Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system

TL;DR

This paper proves that for generic parameters, the Bethe Ansatz provides a complete set of states for the rational, spin-1/2 Richardson--Gaudin model, using a method that does not rely on root distribution assumptions.

Contribution

It introduces a general method to establish Bethe state completeness without relying on the string hypothesis, applicable to a broader class of models.

Findings

Bethe states are complete for generic coupling parameters

The method is independent of Bethe root distribution assumptions

Applicable to wider classes of integrable systems

Abstract

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson--Gaudin system it will be argued that, for generic values of the system's coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.