Higher-order Darboux transformations for the Dirac equation with position-dependent mass at nonvanishing energy
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Higher-order Darboux transformations for the Dirac equation with position-dependent mass at nonvanishing energy Axel Schulze-Halberga Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, IN 46408, USA Received: 27 May 2020 / Accepted: 23 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct Darboux transformations of arbitrary order for the two-dimensional Dirac equation within a position-dependent mass scenario. While we restrict the potential and the mass to a single dimension, the stationary energy remains arbitrary. We use our Darboux transformation to generate several new exactly solvable Dirac systems that admit bound states at nonzero energies.
1 Introduction Graphene, a single layer of carbon atoms that form a two-dimensional lattice of honeycomb structure [1], is an example for a femionic Dirac material [2,3]. Ever since the isolation of graphene, there has been a considerably increasing interest in Dirac materials due to their wide variety of applications that include superfluid phases of 3 He [4], topological insulators [5], high-temperature d-wave semiconductors [6], among many others. The common property of all fermionic Dirac materials is that their charge carriers at low energies behave like massless Dirac fermions, and are therefore governed by the massless Dirac equation. Confining the charge carriers in a Dirac material is a difficult task due to the occurrence of Klein tunneling [7–9] that prevents the formation of bound states. Several methods have been proposed to circumvent this problem, including coupling the system to electric fields [10] or magnetic fields [11], spatial variation of the Fermi velocity [12,13], and the introduction of a position-dependent mass function [14]. Each of these methods requires to solve the respective massless Dirac equation, which in the vast majority of cases is possible only by numerical approximations. Very few systems have been identified that allow for closed-form solutions of the massless Dirac equation (exactly solvable systems), see, for example, [15– 18]. Since such closed-form solutions are interesting in themselves as actual models of nature and can be used in a more very versatile way than numerical solutions, it is of high interest to develop methods for systematically generating them. One of these methods is the Darboux transformation [19] that relates two Dirac equations by mapping their solutions onto each other through differential operators of a certain order. First-order Darboux transformations were introduced in [20] and [21] for the one-dimensional and two-dimensional case, respec-
a e-mail: [email protected] (corresponding author)
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tively. In the special situation of a two-dimensional case at zero energy with one-dimensional potential that especially applies to the case of Dirac materials, arbitrary-order Darboux
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