The super Dirac $\delta$ function and its applications

Said Lantigua, Jonas Maziero

For the relativistic tunneling effect described using Dirac's equation, Winful et al. [Phys. Rev. A 70, 052112 (2004)] presented the deduction of a general result that allows for the determination of the phase time (group delay) as the sum of the particle dwell time inside a potential barrier and of the self-interference delay associated with the incident and reflected wave functions interaction. In this paper, a mathematical model is derived through a construction analogous to the proposal mentioned above, but based on an alternative representation for Dirac's equation. This representation is similar to the one introduced by Ajaib [Found. Phys. 45, 1586 (2015)]. Thus, from the application of this model in the study of the tunneling effect in the absence of an external magnetic field, the influence of spin on the tunneling times is described. More specifically, the tunneling time is obtained as the sum of the dwell times inside the potential barrier for particles with spin up and spin down and the self-interaction time associated with the incident and reflected wave functions for particles with spin up.

Ajoy Kumar Ghatak, S. Lokanathan

André LeClair, Dean J. Robinson

We prove versions of super spin-charge separation for all three of the symmetry groups SU( N ), Sp(2 N ) and SO( N ) of disordered Dirac fermions in 2 + 1 dimensions, which involve the supercurrent algebras and respectively. For certain restricted classes of disordered potentials, the latter supercurrent algebra based conformal field theories can arise as non-trivial low energy fixed points. For all cases with such a fixed point, we compute the density of states exponents as a function of N .

Francisco Correa, Luis Inzunza, Olaf Lechtenfeld

We establish a relationship between the Davey--Stewartson II (DS II) integrable system in $(2{+}1)$ dimensions and quasi-exactly solvable planar interacting Dirac Hamiltonians that exhibit the super-Klein tunneling (SKT) effect. The Dirac interactions are constructed from the real and imaginary parts of breather solutions of the DS II system. In this framework, the SKT effect arises when the energy is tuned to match the constant background of the soliton, while the resulting Dirac Hamiltonians simultaneously support bound states embedded in the continuum. By imposing the SKT boundary conditions, we employ Darboux transformations to construct a general three-parameter family of DS II breather solutions that can be mapped to Dirac Hamiltonians. At the initial soliton time, the corresponding Dirac systems form a massless two-parameter family of Hermitian models with nontrivial electrostatic potentials. As the soliton time evolves, the systems become $\mathcal{PT}$-symmetric and develop a nontrivial imaginary mass term. Finally, when the soliton time is taken to be imaginary, the construction yields Hermitian Dirac systems that lack time-reversal symmetry. In all cases, we identify the emergence of quasi-symmetry transformations that preserve the SKT subspace of states while not commuting with the full Hamiltonian.

Voja Radovanović