# Relativistic Quantum Motion of an Electron in Spinning Cosmic String Spacetime in the Presence of Uniform Magnetic Field and Aharonov-Bohm Potential

Authors: Márcio M. Cunha; Edilberto O. Silva
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

In this manuscript, we study the relativistic quantum mechanics of an electron in external fields in the spinning cosmic string spacetime. We obtain the Dirac equation and write the first- and second-order equations from it, and then, we solve these equations for bound states. We show that there are bound state solutions for the first-order equation Dirac. For the second-order equation, we obtain the corresponding wave functions, which depend on the Kummer functions. Then, we determine the energies of the particle. We examine the behavior of the energies as a function of the physical parameters of the model, such as rotation, curvature, magnetic field, Aharonov-Bohm flux, and quantum numbers. We find that, depending on the values of these parameters, there are energy nonpermissible levels.


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## Body

## 1. Introduction

Symmetry is key ingredient in the description of natural phenomena. The notion of symmetry is an essential feature in several areas of physics. In this context, the well-known Noether’s theorem [1] establishes a connection between symmetry and conservation laws of relevant physical quantities. In quantum mechanics, we often use symmetry to obtain crucial results concerning angular momenta operators [2]. Likewise, symmetry is relevant in the topic of quantum information [3]. Also, symmetry is essential in the framework of relativity [4], and for this reason, it is indispensable in research areas such as particle physics [5] and cosmology [6].A pertinent question in the research areas cited above refers to think about the consequences of symmetry-breaking in a given physical system. It can happen in several contexts, like in phase transitions, for example [7–9]. Another example of symmetry-breaking occurs in solids: we can create topological defects like disclinations and dislocations [10, 11] from the Volterra process [12].Topological defects can emerge in a large number of physical systems covering themes such as liquid crystals [13], graphene physics [14, 15], magnetism [16], and cosmology [17]. Recent studies also have reported the importance of topological defects in Life Sciences [18, 19]. In cosmology, defects in the spacetime topology can be viewed as a possible consequence of the evolution of the early universe, which has suffered phase transitions due to the temperature decreasing and the process of expansion [20, 21].In this contribution, we are involved in studying the topological defect known as a cosmic string. A cosmic string is a linear defect, similar to a flux tube in type-II superconductors [20]. The spacetime around such defect has a conical symmetry, being identical to the case of a disclination [22]. The concept of a cosmic string was introduced in the literature by Kibble [23]. An intriguing facet in this subject refers to the quantum mechanical description of a particle in a region of the spacetime containing this defect. There are several works dealing with this issue, both in the nonrelativistic and relativistic scenarios. Reference [24], for instance, analyzes the hydrogen atom in a spacetime of a cosmic string. Reference [25] considers the problem of a relativistic electron in the presence of both Coulomb and scalar potentials in the cosmic string spacetime. Results about vacuum polarization in a cosmic string spacetime were reported in Ref. [26]. Again, the cosmic string spacetime was considered as a background to examine relativistic oscillators [27–30], quantum phases [31], and fermionic currents [32].A relevant issue in this context consists of taking into consideration the influence of electromagnetic fields in the quantum particle motion. Landau levels [33] and the Aharonov-Bohm effect [34, 35], for instance, are essential ingredients in the investigation of quantum systems even in a flat spacetime. Landau levels are a quantum analog of classical cyclotron motion, while the Aharonov-Bohm effect reveals the significance of the vector potential in the quantum world. Then, studying the contribution of these magnetic interactions to the quantum mechanical description of a system in spacetime having a topological defect is a natural development. In Ref. [36], for example, it was considered the interaction of a point charge with a magnetic field in a spacetime with a distortion. References [37–44] are examples of studies dealing with Landau levels and the Aharonov-Bohm effect in the presence of topological defects. Reference [45] deals with a quantum ring in graphene with a topological defect and a magnetic field. Reference [46] investigates an analog of the Aharonov-Bohm effect for bound states in a spacetime containing a spacelike dislocation. In particular, the inclusion of electromagnetic interactions in the case of a cosmic string background also has been considered. For instance, Ref. [47] analyzes the quantum dynamics of a charged particle in the presence of a magnetic field and scalar potential. Reference [48] examines different configurations of confined magnetic fields and investigates the existence of induced vacuum fermionic currents.On the other hand, we can be interested in analyzing the behavior of rotating systems. It is a pertinent question, since large structures in the Universe, such as black holes and galaxies, have angular momentum due to rotation [49]. Also, it is possible to conceive cosmologic models with rotation [50]. The study of quantum systems in spinning spacetimes also has been reported in the literature [51–53]. Noninertial effects in quantum systems can provide novel theoretical predictions and feasible experimental