Research Article Effects of a Landau-Type Quantization Induced by the Lorentz Symmetry Violation on a Dirac Field

Hindawi Advances in High Energy Physics Volume 2020, Article ID 4208161, 7 pages https://doi.org/10.1155/2020/4208161

R. L. L. Vitória and H. Belich

Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Goiabeiras, 29060-900 Vitória, ES, Brazil

Correspondence should be addressed to R. L. L. Vitória; [email protected]

Received 16 May 2019; Revised 28 August 2019; Accepted 23 October 2019; Published 22 January 2020

Academic Editor: Elias C. Vagenas

Copyright © 2020 R. L. L. Vitória and H. Belich. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP3.

Inspired by the extension of the Standard Model, we analyzed the eects of the spacetime anisotropies on a massive Dirac eld through a nonminimal CPT-odd coupling in the Dirac equation, where we proposed a possible scenario that characterizes the breaking of the Lorentz symmetry which is governed by a background vector eld and induces a Landau-type quantization. en, in order to generalize our system, we introduce a hard-wall potential and, for a particular case, we determine the energy levels in this background. In addition, at the nonrelativistic limit of the system, we investigate the eects of the Lorentz symmetry violation on thermodynamic aspects of the system.

1. Introduction constraints in electrodynamic [11, 12] and on an analogy of the quantum hall conductivity [13]. Recently, Kostelecký and Samuel [1] have shown that in the In recent years, the LS violation has been applied in quantum context of string eld theory, the Lorentz symmetry (LS) vio- mechanics systems. In the nonrelativistic limit, for example, lation governed by tensor elds is natural when the perturba- there are studies in an Aharonov-Bohm-Casher system [14], in tive string vacuum is unstable. Carroll et al. [2], in the context quantum holonomies [15], in a Dirac neutral particle inside a of electrodynamics, have investigated the theoretical and two-dimensional quantum ring [16], on a spin-orbit coupling observational consequences of the presence of a background for a neutral particle [17] and in a system under the in£uence vector eld in the modied Chern-Simons Lagrangean, that of a Rashba-type coupling induced [18]. In the relativistic case, is, in -dimensions, which preserves the gauge symme- there are studies on EPR correlations [19], in geometric try, but(3+1 violates) the Lorentz symmetry. One of the purposes quantum phases [20], in the Landau-He-McKellar-Wilkens of these investigations is the extension of theories and models quantization and bound states solutions for a Coulomb-like which may involve the LS violation with the intention of potential [21], in a quantum scattering [22] and on a scalar eld searching for the underlying physical theories that can answer [23–29]. One point that has not been dealt with in the literature questions that the usual physics cannot. In this sense, the is the relativistic Landau-type quantization induced by the LS Standard Model (SM) has been the target of these extensions violation under a Dirac eld. characterized by the LS violation which culminated in what Recently, thermodynamic properties of quantum systems we know today as the Extended Standard Model (ESM) [3, 4]. have been investigated and, consequently, can be found in the In recent years, the LS violation has been extensively studied literature. Here, we can cite some examples of these studies, in various branches of physics, for example, in magnetic for example, in diatomic molecule systems [30], in a neutral moment generation [5], in Rashba spin-orbit interaction [6], particle system in the presence of topological defects in mag- in Maxwell-Chern-Simons vortices [7], on vortexlike netic cosmic string background [31], in exponential-type congurations [8], in Casimir eect [9, 10], in cosmological molecule potentials [32], on the a 2D charged particle conned 2 Advances in High Energy Physics by a magnetic and Aharonov-Bohm £ux elds under the radial in addition to breaking the Lorentz symmetry, also breaks the scalar power potential [33], on the Dirac and Klein-Gordon CPT symmetry [38, 42]. In the way, the Dirac equation in any oscillators in the anti-de Sitter space [34], on the Klein-Gordon orthogonal system with the nonminimal coupling is [38] oscillator in the frame work of generalized uncertainty prin- ciple [35] and the on the harmonic oscillator in an environ- 3 ln ℎ1ℎ2ℎ3 v ment with a pointlike defect [36]. ̃ Ψ+ =0 Ψ − Ψ − Ψ = 0, In this paper, we investigate the eects of the LS violation 2 ℎ (2) on a Dirac eld, where the spacetime anisotropies are gov- where is the derivative of the corresponding = 1/ℎ erned by the presence of a vector background eld inserted in coordinate system( ) and is the parameter which corre- the Dirac equation via nonminimal coupling that, for a pos- sponds to the scale factorsℎ( ) of this coordinate system. is sible scenery in this background characterized by a particular means that a new contribution to the Dirac equation can stem electromagnetic eld conguration, it is possible to induce a from the coordinate system in an analogous way to the well- relativistic analogue of the Landau quantization. Further, we known coupling between spinors and curvature discussed in analyze the Dirac eld under the in£uence of a hard-wall quantum eld theory in curved space [43]. In this paper, we potential in this LS violation background for a particular case. are working with the Minkowski spacetime in cylindrical In addition, inspired by Refs. [30–36], we investigate the ther- coordinates modynamic properties of our more general system investi- 2 2 2 2 2 2 (3) gated at low energies and analyze the LS violation eects on = − + + + , some thermodynamic magnitudes. then, the scale factors are 0 1 3 and 2 . In this e structure of this paper is as follows: in Section 2 we way, Equation (2) is rewrittenℎ =ℎ as follows=ℎ =1 ℎ = introduce a LS violation background dened by a vector eld Ψ 1 2 Ψ that governs the spacetime anisotropies, and thus, establish a 0 + 1 + Ψ+ possible scenario of the LS violation that induces a relativistic 2 analogue of the Landau quantization; in Section 3 we inserted Ψ → → + 3 +v00 ⋅→ Ψ − 0 →v ⋅ Ψ − 0 → into the system a hard-wall potential and, for a particular case, we determined the relativistic energy levels in this LS violation →v → background; in Section 4, through the Landau-type non-rel- ⋅ × Ψ − Ψ = 0. (4) ativistic energy levels, we investigated some thermodynamic aspects under the eects of the LS violation; in Section 5 we Now, let us consider a LS violation possible scenario where present our conclusions. we have the background eld and the presence of an electric eld given by

