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arXiv:2312.04153 · Chinese Physics B

Off-diagonal approach to the exact solution of quantum integrable systems

Yi Qiao, Junpeng 俊鹏 Cao 曹, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Oct 27, 2023• 2 citations
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Abstract

We investigate the t–W scheme for the anti-ferromagnetic XXX spin chain under both periodic and open boundary conditions. We propose a new parametrization of the eigenvalues of the transfer matrix. Based on it, we obtain the exact solution of the system. By analyzing the distribution of zero roots at the ground state, we obtain the explicit expressions of the eigenfunctions of the transfer matrix and the associated W operator (see Eqs. (10) and (70)) in the thermodynamic limit. We find that the ratio of the quantum determinant with the eigenvalue of W operator for the ground state exhibits exponential decay behavior. Thus this fact ensures that the so-called inversion relation (the t–W relation without the W-term) can be used to study the ground state properties of quantum integrable systems with/without U(1)-symmetry in the thermodynamic limit.

DOI
10.1088/1674-1056/ad0774

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We propose a novel characterization of the exact solution of the one-dimensional supersymmetric t − J model with generic non-diagonal boundary conditions. Instead of using Bethe roots, we parameterize the eigenvalues of transfer matrix in the spin sector with their zero-roots. By constructing the t − W relation, we obtain the explicit forms of the eigenvalues of the system and the homogeneous Bethe ansatz equations. This scheme is universal and can be applied to other quantum integrable systems with or without U ( 1 ) symmetry.
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We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.