Abstract
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 deficit angle.
1. 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 [6–8]. 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 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.
2. 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, are the Pauli matrices, and is the covariant derivative. The cosmic string spacetime is represented by the line element which is where , , and . The parameter is the angular deficit changing in the interval , where is the linear mass density of the cosmic string. Here, 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 is the standard Dirac matrices defined in the Minkowski spacetime:
We use the units ; the line element of the stationary cosmic string spacetime is written as with and and . Here, the parameter is the angular deficit changing in the interval . Considering the components of the metric tensor , 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]
Let us use the following form for the spinor as
Then, we get a couple of differential equations:
Next, we transform the system given above into the form where 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 and , 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 and in the next section.
3. Potential Models
Now we can arrange some different vector potential models which give rise to effective Hamiltonians. We can argue that the discrete spectrum of the Hamiltonian is and ; then [26],
3.1. Constant Magnetic Field
The constant magnetic field vector , where is a real parameter, is perpendicular to the plane. Then, vector potential component can be taken as , and becomes
This superpotential is introduced in [15] where the singular potential models are also discussed. Here, we have considered the one-dimensional system. Hence, functions become
This system shows that should be negative because of the Coulomb’s potential; let be . The solutions of the system (32) which is known as pseudoharmonic potential are already known [27]: where , , and 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 .
As is shown from Figure 1, we can obtain physical potentials for the values of which takes . Energy graph in Figure 2 shows that for the values of (green curve), we can get an increasing energy graph. For the greater values of the number , the lower bound for the increases.
3.2. 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]
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 as a complex function where is a constant and is the complex function. Then, (23) becomes where we can give the superpotential as and let be where 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 energy-dependence results in [31] can give a solvable model. We can use the parameter 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
Then, one can find the energy eigenvalues as
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 , the lowest bound for the again increases as energy increases.
3.3. 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 [32–34]. Let us make a point transformation in (41) given by Then, we can get
Let us choose and ; then, (54) turns into
The solutions of the type of equations like (55) are already known. We can write the complete solutions as [35] where
, and 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 . We get where . 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 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 increases.
4. Extended Potential Models
Using functions given in the previous section, we derive more general partner potentials for each model here.
4.1. 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, is where 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 and the exponential integral function is given by
Then, the partner potentials become
Comparing (68) and (32) gives the unknown constants and as
Then, one can find the solution of (69) which is as where
We note that (69) shares the same energy level given by (34). In Figure 8, one can see that fits an effective potential with a well which is the partner of a Coulombic potential.
4.2. Extended Potential Models: Extended Energy-Dependent Vector and Scalar Potentials
Let us choose the superpotential 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
Then, we can find the where and we use ; the constants are given by
One can find a higher potential barrier as decreases in Figure 9 for the real part of , and for the real part of the partner potential , it shows a potential getting deeper as decreases in Figure 10.
4.3. 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 which has a form where is the unknown function which can be found by the following equations:
Here, we note that
From (85), can be obtained as where
Now, we can find the as where and and .
Figure 11 shows that the partner potentials and behave like which goes to infinity as . We have seen that the potential pictures can be obtained independently from .
5. Conclusions
Using the fundamental concepts of SUSY QM, we have obtained physical solutions for the extended constant magnetic field which leads to a Coulomb problem, energy-dependent hyperbolic potential and nonlinear isotonic potential, which is argued as the position-dependent mass model for a fermion near cosmic string spacetime. It is observed that the restricted values of the angular deficit give reasonable behaviours of the potentials for our models.
Data Availability
The [tex, eps] data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.