2. Landau-Type Quantization Induced by the v v → (5) = 0, → = ;̂ = ,̂ Lorentz Symmetry Violation 2 where const and is a parameter associated to the uni- In this section, we investigated the in£uence of the spacetime form volumetric = . distribution of electric charges. We can note anisotropies under a Dirac eld, where we obtain the relativ- that the possible scenario of the LS violation background eld istic Landau-type energy levels induced by the LS violation conguration given in Equation (5) induces an analogue to eects. Recently, in low energy scenarios, the Landau-type the quantization of Landau [44], that is, we have a vector → → quantization induced by the LS violation has been investigated potential A = →v × which it gives a uniform “magnetic eld” [37, 38]. ese analyses are possible through couplings non- B → A → B =∇× = 0̂ = ̂. In addition, the electric eld con- minimal in wave equations [23, 38–40], where this procedure guration given in Equation (5) has been studied in Landau carries the information that there are privileged directions in levels induced by magnetic dipole moment [45], in Landau the spacetime to which they characterize the LS violation. levels induced by electric dipole moment [46–48] and in LS From the mathematical point of view, this information is given violation possible scenarios [18, 24, 26]. In this way, the Dirac through the presence of constant background elds of vector equation (4) becomes or tensor nature [41]. en, based on Refs. [38, 40], consider the non-minimum coupling dened by (with ) 2 0 1 3 2 v, where is coupling =ℏ= constant,=1 Ψ 1 Ψ Ψ ̃ + + Ψ+ + − Ψ − Ψ = 0. are Dirac→ matrices − 2 2 (6) Assuming stationary state solutions and that the linear mo- 0 0 (1) mentum and angular momentum 0 0 ̂ = ; = ; = = , operators ̂commute= −(/ with) the Hamiltonian operator, = −/we have 0 − − 0 with and are the Pauli matrices, the following solution + = 2 ̃() = is dual electromagnetic tensor, where E (1/2) () and , and v is background − (+(1/2)) + 0 0 (7) vector= − eld= that − governs = the − LS= violation. is nonminimal Ψ, , , = − , coupling is inspired by the gauge sector of the SME [3, 4], where are the eigenvalues of angular which is known as the nonminimal CPT-odd coupling, as it, momentum = 0, and±1, ±2, ±3,... . In this way, by substituting −∞ < < ∞ Advances in High Energy Physics 3

