncommutative spaces [16, 17], minimal length scenario [18, 19], supersymmetry [20, 21], etc. To model the DO, it is necessary inserted into free Dirac equation (DE) the following nonminimal coupling \(\mathbf{p}\rightarrow \mathbf{p}-im_0\omega \beta \mathbf{r}\), where \(m_0\) is the rest mass of the oscillator with angular frequency \(\omega >0\) and **r** is the position 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 literature 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 noninertial effects in relativistic quantum systems also gained relevance and focus of investigations in recent years [40, 41, 42, 43, 44]. In particular, the DE in a rotating frame has several interesting applications, for instance, is applied in physical problems involving spin currents [45, 46], Sagnac and Hall effects [47, 48], chiral symmetry [49], external magnetic fields [50, 51], fullerene molecules [52, 53, 54], nanotubes and carbon nanocons [55, 56], and so on.

In literature, the first formal approach on nonrelativistic 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 theoretically 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, 61, 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 production [65, 66], noncommutative quantum electrodynamics [67], vacuum instability [68], magnetic fields in conical and flat spacetimes [69, 70, 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 \(\varphi \rightarrow \varphi +\varOmega t\), where \(\varOmega \) is the constant angular velocity of the rotating frame, and we use the cosmic string background (gravitational topological defect) described by a deficit angle \(\eta \), where \(\eta =1-4M\) and *M* is the linear mass density of the cosmic string [75, 76, 77]. Now, to include the spin effects, we modify a of the Dirac matrices in the form \(\gamma _2\rightarrow s\gamma _2\), where the parameter *s* (spin parameter) characterizes the two spin states of the DO, with \(s=+1\) for 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 presence 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 \(\hbar =c=G=1\) and the spacetime with signature \((+---)\).

## 2 The rotating electric cosmic string background

*Z*-axis), however, in all other places the curvature is null [75, 76]. This conical singularity is represented by the following curvature tensor

*z*-axis of the curved spacetime, respectively. With these informations about the choice of the local reference frame, we can obtain the one-form connection \(\omega ^a_{\ b}=\omega ^{\ a}_{\mu \ b}(x)dx^\mu \) via Maurer-Cartan equations [75, 85]. In the absence of the torsion (or torque), these equations are written as follows

*d*is the exterior derivative and the symbol \(\wedge \) means the external product. Therefore, the non-null components of the one-form connection are

*z*-axis) [70, 80, 81].

## 3 Relativistic quantum dynamics of the Dirac oscillator in the rotating electric cosmic string background

*E*is the total relativistic energy, \(m_l=\pm 1/2,\pm 3/2,\ldots \) is the orbital magnetic quantum number and \(\varPhi _{AC}=2\pi s\mu \lambda \) is a topological quantum phase, also so-called of AC quantum phase, and \(s=\pm 1\) corresponds to projections of the MDM of the particle along on the

*z*-axis (aligned or unaligned with the spin) [57, 81].

*r*characterizes the two components of the spinor, being that \(r=+1\) describes a particle with spin up (\(s=+1\)) or down (\(s=-1\)) and \(r=-1\) describes a antiparticle with spin up (\(s=+1\)) or down (\(s=-1\)), respectively.

*n*, consequently, the parameter \(\frac{\vert {\bar{\gamma }}_r\vert }{2}-{\bar{E}}_r\) must to be equal to a non-positive integer number \(-n\) (\(n=0,1,2,\ldots \)). Therefore, we obtain from this condition (quantization condition) the following relativistic energy spectrum of the 2D DO in the presence of the AC effect and under the influence of topological, noninertial and spin effects

