The Extension of the Massless Fermion Behaviour in the Cosmic String Spacetime

In this work, we have obtained the solutions of a massless fermion which is under the external magnetic field around a cosmic string for specific three potential models using supersymmetric quantum mechanics. The constant magnetic field, energy-dependent potentials, and position-dependent mass models are investigated for the Dirac Hamiltonians, and an extension of these three potential models and their solutions is also obtained. The energy spectrum and potential graphs for each case are discussed for the α deficit angle.


Introduction
The massless Dirac character of the low energy electrons moving has attracted much interest in physics due to the graphene's important electronic properties [1]. There are a series of studies on the interaction of graphene electrons in perpendicular magnetic fields which have been carried out in order to find a way for confining the charges [2,3]. Graphene and its derivatives (nanotubes, fullerenes) have become a mine of novel technologies and studying various aspects of physics through its many subfields as well as cosmological models on honeycomb branes [4]. Moreover, topological defects which are disorders lead to important effects on the electronic properties of low dimensional systems. A cosmic string and topological defect relationship is studied where the topological defects are located at arbitrary positions on the graphene plane [5]. In line with the efforts to combine general relativity theory with quantum mechanics, these studies are of great interest [6][7][8]. Cosmic strings were introduced by Kibble in 1976 [9], and the geometry of the massless cosmic strings is examined in the large scale limit of the model given in [10]. In relativistic quantum mechanics, Lie algebraic approaches [11], a hydrogen atom in the background of an infinitely thin cosmic string [12], scalar particle dynamics in gravity's rainbow through the space-time of a cosmic string [13], and N = 2 supersymmetric approach to cosmic string dynamics [14] are some recent studies about the topic. For more details about the technique of SUSY, the readers may look at the book [15]. The origin of supersymmetric (SUSY) quantum mechanics is based on the very early development of quantum mechanics by Dirac who found a method to factorize the harmonic oscillator and Schrödinger who noted symmetries in the solutions to his equation [16]. After then, the context of SUSY QM was first studied by Witten [17] and Cooper and Freedman [18]. This theory, which still attracts much attention today, has also application in optics [19] and biophysics [20]; it has a very important place in both relativistic [21] and nonrelativistic quantum mechanics [22,23]. This work points the dynamics of a massless fermion out when it is under the influence of a magnetic field. Using the fundamental aspects of SUSY quantum mechanics, three different potential models are discussed for the different values of a deficit angle. Moreover, supersymmetric extension of these models provides generated unknown and complex potential models. The paper is organized as follows: low dimensional Dirac equation is written in the curved spacetime using cosmic string line element and aspects of SUSY quantum mechanics given in Section 2. Section 3 is devoted to all potential models which are radial, hyperbolic, and nonlinear ones, and their solutions are given. Their energy and potential function graphs are shown. Section 4 includes the extended new quantum mechanical potential models and their solutions. Energy and potential graphs are also shown. We conclude our results in Section 5.

Dirac Hamiltonian for a Cosmic String in the Gravitational Background
A massless fermion dynamics can be represented by the Dirac equation-Weyl equation. Specially, we assume that the particle occurs in the external electromagnetic field in a cosmic string spacetime. Then, the particle is described by Here, σ a are the Pauli matrices, and ∇ μ = ∂ μ + Γ μ is the covariant derivative. The cosmic string spacetime is represented by the line element which is where −∞<ðt, zÞ<∞, r ≥ 0, and 0 ≤ ϕ ≤ 2π. The parameter α = 1 − ð4m/c 2 Þ is the angular deficit changing in the interval ð0, 1, wherem is the linear mass density of the cosmic string.
Here, e a μ are called as the tetrad fields which connect the Riemannian metric tensor to the flat spacetime metric tensor as The tetrads and spinor connections are given, respectively, as where where γ a is the standard Dirac matrices defined in the Minkowski spacetime: We use the units ℏ = c = 1; the line element of the stationary cosmic string spacetime is written as with −∞ < ðt, zÞ<∞ and r ≥ 0 and 0 ≤ ϕ ≤ 2π. Here, the parameter α is the angular deficit changing in the interval ð 0, 1. Considering the components of the metric tensor g μν , and the Christoffel symbols the spin connection components can be calculated using the tetrad components and the Christoffel symbols, which leads to Substituting (17), (7), (11), and (4) in (1), we can obtain [24] iγ 0 ∂Ψ ∂t Advances in High Energy Physics Let us use the following form for the spinor as Putting (19) in (18) gives Then, we get a couple of differential equations: Next, we transform the system given above into the form where ε = E 2 − k 2 and are used. Here, (21) are transformed into (23) with the same energy which also shows that the system is supersymmetric. Let us call two effective Hamiltonians for (23) as H 1 and H 2 , respectively: and the intertwining operators are defined as It is noted that (27) can be used to intertwine the system as [25] Furthermore, one can observe that Let us discuss the exactly solvable potential models for H 1 and H 2 in the next section.