Equation (7) into Equation (6) and considering the matrices where is the cyclotron frequency. Equation (15) given in Equation (1), we obtain the two coupled equations below represents = the / energy levels of the system of a Dirac eld in

1 2 3 2 an anisotropic spacetime, where the anisotropies are governed E 1 1 1 ( −)+ + + − − + − − − − − = 0; by a background vector eld present in a eld conguration 2 2 2 1 2 3 2 which provides a relativistic analogue to Landau quantization. E 1 1 1 ( +)− + + + − + + − + − + = 0. We can note that the eects of the LS violation in£uence the 2 2 2 (8) energy prole of the system through the cyclotron frequency By multiplying rst equation by E and using the second which is dened in terms of the parameters associated with equation we obtain ( + ) the Lorentz symmetry violation, , , and . We can note that 2 1 1 2 1 1 by taking we recover energy of a free Dirac eld in the + + + −+ + −+ 3 + Minkowski →0 spacetime. 2 2 4 2 2 2 1 3 − 2 + E2 − 2 − 2 − + + = 0. 2.1. Nonrelativistic Limit. To analyze the nonrelativistic limit 4 + 2 2 + of the energy levels, we can write Equation (15) in the form (9) 2 1/2 An analogue equation can be found for −. We can simplify E ,, 2 2 1 + (16) the dierential equations for + and −, since the is an eigen- = 1 + + + + , function of 3, whose eigenvalues are . ereby, we write 2 3 =±1 . Hence, we can represent the two dierential where . en, by using the approximation equations =± in= a single radial dierential equation: = (1/2) |in |Equation+ (16) and making E , (1 + ) ≃ 1 + ,, = ,, − 2 2 2 where ,, , we obtain the expression 2 ≪ 2 1 1 2 2 + −+ (1−) − ᐈ ᐈ 2 4 ,, 1ᐈ 1 ᐈ 1 1 1 + 2 2 2 = + + ᐈ + (1 − )ᐈ + + (1 − ) + , E 1 (10) 2 2ᐈ 2 ᐈ 2 2 2 + − − − + (1+) = 0. (17) 2 which represents the Landau-type energy levels of a Dirac We proceed with a change of variables given by 2, particle at low energies in a LS violation possible scenario. and thus, we rewrite the Equation (10) in the form = /2 We can see that the energy prole of the system is in£uenced 2 2 2 by the LS violation through the cyclotron frequency which 1 1 (11) is dened in terms of the parameters associated with the LS 2 + − 2 + − = 0, 4 4 violation background, , , and . For the particular case where we recover the result obtained in the [38]. In where we dene the parameters addition, = by0 making implies in , where we →0 → 0 2 2 2 2 obtain free energy of a Dirac particle in an isotropic 1 E 1 = − − − + (1+) ; environment. 2 2 2 (12) 2 1 = + (1−) . 3. Effects of a Hard-Wall Potential 2 By imposing that when and , we can write in terms of an unknown→ 0 function → 0 →as follows:∞ In this section, we conne the system described in the previous section to a region where the hard-wall potential is present. /2 −(1/2) is conning potential is characterized by the following (13) = | | . boundary condition: en, by substituting Equation (13) into Equation (11), (18) we have the second order dierential equation 0 = 0, which means that the radial wave function vanishes at a 2 xed radius . e hard-wall potential has been studied in ᐈ ᐈ 2 0 2 ᐈ ᐈ ᐈ ᐈ 1 (14) + ᐈ ᐈ + 1 − − ᐈ ᐈ + − = 0. noninertial eects [50–53], in relativistic scalar particle ᐈ ᐈ 2 2 systems in a spacetime with a spacelike dislocation [54], in Equation (14) is called in the literature as the con£uent a geometric approach to conning a Dirac neutral particle hypergeometric equation and the function 1 1 , in analogous way to a quantum dot [55], in a 2 = , ; with and , is the con£uent Landau-Aharonov-Casher system [56] and on a harmonic hypergeometric = | |/2 function+ (1/2) − [49]. It is well= | |known+ 1 that the con- oscillator in an elastic medium with a spiral dislocation [57]. £uent hypergeometric series becomes a polynomial of degree en, let us consider the particular case which 2 , while ≫1 by imposing that , where . With this the 0 and the parameter of the con£uent hypergeometric condition, we obtain =− = 0, 1, 2, 3, . . . function are xed and that the parameter of the con£uent hypergeometric function to be large. In the way, a con£u- E 2 2 ᐈ ᐈ ,, 1ᐈ 1 ᐈ 1 1 (1 + ) =± + + 2 + ᐈ + (1 − )ᐈ + + (1 − ) + , ent hypergeometric function can be written in the form 2ᐈ 2 ᐈ 2 2 2 (15) [54, 58, 59]: 4 Advances in High Energy Physics