*s*, quantum phase \(\varPhi _{AC}\), angular velocity \(\varOmega \) of the rotating frame, and of the deficit angle \(\eta \) generated by cosmic string. In particular, the rotating frame breaks the periodicity of the spectrum, since for \(\varOmega =0\) we have a spectrum with periodicity \(\pm 2\pi \) [81]. Although the uniformly charged wire which generates the electric field of the AC effect be located on the axis of symmetry of the cosmic string, a region forbidden, we see that the energies of the DO has a contribution due to conical singularity generated by the cosmic string. We note that for \(r=s=+1\) the terms of the spectrum (39) are summed, while for \(r=+1\) and \(s=-1\) , the terms are subtracted, consequently, the energies of the particle with spin up are larger than with spin down. On the other hand, we note that for \(r=s=-1\) the terms of the spectrum (39) are summed, while for \(r=-1\) and \(s=+1\) , the terms are subtracted, consequently, the energies (in absolute values) of the antiparticle with spin down are larger than with spin up. Moreover, we verified that the parameters \(\eta \), \(\varOmega \) and \(\varPhi _{AC}\) have the function of increasing the energies of the spectrum, i.e., in the limits \(\eta \rightarrow {0}\) (extremely dense cosmic string, \(\varOmega \rightarrow \infty \) or \(\varPhi _{AC}\rightarrow \infty \), we have \(\vert E^\sigma _{n_s,m_l,s}\vert \rightarrow \infty \).

Now, comparing the spectrum (39) with the literature, we verified that in the absence of the AC effect (\(\varPhi _{AC}=0\)) and of the topological and spin effects (\(\eta =s=+1\)) with \(m_l>0\), we obtain the spectrum of the DO in a rotating frame [86]. Already in the absence of the AC effect (\(\varPhi _{AC}=0\)) and of the topological and noninertial effects (\(\eta =+1\) and \(\varOmega =0\)), we obtain the spectrum of the DO in a flat inertial frame with and without the influence of the spin effects [78, 79, 88]. From the above, we see that the spectrum (39) generalizes some relativistic particular planar cases of the literature when \(\varOmega \), \(\varPhi _{AC}\), *s*, or \(\eta \) are excluded of the system.

## 4 Nonrelativistic limit

*r*, AC quantum phase \(\varPhi _{AC}\), angular velocity \(\varOmega \) of the rotating frame, and of the deficit angle \(\eta \) associated to topology of the conic space, grow infinitely in the limits \(\eta \rightarrow 0\), \(\varOmega \rightarrow \infty \) or \(\varPhi _{AC}\rightarrow \infty \). We see that for \(r=+1\) the terms of the spectrum (46) are summed, while for \(r=-1\) , the terms are subtracted, i.e., the energies of the particle (or QHO) with spin up are larger than with spin down. Now, comparing the spectrum (46) with the literature, we verified that in the absence of the AC effect (\(\varPhi _{AC}=0\)) and of the topological and spin effects (\(\eta =r=+1\)) with \(m_l>0\), we obtain the spectrum of the QHO in a rotating frame [86]. Already in the absence of the AC effect (\(\varPhi _{AC}=0\)) and of the topological and noninertial effects (\(\eta =+1\) and \(\varOmega =0\)), we obtain the spectrum of the OHQ in a flat inertial frame under the influence of the spin effects [78]. From the above, we see that the spectrum (46) generalizes some nonrelativistic particular planar cases of the literature when \(\varOmega \), \(\varPhi _{AC}\),

*r*, or \(\eta \) are excluded of the system.

## 5 Conclusions

In this paper, we study the influence of topological, spin and noninertial effects on the relativistic and nonrelativistic quantum dynamics of the 2D DO in the presence of the AC effect. Next, we determine the relativistic bound-state solutions of the system, given by two-component Dirac spinor and the energy spectrum.In particular, we verify that this spinor is written in terms of the confluent hypergeometric functions and this spectrum depends on the quantum numbers *n* and \(m_l\), parameters *s* and \(\eta \) associated to the topological and spin effects, quantum phase \(\varPhi _{AC}\), and of the angular velocity \(\varOmega \) of the rotating frame. We note that besides of the rotating frame break the periodicity of the spectrum, in the limits \(\eta \rightarrow 0\) (extremely dense cosmic string), \(\varOmega \rightarrow \infty \) or \(\varPhi _{AC}\rightarrow \infty \) the spectrum grows infinitely. On the other hand, we note that the energies of the particle (or DO) with spin up are larger than with spin down. However, we also note that the energies of the antiparticle (or anti-DO) with spin down are larger than with spin up. Now, comparing our energy spectrum with other works, we verified that this spectrum generalizes some relativistic particular planar cases of the literature when \(\varOmega \), \(\varPhi _{AC}\), *s* or \(\eta \) are excluded of the system.

