Dirac Equation on the Torus and Rationally Extended Trigonometric Potentials within Supersymmetric QM

The exact solutions of the (2+1) dimensional Dirac equation on the torus and the new extension and generalization of the trigonometric Poschl-Teller potential families in terms of the torus parameters are obtained. Supersymmetric quantum mechanical techniques are used to get the extended potentials when the inner and outer radii of the torus are both equal and inequal. In addition, using the aspects of the Lie algebraic approaches, the iso(2; 1) algebra is also applied to the system where we have arrived at the spectrum solutions of the extended potentials using the Casimir operator that matches with the results of the exact solutions.


Introduction
There are several potentials which are used in both theoretical and applied physics with exact solutions in non-relativistic quantum mechanics.These potential classes can also give physical results in the successful union of quantum mechanics and special relativity where the Dirac equation has a successfully explanation on the antimatter, spin, and the realistic behavior of atoms [1].On the other hand, the gravitational field effects on some quantum mechanical systems have been studied as an exciting research field [2], [3], [4].From the symmetries of the Dirac equation [5], Hermiticity and uniqueness [6], to the factorization and pseudo-supersymmetry [7], [8], covariant form of the Dirac equation and its aspects are studied.The gravitational background can bring some mathematical difficulties through some geometries.One of them is the solutions of the wave equations on the torus geometry [9], [10], [11], [12].At the same time, in physical applications, such as graphene related ones, it is stated that the curvature of the material can change the electron density of the states.In [9], graphene nanoribbon along the surface of a torus is examined within the long wave approximation where the Dirac equation is solved approximately.Then, the recent study aims to bring a new viewpoint to the problem of the exact solutions for the Dirac equation on the torus which may lead to both applied and theoretical interest in the recent studies.When considering the works which are about the general theory of relativity and the quantum mechanics unification, there is a fact that the curvature of space-time at the position of the atom can affect the spectrum.Therefore, the problem of electron and its perturbed energy spectrum owing to the gravitational field can bring more problems in the solutions through the geometries of the space-time.To our knowledge, there are a few studies about the torus parametrization of the wave equations and exact solutions, then, we have devoted our motivation to find soluble potential models.In this paper, Section I involves the relativistic quantum mechanical wave equation in the gravitational background for a massless fermion where the Fermi velocity is taken as a position dependent function [13].Supersymmetric partner potentials are obtained for the transformed Dirac Hamiltonian.The spectrum and spinor solutions are given when the inner and outer radii of the torus are equal and inequal.In section III, we present the iso(2, 1) Lie algebraic computations for the Dirac Hamiltonians.

Dirac equation on the torus
Formulating the Dirac equation in curved spacetime, it is known that all metrics related by a general coordinate transformation are physically equivalent and physical observables in gravitation field should be invariant under general coordinate transformations.This is known as general covariance principle which necessities transforming a tensor in the flat spacetime as a tensor under general transformations in the curved manifold.The covariant generalization of the Dirac equation to curved space was independently introduced by Weyl and by Fock [14], [15].Then, the Dirac equation can be written in terms of vierbein fields and gravitational spin connection as [14] [ where Γ µ is the spin connection.And Ψ is the spinor which includes electron's wave-functions Ψ = ψ 1 ψ 2 near the Dirac point.The Dirac matrices γ µ in curved spacetime satisfy, Here g µν is the metric tensor and the tetrad(vierbein) frames field is defined as where . The metric for the torus surface is given by In the metric given above, the inner radius of the torus is c, the outer radius is shown by a and u, v ∈ [0, 2π).We can use the spin connection formula which is where Γ λ µσ is the Christoffel symbols.In [9], the symbols Γ λ µσ were given in terms of the variable R(x 1 ) which corresponds to R(v) in our work.So one can obtain them as, and accordingly, For the Dirac matrices we will use σ j Pauli matrices as γ 0 = σ 3 , γ 1 = −iσ 2 , γ 2 = −iσ 1 .We use ( 6) and ( 7) in (1) and we get [9] ( where may not be a constant [13].Using , we obtain where we use v → x for the sake of simplicity.Considering (10), the solutions may take the form Then we have ) (13) can also be expressed as where We may give U (x) as below One can make the coefficient of F ′ 1 (g) as zero and f (x) can be found as Then, using 18), ( 14) becomes where In this work [16], it is shown that if a superpotential W (x) which is a hyperbolic function then, the pair of potentials are given as The authors removed the rational term in V + (x) using a parameter condition C = 2B 2A−1 .In this case, the potential V + (x) is known as generalized Pöschl-Teller potential in the literature with the exact solutions.Now we will discuss the cases depending on the different choices of rational term in W (x) superpotential.Now we shall look at (20) to arrive at a solvable model given below.If we take where V 1 , V 2 , ε are real constants.Hence, our function g(x) satisfying (23) can be found as where C 1 , C 2 are constants and Now we will search for the solutions for our systems in case of the equal and unequal torus radii.

