Fermionic solutions of chiral Gross-Neveu and Bogoliubov-de Gennes systems in nonlinear Schr\"odinger hierarchy

TL;DR

This paper derives exact fermionic solutions for the chiral Gross-Neveu and Bogoliubov-de Gennes systems within the nonlinear Schrödinger hierarchy, revealing the structure of energy gaps and localized fermion bound states.

Contribution

It provides a general expression for solutions across all NLS hierarchy orders and analyzes fermion bound state localization in multi-kink configurations.

Findings

Energy spectrum of the n-th NLS hierarchy has n+1 gaps.

Self-consistent two-kink solutions with phase shifts are constructed.

Fermion bound states localize or delocalize depending on kink separation and phase shifts.

Abstract

The chiral Gross-Neveu model or equivalently the linearized Bogoliubov-de Gennes equation has been mapped to the nonlinear Schr\"odinger (NLS) hierarchy in the Ablowitz-Kaup-Newell-Segur formalism by Correa, Dunne and Plyushchay. We derive the general expression for exact fermionic solutions for all gap functions in the arbitrary order of the NLS hierarchy. We also find that the energy spectrum of the n-th NLS hierarchy generally has n+1 gaps. As an illustration, we present the self-consistent two-complex-kink solution with four real parameters and two fermion bound states. The two kinks can be placed at any position and have phase shifts. When the two kinks are well separated, the fermion bound states are localized around each kink in most parameter region. When two kinks with phase shifts close to each other are placed at distance as short as possible, the both fermion bound states have two peaks at the two kinks, i.e., the delocalization of the bound states occurs.