developments. For instance, quantum phases in rotating systems can emerge, in analogy to the Aharonov-Bohm effect [54, 55]. It is also possible to establish a relation between the Hall effect and the inertial forces [56]. Besides, if a given system is rotating, physical properties like spin transport [57, 58] and electronic structure [59] are modified. A system can present magnetization due to rotation, like in the Barnett effect [60]. While a magnetic field produces a spin-field coupling, resulting in the anomalous Zeeman effect [61], rotation produces an analog effect, due to the spin-rotation coupling [62]. Thus, rotation can contribute similarly to a magnetic field in the dynamics of a quantum system. More, noninertial effects are an interesting issue when the spacetime contains topological defects. In this case, we can include noninertial effects and the topological defects in the quantum mechanical description by employing the same tool. More specifically, we can use a metric tensor to a spinning spacetime with a topological defect [63]. The spacetime of a spinning cosmic string has been considered as background for several problems involving quantum systems [64]. For instance, Ref. [65] deals with the Schrödinger equation in that spacetime. Reference [66] analyzed the bound states for neutral particles in a rotating frame of a cosmic string. Likewise, Ref. [67] investigates rotating effects on a Landau-Aharonov-Casher System in the spacetime of a cosmic string.As we already have mentioned, rotation can present similarities to the electromagnetic fields. This way, it is also an attractive question examining how the electromagnetic interactions affect the particle quantum motion of a rotating system in the presence of a topological defect. Reference [68], for instance, is a recent work dealing with both topological and noninertial effects in the presence of an Aharonov-Bohm potential. Reference [69] addresses the problem of a spinless relativistic particle subjected to a uniform magnetic field in the spinning cosmic string spacetime. The Dirac oscillator in the spacetime of a cosmic string in the presence of the Aharonov-Casher effect was analyzed in Ref. [70]. The presence of noninertial effects also was considered in this context [71]. Reference [72] treats the problem of a charged half-spin particle depicted by the Dirac equation in the presence of a uniform magnetic field in the rotating cosmic string spacetime. A meaningful aspect in this context consists of analyzing how different configurations of magnetic fields affect the quantum particle motion. In particular, in this contribution, we choose to deal with a superposition of two different configurations, due to their fundamental aspects: a uniform magnetic field, which is related to the raising of Landau levels in quantum systems, and an Aharonov-Bohm flux, since it provides a new significance to the role of the electromagnetic interactions in the quantum theory, which can be manifested even for bound states [73]. Besides, several analogs of the AB effect can emerge in spacetimes with topological defects [74–76].In this paper, we study the relativistic quantum mechanics of an electron in the presence of both a uniform magnetic field and Aharonov-Bohm potential in the spinning cosmic string spacetime. In other words, we solve the Dirac equation in this scenario and investigate how the rotation, curvature, and external magnetic fields affect the wave functions and energies of the electron. Although the existence of a previous study deals with the Dirac equation in the presence of noninertial effects and uniform magnetic field, in Section3 of Ref. [72], here, we introduce some new ingredients, as well as a different framework: more explicitly, we have the following contributions: (i) the tetrad basis we adopt leads to a zero affine connection in the flat space, which does not occur in the previous study; (ii) the presence of the Aharonov-Bohm flux tube. We show explicitly that the presence of this effect modifies the energy levels of the particle; (iii) we also study first-order solutions for the Dirac equation and find particular bound state solutions for it; (iv) we employ an ansatz for the solutions whose spinor is an eigenstate of the momentum angular operator J∧z [47].The manuscript is organized as follows. In Section2, we present some algebraic elements necessary to construct the field equations in curved spacetime and write the Dirac equation describing the quantum motion of the electron in the presence of external magnetic fields in the spinning cosmic string background. In Section 3, we deal with first-order solutions and study the existence of isolated solutions for the particular case of a particle at rest. In Section 4, we take our attention to the case when the energy of the particle is different from its rest energy. We map the Dirac equation problem in curved space with minimal coupling into a Sturm-Liouville problem for the upper component of the Dirac spinor, and using an appropriate ansatz, we derive the radial equation. We solve the radial equation and find the wave functions and energies of the particle. We make a detailed discussion of the results and also comparisons with other studies in the literature. In Section 5, we present our conclusions. In our work, we use natural units, ℏ=c=G=1.


## 2. Dirac Equation in the Spinning Cosmic String Spacetime

In this section, we briefly present the tools needed to construct the Dirac equation in the conical spacetime in the presence of noninertial effects. The first step consists of taking a look at the metric tensor characterizing this geometry. Next, we will choose an appropriate tetrad basis and implement the field configuration involved through the performing of a minimal substitution. The spacetime induced by a rotating cosmic string is described by the metric [72, 77, 78].