cos numbers , , and xed. en, from now on, let us consider (19) 1 1, , 0∝ 20( − 2) + − . Dirac particles in a LS violation background with energy 4 2 given in Equation (17), localized and noninteracting, but now By substituting Equations (13) and (19) into Equation in contact with a thermal reservoir at the nite temperature (18), we obtain T. e microscopic states of this system are characterized by

the set of principal quantum numbers {1 2 3 [60]. 2 2 erefore, given a microscopic state , the, energy, ,..., of that} state E 2 2 ᐈ ᐈ ,, 1 2 1ᐈ 1 ᐈ 3 can be written in the form ≈± + + + (1 − ) + + 0 + ᐈ + (1 − )ᐈ + , 2 2ᐈ 2 ᐈ 4 (20) (23) with . Equation (20) represents the energy levels = =1 + , of a Dirac = 0, eld 1, 2, subject . . . to the eects of a Landau-type quanti- 2 zation induced by the LS violation eects and under the eects where of a hard-wall potential. By comparing Equations (15) and 2 (20), we can note that the presence of the hard-wall potential ᐈ ᐈ (24) ᐈ 1 ᐈ 1 in the system modies the relativistic energy levels. is mod- = + ᐈ + (1 − )ᐈ + + (1 − ) + 1 + . ᐈ 2 ᐈ 2 ication can be seen through the degenrescence that is broken erefore, the canonical partition function is given by [31, 60] and the main quantum number that now has a quadratic char- − −∑ =1 + /2 acter in contrast to the previous section. In addition, by taking ( ( )) (25) = = , , ,..., , we obtain the energy spectrum of a Dirac eld subject 1 2 3 →0to a hard-wall conning potential in the Minkowski where −1. Since there are no terms of interaction, the spacetime. multiple =sum is factorized, source damage to a very simple expression, that is, 3.1. Nonrelativistic Limit. In this section, we are interested ∞ in obtaining the nonrelativistic energy levels of a Dirac eld − + /2 ( ( )) 1 subject to the Landau-type quantization induced by the LS = =0 = , (26) violation eects and the hard-wall potential given in Equation where (18). In this sense, we can write Equation (20) as follows ∞ − /2 − + /2 E 1 ,, ( ( )) − (27) = =0 = , 2 2 2 1 − 1 1ᐈ 1 ᐈ 3 2 2 2 ᐈ ᐈ is the partition function of a single Dirac particle with energy ≈ 1+ + + (1−) ++ 0 + ᐈ+ (1−)ᐈ + . 2 2ᐈ 2 ᐈ (21)4 levels given in Equation (17). With the partition function (27) in hand, we can deter- en, by following the same steps from Equation (16) to mine thermodynamic quantities. We start with the free energy Equation (17), we have of Helmhotz by particle, which is given by the expression [60]

