We study quantum integrable models with $GL(3)$ trigonometric $R$-matrix solvable by the nested algebraic Bethe ansatz. We analyze scalar products of generic Bethe vectors and obtain an explicit representation for them in terms of a sum with respect to partitions of Bethe parameters. This representation generalizes known formula for the scalar products in the models with $GL(3)$-invariant $R$-matrix.
In this paper we study the relation between the function $J_{4_1,0}$, which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find $J_{4_1,0}$ up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of $J_{4_1,0}$ to that of Sudler's trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in respon…
The zero modes method is applied in order to get action of the monodromy matrix entries onto off-shell Bethe vectors in quantum integrable models associated with $U_q(\mathfrak{gl}_N)$-invariant $R$-matrices. The action formulas allow to get recurrence relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.
We obtain a determinant representation of normalized scalar products of on-shell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solvability and solve it in terms of determinants of explicitly known matrices.
Construction of Monodromy Matrix in the F- basis and Scalar products in Spin Chains
本文简单地介绍了Maillet和Sanches de Santos最近提出的自旋为1/2的链的因子分解算子理论。我们得到了分解算子的矩阵元素关于Monodromy矩阵元素的显式表达式。我们利用这个结果导出了量子自旋链的一般标积的表达式。本文对前人关于贝特本征态与任意对偶态的标积的测定作了评述。我们还建立了F-基上的标积计算与通常基上的标积计算之间的直接对应。
We calculate the scalar product of Bethe states of the XXZ spin-$\frac{1}{2}$ chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.
