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

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.

💡 Research Summary

The paper establishes a comprehensive analytical framework that connects the chiral Gross‑Neveu (CGN) model – equivalently the linearized Bogoliubov‑de Gennes (BdG) equation – to the entire nonlinear Schrödinger (NLS) hierarchy within the Ablowitz‑Kaup‑Newell‑Segur (AKNS) formalism. Building on the earlier work of Correa, Dunne and Plyushchay, which demonstrated the mapping for the first member of the hierarchy, the authors extend the construction to an arbitrary order n. By formulating the Lax pair for the n‑th NLS flow and exploiting its conserved quantities, they derive a universal expression for the fermionic wavefunction ψ(x) and the associated gap (order‑parameter) function Δ(x) that solves the BdG equation for any admissible Δ(x) belonging to the n‑th hierarchy.

A central result is that the energy spectrum of the n‑th NLS hierarchy generically exhibits n + 1 gaps. This follows from the analytic structure of the scattering data: each additional nonlinear term in the hierarchy introduces an extra pair of branch points in the complex energy plane, thereby opening a new forbidden band. Consequently, the hierarchy provides a systematic way to generate multi‑gap superconducting or charge‑density‑wave backgrounds in one dimension.

To illustrate the abstract construction, the authors present an explicit self‑consistent solution corresponding to a “two‑complex‑kink” configuration. The gap function is parametrized by four real parameters: the positions x₁, x₂ of the two kinks and their respective phase shifts θ₁, θ₂. The solution satisfies the self‑consistency condition Δ(x) = g⟨ψ̄ψ⟩, confirming that it is a genuine stationary configuration of the CGN model. The spectrum contains exactly two bound states (E₁, E₂) residing in the central gap. When the kinks are far apart, each bound state is sharply localized around one kink, reproducing the familiar picture of isolated soliton‑bound fermions. However, as the kinks are brought together and the relative phase difference Δθ = θ₁ − θ₂ approaches zero, the bound‑state wavefunctions develop two comparable peaks at both kink locations. This delocalization signals a coherent hybridization of the two soliton‑induced states, a phenomenon that would be invisible in a single‑gap (n = 1) analysis.

The paper discusses several physical implications. First, the n + 1 gap structure mirrors multi‑band superconductors or charge‑density‑wave systems where several energy windows are depleted, suggesting that the NLS hierarchy could serve as a model for engineered heterostructures with tailored spectral properties. Second, the controllable phase shifts and separation of the kinks provide a knob for tuning the degree of fermion localization, which may be relevant for designing quantum wires or topological defect networks where bound Majorana‑like modes are desired. Third, the analytical tractability of the entire hierarchy opens the door to studying dynamical processes (e.g., kink collisions, quench dynamics) while retaining exact control over the fermionic sector.

In summary, the authors have shown that the AKNS‑based mapping of the CGN/BdG system to the full NLS hierarchy yields a powerful, exact method for constructing multi‑gap, multi‑soliton backgrounds together with their fermionic bound spectra. The work not only generalizes previous first‑order results but also provides concrete examples (the two‑complex‑kink solution) that illuminate how bound states can transition from localized to delocalized regimes as the underlying order‑parameter texture is varied. This framework is poised to impact a broad range of one‑dimensional quantum many‑body problems, from condensed‑matter realizations of exotic superconductivity to nonlinear optics where analogous equations govern pulse propagation in media with complex refractive‑index landscapes.