(1)ds2=dt+adφ2−dr2−α2r2dφ2−dz2,where −∞<z<∞, r≥0, and 0≤φ≤2π. The parameter α is related to the linear mass density μ of the cosmic string through the relation α=1−4μ, and it runs in the interval 0,1. The quantity a=4J is the rotation parameter, with J representing the angular momentum of the spinning cosmic string. A remarkable feature of the spinning cosmic string spacetime is related to the appearance of closed timelike curves (CTCs) [79–81]. Such structures can take place depending on the relation between the radial coordinate, the rotation parameter, and the curvature parameter. Writing the metric (1) in the form
(2)ds2=dt2+2adtdφ−α2r2−a2dφ2−dr2−dz2,such a structure appears if α2r2−a2<0. CTCs are one of the most controversial aspects of modern physics [82]. It is possible to find in the literature works dealing with the concepts of causality [83], chronology protection [84], time travel [85], and flow information [86] in the framework of closed timelike curves. In our study, we consider the quantum motion only in regions such that α2r2−a2>0. More explicitly, we will consider the following restriction on the radial coordinate:
(3)r>aα.This type of restriction on the radial coordinate also occurs, for example, in the case of a spacetime containing a time dislocation, as pointed in Ref. [51].The relativistic quantum dynamics of a spin-1/2 particle interacting with external magnetic fields in the rotating cosmic string spacetime is governed by the Dirac equation
(4)iγμx∇μ+ieAμx−MΨx=0,where M is the mass of the particle and ∇μ is the covariant derivative for fermion fields defined as
(5)∇μ=∂μ+Γμx,and γμx are the Dirac matrices in the rotating cosmic string spacetime, which are defined in terms of the tetrad fields eaμ and Dirac matrices in the flat space γa in the following way:
(6)γμx=eaμxγa,where
(7)γa=γ0,γi,withγ0=100−1,γi=0σi−σi0,are the standard Dirac matrices and σi=σx,σy,σz are the usual Pauli matrices. The matrices of (6) satisfy the following relation:
(8)γμx,γνx=2gμνx.Also, in Equation (5), Γμx is the spin affine connection given by
(9)Γμx=14γaγbeaνx∂μebνx−Γμνσebσx,where Γμνσ are the Christoffel symbols of the second kind and eaμx is the tetrad field. The tetrad basis satisfies the relations
(10)eμaxeνbxηab=gμνx,(11)eμaxebμx=δba,(12)eaμxeνax=δνμ.Here, the Greek letters are used for tensor indices while the Latin letters are denoting Minkowski indices. We use the tetrad basis and its inverse defined as [77]
(13)eμax=10a00cosφ−rαsinφ00sinφrαcosφ00001,(14)eaμx=1asinφrα−acosφrα00cosφsinφ00−sinφrαcosφrα00001.For this choice, it can be shown that the nonvanishing affine connection is given by(15)Γμ=0,0,Γφ,0,withΓφ=i21−α∑z,where
(16)∑z=σz00σz,σz=100−1.By using the tetrad basis (14), the matrices of (6) can be written explicitly as
(17)γt=γ0−aγφ,γ0=100−1,(18)γr=0σr−σr0,γφ=0σφ−σφ0,with
(19)σr=0e−iφe+iφ0,σφ=1rα0−ie−iφie+iφ0,being the Pauli matrices in the curved spacetime.Since we are first interested in studying the solutions of the Dirac equation in its present form (Equation (4)), we need to write the corresponding system of first-order coupled differential equations. Let us assume the time-dependence of the wave functions together with the decomposition of the fermion field in the form
(20)Ψr,φ=e−iEtψ1r,φψ2r,φ,with
(21)ψ1r,φ=ψar,φψbr,φ=eimφf+rieim+1φf−r,(22)ψ2r,φ=ψcr,φψdr,φ=eimφg+rieim+1φg−r.The system we will analyze takes into account the particle is immersed in a region where there is a uniform magnetic field and also the potential due to a thin long solenoid along thez-axis. Having this field configuration in mind, we study the physical implications due to noninertial effects and the Aharonov-Bohm potential on the relativistic Landau quantization. We also take into account the translational invariance of the system along the z-direction, which allows us to eliminate the third direction (pz=z=0), and, consequently, we can consider only the planar motion [87–90]. Here, the four-potential of the electromagnetic field in Equation (4) has only the spatial component, Aμ=0,A, and the potential vector in the Coulomb gauge is specified by
(23)A=0,Aφ,0,with
(24)Aφ=−Aφ,1+Aφ,2,(25)Aφ,1=12αBr2,Aφ,2=ϕe1αr,where B is the magnetic field magnitude, ϕ=Φ/Φ0, Φ is the magnetic flux, and Φ0=2π/e is the quantum of magnetic flux along the solenoid. This configuration provides an superposition of magnetic fields in the z-direction
(26)B=Bz,1+Bz,2,with
(27)B1,z=B,Bz,2=ϕeδrαr,Note that the particle only interacts with the magnetic field due to the potential vectorAφ,1. Here, we are focused on studying the electron motion only in the r≠0 region, so that we can neglect the point interaction Bz,2 and, consequently, consider only regular wave functions.Using the results above, the Dirac equation (4) can be written as
(28)E−Mψ1+σri∂rψ2+σφi∂φ−eAφ−aE−121−ασzψ2=0,(29)E+Mψ2+σri∂rψ1+σφi∂φ−eAφ−aE−121−ασzψ1=0.At this point, we are ready to solve Equations (28) and (29) by considering two distinct circumstances:
(i)
Take our attention to isolated solutions of the first-order Dirac equation by imposing the conditionE=±M(ii)
By imposing the conditionE≠±M, we search the solutions of the second-order Dirac equationWe will show in the next two sections that there are bound state solutions for both cases and discuss their main physical properties. To distinguish each case in (i), in the next section, we use the superscripts (±) to label the quantities corresponding to E=±M.


