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arXiv:1401.4358 · Reviews in Mathematical Physics

COORDINATE BETHE ANSÄTZE FOR NON-DIAGONAL BOUNDARIES

E. Ragoucy

Published Nov 1, 2013• 5 citations
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Abstract

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.

DOI
10.1142/s0129055x13430071

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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.
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al.The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain.We prove this result for the chains with small length.We obtain also an off-shell equation (i.e.satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.