The soliton nature of the super-Klein tunneling effect

TL;DR

The paper develops a bridge between the $(2{+}1)$-dimensional DS II integrable system and planar Dirac Hamiltonians that exhibit super-Klein tunneling (SKT) by interpreting DS II breather solutions as sources of planar Dirac potentials. Using a Darboux transformation on a constant DS II background, the authors construct a three-parameter family of DS II breather solutions that map to Dirac Hamiltonians, with the soliton time $\tau$ acting as a parameter that tunes between Hermitian and $\mathcal{PT}$-symmetric regimes. They demonstrate SKT through exact scattering states at fixed energy and reveal bound states in continuum, including a quasi-symmetry that preserves the SKT subspace. The work provides explicit parametric families of Dirac models with SKT, clarifies the role of energy matching to the soliton background, and connects soliton theory, Darboux symmetry, and quasi-exact solvability in a two-dimensional Dirac setting, offering a framework to engineer SKT-compatible Hamiltonians in related physical systems.

Abstract

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.

Figures (6)

• Figure 1: Plots for $V_\gamma(x,y,\tau)$, $m_\gamma(x,y,\tau)$, for $\tau=1$, $\gamma=0.4$.
• Figure 2: Probability density current of the SKT states. The parameters are taken to be $\tau=2$ and $\gamma=0.4$, while the incident angles are (from left to right) chosen as $\phi=\{-\frac{21}{40}\pi,\frac{\pi}{2},\frac{18}{40}\pi\}$. In each case, the current is affected only in the interaction zone, but it agrees in the two $y$-interacting regions.
• Figure 3: Differences in the probability density for bound states in the $\mathcal{PT}$-symmetric and the Hermitian cases. In the first panel, the density plot for (\ref{['RhoPT']}) is shown. Subsequently, we display the probability density (\ref{['RhoHer']}) with plus and minus signs (in that order) for the Hermitian case. Maximal (minimal) brightness corresponds to maximal (minimal) probability of finding the confined particle described by the bound state. The parameters are $\gamma=0.4$, $\tau=2$ and $z=2.5$.
• Figure 4: The parameter space projected to the $S^2(s_3)$ sphere with $s_3=0$. The vector $\breve{n}$, as given in equation (\ref{['xtranslation']}), defines the rotation axis of $R(a)$. The orbits of $R(a)$, shown in green, represent the equivalence classes of DS II solutions that are related by translations. Red points stand for two equivalent vectors under the action of $R(a)$, corresponding to half the period of the rotation. The class representatives are chosen in the equatorial semicircle defined between points $\breve{n}$ and -$\breve{n}$.
• Figure 5: Different plots of $V_{\gamma,\varphi}(x,y,0)$. From left to right the parameters are $\gamma=0.4$ and $\varphi=\{\tfrac{9}{10}\pi,\,\tfrac{99}{100}\pi,\tfrac{108}{100}\pi\}$.