## 3. Solution of the Equation of Motion toE=±M

To study the existence of isolated solutions of the Dirac equation (4), we must set E=±M in Equations (28) and (29). In literature, such solutions are known to be excluded from the Sturm-Liouville problem. The search for isolated solutions of the Dirac equation has been performed in different physical contexts [91–95]. Since condition (3) establishes a lower limit to the radial coordinate, given by rmin=a/α, the bound state solution must satisfy the normalization condition
(30)∫rmin∞Ψ¯γtΨrdr=∫rmin∞Ψ†Ψrdr−a∫rmin∞Ψ†γ0γφΨrdr,where Ψ¯=Ψ†γ0, which using the ansatz (20) and the matrices (18) and (17), takes the form
(31)∫rmin∞f+r2+f−r2+g+r2+g−r2rdr−aα∫rmin∞f+†rg−r+f−†rg+r+g+†rf−r+g−†rf+rdr.Equations (30) and (31) must be finite to ensure the existence of bound state solutions. By making E=+M in Equations (28) and (29) and using Equations (21) and (22), we get
(32)eiφi∂rψcr,φ+irαeiφi∂φ−eAφ−aMψcr,φ−i2rα1−αeiφψcr,φ=0,(33)e−iφi∂rψdr,φ−irαe−iφi∂φ−eAφ−aMψdr,φ−i2rα1−αe−iφψdr,φ=0,(34)i∂reiφψar,φ+irαeiφi∂φ−eAφ−aMψar,φ−i2αr1−αeiφψa=−2Mψdr,φ,(35)i∂re−iφψbr,φ−irαe−iφi∂φ−eAφ−aMψbr,φ−i2αr1−αe−iφψbr,φ=−2Mψcr,φ,which using Equation (25) can be written as
(36)dg++rdr−Lm+rαg++r+eBr2g++r=0,(37)dg−+rdr+Lm+1+rαg−+r−eBr2g−+r=0,(38)df++rdr−Lm+rαf++r+eBr2f++r=−2Mg−+r,(39)df−+rdr+Lm+1+rαf−+r−eBr2f−+r=2Mg++r,with
(40)Lm+=m−ϕ+aM+121−α,(41)Lm+1+=m+1−ϕ+aM−121−α.The solution of the coupled linear differential equation systems (36)–(39) is given by
(42)f++r=e−eBr2/4rLm+/αa2+a1M−eB2ΩaΓa+,(43)f−+r=eeBr2/4r−Lm+1+/αb2−b1MeB2−ΩbΓb+,(44)g++r=b1e−eBr2/4rLm+/α,(45)g−+r=a1eeBr2/4r−Lm+1+/α,with
(46)Ωa=12αLm++Lm+1+−α,Ωb=12αLm++Lm+1++α,where
(47)Γa+=Γ−Ωa,−12eBr2,(48)Γb+=ΓΩb,12eBr2,are upper incomplete Gamma functions [96], and a1, a2, b1, and b2 are constants. Analyzing the solutions (42) and (44), we note that e−eBr2/4 dominates over rLm+/α for any value of Lm+/α⩾0, in such way both solutions converge when r⟶rmin and r⟶∞. This will not occur for the function eeBr2/4 in the solutions (43) and (45). Moreover, since the incomplete Gamma functions Γa+ and Γb+ always diverge, then the function f++r will only converge when r⟶rmin if a1=0 while the function f−+r will always diverge when r⟶∞ and, therefore, will not be a square-integrable function. Thus, the only allowed solution for the equation systems (36)–(39) results
(49)f++r=a2e−eBr2/4rLm+/α,withLm+α⩾0,r>aα,with f−+r=g++r=g−+r=0, and the requirement a1=b1=b2=0. Solution (49) satisfies Equation (31) and constitutes a bound state solution for the case E=M, i.e., an isolated solution to the Dirac equation (4) in the metric spacetime (1). Note there is another particular solution for this case when we reverse the signal of the magnetic field or the charge, i.e., eB<0. In this case, the only nonzero component of the normalizable solution Ψ is (Equation (43) with b1=0).
