# Why scalar products in the algebraic Bethe ansatz have determinant representation

@article{Belliard2019WhySP, title={Why scalar products in the algebraic Bethe ansatz have determinant representation}, author={Samuel Belliard and Nikita Andreevich Slavnov}, journal={Journal of High Energy Physics}, year={2019} }

Abstract
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.

## 13 Citations

### Scalar products of Bethe vectors in the 8-vertex model

- MathematicsJournal of High Energy Physics
- 2020

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…

### Generating function for scalar products in the algebraic Bethe ansatz

- Mathematics
- 2020

Abstract 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…

### Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin

- Mathematics
- 2022

In this note we construct Q-operators for the spin s open Heisenberg XXX chain with diagonal boundaries in the framework of the quantum inverse scattering method. Following the algebraic Bethe ansatz…

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

- Mathematics
- 2021

We calculate the scalar product of Bethe states of the XXZ spin- 12 chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe…

### Overlap between usual and modified Bethe vectors

- PhysicsTheoretical and Mathematical Physics
- 2021

We consider the overlap of Bethe vectors of the XXX spin chain with a diagonal twist and the modified Bethe vectors with a general twist. We find a determinant representation for this overlap under…

### Determinant form of correlators in high rank integrable spin chains via separation of variables

- Mathematics
- 2020

In this paper we take further steps towards developing the separation of variables program for integrable spin chains with gl(N) symmetry. By finding, for the first time, the matrix elements of the…

### On scalar products and form factors by separation of variables: the antiperiodic XXZ model

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by separation of variables, and the eigenstates can be…

### Correlation functions by separation of variables: the XXX spin chain

- Mathematics
- 2020

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg…

### Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

- MathematicsNuclear Physics B
- 2019

### Density matrices of seniority-zero geminal wavefunctions.

- PhysicsThe Journal of chemical physics
- 2022

Scalar products and density matrix elements of closed-shell pair geminal wavefunctions are evaluated directly in terms of the pair amplitudes, resulting in an analog of Wick's theorem for fermions or…

## References

SHOWING 1-10 OF 40 REFERENCES

### Scalar products of Bethe vectors in models with gl(2|1) symmetry 2. Determinant representation

- Mathematics
- 2016

We study integrable models with gl(2|1) symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters…

### Scalar Products in Twisted XXX Spin Chain. Determinant Representation

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2019

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…

### Scalar product of twisted XXX modified Bethe vectors

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2018

We consider closed XXX spin chains with broken total spin symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on…

### Bethe vectors of GL(3)-invariant integrable models

- Mathematics, Physics
- 2012

We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian 𝒴(gl3)…

### Modified Algebraic Bethe Ansatz: Twisted XXX Case

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

We prove the modified algebraic Bethe Ansatz characterization of the spectral problem for the closed XXX Heisenberg spin chain with an arbitrary twist and arbitrary positive (half)-integer spin at…

### How algebraic Bethe ansatz works for integrable model

- Physics
- 1996

I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated…

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

- Mathematics
- 2015

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…