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arXiv:1601.04771 · Journal of High Energy Physics

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

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

Published May 1, 2016• 4 citations

Abstract

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

DOI: 10.1007/jhep05(2016)119

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An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous 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.

Based on the inhomogeneous T – Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin- 1 2 chain as examples, we show that for a given inhomogeneous T – Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.

The general expression for the local matrix of a quantum chain with the site space in any representation of su(3) is obtained. This is made by generalizing from the fundamental representation and imposing the fulfilment of the Yang - Baxter equation. Then, a non-homogeneous spin chain combining different representations of su(3) is solved by developing a method inspired in the nested Bethe ansatz. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations. A conjecture about the solution of a chain with the site states in different representations of su(n) is presented. The thermodynamic limit of the ground state is calculated.

Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.

Based on the inhomogeneous T–Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin- chain as examples, we show that for a given inhomogeneous T–Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.