- freely available
- re-usable

*Symmetry*
**2014**,
*6*(1),
103-110;
doi:10.3390/sym6010103

^{1}

^{2}

Published: 17 March 2014

## Abstract

**:**We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been explicitly worked out.

## 1. Introduction

In recent years, the massless Dirac equation in (2 + 1) dimensions has drawn a lot of attention, primarily because of its similarity to the equation governing the motion of charge carriers in graphene [1,2]. In view of the fact that electrostatic fields alone cannot provide confinement of the electrons, there have been quite a number of works on exact solutions of the relevant Dirac equation with different magnetic field configurations, for example, square well magnetic barriers [3–5], non-zero magnetic fields in dots [6], decaying magnetic fields [7], solvable magnetic field configurations [8], etc. On the other hand, at the same time, there have been some investigations into the possible role of non-Hermiticity and symmetry [9] in graphene [10–12], optical analogues of relativistic quantum mechanics [13] and relativistic non-Hermitian quantum mechanics [14], photonic honeycomb lattice [15], etc. Furthermore, the (2 + 1)-dimensional Dirac equation with non-Hermitian Rashba and scalar interaction was studied [16]. Here, our objective is to widen the scope of incorporating non-Hermitian interactions in the (2 + 1)-dimensional Dirac equation. We shall introduce η pseudo Hermitian interactions by using imaginary vector potentials. It may be noted that imaginary vector potentials have been studied previously in connection with the localization/delocalization problem [17,18], as well as phase transition in higher dimensions [19]. Furthermore, in the case of the Dirac equation, there are the possibilities of transforming real electric fields to complex magnetic fields and vice versa by the application of a complex Lorentz boost [20]. To be more specific, we shall consider η-pseudo Hermitian interactions [21] within the framework of the (2 + 1)-dimensional massless Dirac equation. In particular, we shall examine the exact bound state solutions in the presence of imaginary magnetic fields arising out of imaginary vector potentials. We shall also obtain the η operator, and it will be shown that the Dirac Hamiltonians are η-pseudo Hermitian.

## 2. The Model

The (2 + 1)-dimensional massless Dirac equation is given by:

Then, from (3), the equations for the components are found to be (in units of ℏ = 1):

#### 2.1. Complex Decaying Magnetic Field

It is now necessary to choose the function, f(x). Our first choice for this function is:

Now, from the second of Equation (7), we obtain:

_{2}(x) in Equation (12) as the complex analogue of the Morse potential whose solutions are well known [22,23]. Using these results, we find:

^{−}

^{x}and ${L}_{n}^{(a)}(t)$ denote generalized Laguerre polynomials. The first point to note here is that for the energy levels to be real, it follows from Equation (13) that the corresponding eigenfunctions are normalizable when the condition k

_{y}≥ 0 holds. For k

_{y}< 0, the wave functions are not normalizable, i.e., no bound states are possible.

Let us now examine the upper component, ϕ_{1}. Since ϕ_{2} is known, one can always use the intertwining relation:

_{1}. Nevertheless, for the sake of completeness, we present the explicit results for ϕ

_{1}. In this case, the potential analogous to Equation (12) reads:

_{1}(x) can be obtained from V

_{2}(x) by the replacement k

_{y}→ k

_{y}− 1, and so, the solutions can be obtained from Equation (13) as:

_{2,}

_{n}

_{+1}= E

_{1,}

_{n}, so that the ground state is a singlet, while the excited ones are doubly degenerate. Similarly, the negative energy states are also paired. In this connection, we would like to note that {H, σ

_{3}} = 0, and consequently, except for the ground state, there is particle hole symmetry. The wave functions for the holes are given by σ

_{3}ψ. The precise structure of the wave functions of the original Dirac equation are as follows (we present only the positive energy solutions):

_{y}→ −k

_{y}, the two potentials V

_{1,2}(x) = W(x) ± W′(x) go one into each other, V

_{1}(x) ↔ V

_{2}(x). Therefore, the solutions are correspondingly interchanged, ϕ

_{1,}

_{n}↔ ϕ

_{2,}

_{n}and E

_{1,}

_{n}↔ E

_{2,}

_{n}, but retain the same functional form as in Equations (13) and (16).

Therefore, we find that it is indeed possible to create bound states with an imaginary vector potential. We shall now demonstrate the above results for a second example.

#### 2.2. Complex Hyperbolic Magnetic Field

Here, we choose f(x), which leads to an effective potential of the complex hyperbolic Rosen–Morse type:

_{1}can be found out using relation Equation (14).

## 3. η-Pseudo Hermiticity

Let us recall that a Hamiltonian is η-pseudo Hermitian if [21]:

We recall that in both the cases considered here, the Hamiltonian is of the form:

_{2}, where

_{2}is the (2 × 2) unit matrix. Then, it can be shown that:

For the magnetic field given by Equation (19), the operator, η, can be found by using relations Equation (26). After a straightforward calculation, it can be shown that the η operator is given by:

## 4. Conclusions

Here, we have studied the (2 + 1)-dimensional massless Dirac equation (we note that if a massive particle of mass m is considered, the energy spectrum in the first example would become ${E}_{n}=c\sqrt{{k}_{y}^{2}+{m}^{2}{c}^{2}-{({k}_{y}-n)}^{2}}$. Similar changes will occur in the second example, too). in the presence of complex magnetic fields, and it has been shown that such magnetic fields can create bound states. It has also been shown that Dirac Hamiltonians in the presence of such magnetic fields are η-pseudo Hermitian. We feel it would be of interest to study the generation of bound states using other types of magnetic fields, e.g., periodic magnetic fields.