2.1
Equal inner an outer radii; a = c First we assume that the superpotential W 1 (x) is a trigonometric function which is and we get the partner potentials (28) It can be seen from ( 27) that in case of an applied special condition a = 1, λ = 1 and x → ix, one can obtain the system (22).In order to find a parameter condition on the rational term, we can simplify (27) as Then, we can find this conditions as Thus we have, which is known as trigonometric Pöschl-Teller potential.On the other hand, we can give the partner potential V + (x) as Here, it is well known that (34) and ( 35) are isospectral partner potentials except the ground-state.Comparing (34) and V 1 (x) in (23), we obtain The solutions of V − (x) are already known as [17] This result shows that when a → ∞, E → 0. Reminding the eigenvalue equation ( 19), we can give Then, the solutions are written as [17] In fact, the complete solutions are not given yet.The spinor solutions can be given as where N 1 is the normalization constant.The solutions F + (x) corresponding the ones for the partner potential V 1,+ (x) can be found.The system can be summarized as follows For (35), and it is reminded that the energy states of these Hamiltonians are given by At the same time, we can find 1,n (x) solutions using (42), When it comes to the solutions ψ 2 (x) of ( 11) which shares the same energy with (10), we give The wavefunction mapping can be given as and we obtain, where Then, we can give V 2 (x) as P 1 and P 2 are real constants and equating ( 48) and (49), g 2 (x) is found to be If the parameters are chosen as and using (30), P 1 and P 2 can be expressed in terms of the torus parameter and hence, one can give the supersymmetric partner potentials for the V 2 (x) which are V ± 2 are and x 2 Now, for the system (11) we can give the solutions given below 2.2 different inner and outer radii; a = c where G(x) is the unknown function and we get In order to make the rational term zero, G(x) can be found as below where B(z; s, w) is the incomplete beta function which is equal to where (70) Here, F 1 (α, β, β ′ , γ; x; y) is the Appel hypergeometric function with two variables.Thus, V 1,− (x) can be given by ( 59) and V + (x) becomes where Similarly, one can obtain the partner potentials for the system given in (11).
3 The iso(2, 1) algebraic approach Let us look at the operators given below and Here, the term U (x, µ ± 1/2) is the modification operator which is used in [18], µ is a constant.Without U (x, µ ± 1/2), these operators are known as those given in the iso(2, 1) algebra but this functional operator, U (x, µ ± 1/2), helps to construct the algebra for the rationally extended potentials.Here, we discuss the extended trigonometric Pöschl-Teller potential within the iso(2, 1) algebra.These operators provide the commutation relations which are given as If J ± and J 3 are used in (75), then, one can get Moreover, (77) can be satisfied in case of chosen U 1 (x) and and U 2 (x) in (85) and the parameter conditions as given below Thus, the energy eigenvalues can be expressed in terms of the parameters given above where one can say that j = n + µ to compare our results with those found in [18].

Conclusions
Our findings in this paper point to the fact that the exact solutions for a given system which is relativistic can be obtained using the similar techniques used in non-relativistic quantum mechanics.Especially, considering the massless particle dynamics, the latest trends in relativisitic quantum mechanics can bring new bound state problems which are not solved yet such as the Dirac equation in a curved space-time which has a toroidal geometry.Because the metric contains a more general trigonometric function which is R(x) = c+a cos x, the Klein-Gordon-like equations obtained from the couple of first order Dirac equations are not familiar which are generally known in relativistic quantum mechanics.In this problem, the Fermi velocity is chosen as a non-constant function which is expressed in terms of the point transformation function in our solutions, after then, solvable potentials are derived using the superpotential suggestions.
In the equal inner and outer radius case, one of the partner potentials is trigonometric Pöschl-Teller potential while the other one is including the rational terms.We have obtained the solutions of the partner potentials for each system (10) and (11).For the different radius values of the torus surface, as a more general case, one of the partner potential is found as not solvable rational function which includes beta function while the other one is trigonometric Pöschl-Teller potential.In the next case, unsolvable partner potential is obtained in tems of the Appel hypergeometric functions.We also note that these unsolvable potentials share the same energy with the trigonometric Pöschl-Teller potential.In the final section of this work, operators of the Lie algebra iso(2, 1) are found in order to express the Casimir operator with the potential functions like the extended trigonometric Pöschl-Teller potentials which are (27) and (28) given in the a = c case.Finally we note that the Dirac equation on the toroidal spacetime problem can lead to obtain more general potential families.