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arXiv:1608.06209 · Journal of High Energy Physics

Bethe ansatz solutions of the τ 2-model with arbitrary boundary fields

Xiaotian Xu, Kun Hao, Tao Yang, Junpeng Cao, Wen‐Li Yang, Kangjie Shi

Published Nov 1, 2016• 8 citations
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Abstract

The quantum τ 2-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T − Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.

DOI
10.1007/jhep11(2016)080

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Off-diagonal Bethe Ansatz solution of the τ2τ_2τ2​-model
The generic quantum τ2τ_2τ2​-model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix (solutions of the recursive functional relations in τjτ_jτj​-hierarchy) with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T-Q relation with polynomial Q-functions. The associated Bethe Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz equations for small number of sites indicate that the inhomogeneous T-Q relation does indeed give the complete spectrum.
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The exact solution of the one-dimensional super-symmetric t-J model under generic integrable boundary conditions is obtained via the Bethe ansatz methods. With the coordinate Bethe ansatz, the corresponding R-matrix and K-matrices are derived for the second eigenvalue problem associated with spin degrees of freedom. It is found that the second eigenvalue problem can be transformed to that of the transfer matrix of the inhomogeneous XXX spin chain, which allows us to obtain the spectrum of the Hamiltonian and the associated Bethe ansatz equations by the off-diagonal Bethe ansatz method.