Planar Dirac equation with radial contact potentials

TL;DR

The paper tackles the planar (2+1) Dirac equation with the most general radial contact interaction supported on a circumference, recasting the problem as a one-dimensional radial Dirac equation using a distributional framework. Four physical parameters, interpretable as a scalar and three Lorentz-vector components, encode the circle-supported interaction and determine matching conditions that connect inner and outer solutions at r=R. The authors derive and analyze bound, scattering, and resonance states, including five physically motivated special cases, and demonstrate how permeability and confinement emerge from the parameter choices, with strong agreement between complex-energy resonances and Wigner time-delay peaks. These results illuminate confinement mechanisms in Dirac materials and point toward potential extensions to Aharonov-Bohm-type setups and higher-dimensional analogs.

Abstract

We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.

Figures (11)

• Figure 1: Graph of the energies of the bound states $E_b$ and the potential intensity $B$ for a purely scalar singular potential concentrated on the circumference $r=R$, when $m=2$ and $R=1$. Bound states can only exist for angular momentum in the range $0\leq l\leq mR$. Supercritical states ($E_b=-m$) are quasi-bound states for $l=0$. For $l=1$, the orange dot corresponds to an impenetrable circumference ($B=-2$), and $E=-m$ corresponds to the supercritical state of a particle confined in the inner region of the circle. Critical states are quasi-bound for $l=0,1$. For $l=2$, the green dot corresponds to an impermeable wall ($B=-2$) and $E=+m$ corresponds to the critical state of a particle confined in the outer region. The curves in the figure have the symmetry $B\to \frac{4}{B}$.
• Figure 2: The complex energy solutions of (\ref{['secscalar']}), corresponding to a singular pure scalar potential of intensity $B$ concentrated on the circumference $r=R$, are represented by points of different colors, for $m=2, R=1$ and $l=1$, and various values of the intensity $B$ in the interval $[-2,2]$. The blue curve represents the locus of the complex solutions to the imaginary part of (\ref{['secscalar']}), which does not depend on the strength $B$. The values $B=\pm 2$ correspond to an impenetrable wall at $r=R$ (with boundary conditions depending on the sign of $B$), and the corresponding real energy solutions are the admissible energies for an (anti)particle confined inside the circular box (except for the energy $0<E_b<+m$), in the case $B=-2$, which corresponds to a bound state confined in the outer region). As $B$ increases in absolute value the complex solutions shift towards the real axis, along the blue curve. The red points correspond to the critical value $B=B_{\text{crit}}$, at which absorption/emission of a bound state occurs at $E=+m$.
• Figure 3: The Wigner time delay $\tau_l$ (blue curves) for scattering states as a function of energy, for some values of the parameter $B$ from (\ref{['escparam']}) for a purely scalar strength ($A_0=A_r=A_\theta=0$, arbitrary $B$), and for the same parameters as in Figure \ref{['PScalarComplex']}. The red points correspond to $\left(E_R,-\frac{1}{E_I}\right)$ where $E=E_R + i E_I$ are the complex energies that solve the equation (\ref{['secscalar']}). We note that the resonant energies $E_R$ approximate very well the sharp peaks (or valleys if $E<-m$) of the Wigner time delay, and $-1/E_I$ provide a scale for the Wigner time delay at the resonances.
• Figure 4: Bound state energy $E_b$ as a function of the singular electric potential strength $A_0$, for three values of the angular momentum $l$, with $m=2$ and $R=1$. A single bound state will result if the curve crosses the horizontal line defined by the value of $A_0$. This curve also shows the symmetry $A_0\to -\frac{4}{A_0}$, as well as the relationships between the values of $A_0$ associated to critical and supercritical states discussed in the text.
• Figure 5: The complex energies $E=E_{R}+i E_{I}$ that are solutions of (\ref{['comel']}), corresponding to purely outgoing scattering states for a pure electrostatic potential ($B=A_r=A_\theta=0$, $A_0\neq 0$), are represented by points of different colors, taking $m=2$, $R=1$ and $l=2$. The complex energies corresponding to each value of $A_0$ are shown in the same color. The blue curve corresponds to all solutions to the real part of (\ref{['comel']}), which do not depend on the value of $A_0$. As the strength $A_0$ varies from $0$ to $\infty$, the corresponding complex energies move from left to right along the blue curve. Due to the $A_0 \to -\frac{4}{A_0}$ symmetry, the same happens as $A_0$ varies from $-\infty$ to $0$. With increasing values of $A_0$, a bound (supercritical) state is captured at $A_0\approx 1.07$ (green dot) and emitted at $A_0\approx 4.828$ (brown dot).