2 2 2 lim ln ln 1 1 1 1 1 1ᐈ 1 ᐈ 3 =− →∞ =− ; ,, 2 ᐈ ᐈ ≈ + + (1 − ) + + 0 + ᐈ + (1 − )ᐈ + . 2 2 2 2 2 2ᐈ 2 ᐈ 4 ᐈ ᐈ ln −/ 1ᐈ 1 ᐈ 1 1 1+ (22) = + ᐈ+ (1−)ᐈ + + (1−) + + 1 − . (28) 2 2ᐈ 2 ᐈ 2 2 2 Equation (22) represents the nonrelativistic energy levels From Equation (28), we can calculate all the thermody- of a Dirac particle under the eects of a Landau-type quanti- namic properties of the system. In particular, entropy by par- zation and conned in a region where a hard-wall potential is ticle as a function of temperature present. Note that the energy prole of the system at low ener- () gies is in£uenced by the LS violation and the hard-wall poten- −/ ln −/ tial by the presence of the parameters , , , and , respectively. −/ (29) 0 =− =− 1 − + , In addition, in contrast to Equation (17), the main quantum 1 − number of the energy spectrum given in Equation (22) has which we can obtain the specic heat quadratic character. In addition, the degeneracy is broken. ese latter two eects are caused by the presence of the hard- 2 −/ wall potential. We can observe that by making into −/ 2 (30) →0 = = . Equation (22) we obtain the energy spectrum of a Dirac par- 1 − ticle under conning eects of a hard-wall potential. In addition, we can also calculate the intern energy by particle as a function of temperature () 1 ln ln 1 (31) 4. Thermodynamic Properties =−2 =− ; 1 1 1 1+ ᐈ ᐈ / = 2 + 2 ᐈ + 2 (1 − )ᐈ + + 2 (1 − ) + 2 + −1 . In order to obtain the thermodynamic magnitudes of a Dirac ᐈ ᐈ ( ) particle in a LS violation background, we converged, rst, to We note that the thermodynamic quantities calculated and dene the partition function from the Landau-type energy dened in Equations (28)–(31) are in£uenced by the LS vio- levels given in Equation (17) considering the quantum lation. is in£uence is due to the presence of the cyclotron Advances in High Energy Physics 5 frequency in the denition of the partition function (27) [2] S. M. Carroll, G. B. Field, and R. Jackiw, “Limits on a Lorentz- which, in turn, gives us the free energy of Helmhotz by particle and parity-violating modication of electrodynamics,” Physical (28) from which we can describe the thermodynamic prop- Review D, vol. 41, no. 4, pp. 1231–1240, 1990. erties of the system, since the cyclotron frequency is dened [3] D. Colladay and V. A. Kostelecký, “CPT violation and the stand- in terms of parameters associated with the LS violation in a ard model,” Physical Review D, vol. 55, no. 11, pp. 6760–6774, possible low energy scenario, , , and . 1997. [4] D. Colladay and V. A. Kostelecký, “Lorentz-violating extension of the standard model,” Physical Review D, vol. 58, no. 11, 5. Conclusion Article ID 116002, 1998. [5] H. Belich, L. P. Collato, T. Costa-Soares, J. A. Helayël-Neto, We have analyzed a Dirac eld in an anisotropic spacetime and M. T. D. Orlando, “Magnetic moment generation from where there is the presence of a background vector eld that non-minimal couplings in a scenario with Lorentz-symmetry governs the LS violation. en, for our analysis, we have cho- violation,” e European Physical Journal C, vol. 62, no. 2, pp. sen a particular case which characterizes a possible scenario 425–432, 2009. of LS violation that induces a relativistic Landau-type quanti- [6] M. A. Ajaib, “Understanding Lorentz violation with Rashba zation, and then, with this information coming from this back- interaction,” International Journal of Modern Physics A, vol. ground, we calculate the relativistic Landau-type energy levels 27, Article ID 1250139, 2012. of a Dirac eld in this possible scenario. In addition, we con- [7] R. Casana, M. M. Ferreira, E. da Hora