Potential Models
Now we can arrange some different A ϕ ðrÞ vector potential models which give rise to effective Hamiltonians. We can argue that the discrete spectrum of the Hamiltonian H 2 is ε 2,n and L − χ 2,0 = 0; then [26], ε 1,n−1 = ε 2,n , where a 0 is a real parameter, is perpendicular to the plane. Then, vector potential component A ϕ ð rÞ can be taken as A ϕ ðrÞ = a 0 r, and WðrÞ becomes This superpotential is introduced in [15] where the singular potential models are also discussed. Here, we have considered the one-dimensional system. Hence, V 1,2 ðrÞ functions become This system shows that a 0 should be negative because of the Coulomb's potential; let a 0 be a 0 = −ν, ν > 0. The solutions of the system (32) which is known as pseudoharmonic potential are already known [27]: 3 Advances in High Energy Physics where μ = ðmðm − αÞ/α 2 Þ + ð1/4Þ, ε = ðν 2 e 2 /α 2 Þ − E, and 1 F 1 ð−n, 2μ + 1, 2 ∈ rÞ are the confluent hypergeometric functions. Let us look at the energy spectrum and the conditions for getting real energy levels. The inside of the square root in (34) must be positive for the real energy spectrum, and The normalization constant can be written as [27] Let us see how the energy and potential functions change with α.
As is shown from Figure 1, we can obtain physical potentials for the values of α which takes 0 < α < 1. Energy graph in Figure 2 shows that for the values of α ≥ 0:1831 (green curve), we can get an increasing energy graph. For the greater values of the number n, the lower bound for the α increases.

Potential Models: Energy-Dependent Vector and Scalar
Potentials. The energy-dependent potentials are one of the some modified problems of the quantum mechanics [28]. The condition on the density distribution is [28] ρ n E n , r As it is seen from the above equation, a positivity condition is The details of the related energy-dependent potentials in supersymmetry can be found in [29]. In this case, we take the vector potential AðrÞ as a complex function where C 1 is a constant and SðrÞ is the complex function. Then, (23) becomes where we can give the superpotential WðrÞ as and let SðrÞ be where A 1 , B 1 , C 1 are the constants. We can get the partner potentials as Here, (45) is known as Scarf II potentials (see [30]). Comparing our system (41), (43), and (44) with the linear energydependence results in [31] can give a solvable model. We can use the parameter A 1 as energy-dependent potential in our calculations as Then, we match our system with the one where linear energy dependency can be found in page 9 in [31] and write the partner potentials as energy-dependent potentials as Advances in High Energy Physics The condition for the real energies is And the solutions become In Figure 3, we have obtained an effective potential graph for the real potential while we can get a potential barrier from the imaginary part. Figure 4 shows the component of the vector potential whose real component is corresponding to another effective potential curve. In Figure 5, we can see that for the greater values of n, the lowest bound for the α again increases as energy increases.

Potential Models: Position-Dependent Mass Model.
The idea of a position-dependent-effective-mass (PDEM) quantum Hamiltonians gains more attention because of the physical applications in graded crystals, quantum dots, and liquid crystals. The hermiticity and the uniqueness of the problem are studied in the region of mathematical physics. For more details, see [32][33][34]. Let us make a point transformation in (41) given by

Advances in High Energy Physics
Then, we can get Let us choose f ðzÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + λz 2 p and SðzÞ = ðiC 1 / ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + λz 2 p Þ − c 1 ; then, (54) turns into The solutions of the type of equations like (55) are already known. We can write the complete solutions as [35] E n = ± ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where s = ð2C 1 e − α ffiffiffi λ p Þ/2α 3/2 , r = −i/2, and P ða,bÞ n ðyÞ are the corresponding Jacobi polynomials. Now, let us examine the SUSY of this model. First, we need to make a point transformation to (55) which is χ 1 ðzÞ = 1/ðð1 + 4λ 2 z 2 Þ 1/4 ÞYðzÞ. We get where A = C 1 e/α. We propose a super potential which is given by Then, for (59), the unknown parameters of the superpotential can be obtained as Figure 6 shows that the energy becomes zero when α = 0:5 and it is decreasing for the greater values of this angular deficit parameter. In Figure 7, the effective potential graph is shown where the potential well becomes more distinct as α increases.

Extended Potential Models
Using V 1 functions given in the previous section, we derive more general partner potentials for each model here.

Extended Potential Models: Constant Magnetic
Field. Let us start with the choice of the superpotential. Here, we want to generate more general potential family of the system given in (32). Now, WðrÞ is where f ðrÞ, gðrÞ are unknown functions. Then, partner potentials can be obtained as Here, we match (63) with (32). If we equate the rational terms to zero in (63) as below then, we obtain where n = −2B and the exponential integral function is given by Then, the partner potentials become Comparing (68) and (32) gives the unknown constants A and B as Then, one can find the solution of (69) which is χ 2,n as where We note that (69) shares the same energy level given by (34). In Figure 8, one can see that V 2 ðrÞ fits an effective potential with a well which is the partner of a Coulombic potential.

Extended Potential Models: Extended Energy-Dependent
Vector and Scalar Potentials. Let us choose the superpotential WðrÞ as So, partner potentials are obtained as In (73), we can equate the terms to zero which are given below Then, we find Once can give the first partner potential as and comparing the coefficients of the hyperbolic functions in (74) with those in (48) gives us and we use a = a 1 + ib 1 ; the constants λ 1,2 are given by One can find a higher potential barrier as α decreases in Figure 9 for the real part of V 1 , and for the real part of the partner potential V 2 , it shows a potential getting deeper as α decreases in Figure 10.

Extended Potential Models: Extended Position-Dependent
Mass Model. The exact solutions of the position-dependent effective mass Schrödinger equation for the nonlinear isotonic oscillator are found in [36]. In this case, we search for a new family of the extension of the nonlinear isotonic oscillator type potentials within the approach mentioned in (55). Now, we take WðzÞ which has a form