Violations of the string hypothesis in the solutions of the Bethe ansatz equations in the XXX-Heisenberg model

We study the equations for the quasi-momenta 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. We also introduce a new set of variables which allows the equations to be written as a single polynomial in one variable. We consider in some data the case of three spins flipped.