Scalar Products Of Bethe Vectors Research Articles

Scalar products of Bethe vectors in the generalized algebraic Bethe ansatz

Scalar products of Bethe vectors in the generalized algebraic Bethe ansatz

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$-invariant integrable models

A class of $\mathfrak{o}_{2n+1}$-invariant quantum integrable models is investigated in the framework of algebraic Bethe ansatz method. A construction of the $\mathfrak{o}_{2n+1}$-invariant Bethe vector is proposed in terms of the Drinfeld currents for the double of Yangian $\mathcal{D}Y(\mathfrak{o}_{2n + 1})$. Action of the monodromy matrix entries onto off-shell Bethe vectors for these models is calculated. Recursion relations for these vectors were obtained. The action formulas can be used to investigate structure of the scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models.

Generating function for scalar products in the algebraic Bethe ansatz

We construct a family of determinant representations for scalar products of Bethe vectors in models with $$ \mathfrak{gl} (3)$$ symmetry. This family is defined by a single generating function containing arbitrary complex parameters but is independent of their specific values. Choosing these parameters in different ways, we can obtain different determinant representations.

Scalar products of Bethe vectors in the 8-vertex model

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.

Scalar products of Bethe vectors in the models with [formula omitted] symmetry

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by gl(m|n) superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.

Continuous representations of scalar products of Bethe vectors

We present families of single determinantal representations of on-shell scalar products of Bethe vectors. Our families of representations are parameterized by a continuous complex variable which can be fixed at one’s convenience. Here we consider Bethe vectors in two versions of the six-vertex model: the case with boundary twists and the case with open boundaries.

Scalar products of Bethe vectors in models with symmetry 2. Determinant representation**Dedicated to the memory of Petr Petrovich Kulish.

We study integrable models with symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.

Scalar products of Bethe vectors in models with symmetry 1. Super-analog of Reshetikhin formula**Dedicated to the memory of Petr Petrovich Kulish.

We study the scalar products of Bethe vectors in integrable models solvable by the nested algebraic Bethe ansatz and possessing symmetry. Using explicit formulas of the monodromy matrix entries’ multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with symmetry.

Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models

We study $\mathfrak{gl}(2|1)$ symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.

Scalar products in GL(3)-based models with trigonometric R-matrix. Determinant representation

We study quantum integrable GL(3)-based models with a trigonometric R-matrix solvable by the nested algebraic Bethe ansatz. We derive a determinant representation for a special case of scalar products of Bethe vectors. This representation allows one to find a determinant formula for the form factor of one of the monodromy matrix entries. We also point out an essential difference between form factors in the models with the trigonometric R-matrix and their analogs in GL(3)-invariant models.

Off-shell scalar products for the XXZ spin chain with open boundaries

In this work we study scalar products of Bethe vectors associated with the XXZ spin chain with open boundary conditions. The scalar products are obtained as solutions of a system of functional equations. The description of scalar products through functional relations follows from a particular map having the reflection algebra as its domain and a function space as the codomain. Within this approach we find a multiple contour integral representation for the scalar products in which the homogeneous limit can be obtained trivially.

Scalar products in models with a GL(3) trigonometric R-matrix: Highest coefficient

We study quantum integrable models with a GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of bilinear combinations of the highest coefficients. We show that there exist two different highest coefficients in the models with a GL(3) trigonometric R-matrix. We obtain various representations for the highest coefficients in terms of sums over partitions. We also prove several important properties of the highest coefficients, which are necessary for evaluating the scalar products.

The algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of the XXX SU(3)-invariant Heisenberg chain.

Highest coefficient of scalar products in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of a bilinear combination of their highest coefficients. We obtain various different representations for the highest coefficient in terms of sums over partitions. We also obtain multiple integral representations for the highest coefficient.