Jr., and A. B. F. Neves, ne the Dirac eld in a region where a hard-wall potential is “Maxwell-Chern-Simons vortices in a CPT-odd Lorentz- present and, for a particular case, we dene the relativistic violating Higgs electrodynamics,” e European Physical Journal energy prole of the system, which is modied due to the C, vol. 74, no. 9, p. 3064, 2014. presence of the hard-wall potential and, consequently, the [8] H. Belich, F. J. L. Leal, H. L. C. Louzada, and M. T. D. Orlando, degeneracy is broken. In both cases, we calculated the energy “Investigation of the KF-type Lorentz-symmetry breaking levels in the low energy limit, where it is possible to notice that gauge models with vortexlike congurations,” Physical Review D, vol. 86, no. 12, Article ID 125037, 2012. there are in£uences of the LS violation on the energy proles of the analyzed systems. [9] M. B. Cruz, E. R. Bezerra de Mello, and A. Y. Petrov, “Casimir With the Landau-type energy levels in a possible scenario eects in Lorentz-violating scalar eld theory,” Physical Review D, vol. 96, no. 4, Article ID 045019, 2017. of LS violation at low energies, through the partition function and, consequently, of the free energy of Helmholtz by particle, [10] M. B. Cruz, E. R. Bezerra de Mello, and A. Y. Petrov, “ermal corrections to the Casimir energy in a Lorentz-breaking scalar we can calculate thermodynamic quantities of this system, eld theory,” Modern Physics Letters A, vol. 33, no. 20, Article ID which are in£uenced by the LS violation background. is 1850115, 2018. in£uence is due to the dependence of the thermodynamic [11] V. A. Kostelecký and M. Mewes, “Cosmological constraints on quantities of the cyclotron frequency, which in turn is dened Lorentz violation in electrodynamics,” Physical Review Letters, in terms of parameters associated with the LS violation. vol. 87, no. 25, Article ID 251304, 2001. [12] V. A. Kostelecký and M. Mewes, “Signals for Lorentz violation Data Availability in electrodynamics,” Physical Review D, vol. 66, no. 5, Article ID 056005, 2002. No data in this paper. [13] L. R. Ribeiro and C. Furtado, E. Passos, “An analogy of the quantum hall conductivity in a Lorentz-symmetry violation setup,” Journal of Physics G: Nuclear and Particle Physics, vol. Conflicts of Interest 39, no. 10, Article ID 105004, 2012. [14] H. Belich, E. O. Silva, M. M. Ferreira Jr., and M. T. D. Orlando, e authors declare that they have no con£icts of interest. “A h a r o n o v -Bohm-Casher problem with a nonminimal Lorentz- violating coupling,” Physical Review D, vol. 83, no. 12, Article ID 125025, 2011. Acknowledgments [15] K. Bakke and H. Belich, “Quantum holonomies based on the Lorentz-violating tensor background,” Journal of Physics G: e authors would like to thank CNPq (Conselho Nacional Nuclear and Particle Physics, vol. 40, no. 6, Article ID 065002, de Desenvolvimento Cientíco e Tecnológico - Brazil) for 2013. nancial support. R. L. L. Vitória was supported by the CNPq [16] K. Bakke and H. Belich, “On the Lorentz symmetry breaking project No. 150538/2018-9. eects on a Dirac neutral particle inside a two-dimensional quantum ring,” e European Physical Journal Plus, vol. 129, no. 7, p. 147, 2014. References [17] H. Belich and K. Bakke, “A spin-orbit coupling for a neutral particle from Lorentz symmetry breaking eects in the CPT- [1] V. A. Kostelecký and S. Samuel, “Spontaneous breaking of odd sector of the Standard Model Extension,” International Lorentz symmetry in string theory,” Physical Review D, vol. 39, Journal of Modern Physics A, vol. 30, no. 22, Article ID 1550136, no. 2, pp. 683–685, 1989. 2015. 6 Advances in High Energy Physics