Finally, we study the nonrelativistic limit of our results. For instance, considering consider that most of the total energy of the system stay concentrated in the rest energy of the particle, we obtain the motion equation of the 2D QHO in the presence of the AC effect and under the influence of topological, spin and noninertial effects. We verified that these topological effects are generated by a conic space and the oscillator has two types of couplings: the spin-orbit coupling, given by \(\mathbf{S\cdot L}\), and the spin-rotation coupling, given by \({\varvec{S}}{{\cdot }}{\varvec{\varOmega }}\). With respect to the nonrelativistic energy spectrum, such spectrum has some similarities with the relativistic case, for instance, depend of *n*, \(m_l\), *s*, \(\varPhi _{AC}\), \(\varOmega \) and \(\eta \), not is a periodic function and the parameters \(\eta \), \(\varOmega \) and \(\varPhi _{AC}\) have the function of increasing the energies. Now, comparing our energy spectrum with other works, we verified that this spectrum generalizes some nonrelativistic particular planar cases of the literature when \(\varOmega \), \(\varPhi _{AC}\), *s* or \(\eta \) are excluded of the system.

## Notes

### Acknowledgements

The author would like to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

## References

- 1.M. Moshinsky, A. Szczepaniak, J. Phys. A: Math. Gen.
**22**, L817 (1989)CrossRefADSGoogle Scholar - 2.A. Boumali, H. Hassanabadi, Eur. Phys. J. Plus
**128**, 124 (2013)CrossRefGoogle Scholar - 3.H. Hassanabadi, S. Sargolzaeipor, B.H. Yazarloo, Few-Body Syst.
**56**, 115 (2015)CrossRefADSGoogle Scholar - 4.J. Munarriz, F. Dominguez-Adame, R.P.A. Lima, Phys. Lett. A
**376**, 3475 (2012)CrossRefADSGoogle Scholar - 5.D.A. Kulikov, I.V. Uvarov, A.P. Yaroshenko, Cent. Eur. J. Phys.
**11**, 1006 (2013). arXiv:1112.4653 [hep-ph]Google Scholar - 6.J. Grineviciute, D. Halderson, Phys. Rev. C
**85**, 054617 (2012). arXiv:1404.4170 [nucl-th]CrossRefADSGoogle Scholar - 7.M. Moshinsky, G. Loyola, Found. Phys.
**23**, 197 (1993)CrossRefADSMathSciNetGoogle Scholar - 8.O.L. de Lange, J. Math. Phys.
**32**, 1296 (1991)CrossRefADSMathSciNetGoogle Scholar - 9.A. Bermudez, M.A. Martin-Delgado, E. Solano, Phys. Rev. A
**76**, 041801 (2007). arXiv:0704.2315 [quant-ph]CrossRefADSGoogle Scholar - 10.
- 11.C. Quimbay, P. Strange, Graphene physics via the Dirac oscillator in (2+1) dimensions (2013). arXiv:1311.2021 [cond-mat.mes-hall]
- 12.A. Boumali, Phys. Scrip.
**90**, 045702 (2015). arXiv:1411.1353 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 13.A. Belouad, A. Jellal, Y. Zahidi, Phys. Lett. A
**380**, 773 (2016). arXiv:1505.08068 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 14.A. Bermudez, M. Martin-Delgado, A. Luis, Phys. Rev. A
**77**, 063815 (2008). arXiv:0802.0577 [quant-ph]CrossRefADSGoogle Scholar - 15.C. Quimbay, P. Strange, Quantum phase transition in the chirality of the (2+1)-dimensional Dirac oscillator (2013). arXiv:1312.5251 [hep-th]
- 16.B.P. Mandal, S.K. Rai, Phys. Lett. A
