Spectrally isomorphic Dirac systems: Graphene in an electromagnetic field

Abstract

We construct the new one-dimensional Dirac Hamiltonians that are spectrally isomorphic (not isospectral) with the known exactly solvable models. Explicit formulas for their spectra and eigenstates are provided. The operators are utilized for description of Dirac fermions in graphene in presence of an inhomogeneous electromagnetic field. We discuss explicit, physically relevant, examples of spectrally isomorphic systems with both non-periodic and periodic electromagnetic barriers. In the latter case, spectrally isomorphic two-and three-gap systems associated with the Ablowitz-Kaup-Newell-Segur hierarchy are considered.

Dirac electron in graphene with magnetic fields arising from first-order intertwining operators

“…The first-order intertwining method presented here generalizes the shape invariant technique that different authors have used previously to describe the behavior of an electron near to the Dirac points in a graphene layer with applied external magnetic fields [23][24][25][26][27][28]. Similarly as Figure 7: Second intertwining for the Morse potential with k = 6α, ν 1 = − 3 2 , ν 2 = − 1 2 , 1 = − 1 2 E − 1 = − 11α 2 2 , 2 = −E − 1 = −11α 2 : (a) generated potential V 2 (x, 2 ) (continuous line) and initial oneṼ 1 (x, 1 ) (dashed line), with energy levels E n (x)| 2 for the ground state (GS, blue) and the excited states n = 1, 2, 3 (red, green, purple); (d) current density for the excited states with the same colors that in (c).…”

Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential

“…For the sake of brevity we omit to show this rigorously, as it would require a similar series of considerations as done above for the function (29). It now follows that the density |ψ + | 2 + |ψ − | 2 also vanishes at the infinities, establishing existence of the norm integral in (17).…”

“…Here, the symbol P n stands for a Jacobi polynomial of degree n. Before we continue, let us point out that the functions (29) do not lead to bound states of our Dirac equation (3) unless the parameters satisfy certain conditions. In particular, existence and positiveness of our norm (17) is not guaranteed in general. In order to find out more about this, let us now analyze the asymptotic behavior of the solution (29) at the infinities.…”

“…This constraint is satisfied since it coincides with our requirement in (38). Since condition (35) is already satisfied, it follows that the stationary energies (38) are associated with solutions that are normalizable in the sense (17). There are infinitely many such bound-state solutions because the constraint (39) is fulfilled for all values of n. Since the closed form of the bound-state solutions is very long, we restrict ourselves to showing only the function ψ 1 .…”

“…While for the vast majority of such external potentials the Dirac equation will not be solvable, there are some exceptions. Such exceptional cases include coupling to scalar potentials [8] [14] [12] [24], as well as to vector potentials [4] [17] [22]. The purpose of this work is to extend the latter context to the case where the external potential depends on the energy.…”