One of us (P. R.) wishes to thank INFN Sezione di Perugia for supporting a visit during which part of this work was carried out. He would also like to thank the Physics Department of the University of Perugia for its hospitality.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

- Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric field effect in atomically thin carbon films. Science
**2004**, 306, 666–669, doi:10.1126/science.1102896. - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Katsnelson, M.I.; Grigorieva, I.V.; Dubonos, S.V.; Firsov, A.A. Two-dimensional gas of massless Dirac fermions in graphene. Nature
**2005**, 438, 197–200, doi:10.1038/nature04233. - De Martino, A.; Dell'Anna, L.; Egger, R. Magnetic confinement of massless dirac fermions in graphene. Phys. Rev. Lett.
**2007**, 98, 066802:1–066802:4. - De Martino, A.; Dell'Anna, L.; Eggert, R. Magnetic barriers and confinement of Dirac-Weyl quasiparticles in graphene. Solid State Commun.
**2007**, 144, 547–550, doi:10.1016/j.ssc.2007.03.062. - Dell'Anna, L.; de Martino, A. Multiple magnetic barriers in graphene. Phys. Rev. B
**2009**, 79, 045420:1–045420:9. - Wang, D.; Jin, G. Bound states of Dirac electrons in a graphene-based magnetic quantum dot. Phys. Lett. A
**2009**, 373, 4082–4085, doi:10.1016/j.physleta.2009.09.007. - Ghosh, T.K. Exact solutions for a Dirac electron in an exponentially decaying magnetic field. J. Phys. Condens. Matter
**2009**, 21, doi:10.1088/0953-8984/21/4/045505. - Kuru, S; Negro, J.M.; Nieto, L.M. Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields. J. Phys. Condens. Matter
**2009**, 21, doi:10.1088/0953-8984/21/45/455305. - Bender, C.M.; Boettcher, S. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett.
**1988**, 80, 5243–5246. - Fagotti, M; Bonati, C.; Logoteta, D.; Marconcini, P.; Macucci, M. Armchair graphene nanoribbons: PT-symmetry breaking and exceptional points without dissipation. Phys. Rev. B
**2011**, 83, 241406:1–241406:4. - Szameit, A.; Rechtsman, M.C.; Bahat-Treidel, O.; Segev, M. PT-Symmetry in heoneycomeb photonic lattices. Phys. Rev. A
**2011**, 84, 021806(R):1–021806(R):5. - Esaki, K.; Sato, M.; Hasebe, K.; Kohmoto, M. Edge states and topological phases in non-Hermitian systems. Phys. Rev. B
**2011**, 84, 205128:1–205128:19. - Longhi, S. Classical simulation of relativistic quantum mechanics in periodic optical structures. Appl. Phys. B
**2011**, 104, 453–468. - Longhi, S. Optical realization of relativistic non-hermitian quantum mechanics. Phys. Rev. Lett.
**2010**, 105, 013903:1–013903:4. - Ramezani, H.; Kottos, T.; Kovanis, V.; Christodoulides, D.N. Exceptional-point dynamics in photoni honeycomb lattices with PT-symmetry. Phys. Rev. A
**2012**, 85, 013818:1–013818:6. - Mandal, B.P.; Gupta, S. Pseudo-hermitian interactions in Dirac theory: Examples. Mod. Phys. Lett. A
**2010**, 25, 1723–1732, doi:10.1142/S0217732310032901. - Hatano, N.; Nelson, D. Localization transitions in non-hermitian quantum mechanics. Phys. Rev. Lett.
**1996**, 77, 570–573, doi:10.1103/PhysRevLett.77.570. - Feinberg, J.; Zee, A. Non-Hermitian localization and delocalization. Phys. Rev. E
**1999**, 59, 6433–6443, doi:10.1103/PhysRevE.59.6433. - Mandal, B.P.; Mourya, B.K.; Yadav, R.K. PT phase transition in higher-dimensional quantum systems. Phys. Lett. A
**2013**, 377, 1043–1046, doi:10.1016/j.physleta.2013.02.023. - Tan, L.Z.; Park, C.-H.; Louie, S.G. Graphene Dirac fermions in one dimensional field profiles; Tansforming magnetic to electric field. Phys. Rev. B
**2010**, 81, 195426:1–195426:8. - Mostafazadeh, A. Pseudo-hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries. J. Math. Phys.
**2002**, 43, 3944–3951, doi:10.1063/1.1489072. - Flügge, S. Practical Quantum Mechanics; Springer-Verlag: Berlin, Germany, 1974.
- Cooper, F.; Khare, A; Sukhatme, U. Supersymmetry in Quantum Mechanics; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2001.
- Milpas, E.; Torres, M.; Murguía, G. Magnetic field barriers in graphene: An analytically solvable model. J. Phys. Condens. Matter
**2011**, 23, 245304:1–245304:7. - Rosen, N.; Morse, P.M. On the vibrations of polyatomic molecules. Phys. Rev.
**1932**, 42, 210–217, doi:10.1103/PhysRev.42.210. - Ahmed, Z. Pseudo-hermiticity of hamiltonians under imaginary shift of the coordinate: Real spectrum of complex potentials. Phys. Lett. A
**2001**, 290, 19–22, doi:10.1016/S0375-9601(01)00622-3.

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license(http://creativecommons.org/licenses/by/3.0/).