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arXiv:1601.04389 · Journal of Statistical Mechanics Theory and Experiment

Exact solution of an su ( n ) spin torus

Kun Hao, Junpeng Cao, Guang-Liang Li, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Jul 1, 2016• 8 citations
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Abstract

Abstract The trigonometric su ( n ) spin chain with anti-periodic boundary condition ( su ( n ) spin torus) is demonstrated to be Yang–Baxter integrable. Based on some intrinsic properties of the R -matrix, certain operator product identities of the transfer matrix are derived. These identities and the asymptotic behavior of the transfer matrix together allow us to obtain the exact eigenvalues in terms of an inhomogeneous T − Q relation via the off-diagonal Bethe Ansatz.

DOI
10.1088/1742-5468/2016/07/073104

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Abstract The trigonometric su ( n ) spin chain with anti-periodic boundary condition ( su ( n ) spin torus) is demonstrated to be Yang–Baxter integrable. Based on some intrinsic properties of the R -matrix, certain operator product identities of the transfer matrix are derived. These identities and the asymptotic behavior of the transfer matrix together allow us to obtain the exact eigenvalues in terms of an inhomogeneous T − Q relation via the off-diagonal Bethe Ansatz.
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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.
By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the trigonometric SU(3) model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T−Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe-type eigenstates which have well-defined homogeneous limit. This exact solution provides a basis for further analyzing the thermodynamic properties and correlation functions of the anisotropic models associated with higher rank algebras.
The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T − Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T − Q relations obeying the operator product identities can be constructed. Numerical results for two-site s = 1 case indicate that an arbitrary choice of the derived T − Q relations is enough to give the complete spectrum of the transfer matrix.