(50)f−+r=b2eeBr2/4r−Lm+1+/α,withLm+1+α⩽0,r>aα.Proceeding in an analogous way, now we makeE=−M in Equations (28) and (29). We find the system of equations
(51)df+−rdr−Lm−rαf+−r+eBr2f+−r=0,(52)df−−rdr+Lm+1−rαf−−r−eBr2f−−r=0,(53)dg+−rdr−Lm−rαg+−r+eBr2g+−r=2Mf−−r,(54)dg−−rdr+Lm+1−rαg−−r−eBr2g−−r=−2Mf+−r,with
(55)Lm−=m−ϕ−aM+121−α,(56)Lm+1−=m+1−ϕ−aM−121−α.The solution of the coupled linear ordinary differential equation systems (51)–(54) is given by
(57)f+−r=c1e−eBr2/4rLm−/α,(58)f−−r=d1eeBr2/4r−Lm+1−/α,(59)g+−r=e−eBr2/4rLm−/α−d1M−eB2ΛcΓc−+d2,(60)g−−r=eeBr2/4r−Lm+1−/αc1MeB2−ΛdΓd−+c2,with
(61)Λc=12αLm−+Lm+1−−α,Λd=12αLm−+Lm+1−+α,where
(62)Γc−=Γ−Λc,−12eBr2,Γd−=ΓΛd,12eBr2.By making the same analysis of the solutions as we have made for the caseE=M, i.e., analyzing the behavior of the functions for r⟶±∞, we find that the only solution that admits bound states is the (59) one. Thus, the appropriate solution for the case E=−M satisfying the normalization condition (31) is given by
(63)g+−r=d2e−eBr2/4rLm−/α,withLm−α⩾0,r>aα,with f+−r=f−−r=g−−r=0 and c1=c2=d1=0. Note that the solutions (49) and (63) are affected by rotation through Equations (40) and (55), respectively. Similarly to the previous case, if eB<0, the only nonzero component of the normalizable solution Ψ is (Equation (60) with c1=0)
(64)g−−r=c2eeBr2/4r−Lm+1−/α,withLm+1−α⩽0,eB<0,r>aα.Moreover, from the discussion of the solutions for each case above, we immediately see that only the first line of the normalization condition (31) is relevant in the current study.


## 4. Solution of the Equation of Motion toE≠±M

In this section, we solve the second-order equation toψ that we find from Equations (28) and (29). The solution of this equation is different from that one calculated in the previous section and allows us to obtain an expression for the particle energies. By isolating ψ2 in Equation (29) and replacing in Equation (28), we are able to write the second-order differential equation for ψ1 as
(65)E2−M2ψ1+∂r2ψ1+1r∂rψ1−1αrσze∂rAφψ1+1α2r2∂φ+ieAφ+i1−α2σz+iaE2ψ1=0.Using the decomposition of the fermion field (21) together with Equations (23)–(25), we obtain the radial equation for f+r(66)d2dr2+1rddr−Lm2α2r2−e2B2r24+km2f+r=0,where
(67)km2=E2−M2+eBαLm+eB,(68)Lm=m−ϕ+1−α2+aE.Equation (66) is the confluent hypergeometric equation, and its solution is well known. Thus, it can be shown that the solution to ψa is
(69)ψar,φ=anmeB21+Lm/α/2eimφrLm/αe−eBr2/4×1F1121+Lmα−k22eB,1+Lmα,12eBr2,where F11a,b,z denotes the confluent hypergeometric function of the first kind or Kummer’s function Ma,b,z and anm the normalization constant. It can be shown that the hypergeometric function F11a,b,z has a divergent behavior for large values of z. Because of this, bound state solutions for Equation (69) are only possible if we impose that this function becomes a polynomial of degree n. For this to be accomplished, we require that 1/2+Lm/2α−k2/2Be=−n, where n∈ℤ∗, with ℤ∗ denoting the set of the nonnegative integers. However, as we can see in Equation (68), the absolute value of the effective angular moment L is defined in terms of the energy E. In this way, to obtain the energy eigenvalues from the above condition, we must consider Lm>0 and Lm<0, respectively, and then solve them for E. By making this, we get
(70)En>=±2neB+M2,(71)En,m<=−aeBα±1αa2e2B2+αQn,m,with the following requirement:
(72)a2e2B2+αQn,m⩾0,where
(73)Qn,m=αeB2n−2αm−ϕ+12+1+αM2.In Equations (70) and (71), the superscripts (>,<) refer to the energies calculated for Lm>0 and Lm<0, respectively. A similar equation for f−r can be immediately obtained by making m⟶m+1 and 1+α/2⟶−1+α/2 in Equation (66). We find
(74)d2dr2+1rddr−Lm+12α2r2−e2B2r24+km+12f−r=0,where
(75)km+12=E2−M2+eBαLm+1−eB,Lm+1=m+1−ϕ−1−α2+aE.The functionψbr,φ is found by solving Equation (74). The result is
(76)ψbr,φ=anmeB21+Lm+1/α/2eim+1φrLm+1/αe−eBr2/4×1F112+Lm+12α+km+122Be,1+Lm+1α,12eBr2.The energies obtained from (74) are
(77)En>=±2eBn+1+M2,(78)En,m+1<=−aeBα±1αa2e2B2+αQn,m+1,with the following requirement:
