The paper derives normalized scalar products of on-shell and off-shell Bethe states using Leonard triples, obtains explicit solutions of Belliard-Slavnov systems, and gives a determinant formula for q-Racah polynomials.
Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
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abstract
The modified algebraic Bethe ansatz, introduced by Cramp\'e and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to $N$ the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with two upper triangular boundaries.
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The $q$-Racah polynomials from scalar products of Bethe states II
The paper derives normalized scalar products of on-shell and off-shell Bethe states using Leonard triples, obtains explicit solutions of Belliard-Slavnov systems, and gives a determinant formula for q-Racah polynomials.