Off-shell Bethe Vectors Research Articles

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

Integrable mathfrak{sl} (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for mathfrak{sl} (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

Recurrence relations for off-shell Bethe vectors in trigonometric integrable models

The zero modes method is applied in order to get the action of the monodromy matrix entries on off-shell Bethe vectors in quantum integrable models associated with -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.

Scalar product for the XXZ spin chain with general integrable boundaries * *Dedicated to the memory of Omar Foda.

We calculate the scalar product of Bethe states of the XXZ spin- 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.

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.

Actions of the monodromy matrix elements onto -invariant Bethe vectors

Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the -invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.

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.

Bethe Vectors for Orthogonal Integrable Models

We consider quantum integrable models associated with $\mathfrak{so}_3$ algebra. We describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this approach we use isomorphism between $R$-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types $B$, $C$, and $D$ series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The action formulas can also be used for calculating the scalar products in the models associated with $\mathfrak{so}_3$ algebra.

Why scalar products in the algebraic Bethe ansatz have determinant representation

We show that the scalar products of on-shell and off-shell Bethe vectors in the algebralic Bethe ansatz solvable models satisfy a system of linear equations. We find solutions to this system for a wide class of integrable models. We also apply our method to the XXX spin chain with broken U(l) symmetry.

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

We consider XXX spin-$1/2$ Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix.

Generalized sℓ(2) Gaudin algebra and corresponding Knizhnik–Zamolodchikov equation

The Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.

A note on mathfrak{g}{mathfrak{l}}_2 -invariant Bethe vectors

We consider mathfrak{g}{mathfrak{l}}_2 -invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries on the twisted off-shell Bethe vectors.

Multiple integral formulae for the scalar product of on-shell and off-shell Bethe vectors in [formula omitted]-invariant models

We study the scalar product Sℓ,m between an on-shell and an off-shell Bethe state in models with SU(3)-invariance, where ℓ and m denote the cardinalities of the two sets of Bethe roots. We construct recursion relations relating Sℓ,m to scalar products of smaller dimension, namely Sℓ−1,m and Sℓ,m−1. Solving these recursion relations we obtain new multiple integral expressions for Sℓ,m, whose integrands are (ℓ+m)×(ℓ+m) determinants, and closely related to the Slavnov determinant expression for the SU(2) scalar product.

Bethe Ansatz for the Universal Weight Function

We consider universal off-shell Bethe vectors given in terms of Drinfeld realization of the algebra $U_q(\widehat{gl}_N)$ [arXiv:math/0610517,arXiv:0711.2819]. We investigate ordering properties of the product of the transfer matrix and these vectors. We derive that these vectors are eigenvectors of the transfer matrix if their Bethe parameters satisfy the universal Bethe equations [arXiv:math-ph/0512037].

Generating Series for Nested Bethe Vectors

We reformulate nested relations between off-shell Uq( b glN) Bethe vectors as a certain equation on generating series of strings of the composed Uq(b gl N ) currents. Using inversion of the generating series we find a new type of hierarchical relations between univer- sal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations (Arnaudon D. et al., Ann. Henri Poincare 7 (2006), 1217-1268, math-ph/0512037) for the parameters of universal Bethe vectors of the algebra Uq(b gl 2 ).

A computation of universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$

We compute weight functions (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in {EKP}.

Weight Functions and Drinfeld Currents

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.

Off-shell Bethe vectors and Drinfeld currents

In this paper we compare two constructions of weight functions (off-shell Bethe vectors) for the quantum affine algebra U q ( g l ̂ N ) . The first construction comes from the algebraic nested Bethe ansatz. The second one is defined in terms of certain projections of products of Drinfeld currents. We show that the two constructions give the same result in tensor products of vector representations of U q ( g l ̂ N ) .