**376**, 2467 (2012). arXiv:1203.2714 [hep-th]CrossRefADSMathSciNetGoogle Scholar - 17.G. Melo, M. Montigny, P. Pompeia, E. Santos, Int. J. Theor. Phys.
**52**, 441 (2013)CrossRefGoogle Scholar - 18.C. Quesne, V.M. Tkachuk, J. Phys. A: Math. Gen.
**38**, 1747 (2005). arXiv:math-ph/0412052 CrossRefADSGoogle Scholar - 19.L. Menculini, O. Panella, P. Roy, Phys. Rev. D
**91**, 045032 (2015). arXiv:1411.5278 [quant-ph]CrossRefADSGoogle Scholar - 20.Int Quesne, J. Mod. Phys. A
**6**, 1567 (1991)CrossRefADSMathSciNetGoogle Scholar - 21.G. Junker, A. Inomata, J. Math. Phys.
**59**, 052301 (2018). arXiv:1712.08759 [quant-ph]CrossRefADSMathSciNetGoogle Scholar - 22.J.A. Franco-Villafañe, E. Sadurní, S. Barkhofen, U. Kuhl, F. Mortessagne, T.H. Seligman, Phys. Rev. Lett.
**111**, 170405 (2013). arXiv:1306.2204 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 23.C.L. Ho, P. Roy, Europhys. Lett.
**124**, 60003 (2019). arXiv:1808.03962 [quant-ph]CrossRefADSGoogle Scholar - 24.R.R.S. Oliveira, R.V. Maluf, C.A.S. Almeida, Ann. Phys.
**400**, 1 (2019). arXiv:1809.03801 [quant-ph]CrossRefADSGoogle Scholar - 25.M. Salazar-Ramírez, D. Ojeda-Guillén, A. Morales-González, V.H. García-Ortega, Eur. Phys. J. Plus
**134**, 8 (2019). arXiv:1806.05329 [math-ph]CrossRefGoogle Scholar - 26.M. Hosseinpour, H. Hassanabadi, M. de Montigny, Eur. Phys. J. C
**79**, 311 (2019). arXiv:1904.05889 [hep-th]CrossRefADSGoogle Scholar - 27.M. Matsuo, J.I. Ieda, E. Saitoh, S. Maekawa, Phys. Rev. B
**84**, 104410 (2011). arXiv:1106.0366 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 28.M.G. Sagnac, C.R. Acad, Sci. (Paris)
**157**, 708 (1913)Google Scholar - 29.S.J. Barnett, Phys. Rev.
**6**, 239 (1915)CrossRefADSGoogle Scholar - 30.A. Einstein, W.J. de Haas, Verh. Dtsch. Phys. Ges.
**17**, 152 (1915)Google Scholar - 31.B. Mashhoon, Phys. Rev. Lett.
**61**, 2639 (1988)CrossRefADSGoogle Scholar - 32.U.R. Fischer, N. Schopohl, Europhys. Lett.
**54**, 502 (2001). arXiv:cond-mat/0004339 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 33.S. Viefers, J. Phys, Condens. Matter
**20**, 123202 (2008). arXiv:0801.4856 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 34.V. Schweikhard, I. Coddington, P. Engels, V.P. Mogendorff, E.A. Cornell, Phys. Rev. Lett.
**92**, 040404 (2004). arXiv:cond-mat/0308582 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 35.N.R. Cooper, N.K. Wilkin, J.M.F. Gunn, Phys. Rev. Lett.
**87**, 120405 (2001). arXiv:cond-mat/0107005 [cond- mat]CrossRefADSGoogle Scholar - 36.J.Q. Shen, S.L. He, Phys. Rev. B
**68**, 195421 (2003). arXiv:quant-ph/0301094 [quant-ph]CrossRefADSGoogle Scholar - 37.J.R. Lima, J. Brandão, M.M. Cunha, F. Moraes, Eur. Phys. J. D
**68**, 94 (2014). arXiv:1405.6633 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 38.N.R. Cooper, Adv. Phys.
