Hirota-tau and Heun-function framework for Dirac vacuum polarization and quantum stabilization of kinks

Abstract

We investigate a modified affine Toda model coupled to matter (ATM) which includes a scalar self-interacting potential and demonstrate that its first-order integro-differential structure, preserving a deformed Noether-topological current correspondence, provides a consistent framework for fermion-soliton interactions. In this formulation, the fermion-soliton energy is proportional to the soliton's topological charge. We show that fermionic back-reaction and the self-interacting scalar critically shape the fermion-kink energy, the in-gap bound-state spectrum, and the fermionic vacuum-polarization energy, yielding well-defined stability minima of the total energy as functions of the fermion and scalar masses and coupling parameters. A key result is that the Heun-equation formalism is necessary to construct nonzero-energy bound and scattering states: unlike the tau-function method, which captures only the zero mode, the Heun approach encodes the full scattering data through local solution matching conditions. These results refine the spectral analysis of deformed integrable models. The stability of soliton-fermion configurations has direct implications for topologically protected states in quantum information and condensed-matter systems.

Figures (5)

• Figure 1: (color online) Scalar kink (blue) in the left and right Figs. The real and imaginary parts of the scattering wave function $u(x)$: Left Fig. $Re[c_1 u_{in}(x) +c_2 u_{ref}(x)]$ (green), $Re[u_{tr}(x)]$ (red), and right Fig. $Im[c_1 u_{in}(x) +c_2 u_{ref}(x)]$ (magenta), $Im[u_{tr}(x)]$ (red). For $\beta =1, M=5, k=1.5, E_1=+5.22$.
• Figure 2: (color online) Dependence of the transmission coefficient $T$ (left Fig.) and the reflection coefficient $R$ (right Fig.) for the fermionic wave function on the fermion momentum $k$ for the fermion masses $M=0.65$ (red), $M=1$ (green) and $M=1.75$ (brown).
• Figure 3: (color online) Phase shift $\delta(k)$ vs $k$ of the scattering states $u$ and $v$ for the fermion mass $M=2.15 \times 10^{-5}$. Note that $\delta(0)-\delta(+\infty) = \frac{\pi}{2}$. The dashed line shows the reference function $\frac{10^{-8}}{k^2}$. Note that the phase shift behaves as $\frac{1}{k^2}$ in the region $k \rightarrow large$.
• Figure 4: (color online) Total energy $E_{tot,\,0}$ (\ref{['Etot01']}) for the kink-fermion system in terms of $M$ in the zero-mode bound state sector with $E_{10}= 0$, $\beta = -1.91$, $Q_{k(\bar{k})} = \frac{1}{2}$ and for various scalar mass $m$ parameters. Note that $m \approx 3.3 M_o$, such that the lowest value of $E_{tot,\,0}$ is attained at $M_o$.
• Figure 5: (color online) Total energy $E_{tot,\,1}=E_{kf} + VPE + E_{11}$ (\ref{['Etot01']}) for the kink-fermion system in terms of $M$ in the bound state sector with $E_{11}= 0.8 M$, $Q_{k(\bar{k})} = \frac{1}{2}$, $\beta = -1.91$, and for various scalar mass $m$ parameters. Note that $m \approx 4 M_o$, such that the lowest value of $E_{tot,\,1}$ is attained at $M_o$.