(79)a2e2B2+αQn,m+1⩾0,where(80)Qn,m+1=αeB2n−2αm+1−ϕ−12+1+αM2.Note that two other radial equations can be derived from the equation forψ2 from Equations (28) and (29). However, there is no need to solve them here because they have both energy and wave functions similar to those obtained above. Let us study the energies (70) and (71). First, we see that the energy En> depends only on the quantum number n and the magnetic field B. If we make n=0 in Equation (70), we get E0>=±M. As mentioned above, since we are solving the problem with the requirement that E≠±M, then these energies are excluded in the present case.Now let us investigate the energy levels we find. In all the energy profiles we address here, we usee=1, M=1. We also use solid lines to represent the energy of the particle and dashed lines for the energy of the antiparticle. In Figure 1, we show the profile of En> for this particular case, where we see clearly that E increases when the magnetic field is increased. The energies (70) and (71) denote the relativistic Landau levels in the present context. These energies can be directly compared with those obtained for the relativistic oscillator (Dirac oscillator) addressed in Ref. [77]. Although that scenario is different from the one we are exploring here, there are similarities between the profiles of the energy levels in both models. For example, the energy (47) of the Ref. [77] depends only on the frequency of the oscillator and the quantum number n. In our case, by defining the cyclotron frequency ωc=eB/M, Equation (70) results
(81)E~n>=±2nMωc+M2,which makes such a similarity clear. We can see that the energies (71) and (78) are the only ones that depend on all the physical parameters involved in the current problem. However, it is possible to verify that such spectra are similar. Because of this, we analyze in detail only the energy (71). One of the objectives in our analysis is to show that the combined effect of the external magnetic field, the Aharonov-Bohm flux, the rotation, and the curvature modifies the energy spectrum of the particle. As mentioned in Section 1, one of the new ingredients in the present model is the inclusion of the Aharonov-Bohm flux tube. Then, let us focus on the physical implications of this. We can access varied forms of energy profiles by attributing different values to the parameters of the model. Figure 2 shows the profile of En,m< as a function of ϕ by considering four different parameter choices. In these four configurations, we consider m=1 and n=0,5,10. For B=1.0, a=0.05 and α=0.01, and the energies are permissible only for values of flux larger than 1.37 (blue line), 1.33 (red line), and 1.30 (green line) (Figure 2(a)). Each value of flux corresponds to the energies E0,1<, E5,1<, and E10,1<, respectively. The values of n are chosen in such a way the separation between the energies becomes more notable. When this configuration is changed to B=10.0, a=1.0 and α=0.9 (Figure 2(b)), the energy levels are more spaced out and a gap between the particle and antiparticle energies emerges. By increasing the magnetic field to B=50.0 and maintaining the other parameters, we notice that the antiparticle energy changes faster than the energy of the particle and the spacing between the energy levels decreases (Figure 2(c)). For increasing B values, these effects tend to increase. In the fourth configuration, we use B=1.0, a=0.5 and α=0.99 (Figure 2(d)). When α approaches 1, we can see the energy E0,1< is only permissible for values of ϕ larger than 0.38 while E5,1< and E10,1< are defined for any value of ϕ. In the four settings displayed in Figure 2, we note that ∣En,m<∣ increases when n and ϕ are increased and the other parameters are kept fixed.Figure 1
Sketch of the energy levelsEn> (Equation (70)) as a function of the magnetic field B for different values of n. The positive energies are represented by solid lines and the negative by dashed lines.Figure 2
Sketch of the energy (Equation (71)) as a function of ϕ for m=1. In panel (a), for B=1.0, a=0.05 and α=0.01. In panel (b), B=10.0, a=1.0 and α=0.9. In panel (c), B=50.0, a=1.0 and α=0.9. In panel (d), B=1.0, a=0.5 and α=0.99. For a particular choice of parameters, ranges of ϕ where the energy of a state with a given n is forbidden appear.
