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

TL;DR

This paper develops a Heun-function framework for analyzing fermion-soliton interactions in a modified affine Toda model, incorporating quantum corrections and revealing the importance of the Heun approach over tau-functions for full spectral data.

Contribution

It introduces a Heun-equation based method to construct nonzero-energy states, extending spectral analysis beyond zero modes in deformed integrable models.

Findings

Heun formalism captures full scattering data, unlike tau-functions.

Quantum corrections influence fermion-kink energy and stability.

Full spectral analysis refines understanding of soliton-fermion systems.

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 evaluate the renormalized energy functional, incorporating one-loop quantum corrections, and perform a variational minimization to determine the configuration that extremizes the functional. The 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, in the semiclassical approximation. 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.