Coupled fermion-kink system in Jackiw-Rebbi model

TL;DR

This study investigates the coupled dynamics of fermions and solitons in the Jackiw-Rebbi model, incorporating back-reaction effects through numerical solutions, and explores the impact on the soliton's shape and energy spectrum.

Contribution

It introduces a self-consistent numerical approach to analyze fermion-soliton interactions including back-reaction, extending previous models that treated the soliton as a fixed background.

Findings

Bound energy spectrum and states determined numerically.

Soliton shape is affected by fermionic back-reaction.

Results align with existing literature for fixed soliton models.

Abstract

In this paper we study Jackiw-Rebbi model, in which a massless fermion is coupled to the kink of $\lambda \phi^4$ theory through a Yukawa interaction. In the original Jackiw-Rebbi model the soliton is prescribed. However, we are interested in the back-reaction of the fermion on the soliton besides the effect of the soliton on the fermion. Also, as a particular example, we consider a minimal supersymmetric kink model, $\mathcal{N}=1$, in ($1+1$) dimensions. In this case, the bosonic self-coupling, $\lambda$, and the Yukawa coupling between fermion and soliton, $g$, have specific relation, $g=\sqrt{\lambda/2}$. As the set of coupled equations of motion of the system is not analytically solvable, we use a numerical method to solve it self-consistently. We obtain the bound energy spectrum, bound states of the system and the corresponding shape of the soliton using a relaxation method, except for the zero mode fermionic state and threshold energies which are analytically solvable. With the aid of these results we are able to show how the soliton is affected in general and supersymmetric cases. The results we obtain are consistent with the ones in the literature, considering the soliton as background.