[18] K. Bakke and H. Belich, “On the influence of a Rashba-type [33] M. Eshghi and H. Mehraban, “Study of a 2D charged particle coupling induced by Lorentz-violating effects on a Landau confined by a magnetic and AB flux fields under the radial scalar system for a neutral particle,” Annals of Physics, vol. 354, pp. power potential,” e European Physical Journal Plus, vol. 132, 1–9, 2015. no. 3, 2017. [19] H. Belich, C. Furtado, and K. Bakke, “Lorentz symmetry [34] B. Hamil and M. Merad, “Dirac and Klein-Gordon oscillators breaking effects on relativistic EPR correlations,” e European on anti-de Sitter space,” e European Physical Journal Plus, Physical Journal C, vol. 75, no. 9, p. 410, 2015. vol. 133, no. 5, 2018. [20] K. Bakke and H. Belich, “Relativistic geometric quantum phases [35] B. Khosropour, “Statistical aspects of the Klein‑Gordon from the Lorentz symmetry violation effects in the CPT-even oscillator in the frame work of GUP,” Indian Journal of Physics, gauge sector of Standard Model Extension,” International vol. 92, no. 1, pp. 43–47, 2017. Journal of Modern Physics A, vol. 30, no. 33, Article ID 1550197, [36] R. L. L. Vitória and H. Belich, “Harmonic oscillator in an 2015. environment with a pointlike defect,” Physica Scripta, vol. 94, [21] K. Bakke and H. Belich, “Relativistic Landau‑He‑McKellar‑ no. 12, Article ID 125301, 2019. Wilkens quantization and relativistic bound states solutions for [37] K. Bakke and H. Belich, “A Landau-type quantization from a a Coulomb-like potential induced by the Lorentz symmetry Lorentz symmetry violation background with crossed electric breaking effects,” Annals of Physics, vol. 333, pp. 272–281, and magnetic fields,” Journal of Physics G: Nuclear and Particle 2013. Physics, vol. 42, no. 9, Article ID 095001, 2015. [22] H. F. Mota, H. Belich, and K. Bakke, “Relativistic quantum [38] K. Bakke and H. Belich, Spontaneous Lorentz Symmetry scattering yielded by Lorentz symmetry breaking effects,” Violation and Low Energy Scenarios, LAMBERT Academic International Journal of Modern Physics A, vol. 32, no. 23-24, Publishing, Saarbrücken, 2015. Article ID 1750140, 2017. [39] H. Belich, T. Costa-Soares, M. M. Ferreira, and J. A. Helayël-Neto [23] K. Bakke and H. Belich, “On a relativistic scalar particle subject Jr., “Non-minimal coupling to a Lorentz-violating background to a Coulomb-type potential given by Lorentz symmetry and topological implications,” e European Physical Journal C, breaking effects,” Annals of Physics, vol. 360, pp. 596–604, 2015. vol. 41, no. 3, pp. 421–426, 2005. [24] K. Bakke and H. Belich, “On the harmonic-type and linear-type [40] K. Bakke and H. Belich, “On the influence of a Coulomb-like confinement of a relativistic scalar particle yielded by Lorentz potential induced by the Lorentz symmetry breaking effects on symmetry breaking effects,” Annals of Physics, vol. 373, pp. the harmonic oscillator,” e European Physical Journal Plus, 115–122, 2016. vol. 127, no. 9, 2012. [25] R. L. L. Vitória, H. Belich, and K. Bakke, “A relativistic quantum [41] H. Belich, T. Costa-Soares, M. A. Santos, and M. T. D. Orlando, oscillator subject to a Coulomb-type potential induced by effects “Violação da simetria de Lorentz,” Revista Brasileira de Ensino of the violation of the Lorentz symmetry,” e European Physical de Física, vol. 29, no. 1, pp. 57–64, 2007. Journal Plus, vol. 132, no. 1, 2017. [42] R. Lehnert, “CPT and Lorentz-invariance violation,” Hyperfine [26] R. L. L. Vitória, H. Belich, and K. Bakke, “Coulomb-type Interactions, vol. 193, no. 1–3, pp. 275–281, 2009. interaction under Lorentz symmetry breaking effects,” Advances [43] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved in High Energy Physics, vol. 2017, Article ID 6893084, 5 pages, Space, Cambridge University Press, Cambridge, UK, 1982. 2017. [44] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: e [27] R. L. L. Vitória, K. Bakke, and H. Belich, “On the effects of Nonrelativistic eory, Pergamon, Oxford, 3rd edition, 1977. the Lorentz symmetry violation yielded by a tensor field on the interaction of a scalar particle and a Coulomb-type field,” [45] K. Bakke and C. Furtado, “Bound states for neutral particles in a Annals of Physics, vol. 399, pp. 117–123, 2018. rotating frame in the cosmic string spacetime,” Physical Review D, vol. 82, no. 8, Article ID 084025, 2010. [28] R. L. L. Vitória and H. Belich, “Effects of a linear central potential induced by the Lorentz symmetry violation on the [46] C. Furtado, J. R. Nascimento, and L. R. Ribeiro, “Landau Klein-Gordon oscillator,” e European Physical Journal C, vol. quantization of neutral particles in an external field,” Physics 78, no. 12, 2018. Letters A, vol. 358, no. 5-6, pp. 336–338, 2006. [29] R. L. L. Vitória and H. Belich, “A central potential with a massive [47] L. R. Ribeiro, C. Furtado, and J. R. Nascimento, “Landau levels scalar field in a Lorentz symmetry violation environment,” analog to electric dipole,” Physics Letters A, vol. 348, no. 3–6, Advances in High Energy Physics, vol. 2019, Article ID 1248393, pp. 135–140, 2006. 13 pages, 2019. [48] K. Bakke, L. R. Ribeiro, and C. Furtado, “Landau quantization [30] X.-Q. Song, C.-W. Wang, and C.-S. Jia, “ ermodynamic for an induced electric dipole in the presence of topological properties for the sodium dimer,” Chemical Physics Letters, vol. defects,” Central European Journal of Physics, vol. 8, no. 6, pp. 673, pp. 50–55, 2017. 893–899, 2010. [31] H. Hassanabadi and M. Hosseinpour, “ ermodynamic [49] G. B. Aren and H. J. Weber, Mathematical Methods for properties of neutral particle in the presence of topological Physicists, Elsevier Academic Press, New York, 6th edition, defects in magnetic cosmic string background,” e European 2005. Physical Journal C, vol. 76, no. 10, 2016. [50] L. B. Castro, “Noninertial effects on the quantum dynamics of [32] A. N. Ikot, B. C. Lutfuoglu, M. I. Ngwueke, M. E. Udoh, S. scalar bosons,” e European Physical Journal C, vol. 76, no. Zare, and H. Hassanabadi, “Klein‑Gordon equation particles in 2, 2016. exponential-type molecule potentials and their thermodynamic [51] R. L. L. Vitória, C. Furtado, and K. Bakke, “Linear confinement properties in D dimensions,” e European Physical Journal of a scalar particle in a Gödel-type spacetime,” e European Plus, vol. 131, no. 12, 2016. Physical Journal C, vol. 78, no. 1, 2018. Advances in High Energy Physics 7