**57**, 539 (2008). arXiv:0810.4398 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 39.L.H. Lu, Y.Q. Li, Phys. Rev. A
**76**, 023410 (2007). arXiv:cond-mat/0701481 [cond-mat.other]CrossRefADSGoogle Scholar - 40.B.Q. Wang, Z.W. Long, C.Y. Long, S.R. Wu, Mod. Phys. Lett. A
**33**, 1850025 (2018)CrossRefADSGoogle Scholar - 41.K. Konno, R. Takahashi, Phys. Rev. D
**85**, 061502 (2012). arXiv:1201.5188v1 [gr-qc]CrossRefADSGoogle Scholar - 42.L.C.N. Santos, C.C. Barros, Eur. Phys. J. C
**78**, 13 (2018). arXiv:1801.01024 [hep-th]CrossRefADSGoogle Scholar - 43.M. Hosseinpour, H. Hassanabadi, Eur. Phys. J. Plus
**130**, 236 (2015). arXiv:1505.00096 [hep-th]CrossRefGoogle Scholar - 44.
- 45.Ö.F. Dayi, E. Yunt, Ann. Phys.
**390**, 143 (2018). arXiv:1705.07590v1 [math-ph]CrossRefADSGoogle Scholar - 46.M. Matsuo, J.I. Ieda, E. Saitoh, S. Maekawa, Phys. Rev. Lett.
**106**, 076601 (2011). arXiv:1009.5424 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 47.J. Anandan, Phys. Rev. D
**24**, 338 (1981)CrossRefADSGoogle Scholar - 48.
- 49.M.N. Chernodub, S. Gongyo, J. High Energy Phys.
**2017**, 136 (2017). arXiv:1611.02598 [hep-th]CrossRefGoogle Scholar - 50.Y. Liu, I. Zahed, Phys. Rev. D
**98**, 014017 (2018). arXiv:1710.02895 [hep-ph]CrossRefADSMathSciNetGoogle Scholar - 51.M.N. Chernodub, S. Gongyo, Phys. Rev. D
**96**, 096014 (2017). arXiv:1706.08448 [hep-th]CrossRefADSGoogle Scholar - 52.E. Cavalcante, J. Carvalho, C. Furtado, Eur. Phys. J. Plus
**131**, 288 (2016)CrossRefGoogle Scholar - 53.J. Gonzalez, F. Guinea, M.A.H. Vozmediano, Nucl. Phys. B
**406**, 771 (1993)CrossRefADSGoogle Scholar - 54.D.V. Kolesnikov, V.A. Osipov, Eur. Phys. J. B
**49**, 465 (2006). arXiv:cond-mat/0510636 [cond-mat.mtrl-sci]CrossRefADSGoogle Scholar - 55.M.M. Cunha, J. Brandão, J.R. Lima, F. Moraes, Eur. Phys. J. B
**88**, 288 (2015)CrossRefADSGoogle Scholar - 56.F.A. Gomes, V.B. Bezerra, J.R.F. de Lima, F.J.S. Moraes, Eur. Phys. J. B
**92**, 41 (2019). arXiv:1806.01324 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 57.Y. Aharonov, A. Casher, Phys. Rev. Lett.
**53**, 319 (1984)CrossRefADSMathSciNetGoogle Scholar - 58.A. Cimmino, G.I. Opat, A.G. Klein, H. Kaiser, S.A. Werner, M. Arif, R. Clothier, Phys. Rev. Lett.
**63**, 380 (1989)CrossRefADSGoogle Scholar - 59.K. Sangster, E.A. Hinds, S.M. Barnett, E. Riis, Phys. Rev. Lett.
**71**, 3641 (1993)CrossRefADSGoogle Scholar - 60.C.R. Hagen, Phys. Rev. Lett.
**64**, 2347 (1990)CrossRefADSMathSciNetGoogle Scholar - 61.S. Bruce, L. Roa, C. Saavedra, A.B. Klimov, Phys Rev. A
**60**, R1 (1999). arXiv:quant-ph/9905074 CrossRefADSGoogle Scholar - 62.
- 63.N.M.R. Peres, E.V. Castro, J. Phys. Condens. Matter
