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arXiv:1309.6165 · Symmetry Integrability and Geometry Methods and Applications

Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz

S. Belliard

Published Nov 22, 2013• 97 citations

Abstract

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.

DOI: 10.3842/sigma.2013.072

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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.

The modified algebraic Bethe ansatz, introduced by Crampé and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-12 chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to N the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-12 chain on the segment with two upper triangular boundaries.

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.

We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.

Abstract In this note we construct Q-operators for the spin s open Heisenberg XXX chain with diagonal boundaries in the framework of the quantum inverse scattering method. Following the algebraic Bethe ansatz we diagonalise the introduced Q-operators using the fundamental commutation relations. By acting on Bethe off-shell states and explicitly evaluating the trace in the auxiliary space we compute the eigenvalues of the Q-operators in terms of Bethe roots and show that the unwanted terms vanish if the Bethe equations are satisfied.