On zero-remainder conditions in the Bethe ansatz

TL;DR

This paper proves that physical solutions to the Heisenberg spin chain Bethe ansatz are characterized by two zero-remainder conditions, providing an alternative proof of an existing algorithm and addressing its minimality.

Contribution

It introduces a new characterization of solutions via zero-remainder conditions, bridging criteria and resolving the minimality issue of a recent algorithm.

Findings

Physical solutions characterized by zero-remainder conditions

Bridging different solution criteria

Resolving minimality of the $QQ$ relations algorithm

Abstract

We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, yielding an alternative proof of a recently devised algorithm based on $QQ$ relations, and solving its minimality issue.