Fermions on one or fewer Kinks

TL;DR

This paper analyzes the spectrum of fermion bound states on a Z_2 kink, revealing multiple bound states and their behavior in kink-antikink backgrounds, with both analytical and numerical results.

Contribution

It provides a complete spectrum of fermion bound states on a Z_2 kink and proves the existence of at least one bound state in kink-antikink configurations.

Findings

Number of bound states depends on fermion and scalar masses

Bound state energy approaches zero as kink-antikink separation increases

Numerical methods explicitly identify low-lying bound states

Abstract

We find the full spectrum of fermion bound states on a Z_2 kink. In addition to the zero mode, there are int[2 m_f/m_s] bound states, where m_f is the fermion and m_s the scalar mass. We also study fermion modes on the background of a well-separated kink-antikink pair. Using a variational argument, we prove that there is at least one bound state in this background, and that the energy of this bound state goes to zero with increasing kink-antikink separation, 2L, and faster than e^{-a2L} where a = min(m_s, 2 m_f). By numerical evaluation, we find some of the low lying bound states explicitly.