(a)(b)(c)(d)Another interesting configuration is the behavior ofEn,m< as a function of ϕ for n=1, m=1, B=2.0, a=0.1, and some values of α (Figure 3). This configuration shows that when α is close to 1, the energy is defined to all the values of ϕ in the considered range. However, for small values of α, a range of ϕ where the energy is nonpermissible arises. In this process, the most pronounced variations occur in the antiparticle energy.Figure 3
Sketch of the energy (Equation (71)) as a function of ϕ for n=1, m=1, B=2, a=0.1, and different values of α. When α approaches 1, the energy is permissible for all values of ϕ>0. For small values of α, the energy is permissible only from a threshold value of ϕ. For α=0.01 (blue line) and α=0.03 (red line), these threshold values are ϕ=0.50 and α=1.12, respectively.We can also observe interesting effects in the profile ofEn,m< as a function of B. For this case, we consider n=1, m=1, a=0.1, and α=0.5 and investigate it for two sets of values of ϕ as illustrated in Figure 4. By carefully studying this configuration, we can observe several effects for values of ϕ smaller than 0.99 (Figure 4(a)). For ϕ=0.10 (blue line) and ϕ=0.60 (red line), the energy tends to decrease when B is increased while for ϕ=0.65 (green line), the energy decreases when B is increased to 5.0. At this value, an overlap between particle and antiparticle energy occurs. The energy increases for values of B higher than 5.0. For ϕ=0.68 (brown line), the energy of the particle tends to decrease when B is increased to 6.0 and then the behavior remains approximately stable. When we look at the energy of the antiparticle, we observe the opposite process. By increasing B to ϕ=4.0, the energy remains practically stable and then tends to increase as B is increased. When ϕ is changed to 0.99 (orange line), both the particle and antiparticle energies increase when B is increased. However, the variation on the antiparticle energy is larger. On the other hand, when we consider different values of ϕ higher than 1.0, the profiles are similar (Figure 4(b)). In this case, En,m< increases when both ϕ and B are increased.Figure 4
Sketch of the energy (Equation (71)) as a function of B for n=1, m=1, a=0.1, and α=0.5. In panel (a), we consider values of ϕ smaller than 1 while in panel (b), we use larger values. The physical implications due to the variation of ϕ on the energy spectrum are more evident for values of ϕ about 0.65. For values of ϕ larger than 1, the curves assume the same profile.
(a)(b)In Figure5, we investigate the profile of En,m< as a function of B for n=1, m=1, a=0.1, and different values of α by considering the cases where there is no Aharonov-Bohm flux, and also, considering it is present. In Figure 5(a), we consider ϕ=5.0. We can see that the presence of Aharonov-Bohm flux reveals that the energies are well defined for the considered α values. In the absence of Aharonov-Bohm flux, there are intervals of B associated with nonpermissible energies (Figure 5(b)). For values of α close to 1.0, the changes in the energies are not too evident, while for smaller values of α, the modifications in the energy are more evident, being the antiparticle energy the most affected. In both cases, the energy increases when B increases, and α decreases.Figure 5
Sketch of the energy (Equation (71)) as a function of B for n=1, m=1, a=0.1, and different values of α. In panel (a), ϕ=5, and in panel (b), ϕ=0. With the presence of Aharonov-Bohm flux, the energy of the state for a particular value of alpha is permissible for any value of B. In the absence of Aharonov-Bohm flux, the energy of a state with a particular value of α is nonpermissible for some values of B (blue and red lines on panel (b)). For ever decreasing α values, the antiparticle energy is more impacted. When α approaches 1 and the flux is zero, the modification in the energy levels is small (see the inset in panel (b)).
(a)(b)It is also important to examine the energy as a function ofα, for some values of n. Figure 6 illustrates this situation. For m=1, B=0.1, a=0.1, and ϕ=0.1, there are intervals of α leading to forbidden energies. This effect becomes more notable to larger values of n. Also, the energy increases when n increases. It is also interesting to explore the energy profiles by performing energy sketches as a function of two quantities of the model. For a given set of fixed parameters, for example, a=0.5, α=0.5, B=1.0, and ϕ=1, we have the profile of the energy levels as a function of n and m (Figure 7). We can clearly see that ∣En,m<∣ increases with n and m. The green solid bars denote the discrete energy values for a given m and n. On the other hand, when we investigate the behavior of (71) as a function of α and a for specific values of the other parameters, we see that the negative spectrum changes more rapidly when compared with the positive one (Figure 8). In the positive spectrum, both rotation and curvature lead to a linear change, except in the region with α<0.3 and arbitrary a. In the negative spectrum, we see that the curvature effects are more predominant in the region where α has values smaller than 0.2. In this region, any variation in the rotation parameter implies in an abrupt change in the energy spectrum. Modifications in the energies for small values of α are an expected manifestation in our analysis. Its physical implication is inherent in metric (1) and is an immediate consequence of the topological cone, since the curvature becomes larger for smaller values of α.Figure 6
Sketch of the energy (Equation (71)) as a function of α for m=1, B=0.1, a=0.1, ϕ=0.1, and different values of n. For values of α smaller than 0.25, the energy level E0,1 (blue line) is nonpermissible. The same occurs with the energy levels E3,1 (red line) and E5,1 (green line), where they are permissible only for α>0.19 and α>0.16, respectively.Figure 7
Sketch of the energy (Equation (71)) as a function of n and m for a=0.5, α=0.5, B=1, and ϕ=1.Figure 8
Sketch of the energy (Equation (71)) as a function of α and a for B=4, n=1, ϕ=2, and m=1.When we study the profile ofEn,m< as a function of ϕ and a, besides the characteristics displayed in Figures 2 and 3, we also see modifications due to noninertial effects (Figure 9). For this particular case, we see that for small values of ϕ and a, some energy levels are forbidden. This characteristic is observed for n=1, m=1, and α=0.1 (Figure 9(a)). By changing α to 0.9, this effect disappears, and the energies are set to all values of ϕ and a in the considered range (Figure 9(b)).Figure 9
Sketch of the energy (Equation (71)) as a function of ϕ and a for n=1 and m=1. In panel (a), α=0.1, and in panel (b), α=0.9.