[52] L. C. N. Santos and C. C. Barros Jr., “Relativistic quantum motion of spin-0 particles under the influence of noninertial effects in the cosmic string spacetime,” e European Physical Journal C, vol. 78, no. 1, 2018. [53] K. Bakke, “Torsion and noninertial effects on a nonrelativistic Dirac particle,” Annals of Physics, vol. 346, pp. 51–58, 2014. [54] R. L. L. Vitória and K. Bakke, “Aharonov-Bohm effect for bound states in relativistic scalar particle systems in a spacetime with a spacelike dislocation,” International Journal of Modern Physics D, vol. 27, no. 2, Article ID 1850005, 2018. [55] K. Bakke, “A geometric approach to confining a Dirac neutral particle in analogous way to a quantum dot,” e European Physical Journal B, vol. 85, no. 10, 2012. [56] K. Bakke, “On the rotating effects and the Landau-Aharonov- Casher system subject to a hard-wall confining potential in the cosmic string spacetime,” International Journal of eoretical Physics, vol. 54, no. 7, pp. 2119–2126, 2015. [57] A. V. D. M. Maia and K. Bakke, “Harmonic oscillator in an elastic medium with a spiral dislocation,” Physica B: Condensed Matter, vol. 531, pp. 213–215, 2018. [58] M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1965. [59] K. Bakke, “Rotating effects on the Dirac oscillator in the cosmic string spacetime,” General Relativity and Gravitation, vol. 45, no. 10, pp. 1847–1859, 2013. [60] S. Salinas, Introdução à Física Estatística, Edusp, São Paulo, Brazil, 2nd edition, 2005. Journal of International Journal of Advances in The Scientifc Applied Bionics Chemistry Engineering World Journal Geophysics and Biomechanics Hindawi Hindawi Hindawi Publishing Corporation Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

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