Abstract
The quantum $\tau_2$-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.
Highlights
The Bethe Ansatz solution of the periodic τ2-model with generic sitedependent inhomogeneity was obtained by constructing an inhomogeneous T − Q relation with polynomial Q-functions (i.e. off-diagonal Bethe Ansatz method (ODBA) [19,20,21,22,23,24]), which provides a perspective to investigate the τ2-model with generic open boundary condition
By introducing an off-diagonal term in the conventional T − Q relation, we obtain the spectrum of the generic open τ2-model and the associated Bethe Ansatz equations
The quadratic relation (2.11) and reflection equations (2.12) and (2.13) lead to the fact that the transfer matrix t(u) of the τ2model with different spectral parameters are mutually commutative [31], i.e., [t(u), t(v)] = 0, which ensures the integrability of the model by treating t(u) as the generating functional of the conserved quantities
Summary
Let us fix an odd integer p such that p ≥ 3, and let V be a p-dimensional vector space (i.e. the local Hilbert space) with an orthonormal basis {|m |m ∈ Zp}. The R-matrix satisfies the quantum Yang-Baxter equation (QYBE) [29, 30] and becomes some projectors when the spectral parameter u takes some special values as Antisymmetric-fusion conditions : R(−η) = −2 sinh ηP (−), Symmetric-fusion conditions : R(η) = 2 sinh η Diag(cosh η, 1, 1, cosh η) P (+),. In order to construct the associated open spin chain, let us introduce the L(u) in the form of. The quadratic relation (2.11) and (dual) reflection equations (2.12) and (2.13) lead to the fact that the transfer matrix t(u) of the τ2model with different spectral parameters are mutually commutative [31], i.e., [t(u), t(v)] = 0, which ensures the integrability of the model by treating t(u) as the generating functional of the conserved quantities.
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