(a)(b)To complete our analysis, we investigate the profile of the energy (71) as a function of magnetic field B and the magnetic flux through the solenoid, ϕ (Figure 10). Similarly to Figure 8, by fixing the other parameters, we see that the energy of the antiparticle varies more rapidly when compared to the energy of the particle (Figure 10). Clearly, we observe that the energy of the particle varies very slowly throughout the region of flux and magnetic field. As a final commentary, we clarify that the cases discussed in Figures 7, 8, and 10 can be investigated for other fixed parameter values. In this way, it can be shown that there are forbidden energies, depending on the values of the parameters considered. In general, this occurs when both the α parameter and the rotation parameter a are smaller than 0.3 and the other parameters assuming higher values than those we use here.Figure 10
Sketch of the energy (Equation (71)) as a function of B and ϕ for a=1, α=0.5, m=1, and n=1.


## 5. Conclusions

In the present manuscript, we have addressed the problem of the relativistic quantum motion of an electron in the spinning cosmic string background considering the presence of a uniform magnetic field and the Aharonov-Bohm potential. We have shown that this combination of potentials allows bound state configurations in the scenario of first-order solutions as well as in the case of second-order solutions of the Dirac equation. It is worth noting the role played by the two different terms in the vector potential. As already known in the literature, we have shown that the uniform field is responsible for a behavior analog to a harmonic oscillator, which leads to the relativistic Landau quantization. Also, we saw the Aharonov-Bohm flux contributes to the angular momentum and energy spectrum of the particle. In the case of first-order solutions, which were obtained by solving Equations (49) and (63) for E=+M and E=−M, respectively, the oscillator-like behavior provided by the uniform magnetic field guarantees the convergent first-order solutions and, consequently, the existence of bound states. The isolated solutions obtained (Equations (49) and (63)) are particular solutions of the Dirac equation (4).We have also studied the more general problem by solving the second-order equation implied by Equations (28) and (29) for the upper component of the Dirac spinor for E≠±M. Using appropriate solutions (Equation (20)), we have derived the radial equation and shown that its solution depends on the Kummer functions. From these functions, we have extracted the expression for the energy levels of the particle (Equations (70) and (71)). For the field configuration considered, we have found that the effective angular momentum of the electron depends on its energy and the Aharonov-Bohm flux tube. Besides, the potential vector corresponding to the uniform field leads to a charged oscillator. Thus, such field superposition provides distinct effects on the motion of the particle. Additionally, in some cases, the rotation produces a combined effect with both the uniform magnetic field and the curvature (see Equation (71)). We have shown that the energy levels of the particle and antiparticle depend on the values of the physical parameters involved. In the case of energy (71), its validity is conditioned to Equation (72). Depending on the choice we make for the parameters, we can obtain forbidden energies. Sketches in Figures 7, 8, and 10 illustrate the profiles of the particle and antiparticle energies and show that they belong to the same spectrum. The effects of curvature and rotation are more evident when α<0.3, being the antiparticle energy the most affected. Summarizing our results, we can state that the rotation breaks the symmetry in the energies (71) and (78) about En,m<=0 and En,m+1<=0, respectively, while the energies (70) and (77) are symmetrical. When a⟶0 or B⟶0 in (71) and (78), this symmetry tends to be recovered. In the case of a strong magnetic field, the separation between the particle and antiparticle energies increases in comparison with the case of a weak field. When a is large and α⟶0, the energies increase rapidly. We saw also that the inclusion of the Aharonov-Bohm flux can produce notable modifications in the energy profile. For instance, it can affect even the behavior of the energy as a function of the uniform magnetic field. As a final comment, we would like to emphasize that the model studied in this article generalizes others found in the literature, such as those of Refs. [68, 72] for the case including a superposition of external magnetic fields and the investigation of isolated solutions of the Dirac equation. Furthermore, we present a detailed discussion on the energy levels of the particle which, in general, is not found in the literature.



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