Abstract
The generic quantum τ 2-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 τ j -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.
Highlights
Of sites [14,15,16,17,18], in which there is no simple polynomial solutions of the Q-function in terms of Baxter’s T − Q relation
The eigenvalues of the corresponding transfer matrix with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T − Q relation with polynomial Qfunctions
By including an extra off-diagonal term in the T − Q relation, we show that the eigenvalues of the generic τ2 transfer matrix can be expressed explicitly in terms of a trigonometric polynomial Q function and a proper set of Bethe Ansatz equations can be derived
Summary
With each site n of the quantum chain, the associated L-operator Ln(u) ∈ End(C2 ⊗ V) defined in the most general cyclic representation of Uq(sl2), is given by [1]. The transfer matrix t(u) of the τ2-model with periodic boundary condition is given by the partial trace of the monodromy matrix T (u) in the auxiliary space, namely, t(u) = tr (T (u)) = A(u) + D(u). The quadratic relation (2.11) leads to the fact that the transfer matrices with different spectral parameters are mutually commutative [19], i.e., [t(u), t(v)] = 0, which guarantees the integrability of the model by treating t(u) as the generating functional of the conserved quantities.
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