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arXiv:1312.4770 · Journal of High Energy Physics

Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries

Junpeng Cao, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Apr 1, 2014• 37 citations
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Abstract

The nested off-diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T − Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.

DOI
10.1007/jhep04(2014)143

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The nested off-diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T − Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.
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The anisotropic four-component spin chain with integrable off-diagonal boundary terms is studied by the nested off-diagonal Bethe ansatz method. Based on the intrinsic properties of the associated R-matrix, the recursive operator product identities of the fused transfer matrices are obtained by using the fusion. The asymptotic behaviors and the values of fused transfer matrices at certain special points are derived explicitly. The nested inhomogeneous T − Q relations of the system are constructed, and the self-consistency of these relations is demonstrated. The method and the results can be applied to study the eigenvalue problems of multicomponent integrable models with open boundaries.
The nested off-diagonal Bethe ansatz is generalized to study the quantum spin chain associated with the SUq(3) R-matrix and generic integrable non-diagonal boundary conditions. By using the fusion technique, certain closed operator identities among the fused transfer matrices at the inhomogeneous points are derived. The corresponding asymptotic behaviors of the transfer matrices and their values at some special points are given in detail. Based on the functional analysis, a nested inhomogeneous T–Q relations and Bethe ansatz equations of the system are obtained. These results can be naturally generalized to cases related to the SUq(n) algebra.
Abstract The off-diagonal Bethe ansatz method is generalized to the integrable model associated with thesp(4) (orC2) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the sameT−Qrelations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneousT−Qrelations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to genericsp(2n) (i.e.,Cn) integrable model.
We study the exact solutions of quantum integrable model associated with the Cn Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the C3 as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous T−Q relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the Cn model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.