Exact quantum numbers of collapsed and non-collapsed 2-string solutions in the Heisenberg spin chain

TL;DR

This paper rigorously derives quantum numbers for all Bethe-ansatz solutions in the spin-1/2 XXX chain, proving completeness and providing exact counts for certain solutions, thus advancing the understanding of quantum integrable systems.

Contribution

It provides a complete, rigorous derivation of quantum numbers for all solutions of the Bethe-ansatz equations in the spin-1/2 XXX chain, including real and complex solutions, and confirms the completeness of the Bethe ansatz.

Findings

Derived quantum numbers for all Bethe-ansatz solutions

Proved the completeness of the Bethe ansatz for the sector

Provided an exact count of collapsed 2-string solutions

Abstract

Every solution of the Bethe-ansatz equations (BAE) is characterized by a set of quantum numbers, by which we can evaluate it numerically. However, no general rule is known how to give quantum numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length $N$. Here we obtain them both for real and complex solutions. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., 2-string solutions) in the sector: $2[ (N-1)/2 - (N/\pi) \tan^{-1}(\sqrt{N-1})]$ in terms of Gauss' symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.