Violations of the String Hypothesis in the Solutions of the Bethe Ansatz Equations in the XXX-Heisenberg

TL;DR

This paper demonstrates that the traditional string hypothesis fails to accurately predict the solutions of Bethe ansatz equations in the XXX-Heisenberg model for large system sizes, revealing discrepancies in the expected number of solutions.

Contribution

The authors analytically show the failure of the string hypothesis for large N and introduce a new variable set simplifying the equations to a single polynomial.

Findings

String hypothesis incorrectly predicts solutions for N>21.

Two complex solutions become real solutions, altering the solution count.

A new variable simplifies Bethe equations to a single polynomial.

Abstract

We study the equations for the quasi-momenta which characterize the wave-functions in the Bethe ansatz for the XXX-Heisenberg model. We show in a simple analytical fashion, that the usual ``string hypothesis" incorrectly predicts the number of real solutions and the number of complex solutions for $N>21$ in the sector with two spins flipped, confirming the work of Essler et al. Two complex pair solutions drop out and form two additional real pair solutions. We also introduce a new set of variables which allows the equations to be written as a single polynomial equation in one variable. We consider in some detail the case of three spins flipped.