A note on the Bethe ansatz solution of the supersymmetric t-J model

TL;DR

The paper addresses the equivalence of three Bethe ansatz formulations for the supersymmetric $t$-$J$ chain and the matching transfer-matrix eigenvalues. It introduces a direct polynomial approach using $p(z)$ and $q(z)$ to map Bethe roots across different gradings under twisted boundary conditions. The main contributions are a simple, direct proof of the mutual equivalence of the three BA equations and a proof that the corresponding transfer-matrix eigenvalues agree across all gradings. This polynomial method provides a transparent, broadly applicable technique and complements residues-based proofs, with potential applicability to related integrable systems such as the Hubbard model.

Abstract

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.