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

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.

arXiv comment18 pages. revised to amplify credit, in plain tex with macropackage psbox (available from the net)

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