Hindawi Advances in High Energy Physics Volume 2020, Article ID 4208161, 7 pages https://doi.org/10.1155/2020/4208161 Research Article Effects of a Landau-Type Quantization Induced by the Lorentz Symmetry Violation on a Dirac Field 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.