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arXiv:cond-mat/0107260 · Journal of Physics A Mathematical and General

Non-regular eigenstate of theXXXmodel as some limit of the Bethe state

Tetsuo Deguchi

Published Nov 13, 2001• 9 citations

Abstract

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU (2) spin-lowering operator. It has been pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure is possible. In this paper, we discuss it explicitly at the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions.

DOI: 10.1088/0305-4470/34/46/301

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Bethe ansatz goes back to 1931, when H. Bethe invented it to solve some one-dimensional models, such as XXX spin chain, proposed by W. Heisenberg in 1928. Although it is a very powerful method to compute eigenvalues and eigenvectors of the corresponding Hamiltonian, it can be applied only for very specific boundary conditions: periodic boundary ones, and so-called open-diagonal boundary ones. After reviewing this method, we will present a generalization of it that applies also to open-triangular boundary conditions. This short note presents only the basic ideas of the technique, and does not attend to give a general overview of the subject. Interested readers should refer to the original papers and references therein.