**19**, 406231 (2007). arXiv:0709.0646 [cond-mat.mes-hall]CrossRefGoogle Scholar - 64.Z.Z. Alisultanov, M.S. Reis, Europhys. Lett.
**113**, 28004 (2016). arXiv:1510.06687 [cond-mat.mes-hall]CrossRefADSGoogle Scholar - 65.A.I. Nikishov, Sov. Phys. JETP
**30**, 660 (1970)ADSGoogle Scholar - 66.S.P. Kim, D.N. Page, Phys. Rev. D
**75**, 045013 (2007)CrossRefADSGoogle Scholar - 67.N. Chair, M.M. Sheikh-Jabbari, Phys. Lett. B
**504**, 141 (2001)CrossRefADSMathSciNetGoogle Scholar - 68.S.P. Gavrilov, D.M. Gitman, Phys. Rev. D
**53**, 7162 (1996)CrossRefADSGoogle Scholar - 69.E.R.B. de Mello, J. High Energy Phys.
**2004**, 016 (2004)CrossRefGoogle Scholar - 70.R. R. S. Oliveira, R. V. Maluf and C. A. S. Almeida, Exact solutions of the Dirac oscillator under the influence of the Aharonov-Casher effect in the cosmic string background (2018). arXiv:1810.11149 [quant-ph]
- 71.R.R.S. Oliveira, M.F. Sousa, Braz. J. Phys.
**49**, 315 (2019)CrossRefADSGoogle Scholar - 72.A.Y. Silenko, Russ. Phys. J.
**48**, 788 (2005)CrossRefGoogle Scholar - 73.K. Bakke, Open Phys.
**10**, 1089 (2012)CrossRefADSGoogle Scholar - 74.R.P. Martínez-y Romero, H.N. Núnez-Yépez, A.L. Salas-Brito, Eur. J. Phys.
**16**, 135 (1995)CrossRefGoogle Scholar - 75.K. Bakke, Eur. Phys. J. Plus
**127**, 82 (2012)CrossRefADSGoogle Scholar - 76.
- 77.A. Vilenkin, E.P.S. Shellard,
*Cosmic Strings and Other Topological Defects*(Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar - 78.F.M. Andrade, E.O. Silva, Europhys. Lett.
**108**, 30003 (2014). arXiv:1406.3249 [hep-th]CrossRefADSGoogle Scholar - 79.F.M. Andrade, E.O. Silva, Eur. Phys. J. C
**74**, 3187 (2014). arXiv:1403.4113 [hep-th]CrossRefADSGoogle Scholar - 80.J. Carvalho, C. Furtado, F. Moraes, Phys. Rev. A
**84**, 032109 (2011)CrossRefADSGoogle Scholar - 81.K. Bakke, C. Furtado, Ann. Phys.
**336**, 489 (2013)CrossRefADSGoogle Scholar - 82.M. Rojas, C. Filgueiras, J. Brandão, F. Moraes, Phys. Lett. A
**382**, 432 (2018). arXiv:1706.06421v1 [cond-mat.other]CrossRefADSMathSciNetGoogle Scholar - 83.M.O. Katanaev, I.V. Volovich, Ann. Phys.
**216**, 1 (1992)CrossRefADSGoogle Scholar - 84.H. Kleinert,
*Gauge Fields in Condensed Matter*, vol. 2 (World Scientific, Singapore, 1989)CrossRefzbMATHGoogle Scholar - 85.K. Bakke, C. Furtado, Phys. Rev. D
**82**, 084025 (2010)CrossRefADSGoogle Scholar - 86.P. Strange, L.H. Ryder, Phys. Lett. A
**380**, 3465 (2016)CrossRefADSMathSciNetGoogle Scholar - 87.W. Greiner,
*Relativistic Quantum Mechanics: Wave Equations*, vol. 3 (Springer, Berlin, 2000)CrossRefzbMATHGoogle Scholar - 88.V.M. Villalba, Phys. Rev. A
**49**, 586 (1994)CrossRefADSMathSciNetGoogle Scholar - 89.F.W. Hehl, W.T. Ni, Phys. Rev. D
**42**, 2045 (1990)CrossRefADSGoogle Scholar - 90.M. Abramowitz, I.A. Stegum,
*Handbook of Mathematical Functions*(Dover Publications Inc., New York, 1965)Google Scholar - 91.F.M. Andrade, E.O. Silva, M. Pereira, Phys. Rev. D
**85**, 041701 (2012). arXiv:1112.0265 [quant-ph]CrossRefADSGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}